http://wiki.math.uwaterloo.ca/statwiki/api.php?action=feedcontributions&user=H4lyu&feedformat=atomstatwiki - User contributions [US]2022-08-12T04:06:15ZUser contributionsMediaWiki 1.28.3http://wiki.math.uwaterloo.ca/statwiki/index.php?title=Deep_Exploration_via_Bootstrapped_DQN&diff=31013Deep Exploration via Bootstrapped DQN2017-11-21T17:51:07Z<p>H4lyu: /* Bootstrapping [2,3] */</p>
<hr />
<div>== Details ==<br />
<br />
'''Title''': Deep Exploration via Bootstrapped DQN<br />
<br />
'''Authors''': Ian Osband {1,2}, Charles Blundell {2}, Alexander Pritzel {2}, Benjamin Van Roy {1}<br />
<br />
'''Organisations''':<br />
# Stanford University<br />
# Google Deepmind<br />
<br />
'''Conference''': NIPS 2016<br />
<br />
'''URL''': [https://papers.nips.cc/paper/6501-deep-exploration-via-bootstrapped-dqn papers.nips.cc]<br />
<br />
'''Online code sources'''<br />
* [https://github.com/iassael/torch-bootstrapped-dqn github.com/iassael/torch-bootstrapped-dqn]<br />
<br />
This summary contains background knowledge from Section 2-7 (except Section 5). Feel free to skip if you already know.<br />
<br />
== Intro to Reinforcement Learning ==<br />
<br />
In reinforcement learning, an agent interacts with an environment with the goal to maximize its long term reward. A common application of reinforcement learning is to the [https://en.wikipedia.org/wiki/Multi-armed_bandit multi armed bandit problem]. In a multi armed bandit problem, there is a gambler and there are $n$ slot machines, and the gambler can choose to play any specific slot machine at any time. All the slot machines have their own probability distributions by which they churn out rewards, but this is unknown to the gambler. So the question is, how can the gambler learn how to get the maximum long term reward?<br />
<br />
There are two things the gambler can do at any instance: either he can try a new slot machine, or he can play the slot machine he has tried before (and he knows he will get some reward). However, even though trying a new slot machine feels like it would bring less reward to the gambler, it is possible that the gambler finds out a new slot machine that gives a better reward than the current best slot machine. This is the dilemma of '''exploration vs exploitation'''. Trying out a new slot machine is '''exploration''', while redoing the best move so far is '''exploiting''' the currently understood perception of the reward.<br />
<br />
[[File:multiarmedbandit.jpg|thumb|Source: [https://blogs.mathworks.com/images/loren/2016/multiarmedbandit.jpg blogs.mathworks.com]]]<br />
<br />
There are many strategies to approach this '''exploration-exploitation dilemma'''. Some [https://web.stanford.edu/class/msande338/lec9.pdf common strategies] for optimizing in an exploration-exploitation setting are Random Walk, Curiosity-Driven Exploration, and Thompson Sampling. A lot of these approaches are provably efficient, but assume that the state space is not very large. For instance, the approach called Curiosity-Driven Exploration aims to take actions that lead to immediate additional information. This requires the model to search “every possible cell in the grid” which is not desirable if state space is very large. Strategies for large state spaces often just either ignore exploration, or do something naive like $\epsilon$-greedy, where you exploit with $1-\epsilon$ probability and explore "randomly" in rest of the cases. The general idea to tackle large or continuous state spaces is by value function approximation. An empirically tested strategy is Value Function Approximation using Fourier Basis [16]. It has also proven to perform well compared to radial basis functions and the polynomial basis, which are the two most popular fixed bases for linear value function approximation. <br />
<br />
This paper tries to use a Thompson sampling like approach to make decisions.<br />
<br />
== Thompson Sampling<sup>[[#References|[1]]]</sup> ==<br />
<br />
In Thompson sampling, our goal is to reach a belief that resembles the truth. Let's consider a case of coin tosses (2-armed bandit). Suppose we want to be able to reach a satisfactory pdf for $\mathbb{P}_h$ (heads). Assuming that this is a Bernoulli bandit problem, i.e. the rewards are $0$ or $1$, we can start off with $\mathbb{P}_h^{(0)}=\beta(1,1)$. The $\beta(x,y)$ distribution is a very good choice for a possible pdf because it works well for Bernoulli rewards. Further $\beta(1,1)$ is the uniform distribution $\mathbb{N}(0,1)$.<br />
<br />
Now, at every iteration $t$, we observe the reward $R^{(t)}$ and try to make our belief close to the truth by doing a Bayesian computation. Assuming $p$ is the probability of getting a heads,<br />
<br />
$$<br />
\begin{align*}<br />
\mathbb{P}(R|D) &\propto \mathbb{P}(D|R) \cdot \mathbb{P}(R) \\<br />
\mathbb{P}_h^{(t+1)}&\propto \mbox{likelihood}\cdot\mbox{prior} \\<br />
&\propto p^{R^{(t)}}(1-p)^{R^{(t)}} \cdot \mathbb{P}_h^{(t)} \\<br />
&\propto p^{R^{(t)}}(1-p)^{R^{(t)}} \cdot \beta(x_t, y_t) \\<br />
&\propto p^{R^{(t)}}(1-p)^{R^{(t)}} \cdot p^{x_t-1}(1-p)^{y_t-1} \\<br />
&\propto p^{x_t+R^{(t)}-1}(1-p)^{y_t+R^{(t)}-1} \\<br />
&\propto \beta(x_t+R^{(t)}, y_t+R^{(t)})<br />
\end{align*}<br />
$$<br />
<br />
[[File:thompson sampling coin example.png|thumb||||600px|Source: [https://www.quora.com/What-is-Thompson-sampling-in-laymans-terms Quora]]]<br />
<br />
This means that with successive sampling, our belief can become better at approximating the truth. There are similar update rules if we use a non Bernoulli setting, say, Gaussian. In the Gaussian case, we start with $\mathbb{P}_h^{(0)}=\mathbb{N}(0,1)$ and given that $\mathbb{P}_h^{(t)}\propto\mathbb{N}(\mu, \sigma)$ it is possible to show that the update rule looks like<br />
<br />
$$<br />
\mathbb{P}_h^{(t+1)} \propto \mathbb{N}\bigg(\frac{t\mu+R^{(t)}}{t+1},\frac{\sigma}{\sigma+1}\bigg)<br />
$$<br />
<br />
=== How can we use this in reinforcement learning? ===<br />
<br />
We can use this idea to decide when to explore and when to exploit. We start with an initial belief, choose an action, observe the reward and based on the kind of reward, we update our belief about what action to choose next.<br />
<br />
== Bootstrapping <sup>[[#References|[2,3]]]</sup> ==<br />
<br />
This idea may be unfamiliar to some people, so I thought it would be a good idea to include this. In statistics, bootstrapping is a method to generate new samples from a given sample. Suppose that we have a given population, and we want to study a property $\theta$ of the population. So, we just find $n$ sample points (sample $\{D_i\}_{i=1}^n$), calculate the estimator of the property, $\hat{\theta}$, for these $n$ points, and make our inference. <br />
<br />
If we later wish to find some property related to the estimator $\hat{\theta}$ itself, e.g. we want a bound of $\hat{\theta}$ such that $\delta_1 \leq \hat{\theta} \leq \delta_2$ with a confidence of $c=95%$, then we can use bootstrapping for this.<br />
<br />
Using bootstrapping, we can create a new sample $\{D'_i\}_{i=1}^{n'}$ by '''randomly sampling $n'$ times from $D$, with replacement'''. So, if $D=\{1,2,3,4\}$, a $D'$ of size $n'=10$ could be $\{1,4,4,3,2,2,2,1,3,4\}$. We do this a sufficient $k$ number of times, calculate $\hat{\theta}$ each time, and thus get a distribution $\{\hat{\theta}_i\}_{i=1}^k$. Now, we can choose the $100\cdot c$<sup>th</sup> and $100\cdot(1-c)$<sup>th</sup> percentile of this distribution, (let them be $\hat{\theta}_\alpha$ and $\hat{\theta}_\beta$ respectively) and say<br />
<br />
$$\hat{\theta}_\alpha \leq \hat{\theta} \leq \hat{\theta}_\beta, \mbox{with confidence }c$$<br />
<br />
== Why choose bootstrap and not dropout? ==<br />
<br />
There is previous work<sup>[[#References|[4]]]</sup> that establishes dropout as a good way to train NNs on a posterior such that the trained NN works like a function approximator that is close to the actual posterior. But, there are several problems with the predictions of this trained NN. The figures below are from the appendix of this paper. The left image is the NN trained by the authors of this paper on a sample noisy distribution and the right image is from the accompanying web demo from [[#References|[4]]], where the authors of [[#References|[4]]] show that their NN converges around the mean with a good confidence.<br />
<br />
[[File:dropout_results.png|thumb||center||700px|Source: this paper's appendix]]<br />
<br />
According to the authors of this paper,<br />
# Even though [[#References|[4]]] says that dropout converges arond the mean, their experiment actually behaves weirdly around a reasonable point like $x=0.75$. They think that this happens because dropout only affects the region local to the original data.<br />
# Samples from the NN trained on the original data do not look like a reasonable posterior (very spiky).<br />
# The trained NN collapses to zero uncertainty at the data points from the original data.<br />
<br />
== Q Learning and Deep Q Networks <sup>[[#References|[5]]]</sup> ==<br />
<br />
At any point of time, our rewards dictate what our actions should be. Also, in general, we want good long term rewards. For example, if we are playing a first person shooter game, it is a good idea to go out of cover to kill an enemy, even if some health is lost. Similarly, in reinforcement learning, we want to maximize our long term reward. So if at each time $t$, the reward is $r_t$, then a naive way is to say we want to maximise<br />
<br />
$$<br />
R_t = \sum_{i=0}^{\infty}r_t<br />
$$<br />
<br />
But, this reward is unbounded. So technically it could tend to $\infty$ in a lot of the cases. This is why we use a '''discounted reward'''.<br />
<br />
$$<br />
R_t = \sum_{i=0}^{\infty}\gamma^t r_t<br />
$$<br />
<br />
Here, we take $0\leq \gamma \lt 1$. So, what this means is that we value our current reward the most ($r_0$ has a coefficient of $1$), but we also consider the future possible rewards. So if we had two choices: get $+4$ now and $0$ at all other timesteps, or get $-2$ now and $+2$ after $3$ timesteps for $20$ timesteps, we choose the latter ($\gamma=0.9$). This is because $(+4) < (-2)+0.9^3(2+0.9\cdot2+\cdots+0.9^{19}\cdot2)$.<br />
<br />
<br />
A '''policy''' $\pi: \mathbb{S} \rightarrow \mathbb{A}$ is just a function that tells us what action to take in a given state $s\in \mathbb{S}$. Our goal is to find the best policy $\pi^*$ that maximises the reward from a given state $s$. So, a '''value function''' is defined from $s$ (which the agent is in, at timestep $t$) and following the policy $\pi$ as $V^\pi(s) = \mathbb{E}[R_t]$. The optimal value function is then simply<br />
<br />
$$<br />
V^*(s)=\displaystyle\max_{\pi}V^\pi(s)<br />
$$<br />
<br />
For convenience however, it is better to work with the '''Q function''' $Q: \mathbb{S}\times\mathbb{A} \rightarrow \mathbb{R}$. $Q$ is defined similarly as $V$. It is the expected return after taking an action $a$ in the given state $s$. So, $Q^\pi(s,a)=\mathbb{E}[R_t|s,a]$. The optimal $Q$ function is<br />
<br />
$$<br />
Q^*(s,a)=\displaystyle\max_{\pi}Q^\pi(s,a)<br />
$$<br />
<br />
Suppose that we know $Q^*$. Then, if we know that we are supposed to start at $s$ and take an action $a$ right now, what is the best course of action from the next time step? We just choose the optimal action $a'$ at the next state $s'$ that we reach. The optimal action $a'$ at state $s'$ is simply the argument $a_x$ that maximises our $Q^*(s',\cdot)$.<br />
<br />
$$<br />
a'=\displaystyle\arg\max_{a_x} Q^*(s',a_x)<br />
$$<br />
<br />
So, our best expected reward from $s$ taking action $a$ is $\mathbb{E}[r_t+\gamma\mathbb{E}[R_{t+1}]]$. This is known as the '''Bellman equation''' in optimal control problem (By the way, its continuous form is called '''Hamilton-Jacobi-Bellman equation''' or HJB equation, which is a very important partial differential equation):<br />
<br />
$$<br />
Q^*(s,a)=\mathbb{E}[r_t+\gamma \displaystyle\max_{a_x} Q^*(s',a_x)]<br />
$$<br />
<br />
In Q learning, we use a deep neural network with weights $\theta$ as a function approximator for $Q^*$, since Bellman equation is indeed a non-linear PDE and very difficult to solve numerically. The '''naive way''' to do this is to design a deep neural network that takes as input the state $s$ and action $a$, and produces an approximation to $Q^*$. <br />
<br />
* Suppose our neural net weights are $\theta_i$ at iteration $i$.<br />
* We want to train our neural net on the case when we are at $s$, take action $a$, get reward $r$, and reach $s'$.<br />
* To find out what action is best from $s'$, i.e. $a'$, we have to simulate all actions from $s'$. We can do this after we complete this iteration, then run $s',a_x$ for all $a_x\in\mathbb{A}$. But, we don't know how to complete this iteration without knowing this $a'$. So, another way is to simulate all actions from $s'$ using last known set of weights $\theta_{i-1}$. We just simulate state $s'$, action $a_x$ for all $a_x\in\mathbb{A}$ from the previous state and get $Q^*(s',a_x;\theta_{i-1})$. ('''Note''' that some papers do not use the set of weights from the previous iteration $\theta_{i-1}$. Instead they fix the weights for finding the best action for every $\tau$ steps to $\theta^-$, and do $Q^*(s',a_x;\theta^-)$ for $a_x\in\mathbb{A}$ and use this for the target value.)<br />
* Now we can compute our loss function using the Bellman equation, and backpropagate.<br />
$$<br />
\mbox{loss}=\mbox{target}-\mbox{prediction}=(r+\displaystyle\max_{a_x}Q^*(s',a_x;\theta_{i-1}))-Q^*(s,a;\theta_i)<br />
$$<br />
<br />
The '''problem''' with this approach is that at every iteration $i$, we have to do $|\mathbb{A}|$ forward passes on the previous set of weights $\theta_{i-1}$ to find out the best action $a'$ at $s'$. This becomes infeasible quickly with more possible actions.<br />
<br />
Authors of [[#References|[5]]] therefore use another kind of architecture. This architecture takes as input the state $s$, and computes the values $Q^*(s,a_x)$ for $a_x\in\mathbb{A}$. So there are $|\mathbb{A}|$ outputs. This basically parallelizes the forward passes so that $r+\displaystyle\max_{a_x}Q^*(s',a_x;\theta_{i-1})$ can be done with just a single pass through the outputs.<br />
<br />
<br />
[[File:DQN_arch.png|thumb||||600px|Source: [https://leonardoaraujosantos.gitbooks.io/artificial-inteligence/content/image_folder_7/DQNBreakoutBlocks.png leonardoaraujosantos.gitbooks.io]]]<br />
<br />
'''Note:''' When I say state $s$ as an input, I mean some representation of $s$. Since the environment is a partially observable MDP, it is hard to know $s$. So, we can for example, apply a CNN on the frames and get an idea of what the current state is. We pass this output to the input of the DNN (DNN is the fully connected layer for the CNN then).<br />
<br />
=== Experience Replay ===<br />
<br />
Authors of this paper borrow the concept of experience replay from [[#References|[5,6]]]. In experience replay, we do training in episodes. In each episode, we play and store consecutive $(s,a,r,s')$ tuples in the experience replay buffer. Then after the play, we choose random samples from this buffer and do our training.<br />
<br />
<br />
Advantages of experience replay over simple online Q learning<sup>[[#References|[5]]]</sup>:<br />
* '''Better data efficiency''': It is better to use one transition many times to learn again and again, rather than just learn once from it.<br />
* Learning from consecutive samples is difficult because of correlated data. Experience replay breaks this correlation.<br />
* Online learning means the input is decided by the previous action. So, if the maximising action is to go left in some game, next inputs would be about what happens when we go left. This can cause the optimiser to get stuck in a feedback loop, or even diverge, as [[#Reference|[7]]] points out.<br />
<br />
== Double Q Learning ==<br />
<br />
=== Problem with Q Learning<sup>[[#References|[8]]]</sup> ===<br />
<br />
For a simple neural network, each update tries to shift the current $Q^*$ estimate to a new value:<br />
<br />
$$<br />
Q^*(s,a) \leftarrow r+\gamma\displaystyle\max_{a_x}Q^*(s',a_x)<br />
$$<br />
<br />
Suppose the neural net has some inherent noise $\epsilon$. So, the neural net actually stores a value $\mathbb{Q}^*$ given by<br />
<br />
$$<br />
\mathbb{Q}^* = Q^*+\epsilon<br />
$$<br />
<br />
Even if $\epsilon$ has zero mean in the beginning, using the $\max$ operator at the update steps will start propagating $\gamma\cdot\max \mathbb{Q}^*$. This leads to a non zero mean subsequently. The problem is that "max causes overestimation because it does not preserve the zero-mean property of the errors of its operands." ([[#References|[8]]]) Thus, Q learning is more likely to choose overoptimistic values.<br />
<br />
=== How does Double Q Learning work? <sup>[[#References|[9]]]</sup> ===<br />
<br />
The problem can be solved by using two sets of weights $\theta$ and $\Theta$. The $\mbox{target}$ can be broken up as<br />
<br />
$$<br />
\mbox{target} = r+\displaystyle\max_{a_x}Q^*(s',a_x;\theta) = r+Q^*(s',\displaystyle\arg\max_{a_x}Q^*(s',a_x;\theta);\theta) = r+Q^*(s',a';\theta)<br />
$$<br />
<br />
Using double Q learning, we '''select''' the best action using current weights $\theta$ and '''evaluate''' the $Q^*$ value to decide the target value using $\Theta$.<br />
<br />
$$<br />
\mbox{target} = r+Q^*(s',\displaystyle\arg\max_{a_x}Q^*(s',a_x;\theta);\Theta) = r+Q^*(s',a';\Theta)<br />
$$<br />
<br />
This makes the evaluation fairer.<br />
<br />
=== Double Deep Q Learning ===<br />
<br />
[[#References|[9]]] further talks about how to use this for deep learning without much additional overhead. The suggestion is to use $\theta^-$ as $\Theta$.<br />
<br />
$$<br />
\mbox{target} = r+Q^*(s',\displaystyle\arg\max_{a_x}Q^*(s',a_x;\theta);\theta^-) = r+Q^*(s',a';\theta^-)<br />
$$<br />
<br />
== Bootstrapped DQN ==<br />
<br />
The authors propose an architecture that has a shared network and $K$ bootstrap heads. So, suppose our experience buffer $E$ has $n$ data points, where each datapoint is a $(s,a,r,s')$ tuple. Each bootstrap head trains on a different buffer $E_i$, where each $E_i$ has been constructed by sampling $n$ data points from the original experience buffer $E$ with replacement ('''bootstrap method''').<br />
<br />
<br />
Because each of the heads train on a different buffer, they model a different $Q^*$ function (say $Q^*_k$). Now, for each episode, we first choose a specific $Q^*_k=Q^*_s$. This $Q^*_s$ helps us create the experience buffer for the episode. From any state $s_t$, we populate the experience buffer by choosing the next action $a_t$ that maximises $Q^*_s$. (similar to '''Thompson Sampling''')<br />
<br />
$$<br />
a_t = \displaystyle\arg\max_a Q^*_s(s_t,a_t)<br />
$$<br />
<br />
Also, along with $s_t,a_t,r_t,s_{t+1}$, they push a bootstrap mask $m_t$. This mask is basically is a binary vector of size $K$, and it tells which $Q_k$ should be affected by this datapoint, if it is chosen as a training point. So, for example, if $K=5$ and there is a experience tuple $(s_t,a_t,r_t,s_{t+1},m_t)$ where $m_t=(0,1,1,0,1)$, then $(s_t,a_t,r_t,s_{t+1})$ should only affect $Q_2,Q_3$ and $Q_5$.<br />
<br />
<br />
So, at each iteration, we just choose few points from this buffer and train the respective $Q_{(\cdot)}$ based on the bootstrap masks.<br />
<br />
=== How to generate masks? ===<br />
<br />
Masks are created by sampling from the '''masking distribution'''. Now, there are many ways to choose this masking distribution:<br />
<br />
* If for each datapoint $D_i$ ($i=1$ to $n$), we mask from $\mbox{Bernoulli}(0.5)$, this will roughly allow us to have half the points from the original buffer. To get to size $n$, we duplicate these points by doubling the weights for each datapoint. This essentially gives us a '''double or nothing''' bootstrap<sup>[[#References|[10]]]</sup>.<br />
* If the mask is $(1, 1 \cdots 1)$, then this becomes an '''ensemble learning''' method.<br />
* $m_t~\mbox{Poi}(1)$ (poisson distribution)<br />
* $m_t[k]~\mbox{Exp}(1)$ (exponential distribution)<br />
<br />
For this paper's results, the authors used a $\mbox{Bernoulli}(p)$ distribution.<br />
<br />
== Related Work ==<br />
<br />
The authors mention the method described in [[#References|[11]]]. The authors of [[#References|[11]]] talk about the principle of "optimism in the face of uncertainty" and modify the reward function to encourage state-action pairs that have not been seen often:<br />
<br />
$$<br />
R(s,a) \leftarrow R(s,a)+\beta\cdot\mbox{novelty}(s,a)<br />
$$<br />
<br />
According to the authors, [[#References|[11]]]'s DQN algorithm relies on a lot of hand tuning and is only good for non stochastic problems. The authors further compare their results to [[#References|[11]]]'s results on Atari.<br />
<br />
<br />
The authors also mention an existing algorithm PSRL<sup>[[#References|[12,13]]]</sup>, or posterior sampling based RL. However, this algorithm requires a solved MDP, which is not feasible for large systems. Bootstrapped DQN approximates this idea by sampling from approximate $Q^*$ functions.<br />
<br />
<br />
Further, the authors mention that the work in [[#References|[12,13]]] has been followed by RLSVI<sup>[[#Reference|[14]]]</sup> which solves the problem for linear cases.<br />
<br />
== Deep Exploration: Why is Bootstrapped DQN so good at it? ==<br />
<br />
The authors consider a simple example to demonstrate the effectiveness of bootstrapped DQN at deep exploration.<br />
<br />
[[File:deep_exploration_example.png|thumb||center||700px|Source: this paper, section 5.1]]<br />
<br />
<br />
<br />
In this example, the agent starts at $s_2$. There are $N$ steps, and $N+9$ timesteps to generate the experience buffer. The agent is said to have learned the optimal policy if it achieves the best possible reward of $10$ (go to the rightmost state in $N-1$ timesteps, then stay there for $10$ timesteps), for at least $100$ such episodes. The results they got:<br />
<br />
[[File:deep_exploration_results.png|thumb||center||700px|Source: this paper, section 5.1]]<br />
<br />
<br />
<br />
The blue dots indicate when the agent learnt the optimal policy. If this took more than $2000$ episodes, they indicate it with a red dot. Thompson DQN is DQN with posterior sampling at every timestep. Ensemble DQN is same as bootstrapped DQN except that the mask is all $(1,1 \cdots 1)$. It is evident from the graphs that bootstrapped DQN can achieve deep exploration better than these two methods, and DQN.<br />
<br />
=== But why is it better? ===<br />
<br />
The authors say that this is because bootstrapped DQN constructs different approximations to the posterior $Q^*$ with the same initial data. This diversity of approximations is because of random initalization of weights for the $Q^*_k$ heads. This means that these heads start out trying random actions (because of diverse random initial $Q^*_k$), but when some head finds a good state and generalises to it, some (but not all) of the heads will learn from it, because of the bootstrapping. Eventually other heads will either find other good states, or end up learning the best good states found by the other heads.<br />
<br />
<br />
So, the architecture explores well and once a head achieves the optimal policy, eventually, all heads achieve the policy.<br />
<br />
== Results ==<br />
<br />
The authors test their architecture on 49 Atari games. They mention that there has been recent work to improve the performance of DDQNs, but those are tweaks whose intentions are orthogonal to this paper's idea. So, they don't compare their results with them.<br />
<br />
=== Scale: What values of $K$, $p$ are best? ===<br />
<br />
[[File:scale_k_p.png|thumb||center||800px|Source: this paper, section 6.1]]<br />
<br />
Recall that $K$ is the number of bootstrap heads and $p$ is the parameter for the masking distribution (Bernoulli). The authors say that around $K=10$, the performance reaches close to the peak, so it should be good.<br />
<br />
<br />
$p$ also represents the amount of data sharing. This is because lesser $p$ means there is lesser chance (due to the Bernoulli distribution) that the corresponding datapoint is taken into the bootstrapped dataset $D_i$. So, lesser $p$ means more identical datapoints, hence more heads share their datapoints. However, the value of $p$ doesn't seem to affect the rewards achieved over time. The authors give the following reasons for it:<br />
<br />
* The heads start with random weights for $Q^*$, so the targets (which use $Q^*$) turn out to be different. So the update rules are different.<br />
* Atari is deterministic.<br />
* Because of the initial diversity, the heads will learn differently even if they predict the same action for the given state.<br />
<br />
$p=1$ is the value they use finally, because this reduces the no. of identical datapoints and reduces time.<br />
<br />
=== Performance on Atari ===<br />
<br />
In general, the results tell us that bootstrapped DQN achieves better results.<br />
<br />
[[File:atari_results_bootstrapped_dqn.png|thumb||center||800px|Source: this paper, section 6.2]]<br />
<br />
The authors plot the improvement they achieved with bootstrapped DQN with the games. They define '''improvement''' to be $x$ if bootstrapped DQN achieves a better result than DQN in $\frac{1}{x}$ frames.<br />
<br />
[[File:bdqn_improvement.png|thumb||center||1000px|Source: this paper, section 6.2]]<br />
<br />
<br />
The authors say that bootstrapped DQN doesn't work good on all Atari games. They point out that there are some challenging games, where exploration is key but bootstrapped DQN doesn't do good enough (but does better than DQN). Some of these games are Frostbite and Montezuma’s Revenge. They say that even better exploration may help, but also point out that there may be other problems like: network instability, reward clipping and temporally extended rewards.<br />
<br />
=== Improvement: Highest Score Reached & how fast is this high score reached? ===<br />
<br />
The authors plot the improvement graphs after 20m and 200m frames.<br />
<br />
[[File:cumulative_rewards_bdqn.png|thumb||center||700px|Source: this paper, section 6.3]]<br />
<br />
=== Visualisation of Results ===<br />
<br />
One of the authors' [https://www.youtube.com/playlist?list=PLdy8eRAW78uLDPNo1jRv8jdTx7aup1ujM youtube playlist] can be found online.<br />
<br />
<br />
The authors also point out that just purely using bootstrapped DQN as an exploitative strategy is pretty good by itself, better than vanilla DQN. This is because of the deep exploration capabilities of bootstrapped DQN, since it can use the best states it knows and also plan to try out states it doesn't have any information about. Even in the videos, it can be seen that the heads agree at all the crucial decisions, but stay diverse at other less important steps.<br />
<br />
== Critique ==<br />
<br />
It would be very interesting and a great addition to the the experimental section of the paper, if the authors would have compared with asynchronous methods of exploration of the state space first introduced in [[#References|[15]]]. The authors unfortunately only compared their DQN with the original DQN and not all the other variations in the literature, and justified it by saying that their idea was "orthogonal" to these improvements.<br />
<br />
=== Different way to do exploration-exploitation? ===<br />
<br />
Instead of choosing the next action $a_t$ that maximises $Q^*_s$, they could have chosen different actions $a_i$ with probabilities<br />
<br />
$$<br />
\mathbb{P}(s_t,a_i) = \frac{Q^*_s(s_t,a_i)}{\displaystyle \sum_{i=1}^{|\mathbb{A}|} Q^*_s(s_t,a_i)}<br />
$$<br />
<br />
According to me, this is closer to Thompson Sampling.<br />
<br />
=== Why use Bernoulli? ===<br />
<br />
The choice of having a Bernoulli masking distribution eventually doesn't help them at all, since the algorithm does good because of the initial diversity. Maybe they can use some other masking distribution?<br />
<br />
=== Unanswered Questions & Miscellaneous ===<br />
* The Thompson DQN is not preferred because other randomized value functions can implement settings similar to Thompson sampling without the need for an intractable exact posterior update and also by working around the computational issue with Thompson Sampling: resampling every time step. Perhaps the authors could have explored Temporal Difference learning which is an attempt at combining Dynamic Programming and Monte Carlo methods.<br />
* The actual algorithm is hidden in the appendix. It could have been helpful if it were in the main paper.<br />
<br />
== References ==<br />
<br />
# [https://bandits.wikischolars.columbia.edu/file/view/Lecture+4.pdf Learning and optimization for sequential decision making, Columbia University, Lec 4]<br />
# [https://www.thoughtco.com/what-is-bootstrapping-in-statistics-3126172 Thoughtco, What is bootstrapping in statistics?]<br />
# [https://ocw.mit.edu/courses/mathematics/18-05-introduction-to-probability-and-statistics-spring-2014/readings/MIT18_05S14_Reading24.pdf Bootstrap confidence intervals, Class 24, 18.05, MIT Open Courseware]<br />
# [https://arxiv.org/abs/1506.02142 Yarin Gal and Zoubin Ghahramani. Dropout as a bayesian approximation: Representing model uncertainty in deep learning. arXiv preprint arXiv:1506.02142, 2015.]<br />
# [https://www.cs.toronto.edu/~vmnih/docs/dqn.pdf Mnih et al., Playing Atari with Deep Reinforcement Learning, 2015]<br />
# Long-Ji Lin. Reinforcement learning for robots using neural networks. Technical report, DTIC Document, 1993.<br />
# John N Tsitsiklis and Benjamin Van Roy. An analysis of temporal-difference learning with function approximation. Automatic Control, IEEE Transactions on, 42(5):674–690, 1997.<br />
# S. Thrun and A. Schwartz. Issues in using function approximation for reinforcement learning, 1993.<br />
# [https://arxiv.org/pdf/1509.06461.pdf Hado Van Hasselt, Arthur Guez, and David Silver. Deep reinforcement learning with double q-learning, 2015.]<br />
# [https://pdfs.semanticscholar.org/d623/c2cbf100d6963ba7dafe55158890d43c78b6.pdf Dean Eckles and Maurits Kaptein, Thompson Sampling with the Online Bootstrap, 2014, Pg 3]<br />
# [https://arxiv.org/abs/1507.00814 Bradly C. Stadie, Sergey Levine, Pieter Abbeel, Incentivizing Exploration In Reinforcement Learning With Deep Predictive Models, 2015.]<br />
# Ian Osband, Daniel Russo, and Benjamin Van Roy. (More) efficient reinforcement learning via posterior sampling, NIPS 2013.<br />
# Ian Osband and Benjamin Van Roy. Model-based reinforcement learning and the eluder dimension, NIPS 2014.<br />
# [https://arxiv.org/abs/1402.0635 Ian Osband, Benjamin Van Roy, Zheng Wen, Generalization and Exploration via Randomized Value Functions, 2014.]<br />
# Mnih, Volodymyr, et al. "Asynchronous methods for deep reinforcement learning." International Conference on Machine Learning. 2016.<br />
# George Konidaris, Sarah Osentoski, and Philip Thomas. 2011. Value function approximation in reinforcement learning using the fourier basis. In Proceedings of the Twenty-Fifth AAAI Conference on Artificial Intelligence (AAAI'11). AAAI Press 380-385.<br />
<br />
<br />
Other helpful links (unsorted):<br />
* [http://pemami4911.github.io/paper-summaries/deep-rl/2016/08/16/Deep-exploration.html pemami4911.github.io]<br />
* [http://www.stat.yale.edu/~pollard/Courses/241.fall97/Poisson.pdf Poisson Approximations]<br />
<br />
== Appendix ==<br />
<br />
=== Algorithm for Bootstrapped DQN ===<br />
The appendix lists the following algorithm. Periodically, the replay buffer is played back to update value function network Q.<br />
<br />
[[File:alg1.PNG|thumb||left||700px|Source: this paper's appendix]]</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Deep_Exploration_via_Bootstrapped_DQN&diff=30941Deep Exploration via Bootstrapped DQN2017-11-21T02:46:38Z<p>H4lyu: /* Q Learning and Deep Q Networks [5] */</p>
<hr />
<div>== Details ==<br />
<br />
'''Title''': Deep Exploration via Bootstrapped DQN<br />
<br />
'''Authors''': Ian Osband {1,2}, Charles Blundell {2}, Alexander Pritzel {2}, Benjamin Van Roy {1}<br />
<br />
'''Organisations''':<br />
# Stanford University<br />
# Google Deepmind<br />
<br />
'''Conference''': NIPS 2016<br />
<br />
'''URL''': [https://papers.nips.cc/paper/6501-deep-exploration-via-bootstrapped-dqn papers.nips.cc]<br />
<br />
'''Online code sources'''<br />
* [https://github.com/iassael/torch-bootstrapped-dqn github.com/iassael/torch-bootstrapped-dqn]<br />
<br />
This summary contains background knowledge from Section 2-7 (except Section 5). Feel free to skip if you already know.<br />
<br />
== Intro to Reinforcement Learning ==<br />
<br />
In reinforcement learning, an agent interacts with an environment with the goal to maximize its long term reward. A common application of reinforcement learning is to the [https://en.wikipedia.org/wiki/Multi-armed_bandit multi armed bandit problem]. In a multi armed bandit problem, there is a gambler and there are $n$ slot machines, and the gambler can choose to play any specific slot machine at any time. All the slot machines have their own probability distributions by which they churn out rewards, but this is unknown to the gambler. So the question is, how can the gambler learn how to get the maximum long term reward?<br />
<br />
There are two things the gambler can do at any instance: either he can try a new slot machine, or he can play the slot machine he has tried before (and he knows he will get some reward). However, even though trying a new slot machine feels like it would bring less reward to the gambler, it is possible that the gambler finds out a new slot machine that gives a better reward than the current best slot machine. This is the dilemma of '''exploration vs exploitation'''. Trying out a new slot machine is '''exploration''', while redoing the best move so far is '''exploiting''' the currently understood perception of the reward.<br />
<br />
[[File:multiarmedbandit.jpg|thumb|Source: [https://blogs.mathworks.com/images/loren/2016/multiarmedbandit.jpg blogs.mathworks.com]]]<br />
<br />
There are many strategies to approach this '''exploration-exploitation dilemma'''. Some [https://web.stanford.edu/class/msande338/lec9.pdf common strategies] for optimizing in an exploration-exploitation setting are Random Walk, Curiosity-Driven Exploration, and Thompson Sampling. A lot of these approaches are provably efficient, but assume that the state space is not very large. For instance, the approach called Curiosity-Driven Exploration aims to take actions that lead to immediate additional information. This requires the model to search “every possible cell in the grid” which is not desirable if state space is very large. Strategies for large state spaces often just either ignore exploration, or do something naive like $\epsilon$-greedy, where you exploit with $1-\epsilon$ probability and explore "randomly" in rest of the cases.<br />
<br />
This paper tries to use a Thompson sampling like approach to make decisions.<br />
<br />
== Thompson Sampling<sup>[[#References|[1]]]</sup> ==<br />
<br />
In Thompson sampling, our goal is to reach a belief that resembles the truth. Let's consider a case of coin tosses (2-armed bandit). Suppose we want to be able to reach a satisfactory pdf for $\mathbb{P}_h$ (heads). Assuming that this is a Bernoulli bandit problem, i.e. the rewards are $0$ or $1$, we can start off with $\mathbb{P}_h^{(0)}=\beta(1,1)$. The $\beta(x,y)$ distribution is a very good choice for a possible pdf because it works well for Bernoulli rewards. Further $\beta(1,1)$ is the uniform distribution $\mathbb{N}(0,1)$.<br />
<br />
Now, at every iteration $t$, we observe the reward $R^{(t)}$ and try to make our belief close to the truth by doing a Bayesian computation. Assuming $p$ is the probability of getting a heads,<br />
<br />
$$<br />
\begin{align*}<br />
\mathbb{P}(R|D) &\propto \mathbb{P}(D|R) \cdot \mathbb{P}(R) \\<br />
\mathbb{P}_h^{(t+1)}&\propto \mbox{likelihood}\cdot\mbox{prior} \\<br />
&\propto p^{R^{(t)}}(1-p)^{R^{(t)}} \cdot \mathbb{P}_h^{(t)} \\<br />
&\propto p^{R^{(t)}}(1-p)^{R^{(t)}} \cdot \beta(x_t, y_t) \\<br />
&\propto p^{R^{(t)}}(1-p)^{R^{(t)}} \cdot p^{x_t-1}(1-p)^{y_t-1} \\<br />
&\propto p^{x_t+R^{(t)}-1}(1-p)^{y_t+R^{(t)}-1} \\<br />
&\propto \beta(x_t+R^{(t)}, y_t+R^{(t)})<br />
\end{align*}<br />
$$<br />
<br />
[[File:thompson sampling coin example.png|thumb||||600px|Source: [https://www.quora.com/What-is-Thompson-sampling-in-laymans-terms Quora]]]<br />
<br />
This means that with successive sampling, our belief can become better at approximating the truth. There are similar update rules if we use a non Bernoulli setting, say, Gaussian. In the Gaussian case, we start with $\mathbb{P}_h^{(0)}=\mathbb{N}(0,1)$ and given that $\mathbb{P}_h^{(t)}\propto\mathbb{N}(\mu, \sigma)$ it is possible to show that the update rule looks like<br />
<br />
$$<br />
\mathbb{P}_h^{(t+1)} \propto \mathbb{N}\bigg(\frac{t\mu+R^{(t)}}{t+1},\frac{\sigma}{\sigma+1}\bigg)<br />
$$<br />
<br />
=== How can we use this in reinforcement learning? ===<br />
<br />
We can use this idea to decide when to explore and when to exploit. We start with an initial belief, choose an action, observe the reward and based on the kind of reward, we update our belief about what action to choose next.<br />
<br />
== Bootstrapping <sup>[[#References|[2,3]]]</sup> ==<br />
<br />
This idea may be unfamiliar to some people, so I thought it would be a good idea to include this. In statistics, bootstrapping is a method to generate new samples from a given sample. Suppose that we have a given population, and we want to study a measure $\theta$. So, we just find $n$ sample points (sample $\{D_i\}_{i=1}^n$), calculate this measure $\hat{\theta}$ for these $n$ points, and make our inference. <br />
<br />
If we later wish to find a better bound on $\hat{\theta}$, i.e. suppose we want to say that $\delta_1 \leq \hat{\theta} \leq \delta_2$ with a confidence of $c$, then we can use bootstrapping for this.<br />
<br />
Using bootstrapping, we can create a new sample $\{D'_i\}_{i=1}^{n'}$ by '''randomly sampling $n'$ times from $D$, with replacement'''. So, if $D=\{1,2,3,4\}$, a $D'$ of size $n'=10$ could be $\{1,4,4,3,2,2,2,1,3,4\}$. We do this a sufficient $k$ number of times, calculate $\hat{\theta}$ each time, and thus get a distribution $\{\hat{\theta}_i\}_{i=1}^k$. Now, we can choose the $100\cdot c$<sup>th</sup> and $100\cdot(1-c)$<sup>th</sup> percentile of this distribution, (let them be $\hat{\theta}_\alpha$ and $\hat{\theta}_\beta$ respectively) and say<br />
<br />
$$\hat{\theta}_\alpha \leq \hat{\theta} \leq \hat{\theta}_\beta, \mbox{with confidence }c$$<br />
<br />
== Why choose bootstrap and not dropout? ==<br />
<br />
There is previous work<sup>[[#References|[4]]]</sup> that establishes dropout as a good way to train NNs on a posterior such that the trained NN works like a function approximator that is close to the actual posterior. But, there are several problems with the predictions of this trained NN. The figures below are from the appendix of this paper. The left image is the NN trained by the authors of this paper on a sample noisy distribution and the right image is from the accompanying web demo from [[#References|[4]]], where the authors of [[#References|[4]]] show that their NN converges around the mean with a good confidence.<br />
<br />
[[File:dropout_results.png|thumb||center||700px|Source: this paper's appendix]]<br />
<br />
According to the authors of this paper,<br />
# Even though [[#References|[4]]] says that dropout converges arond the mean, their experiment actually behaves weirdly around a reasonable point like $x=0.75$. They think that this happens because dropout only affects the region local to the original data.<br />
# Samples from the NN trained on the original data do not look like a reasonable posterior (very spiky).<br />
# The trained NN collapses to zero uncertainty at the data points from the original data.<br />
<br />
== Q Learning and Deep Q Networks <sup>[[#References|[5]]]</sup> ==<br />
<br />
At any point of time, our rewards dictate what our actions should be. Also, in general, we want good long term rewards. For example, if we are playing a first person shooter game, it is a good idea to go out of cover to kill an enemy, even if some health is lost. Similarly, in reinforcement learning, we want to maximize our long term reward. So if at each time $t$, the reward is $r_t$, then a naive way is to say we want to maximise<br />
<br />
$$<br />
R_t = \sum_{i=0}^{\infty}r_t<br />
$$<br />
<br />
But, this reward is unbounded. So technically it could tend to $\infty$ in a lot of the cases. This is why we use a '''discounted reward'''.<br />
<br />
$$<br />
R_t = \sum_{i=0}^{\infty}\gamma^t r_t<br />
$$<br />
<br />
Here, we take $0\leq \gamma \lt 1$. So, what this means is that we value our current reward the most ($r_0$ has a coefficient of $1$), but we also consider the future possible rewards. So if we had two choices: get $+4$ now and $0$ at all other timesteps, or get $-2$ now and $+2$ after $3$ timesteps for $20$ timesteps, we choose the latter ($\gamma=0.9$). This is because $(+4) < (-2)+0.9^3(2+0.9\cdot2+\cdots+0.9^{19}\cdot2)$.<br />
<br />
<br />
A '''policy''' $\pi: \mathbb{S} \rightarrow \mathbb{A}$ is just a function that tells us what action to take in a given state $s\in \mathbb{S}$. Our goal is to find the best policy $\pi^*$ that maximises the reward from a given state $s$. So, a '''value function''' is defined from $s$ (which the agent is in, at timestep $t$) and following the policy $\pi$ as $V^\pi(s) = \mathbb{E}[R_t]$. The optimal value function is then simply<br />
<br />
$$<br />
V^*(s)=\displaystyle\max_{\pi}V^\pi(s)<br />
$$<br />
<br />
For convenience however, it is better to work with the '''Q function''' $Q: \mathbb{S}\times\mathbb{A} \rightarrow \mathbb{R}$. $Q$ is defined similarly as $V$. It is the expected return after taking an action $a$ in the given state $s$. So, $Q^\pi(s,a)=\mathbb{E}[R_t|s,a]$. The optimal $Q$ function is<br />
<br />
$$<br />
Q^*(s,a)=\displaystyle\max_{\pi}Q^\pi(s,a)<br />
$$<br />
<br />
Suppose that we know $Q^*$. Then, if we know that we are supposed to start at $s$ and take an action $a$ right now, what is the best course of action from the next time step? We just choose the optimal action $a'$ at the next state $s'$ that we reach. The optimal action $a'$ at state $s'$ is simply the argument $a_x$ that maximises our $Q^*(s',\cdot)$.<br />
<br />
$$<br />
a'=\displaystyle\arg\max_{a_x} Q^*(s',a_x)<br />
$$<br />
<br />
So, our best expected reward from $s$ taking action $a$ is $\mathbb{E}[r_t+\gamma\mathbb{E}[R_{t+1}]]$. This is known as the '''Bellman equation''' in optimal control problem (By the way, its continuous form is called '''Hamilton-Jacobi-Bellman equation''' or HJB equation, which is a very important partial differential equation):<br />
<br />
$$<br />
Q^*(s,a)=\mathbb{E}[r_t+\gamma \displaystyle\max_{a_x} Q^*(s',a_x)]<br />
$$<br />
<br />
In Q learning, we use a deep neural network with weights $\theta$ as a function approximator for $Q^*$, since Bellman equation is indeed a non-linear PDE and very difficult to solve numerically. The '''naive way''' to do this is to design a deep neural network that takes as input the state $s$ and action $a$, and produces an approximation to $Q^*$. <br />
<br />
* Suppose our neural net weights are $\theta_i$ at iteration $i$.<br />
* We want to train our neural net on the case when we are at $s$, take action $a$, get reward $r$, and reach $s'$.<br />
* To find out what action is best from $s'$, i.e. $a'$, we have to simulate all actions from $s'$. We can do this after we complete this iteration, then run $s',a_x$ for all $a_x\in\mathbb{A}$. But, we don't know how to complete this iteration without knowing this $a'$. So, another way is to simulate all actions from $s'$ using last known set of weights $\theta_{i-1}$. We just simulate state $s'$, action $a_x$ for all $a_x\in\mathbb{A}$ from the previous state and get $Q^*(s',a_x;\theta_{i-1})$. ('''Note''' that some papers do not use the set of weights from the previous iteration $\theta_{i-1}$. Instead they fix the weights for finding the best action for every $\tau$ steps to $\theta^-$, and do $Q^*(s',a_x;\theta^-)$ for $a_x\in\mathbb{A}$ and use this for the target value.)<br />
* Now we can compute our loss function using the Bellman equation, and backpropagate.<br />
$$<br />
\mbox{loss}=\mbox{target}-\mbox{prediction}=(r+\displaystyle\max_{a_x}Q^*(s',a_x;\theta_{i-1}))-Q^*(s,a;\theta_i)<br />
$$<br />
<br />
The '''problem''' with this approach is that at every iteration $i$, we have to do $|\mathbb{A}|$ forward passes on the previous set of weights $\theta_{i-1}$ to find out the best action $a'$ at $s'$. This becomes infeasible quickly with more possible actions.<br />
<br />
Authors of [[#References|[5]]] therefore use another kind of architecture. This architecture takes as input the state $s$, and computes the values $Q^*(s,a_x)$ for $a_x\in\mathbb{A}$. So there are $|\mathbb{A}|$ outputs. This basically parallelizes the forward passes so that $r+\displaystyle\max_{a_x}Q^*(s',a_x;\theta_{i-1})$ can be done with just a single pass through the outputs.<br />
<br />
<br />
[[File:DQN_arch.png|thumb||||600px|Source: [https://leonardoaraujosantos.gitbooks.io/artificial-inteligence/content/image_folder_7/DQNBreakoutBlocks.png leonardoaraujosantos.gitbooks.io]]]<br />
<br />
'''Note:''' When I say state $s$ as an input, I mean some representation of $s$. Since the environment is a partially observable MDP, it is hard to know $s$. So, we can for example, apply a CNN on the frames and get an idea of what the current state is. We pass this output to the input of the DNN (DNN is the fully connected layer for the CNN then).<br />
<br />
=== Experience Replay ===<br />
<br />
Authors of this paper borrow the concept of experience replay from [[#References|[5,6]]]. In experience replay, we do training in episodes. In each episode, we play and store consecutive $(s,a,r,s')$ tuples in the experience replay buffer. Then after the play, we choose random samples from this buffer and do our training.<br />
<br />
<br />
Advantages of experience replay over simple online Q learning<sup>[[#References|[5]]]</sup>:<br />
* '''Better data efficiency''': It is better to use one transition many times to learn again and again, rather than just learn once from it.<br />
* Learning from consecutive samples is difficult because of correlated data. Experience replay breaks this correlation.<br />
* Online learning means the input is decided by the previous action. So, if the maximising action is to go left in some game, next inputs would be about what happens when we go left. This can cause the optimiser to get stuck in a feedback loop, or even diverge, as [[#Reference|[7]]] points out.<br />
<br />
== Double Q Learning ==<br />
<br />
=== Problem with Q Learning<sup>[[#References|[8]]]</sup> ===<br />
<br />
For a simple neural network, each update tries to shift the current $Q^*$ estimate to a new value:<br />
<br />
$$<br />
Q^*(s,a) \leftarrow r+\gamma\displaystyle\max_{a_x}Q^*(s',a_x)<br />
$$<br />
<br />
Suppose the neural net has some inherent noise $\epsilon$. So, the neural net actually stores a value $\mathbb{Q}^*$ given by<br />
<br />
$$<br />
\mathbb{Q}^* = Q^*+\epsilon<br />
$$<br />
<br />
Even if $\epsilon$ has zero mean in the beginning, using the $\max$ operator at the update steps will start propagating $\gamma\cdot\max \mathbb{Q}^*$. This leads to a non zero mean subsequently. The problem is that "max causes overestimation because it does not preserve the zero-mean property of the errors of its operands." ([[#References|[8]]]) Thus, Q learning is more likely to choose overoptimistic values.<br />
<br />
=== How does Double Q Learning work? <sup>[[#References|[9]]]</sup> ===<br />
<br />
The problem can be solved by using two sets of weights $\theta$ and $\Theta$. The $\mbox{target}$ can be broken up as<br />
<br />
$$<br />
\mbox{target} = r+\displaystyle\max_{a_x}Q^*(s',a_x;\theta) = r+Q^*(s',\displaystyle\arg\max_{a_x}Q^*(s',a_x;\theta);\theta) = r+Q^*(s',a';\theta)<br />
$$<br />
<br />
Using double Q learning, we '''select''' the best action using current weights $\theta$ and '''evaluate''' the $Q^*$ value to decide the target value using $\Theta$.<br />
<br />
$$<br />
\mbox{target} = r+Q^*(s',\displaystyle\arg\max_{a_x}Q^*(s',a_x;\theta);\Theta) = r+Q^*(s',a';\Theta)<br />
$$<br />
<br />
This makes the evaluation fairer.<br />
<br />
=== Double Deep Q Learning ===<br />
<br />
[[#References|[9]]] further talks about how to use this for deep learning without much additional overhead. The suggestion is to use $\theta^-$ as $\Theta$.<br />
<br />
$$<br />
\mbox{target} = r+Q^*(s',\displaystyle\arg\max_{a_x}Q^*(s',a_x;\theta);\theta^-) = r+Q^*(s',a';\theta^-)<br />
$$<br />
<br />
== Bootstrapped DQN ==<br />
<br />
The authors propose an architecture that has a shared network and $K$ bootstrap heads. So, suppose our experience buffer $E$ has $n$ datapoints, where each datapoint is a $(s,a,r,s')$ tuple. Each bootstrap head trains on a different buffer $E_i$, where each $E_i$ has been constructed by sampling $n$ datapoints from the original experience buffer $E$ with replacement ('''bootstrap method''').<br />
<br />
<br />
Because each of the heads train on a different buffer, they model a different $Q^*$ function (say $Q^*_k$). Now, for each episode, we first choose a specific $Q^*_k=Q^*_s$. This $Q^*_s$ helps us create the experience buffer for the episode. From any state $s_t$, we populate the experience buffer by choosing the next action $a_t$ that maximises $Q^*_s$. (similar to '''Thompson Sampling''')<br />
<br />
$$<br />
a_t = \displaystyle\arg\max_a Q^*_s(s_t,a_t)<br />
$$<br />
<br />
Also, along with $s_t,a_t,r_t,s_{t+1}$, they push a bootstrap mask $m_t$. This mask is basically is a binary vector of size $K$, and it tells which $Q_k$ should be affected by this datapoint, if it is chosen as a training point. So, for example, if $K=5$ and there is a experience tuple $(s_t,a_t,r_t,s_{t+1},m_t)$ where $m_t=(0,1,1,0,1)$, then $(s_t,a_t,r_t,s_{t+1})$ should only affect $Q_2,Q_3$ and $Q_5$.<br />
<br />
<br />
So, at each iteration, we just choose few points from this buffer and train the respective $Q_{(\cdot)}$ based on the bootstrap masks.<br />
<br />
=== How to generate masks? ===<br />
<br />
Masks are created by sampling from the '''masking distribution'''. Now, there are many ways to choose this masking distribution:<br />
<br />
* If for each datapoint $D_i$ ($i=1$ to $n$), we mask from $\mbox{Bernoulli}(0.5)$, this will roughly allow us to have half the points from the original buffer. To get to size $n$, we duplicate these points by doubling the weights for each datapoint. This essentially gives us a '''double or nothing''' bootstrap<sup>[[#References|[10]]]</sup>.<br />
* If the mask is $(1, 1 \cdots 1)$, then this becomes an '''ensemble learning''' method.<br />
* $m_t~\mbox{Poi}(1)$ (poisson distribution)<br />
* $m_t[k]~\mbox{Exp}(1)$ (exponential distribution)<br />
<br />
For this paper's results, the authors used a $\mbox{Bernoulli}(p)$ distribution.<br />
<br />
== Related Work ==<br />
<br />
The authors mention the method described in [[#References|[11]]]. The authors of [[#References|[11]]] talk about the principle of "optimism in the face of uncertainty" and modify the reward function to encourage state-action pairs that have not been seen often:<br />
<br />
$$<br />
R(s,a) \leftarrow R(s,a)+\beta\cdot\mbox{novelty}(s,a)<br />
$$<br />
<br />
According to the authors, [[#References|[11]]]'s DQN algorithm relies on a lot of hand tuning and is only good for non stochastic problems. The authors further compare their results to [[#References|[11]]]'s results on Atari.<br />
<br />
<br />
The authors also mention an existing algorithm PSRL<sup>[[#References|[12,13]]]</sup>, or posterior sampling based RL. However, this algorithm requires a solved MDP, which is not feasible for large systems. Bootstrapped DQN approximates this idea by sampling from approximate $Q^*$ functions.<br />
<br />
<br />
Further, the authors mention that the work in [[#References|[12,13]]] has been followed by RLSVI<sup>[[#Reference|[14]]]</sup> which solves the problem for linear cases.<br />
<br />
== Deep Exploration: Why is Bootstrapped DQN so good at it? ==<br />
<br />
The authors consider a simple example to demonstrate the effectiveness of bootstrapped DQN at deep exploration.<br />
<br />
[[File:deep_exploration_example.png|thumb||center||700px|Source: this paper, section 5.1]]<br />
<br />
<br />
<br />
In this example, the agent starts at $s_2$. There are $N$ steps, and $N+9$ timesteps to generate the experience buffer. The agent is said to have learned the optimal policy if it achieves the best possible reward of $10$ (go to the rightmost state in $N-1$ timesteps, then stay there for $10$ timesteps), for at least $100$ such episodes. The results they got:<br />
<br />
[[File:deep_exploration_results.png|thumb||center||700px|Source: this paper, section 5.1]]<br />
<br />
<br />
<br />
The blue dots indicate when the agent learnt the optimal policy. If this took more than $2000$ episodes, they indicate it with a red dot. Thompson DQN is DQN with posterior sampling at every timestep. Ensemble DQN is same as bootstrapped DQN except that the mask is all $(1,1 \cdots 1)$. It is evident from the graphs that bootstrapped DQN can achieve deep exploration better than these two methods, and DQN.<br />
<br />
=== But why is it better? ===<br />
<br />
The authors say that this is because bootstrapped DQN constructs different approximations to the posterior $Q^*$ with the same initial data. This diversity of approximations is because of random initalization of weights for the $Q^*_k$ heads. This means that these heads start out trying random actions (because of diverse random initial $Q^*_k$), but when some head finds a good state and generalises to it, some (but not all) of the heads will learn from it, because of the bootstrapping. Eventually other heads will either find other good states, or end up learning the best good states found by the other heads.<br />
<br />
<br />
So, the architecture explores well and once a head achieves the optimal policy, eventually, all heads achieve the policy.<br />
<br />
== Results ==<br />
<br />
The authors test their architecture on 49 Atari games. They mention that there has been recent work to improve the performance of DDQNs, but those are tweaks whose intentions are orthogonal to this paper's idea. So, they don't compare their results with them.<br />
<br />
=== Scale: What values of $K$, $p$ are best? ===<br />
<br />
[[File:scale_k_p.png|thumb||center||800px|Source: this paper, section 6.1]]<br />
<br />
Recall that $K$ is the number of bootstrap heads and $p$ is the parameter for the masking distribution (Bernoulli). The authors say that around $K=10$, the performance reaches close to the peak, so it should be good.<br />
<br />
<br />
$p$ also represents the amount of data sharing. This is because lesser $p$ means there is lesser chance (due to the Bernoulli distribution) that the corresponding datapoint is taken into the bootstrapped dataset $D_i$. So, lesser $p$ means more identical datapoints, hence more heads share their datapoints. However, the value of $p$ doesn't seem to affect the rewards achieved over time. The authors give the following reasons for it:<br />
<br />
* The heads start with random weights for $Q^*$, so the targets (which use $Q^*$) turn out to be different. So the update rules are different.<br />
* Atari is deterministic.<br />
* Because of the initial diversity, the heads will learn differently even if they predict the same action for the given state.<br />
<br />
$p=1$ is the value they use finally, because this reduces the no. of identical datapoints and reduces time.<br />
<br />
=== Performance on Atari ===<br />
<br />
In general, the results tell us that bootstrapped DQN achieves better results.<br />
<br />
[[File:atari_results_bootstrapped_dqn.png|thumb||center||800px|Source: this paper, section 6.2]]<br />
<br />
The authors plot the improvement they achieved with bootstrapped DQN with the games. They define '''improvement''' to be $x$ if bootstrapped DQN achieves a better result than DQN in $\frac{1}{x}$ frames.<br />
<br />
[[File:bdqn_improvement.png|thumb||center||1000px|Source: this paper, section 6.2]]<br />
<br />
<br />
The authors say that bootstrapped DQN doesn't work good on all Atari games. They point out that there are some challenging games, where exploration is key but bootstrapped DQN doesn't do good enough (but does better than DQN). Some of these games are Frostbite and Montezuma’s Revenge. They say that even better exploration may help, but also point out that there may be other problems like: network instability, reward clipping and temporally extended rewards.<br />
<br />
=== Improvement: Highest Score Reached & how fast is this high score reached? ===<br />
<br />
The authors plot the improvement graphs after 20m and 200m frames.<br />
<br />
[[File:cumulative_rewards_bdqn.png|thumb||center||700px|Source: this paper, section 6.3]]<br />
<br />
=== Visualisation of Results ===<br />
<br />
One of the authors' [https://www.youtube.com/playlist?list=PLdy8eRAW78uLDPNo1jRv8jdTx7aup1ujM youtube playlist] can be found online.<br />
<br />
<br />
The authors also point out that just purely using bootstrapped DQN as an exploitative strategy is pretty good by itself, better than vanilla DQN. This is because of the deep exploration capabilities of bootstrapped DQN, since it can use the best states it knows and also plan to try out states it doesn't have any information about. Even in the videos, it can be seen that the heads agree at all the crucial decisions, but stay diverse at other less important steps.<br />
<br />
== Critique ==<br />
<br />
It would be very interesting and a great addition to the the experimental section of the paper, if the authors would have compared with asynchronous methods of exploration of the state space first introduced in [[#References|[15]]]. The authors unfortunately only compared their DQN with the original DQN and not all the other variations in the literature, and justified it by saying that their idea was "orthogonal" to these improvements.<br />
<br />
=== Different way to do exploration-exploitation? ===<br />
<br />
Instead of choosing the next action $a_t$ that maximises $Q^*_s$, they could have chosen different actions $a_i$ with probabilities<br />
<br />
$$<br />
\mathbb{P}(s_t,a_i) = \frac{Q^*_s(s_t,a_i)}{\displaystyle \sum_{i=1}^{|\mathbb{A}|} Q^*_s(s_t,a_i)}<br />
$$<br />
<br />
According to me, this is closer to Thompson Sampling.<br />
<br />
=== Why use Bernoulli? ===<br />
<br />
The choice of having a Bernoulli masking distribution eventually doesn't help them at all, since the algorithm does good because of the initial diversity. Maybe they can use some other masking distribution?<br />
<br />
=== Unanswered Questions & Miscellaneous ===<br />
* Why does Thompson DQN perform poorly?<br />
* The actual algorithm is hidden in the appendix. It could have been helpful if it were in the main paper.<br />
<br />
== References ==<br />
<br />
# [https://bandits.wikischolars.columbia.edu/file/view/Lecture+4.pdf Learning and optimization for sequential decision making, Columbia University, Lec 4]<br />
# [https://www.thoughtco.com/what-is-bootstrapping-in-statistics-3126172 Thoughtco, What is bootstrapping in statistics?]<br />
# [https://ocw.mit.edu/courses/mathematics/18-05-introduction-to-probability-and-statistics-spring-2014/readings/MIT18_05S14_Reading24.pdf Bootstrap confidence intervals, Class 24, 18.05, MIT Open Courseware]<br />
# [https://arxiv.org/abs/1506.02142 Yarin Gal and Zoubin Ghahramani. Dropout as a bayesian approximation: Representing model uncertainty in deep learning. arXiv preprint arXiv:1506.02142, 2015.]<br />
# [https://www.cs.toronto.edu/~vmnih/docs/dqn.pdf Mnih et al., Playing Atari with Deep Reinforcement Learning, 2015]<br />
# Long-Ji Lin. Reinforcement learning for robots using neural networks. Technical report, DTIC Document, 1993.<br />
# John N Tsitsiklis and Benjamin Van Roy. An analysis of temporal-difference learning with function approximation. Automatic Control, IEEE Transactions on, 42(5):674–690, 1997.<br />
# S. Thrun and A. Schwartz. Issues in using function approximation for reinforcement learning, 1993.<br />
# [https://arxiv.org/pdf/1509.06461.pdf Hado Van Hasselt, Arthur Guez, and David Silver. Deep reinforcement learning with double q-learning, 2015.]<br />
# [https://pdfs.semanticscholar.org/d623/c2cbf100d6963ba7dafe55158890d43c78b6.pdf Dean Eckles and Maurits Kaptein, Thompson Sampling with the Online Bootstrap, 2014, Pg 3]<br />
# [https://arxiv.org/abs/1507.00814 Bradly C. Stadie, Sergey Levine, Pieter Abbeel, Incentivizing Exploration In Reinforcement Learning With Deep Predictive Models, 2015.]<br />
# Ian Osband, Daniel Russo, and Benjamin Van Roy. (More) efficient reinforcement learning via posterior sampling, NIPS 2013.<br />
# Ian Osband and Benjamin Van Roy. Model-based reinforcement learning and the eluder dimension, NIPS 2014.<br />
# [https://arxiv.org/abs/1402.0635 Ian Osband, Benjamin Van Roy, Zheng Wen, Generalization and Exploration via Randomized Value Functions, 2014.]<br />
# Mnih, Volodymyr, et al. "Asynchronous methods for deep reinforcement learning." International Conference on Machine Learning. 2016.<br />
Other helpful links (unsorted):<br />
* [http://pemami4911.github.io/paper-summaries/deep-rl/2016/08/16/Deep-exploration.html pemami4911.github.io]<br />
* [http://www.stat.yale.edu/~pollard/Courses/241.fall97/Poisson.pdf Poisson Approximations]<br />
<br />
== Appendix ==<br />
<br />
=== Algorithm for Bootstrapped DQN ===<br />
The appendix lists the following algorithm. Periodically, the replay buffer is played back to update value function network Q.<br />
<br />
[[File:alg1.PNG|thumb||left||700px|Source: this paper's appendix]]</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Unsupervised_Domain_Adaptation_with_Residual_Transfer_Networks&diff=30938Unsupervised Domain Adaptation with Residual Transfer Networks2017-11-21T02:02:42Z<p>H4lyu: /* Conclusion */</p>
<hr />
<div>== Introduction ==<br />
'''Domain Adaptation''' [https://en.wikipedia.org/wiki/Domain_adaptation]is a problem in machine learning which involves taking a model which has been trained on a source domain, and applying this to a different (but related) target domain. '''Unsupervised domain adaptation''' refers to the situation in which the source data is labelled, while the target data is (predominantly) unlabeled. The problem at hand is then finding ways to generalize the learning on the source domain to the target domain. In the age of deep networks this problem has become particularly salient due to the need for vast amounts of labeled training data, in order to reap the benefits of deep learning. Manual generation of labeled data is often prohibitive, and in absence of such data networks are rarely performant. The attempt to circumvent this drought of data typically necessitates the gathering of "off-the-shelf" data sets, which are tangentially related and contain labels, and then building models in these domains. The fundamental issue that unsupervised domain adaptation attempts to address is overcoming the inherent shift in distribution across the domains, without the ability to observe this shift directly.<br />
<br />
This paper proposes a method for unsupervised domain adaptation which relies on three key components: <br />
# A kernel-based penalty to ensure that the abstract representations generated by the networks hidden layers are similar between the source and the target data; <br />
# An entropy based penalty on the target classifier, which exploits the entropy minimization principle; and <br />
# A residual network structure is appended, which allows the source and target classifiers to differ by a (learned) residual function, thus relaxing the shared classifier assumption which is traditionally made.<br />
<br />
This method outperforms state-of-the-art techniques on common benchmark datasets, and is flexible enough to be applied in most feed-forward neural networks.<br />
<br />
[[File:Source-and-Target-Domain-Office-31-Backpack.png|thumb|right|The Office-31 Dataset Images for Backpack. Shows the variation in the source and target domains to motivate why these methods are important.]] <br />
=== Working Example (Office-31) === <br />
In order to assist in the understanding of the methods, it is helpful to have a tangible sense of the problem front of mind. The Domain Adaptation Project [https://people.eecs.berkeley.edu/~jhoffman/domainadapt/] provides data sets which are tailored to the problem of unsupervised domain adaptation. One of these data sets (which is later used in the experiments of this paper) has images which are labeled based on the Amazon product page for the various items. There are then corresponding pictures taken either by webcams or digital SLR cameras. The goal of unsupervised domain adaptation on this data set would be to take any of the three image sources as the source domain, and transfer a classifier to the other domain; see the example images to understand the differences.<br />
<br />
One can imagine that, while it is likely easy to scrape labeled images from Amazon, it is likely far more difficult to collect labeled images from webcam or DSLR pictures directly. The ultimate goal of this method would be to train a model to recognize a picture of a backpack taken with a webcam, based on images of backpacks scraped from Amazon (or similar tasks).<br />
<br />
== Related Work ==<br />
Broadly speaking, the problem of domain adaptation mitigates manual labeling of data in areas such as machine learning, computer vision, and natural language processing. The general goal of domain adaptation is to reduce the discrepancy in probability distributions between the source and target domains.<br />
<br />
Research into the use of Deep Neural Networks for the purpose of domain adaptation has suggested that, while networks learn abstract feature representations which can reduce the discrepancy across domains, it is not possible to wholly remove it [http://www.icml-2011.org/papers/342_icmlpaper.pdf], [https://arxiv.org/pdf/1412.3474.pdf]. Further work has been done to design networks which adapt traditional deep nets (typically CNNs) to specifically address the problems posed by domain adaptation, these methods all only address the issue of feature adaptation [https://arxiv.org/pdf/1502.02791.pdf], [https://arxiv.org/pdf/1409.7495.pdf], [https://people.eecs.berkeley.edu/~jhoffman/papers/Tzeng_ICCV2015.pdf]. That is, they all assume that the target and source classifiers are shared between domains. <br />
<br />
The authors drew particular motivation from He et al. [https://arxiv.org/abs/1512.03385] with the proposed structure of residual networks. Combining the insights from the ResNet architecture, in addition to previous work that had leveraged classifier adaptation (in the context where some target data is labeled) [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.130.8224&rep=rep1&type=pdf], [http://www.machinelearning.org/archive/icml2009/papers/445.pdf], [http://ieeexplore.ieee.org/document/5539870/] the authors develop their proposed network.<br />
<br />
== Residual Transfer Networks ==<br />
Generally, in an unsupervised domain adaptation problem, we are dealing with a set $\mathcal{D}_s$ (called the source domain) which is defined by $\{(x_i^s, y_i^s)\}_{i=1}^{n_s}$. That is the set of all labeled input-output pairs in our source data set. We denote the number of source elements by $n_s$. There is a corresponding set $\mathcal{D}_t = \{(x_i^t)\}_{i=1}^{n_t}$ (the target domain), consisting of unlabeled input values. There are $n_t$ such values. <br />
[[File:RTN-Structure.png|thumb|left|upright|The overarching structure of the RTN. Consists of an existing network, to which a bottleneck, MMD block, and residual block is appended.]]<br />
We can think of $\mathcal{D}_s$ as being sampled from some underlying distribution $p$, and $\mathcal{D}_t$ as being sampled from $q$. Generally we have that $p \neq q$, partially motivating the need for domain adaptation methods. <br />
<br />
We can consider the classifiers $f_s(\underline{x})$ and $f_t(\underline{x})$, for the source domain and target domain respectively. It is possible to learn $f_s$ based on the sample $\mathcal{D}_s$. Under the '''shared classifier assumption''' it would be the case that $f_s(\underline{x}) = f_t(\underline{x})$, and thus learning the source classifier is enough. This method relaxes this assumption, assuming that in general $f_s \neq f_t$, and attempting to learn both.<br />
<br />
The example network extends deep convolutional networks (in this case AlexNet [http://vision.stanford.edu/teaching/cs231b_spring1415/slides/alexnet_tugce_kyunghee.pdf]) to '''Residual Transfer Networks''', the mechanics of which are outlined below. Recall that, if $L(\cdot, \cdot)$ is taken to be the cross-entropy loss function, then the empirical error of a CNN on the source domain $\mathcal{D}_s$ is given by:<br />
<br />
<center><br />
<math display="block"><br />
\min_{f_s} \frac{1}{n_s} \sum_{i=1}^{n_s} L(f_s(x_i^s), y_i^s)<br />
</math> <br />
</center><br />
<br />
In a standard implementation, the CNN optimizes over the above loss. This will be the starting point for the RTN.<br />
<br />
=== Structural Overview ===<br />
The model proposed in this paper extends existing CNN's and alters the loss function that is optimized over. While each of these components is discussed in depth below, the overarching architecture involves four components:<br />
<br />
# An existing deep model. While this can be any model, in theory, the authors leverage AlexNet in practice.<br />
# A bottleneck layer, used to reduce the dimensionality of the learned abstract feature space, directly after the existing network.<br />
# An MMD block, with the expressed intention of feature adaptation.<br />
# A residual block, with the expressed intention of classifier adaptation. <br />
<br />
This structure is then optimized against a loss function which combines the standard cross-entropy penalty with MMD and target entropy penalties, yielding the proposed Residual Transfer Network (RTN) structure.<br />
<br />
=== Feature Adaptation ===<br />
Feature adaptation refers to the process in which the features which are learned to represent the source domain are made applicable to the target domain. Broadly speaking a CNN works to generate abstract feature representations of the distribution that the inputs are sampled from. It has been found that using these deep features can reduce, but not remove, cross-domain distribution discrepancy, hence the need for feature adaptation. It is important to note that CNN's transfer from general to specific features as the network gets deeper. In this light, the discrepancy between the feature representation of the source and the target will grow through a deeper convolutional net. As such a technique for forcing these distributions to be similar is needed.<br />
<br />
In particular the authors of this paper impose a bottleneck layer (call it $fc_b$) which is included after the final convolutional layer of AlexNet. This dense layer is connected to an additional dense layer $fc_c$, (which will serve as the target classification layer). They then compute the tensor product between the activations of the layers, performing "lossless multi-layer feature fusion". That is for the source domain they define $z_i^s \overset{\underset{\mathrm{def}}{}}{=} x_i^{s,fc_b}\otimes x_i^{s,fc_c}$ and for the target domain, $z_i^t \overset{\underset{\mathrm{def}}{}}{=} x_i^{t,fc_b}\otimes x_i^{t,fc_c}$. The authors then employ feature adaptation by means of Maximum Mean Discrepancy, between the source and target domains, on these fusion features.<br />
<br />
[[File:RTN-MMD-Block.png|right|thumb|The Maximum Mean Discrepancy Block (MMD) included in the RTN. The outputs of $fc_b$ and $fc_c$ are fused through a tensor product, and then passed through the MMD penalty, ensuring distributional similarity.]]<br />
<br />
==== Maximum Mean Discrepancy ==== <br />
The Maximum Mean Discrepancy (MMD) is a Kernel method involes mapping to a Reproducing Kernel Hilbert Space (RKHS) [https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space]. Denote the RKHS $\mathcal{H}_K$ with a characteristic kernel $K$. We then define the '''mean embedding''' of a distribution $p$ in $\mathcal{H}_K$ to be the unique element $\mu_K(p)$ such that $\mathbf{E}_{x\sim p}f(x) = \langle f(x), \mu_K(p)\rangle_{\mathcal{H}_K}$ for all $f \in \mathcal{H}_K$. Now, if we take $\phi: \mathcal{X} \to \mathcal{H}_K$, then we can define the MMD between two distributions $p$ and $q$ as follows:<br />
<br />
<center><br />
<math display="block"><br />
d_k(p, q) \overset{\underset{\mathrm{def}}{}}{=} ||\mathbf{E}_{x\sim p}(\phi(x^s)) - \mathbf{E}_{x\sim q}(\phi(x^t))||_{\mathcal{H}_K}<br />
</math><br />
</center><br />
<br />
Effectively, the MMD will compute the self-similarity of $p$ and $q$, and subtract twice the cross-similarity between the distributions: $\widehat{\text{MMD}}^2 = \text{mean}(K_{pp}) + \text{mean}(K_{qq}) - 2\times\text{mean}(K_{pq})$. From here we can infer that $p$ and $q$ are equivalent distributions if and only if the $\text{MMD} = 0$. If we then wish to force two distributions to be similar, this becomes a minimization problem over the MMD.<br />
<br />
Two important notes:<br />
# The RKHS, and as such MMD, depend on the choice of the kernel;<br />
# Computing the MMD efficiently requires an unbiased estimate of the MMD (as outlined [https://arxiv.org/pdf/1502.02791.pdf]).<br />
<br />
==== MMD for Feature Adaptation in the RTN ====<br />
The authors wish to minimize the MMD between the fusion features outlined above derived from the source and target domains. Concretely this amounts to forcing the distribution of the abstract representation of the source domain $\mathcal{D}_s$ to be similar to the distribution of the abstract representation of the target domain $\mathcal{D}_t$. Performing this optimization over the fused features between the $fb_b$ and $fb_c$ forces each of those layers towards similar distributions.<br />
<br />
Practically this involves an additional penalty function given by the following:<br />
<br />
<center><br />
<math display="block"><br />
D_{\mathcal{L}}(\mathcal{D}_s, \mathcal{D}_t) = \sum_{i,j=1}^{n_s} \frac{k(z_i^s, z_j^s)}{n_s^2} + \sum_{i,j=1}^{n_t} \frac{k(z_i^t, z_j^t)}{n_t^2} + \sum_{i=1}^{n_s}\sum_{j=1}^{n_t} \frac{k(z_i^s, z_j^t)}{n_sn_t} <br />
</math><br />
</center><br />
<br />
Where the characteristic kernel $k(z, z')$ is the Gaussian kernel, defined on the vectorization of tensors, with bandwidth parameter $b$. That is: $k(z, z') = \exp(-||vec(z) - vec(z')||^2/b)$.<br />
<br />
=== Classifier Adaptation ===<br />
In traditional unsupervised domain adaptation there is a '''shared-classifier assumption''' which is made. In essence, if $f_s(x)$ represents the classifier on the source domain, and $f_t(x)$ represents the classifier on the target domain then this assumption simply states that $f_s = f_t$. While this may seem to be a reasonable assumption at first glance, it is problematic largely in that this is an assumption that is incredibly difficult to check. If it could be readily confirmed that the source and target classifiers could be shared, then the problem of domain adaptation would be largely trivialized. Instead, the authors here relax this assumption slightly. They postulate that instead of being equivalent, the source and target classifier differ by some perturbation function $\Delta f$. The general idea is that, by assuming $f_S(x) = f_T(x) + \Delta f(x)$, where $f_S$ and $f_T$ correspond to the source and target classifiers, pre-activation, and $\Delta f(x)$ is some residual function.<br />
<br />
The authors then suggest using residual blocks, as popularized by the ResNet framework [https://arxiv.org/pdf/1512.03385.pdf], to learn this residual function.<br />
<br />
[[File:Residual-Block-vs-DNN.png|thumb|left|A comparison of a standard Deep Neural Network block which is designed to fit a function H(x) compared to a residual block which fits H(x) as the sum of the input, x, and a learned residual function, F(X).]]<br />
==== Residual Networks Framework ==== <br />
A (Deep) Residual Network, as proposed initially in ResNet, employs residual blocks to assist in the learning process, and were a key component of being able to train extraordinarily deep networks. The Residual Network is comprised largely in the same manner as standard neural networks, with one key difference, namely the inclusion of residual blocks - sets of layers which aim to estimate a residual function in place of estimating the function itself. <br />
<br />
That is, if we wish to use a DNN to estimate some function $h(x)$, a residual block will decompose this to $h(x) = F(x) + x$. The layers are then used to learn $F(x)$, and after the layers which aim to learn this residual function, the input $x$ is recombined through element-wise addition, to form $h(x) = F(x) + x$. This was initially proposed as a manner to allow for deeper networks to be effectively trained, but has since used in novel contexts.<br />
<br />
==== Residual Blocks in the RTN ====<br />
[[File:RTN-Residual-Block.png|thumb|right|The Structure of the Residual Block in the RTN framework. The block relies on two additional dense layers following the target classifier in an attempt to learn the residual difference between the source and target classifiers.]] The authors leverage residual blocks for the purpose of classifier adaptation. Operating under the assumption that the source and target classifiers differ by an arbitrary perturbation function, $f(x)$, the authors add an additional set of densely connected layers which the source data will flow through. In particular, the authors take the $fc_c$ layer above as the desired target classifier. For the source data an additional set of layers ($fc-1$ and $fc-2$) are added following $fc_c$, which are connected as a residual block. The output of the classifier layer is then added back to the output of the residual block in order to form the source classifier.<br />
<br />
It is necessary to note that in this case the output from $fc_c$ passes the non-activated (i.e. pre-softmax activation) to the element-wise addition, the result of which is passed through the activation layer, yielding the source prediction. In the provided diagram, we have that $f_S(x)$ represents the non-activated output from the additive layer in the residual block; $f_T(x)$ represents the non-activated output from the target classifier; and $fc-1$/$fc-2$ are used to learn the perturbation function $\Delta f(x)$.<br />
<br />
==== Entropy Minimization ====<br />
In addition to the residual blocks, the authors make use of the '''entropy minimization principle''' [http://www.iro.umontreal.ca/~lisa/pointeurs/semi-supervised-entropy-nips2004.pdf] to further refine the classifier adaptation. In particular, by minimizing the entropy of the target classifier (or more correctly, the entropy of the class conditional distribution $f_j^t(x_i^t) = p(y_i^t = j \mid x_i^t; f_t)$), low-density separation between the classes is encouraged. '''Low-Density Separation''' is a concept used predominantly in semi-supervised learning, which in essence tries to draw class decision boundaries in regions where there are few data points (labeled or unlabeled). The above paper leverages an entropy regularization scheme to achieve the goal low-density separation goal; this is adopted here to the case of unsupervised domain adaptation.<br />
<br />
In practice this amounts to adding a further penalty based on the entropy of the class conditional distribution. In particular, if $H(\cdot)$ is defined to be the entropy function, such that $H(f_t(x_i^t)) = - \sum_{j=1}^c f_j^t(x_i^t)\log f_j^t(x_i^t)$, where $c$ is the number of classes and $f_j^t(x_i^t)$ represents the probability of predicting class $j$ for point $x_i^t$, then over the target domain $\mathcal{D}_t$ we define the entropy penalty to be:<br />
<br />
<center><br />
<math display="block"><br />
\frac{1}{n_t} \sum_{i=1}^{n_t} H(f_t(x_i^t))<br />
</math><br />
</center><br />
<br />
The combination of the residual learning and the entropy penalty, the authors hypothesize will enable effective classifier adaptation.<br />
<br />
=== Residual Transfer Network ===<br />
The combination of the MMD loss introduced in feature adaptation, the residual block introduced in classifier adaptation, and the application of the entropy minimization principle cumulates in the Residual Transfer Network proposed by the authors. The model will be optimized according to the following loss function, which combines the standard cross-entropy, MMD penalty, and entropy penalty:<br />
<br />
<center><br />
<math display="block"><br />
\min_{f_s = f_t + \Delta f} \underbrace{\left(\frac{1}{n_s} \sum_{i=1}^{n_s} L(f_s(x_i^s), y_i^s)\right)}_{\text{Typical Cross-Entropy}} + \underbrace{\frac{\gamma}{n_t}\left(\sum_{i=1}^{n_t} H(f_t(x_i^t)) \right)}_{\text{Target Entropy Minimization}} + \underbrace{\lambda\left(D_{\mathcal{L}}(\mathcal{D}_s, \mathcal{D}_t)\right)}_{\text{MMD Penalty}}<br />
</math><br />
</center><br />
<br />
Where we take $\gamma$ and $\lambda$ to be tradeoff parameters between the entropy penalty and the MMD penalty. As classifier adaptation proposed in this paper and feature adaptation studied in [5, 6] are tailored to adapt different layers of deep networks, they are expected to complement each other and to establish better performance.<br />
<br />
The full network, which is trained subject to the above optimization problem, thus takes on the following structure.<br />
<br />
[[File:rtn-full-paper-structure.png||center|alt=The Structure of the RTN]]<br />
<br />
== Experiments == <br />
<br />
=== Set-up ===<br />
The performance of RTN was jointly compared across two key data sets in the area of Unsupervised Domain Adaptation. Specifically, Office-31 (discussed in the introduction) and Office-Caltech (maintained by the same project group). Office-31 is comprised of images from 3 sources, Amazon ('''A'''), Webcam ('''W'''), and DSLR ('''D'''), of 31 different objects. Office-Caltech is derived by considering 10 classes common to both the Office-31 and the Caltech data sets, thus providing further adaptation possibilities. This provides 6 Transfer Tasks on the 31 classes of Office-31 ($\{(A,W), (A,D), (W,A), (W,D), (D,A), (D,W)\}$) and 12 Transfer Tasks on the 10 classes of Office-Caltech ($\{(A,W), (A,D), (A,C), (W,A), (W,D), (W,C), (D,A), (D,W), (D,C), (C,A), (C,W), (C,D)\}$).<br />
<br />
The authors then compare the results on the 18 different adaptation tasks against 6 other models. In order to determine the efficacy of the various contributions outlined in the paper they perform an ablation study, evaluating variants of the RTN. Specifically, they consider the RTN with only the MMD module ('''RTN (mmd)'''), the RTN with the MMD module and the entropy minimization ('''RTN (mmd+ent)'''), and the complete RTN ('''RTN (mmd+ent+res)'''). The experiments leverage all the labeled training data and compute accuracy across all unlabeled domain data. The parameters of the model (i.e. $\gamma$, and $\lambda$) are fixed based on a single validation point on the transfer task $\mathbf{A}\to\mathbf{W}$. These parameters are then maintained across all transfer tasks. <br />
<br />
As for specification details, the authors use mini-batch SGD, with momentum $0.9$, and with the learning rate adjusted based on $\eta_p = \frac{\eta_0}{(1 + \alpha p)^\beta}$, where $p$ indicates the portion of training completed (linear from $0$ to $1$), $\eta_0 = 0.01$, $\alpha = 10$ and $\beta = 0.75$, which was optimized for low error on the source. The MMD and entropy parameters, set as above, were maintained at $\lambda = 0.3$ and $\gamma - 0.3$.<br />
<br />
=== Results ===<br />
[[File:table-1-results.PNG|thumb|right|Results from the Office-31 Experiment]][[File:table-2-results.PNG|thumb|right|Results from the Office-Caltech Experiment]]<br />
In aggregate, the network outperformed all comparison methods, across all transfer tasks. Broadly speaking the network saw the largest increases in accuracy on the hard transfer tasks (for instance $\mathbf{A} \to \mathbf{C}$), where the source-domain discrepancy is large. The authors take this to mean that the proposed model learns "more adaptive classifiers and transferable features for safer domain adaptation." They further indicate that standard deep learning techniques (i.e. just AlexNet) perform similarly to standard shallow techniques (TCA and GFK). Deep-transfer methods which focus on feature adaptation perform significantly better than the standard methods. The proposed RTN, which adds in additional considerations for classifier adaptation, performs even better.<br />
<br />
In addition, the ablation study found a number of interesting results:<br />
# The RTN (mmd) outperforms DAN, which is founded on a similar method, but contains multiple MMD penalties (one for each layer instead of on a bottleneck), and is as such less computationally efficient;<br />
# The addition of the entropy penalty [RTN (mmd+ent)] provides significant marginal benefit over the previous RTN (mmd);<br />
# The full RTN [RTN (mmd+ent+res)] performs the best of all variants, by diminishing returns are seen over the addition of the entropy penalty.<br />
<br />
Overall the authors claim that the RTN (mmd+ent+res) is now regarded as state-of-the-art for unsupervised domain adaptation.<br />
<br />
=== Discussion ===<br />
[[File:t-sne-embeddings.png|thumb|left|t-SNE Embeddings Comparing the Performance of DAN and RTN]] <br />
[[File:mean-sd-layer-outputs.png|thumb|right|The Mean and Standard Deviations of the outputs from the Source Classifier, Target Classifier, and Residual Functions. As expected, the residual function provides a small, but non-zero, contribution.]] <br />
[[File:gamma-tradeoff.png|thumb|left|The accuracy of tests by varying the parameter $\gamma$. We first see an increase in accuracy up to an ideal point, before having the accuracy fall again.]]<br />
[[File:classifier-shift.png|thumb|right|The corresponding weights of the classifier layers, if trained on the labeled source and target data, exhibiting the differences which exist between the two classifiers in an ideal state. ]]<br />
<br />
==== Visualizing Predictions (Versus DAN) ====<br />
DAN uses a similar method for feature adaptation but neglects any attempt at classifier adaptation (i.e. it makes the shared-classifier assumption). In order to demonstrate that this leads to the worse performance, the authors provide images showing the t-SNE embeddings by DAN and RTN on the transfer task $\mathbf{A} \to \mathbf{W}$. The images show that the target categories are not well discriminated by the source classifier, suggesting a violation of the shared-classifier assumption. Conversely, the target classifier for the RTN exhibits better discrimination.<br />
<br />
==== Layer Responses and Classifier Shift ==== <br />
The authors further consider the mean and standard deviation of the outputs of $f_S(x)$, $f_T(x)$ and $\Delta f(x)$ to consider the relative contributions of the different components. As expected, $\Delta f(x)$ provides a small (though non-zero) contribution to the learned source classifier. This provides some merit to the idea of residual learning on the classifiers. <br />
<br />
In addition, the authors train classifiers on the source and target data, with labels present, and compare the realized weights. This is used to test how different the ideal weights are on separate classifiers. The results suggest that there is, in fact, a discrepancy between the classifiers, further motivating the use of tactics to avoid the shared-classifier assumption. <br />
<br />
==== Parameter Sensitivity ==== <br />
Lastly, the authors test the sensitivity of these results against the parameter $\gamma$. They run this test on $\mathbf{A}\to\mathbf{W}$ in addition to $\mathbf{C}\to\mathbf{W}$, varying the parameter from $0.01$ to $1.0$. They find that, on both tasks, the increase of the parameter initially improves accuracy, before seeing a drop-off.<br />
<br />
== Conclusion ==<br />
This paper presented a novel approach to unsupervised domain adaptation which relaxed assumptions made by previous models with regard to the shared nature of classifiers. Like previous models this proposed network leverages feature adaptation by matching the distributions of features across the domains. In addition, using a residual network and entropy minimization tactic, the target classifier is allowed to differ from the source classifier by implementing a new residual transfer module as the bridge. In particular, this approach allows for easy integration into existing networks, and can be implemented with any standard deep learning software.<br />
<br />
For follow-up considerations, the authors propose looking for adaptations which may be useful in the semi-supervised domain adaptation problem.<br />
<br />
== Critique ==<br />
While the paper presents a clear approach, which empirically attains great results on the desired tasks, I question the benefit to the residual block that is employed. The results of the ablation study seem to suggest that the majority of the benefits can be derived from using the MMD and Entropy penalties. The residual block appears to add marginal, perhaps insignificant contributions to the outcome. Despite this, the use of MMD loss is not novel, and the entropy loss is less well documented, and less thoroughly explored. Perhaps a different set of ablations would have indicated that the three parts, indeed, are equally effective (and the diminishing returns stems from stacking the three methods), but as it is presented, I question the utility of the final structure versus a less complicated, less novel approach.<br />
<br />
==References==<br />
# https://en.wikipedia.org/wiki/Domain_adaptation<br />
# https://people.eecs.berkeley.edu/~jhoffman/domainadapt/<br />
# Glorot, Xavier, Antoine Bordes, and Yoshua Bengio. "Domain adaptation for large-scale sentiment classification: A deep learning approach." Proceedings of the 28th international conference on machine learning (ICML-11). 2011.<br />
# Tzeng, Eric, et al. "Deep domain confusion: Maximizing for domain invariance." arXiv preprint arXiv:1412.3474 (2014).<br />
# Long, Mingsheng, et al. "Learning transferable features with deep adaptation networks." International Conference on Machine Learning. 2015.<br />
# Ganin, Yaroslav, and Victor Lempitsky. "Unsupervised domain adaptation by backpropagation." International Conference on Machine Learning. 2015.<br />
# Tzeng, Eric, et al. "Simultaneous deep transfer across domains and tasks." Proceedings of the IEEE International Conference on Computer Vision. 2015.<br />
# He, Kaiming, et al. "Deep residual learning for image recognition." Proceedings of the IEEE conference on computer vision and pattern recognition. 2016.<br />
# Yang, Jun, Rong Yan, and Alexander G. Hauptmann. "Cross-domain video concept detection using adaptive svms." Proceedings of the 15th ACM international conference on Multimedia. ACM, 2007.<br />
# Duan, Lixin, et al. "Domain adaptation from multiple sources via auxiliary classifiers." Proceedings of the 26th Annual International Conference on Machine Learning. ACM, 2009.<br />
# Duan, Lixin, et al. "Visual event recognition in videos by learning from web data." IEEE Transactions on Pattern Analysis and Machine Intelligence 34.9 (2012): 1667-1680.<br />
# http://vision.stanford.edu/teaching/cs231b_spring1415/slides/alexnet_tugce_kyunghee.pdf<br />
# https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space<br />
#Long, Mingsheng, et al. "Learning transferable features with deep adaptation networks." International Conference on Machine Learning. 2015.<br />
#He, Kaiming, et al. "Deep residual learning for image recognition." Proceedings of the IEEE conference on computer vision and pattern recognition. 2016.<br />
# Grandvalet, Yves, and Yoshua Bengio. "Semi-supervised learning by entropy minimization." Advances in neural information processing systems. 2005.<br />
<br />
Expert review from the NIPS community can be found in https://media.nips.cc/nipsbooks/nipspapers/paper_files/nips29/reviews/99.html.<br />
<br />
Implementation Example: https://github.com/thuml/Xlearn</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Unsupervised_Domain_Adaptation_with_Residual_Transfer_Networks&diff=30937Unsupervised Domain Adaptation with Residual Transfer Networks2017-11-21T01:59:38Z<p>H4lyu: /* Residual Transfer Network */</p>
<hr />
<div>== Introduction ==<br />
'''Domain Adaptation''' [https://en.wikipedia.org/wiki/Domain_adaptation]is a problem in machine learning which involves taking a model which has been trained on a source domain, and applying this to a different (but related) target domain. '''Unsupervised domain adaptation''' refers to the situation in which the source data is labelled, while the target data is (predominantly) unlabeled. The problem at hand is then finding ways to generalize the learning on the source domain to the target domain. In the age of deep networks this problem has become particularly salient due to the need for vast amounts of labeled training data, in order to reap the benefits of deep learning. Manual generation of labeled data is often prohibitive, and in absence of such data networks are rarely performant. The attempt to circumvent this drought of data typically necessitates the gathering of "off-the-shelf" data sets, which are tangentially related and contain labels, and then building models in these domains. The fundamental issue that unsupervised domain adaptation attempts to address is overcoming the inherent shift in distribution across the domains, without the ability to observe this shift directly.<br />
<br />
This paper proposes a method for unsupervised domain adaptation which relies on three key components: <br />
# A kernel-based penalty to ensure that the abstract representations generated by the networks hidden layers are similar between the source and the target data; <br />
# An entropy based penalty on the target classifier, which exploits the entropy minimization principle; and <br />
# A residual network structure is appended, which allows the source and target classifiers to differ by a (learned) residual function, thus relaxing the shared classifier assumption which is traditionally made.<br />
<br />
This method outperforms state-of-the-art techniques on common benchmark datasets, and is flexible enough to be applied in most feed-forward neural networks.<br />
<br />
[[File:Source-and-Target-Domain-Office-31-Backpack.png|thumb|right|The Office-31 Dataset Images for Backpack. Shows the variation in the source and target domains to motivate why these methods are important.]] <br />
=== Working Example (Office-31) === <br />
In order to assist in the understanding of the methods, it is helpful to have a tangible sense of the problem front of mind. The Domain Adaptation Project [https://people.eecs.berkeley.edu/~jhoffman/domainadapt/] provides data sets which are tailored to the problem of unsupervised domain adaptation. One of these data sets (which is later used in the experiments of this paper) has images which are labeled based on the Amazon product page for the various items. There are then corresponding pictures taken either by webcams or digital SLR cameras. The goal of unsupervised domain adaptation on this data set would be to take any of the three image sources as the source domain, and transfer a classifier to the other domain; see the example images to understand the differences.<br />
<br />
One can imagine that, while it is likely easy to scrape labeled images from Amazon, it is likely far more difficult to collect labeled images from webcam or DSLR pictures directly. The ultimate goal of this method would be to train a model to recognize a picture of a backpack taken with a webcam, based on images of backpacks scraped from Amazon (or similar tasks).<br />
<br />
== Related Work ==<br />
Broadly speaking, the problem of domain adaptation mitigates manual labeling of data in areas such as machine learning, computer vision, and natural language processing. The general goal of domain adaptation is to reduce the discrepancy in probability distributions between the source and target domains.<br />
<br />
Research into the use of Deep Neural Networks for the purpose of domain adaptation has suggested that, while networks learn abstract feature representations which can reduce the discrepancy across domains, it is not possible to wholly remove it [http://www.icml-2011.org/papers/342_icmlpaper.pdf], [https://arxiv.org/pdf/1412.3474.pdf]. Further work has been done to design networks which adapt traditional deep nets (typically CNNs) to specifically address the problems posed by domain adaptation, these methods all only address the issue of feature adaptation [https://arxiv.org/pdf/1502.02791.pdf], [https://arxiv.org/pdf/1409.7495.pdf], [https://people.eecs.berkeley.edu/~jhoffman/papers/Tzeng_ICCV2015.pdf]. That is, they all assume that the target and source classifiers are shared between domains. <br />
<br />
The authors drew particular motivation from He et al. [https://arxiv.org/abs/1512.03385] with the proposed structure of residual networks. Combining the insights from the ResNet architecture, in addition to previous work that had leveraged classifier adaptation (in the context where some target data is labeled) [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.130.8224&rep=rep1&type=pdf], [http://www.machinelearning.org/archive/icml2009/papers/445.pdf], [http://ieeexplore.ieee.org/document/5539870/] the authors develop their proposed network.<br />
<br />
== Residual Transfer Networks ==<br />
Generally, in an unsupervised domain adaptation problem, we are dealing with a set $\mathcal{D}_s$ (called the source domain) which is defined by $\{(x_i^s, y_i^s)\}_{i=1}^{n_s}$. That is the set of all labeled input-output pairs in our source data set. We denote the number of source elements by $n_s$. There is a corresponding set $\mathcal{D}_t = \{(x_i^t)\}_{i=1}^{n_t}$ (the target domain), consisting of unlabeled input values. There are $n_t$ such values. <br />
[[File:RTN-Structure.png|thumb|left|upright|The overarching structure of the RTN. Consists of an existing network, to which a bottleneck, MMD block, and residual block is appended.]]<br />
We can think of $\mathcal{D}_s$ as being sampled from some underlying distribution $p$, and $\mathcal{D}_t$ as being sampled from $q$. Generally we have that $p \neq q$, partially motivating the need for domain adaptation methods. <br />
<br />
We can consider the classifiers $f_s(\underline{x})$ and $f_t(\underline{x})$, for the source domain and target domain respectively. It is possible to learn $f_s$ based on the sample $\mathcal{D}_s$. Under the '''shared classifier assumption''' it would be the case that $f_s(\underline{x}) = f_t(\underline{x})$, and thus learning the source classifier is enough. This method relaxes this assumption, assuming that in general $f_s \neq f_t$, and attempting to learn both.<br />
<br />
The example network extends deep convolutional networks (in this case AlexNet [http://vision.stanford.edu/teaching/cs231b_spring1415/slides/alexnet_tugce_kyunghee.pdf]) to '''Residual Transfer Networks''', the mechanics of which are outlined below. Recall that, if $L(\cdot, \cdot)$ is taken to be the cross-entropy loss function, then the empirical error of a CNN on the source domain $\mathcal{D}_s$ is given by:<br />
<br />
<center><br />
<math display="block"><br />
\min_{f_s} \frac{1}{n_s} \sum_{i=1}^{n_s} L(f_s(x_i^s), y_i^s)<br />
</math> <br />
</center><br />
<br />
In a standard implementation, the CNN optimizes over the above loss. This will be the starting point for the RTN.<br />
<br />
=== Structural Overview ===<br />
The model proposed in this paper extends existing CNN's and alters the loss function that is optimized over. While each of these components is discussed in depth below, the overarching architecture involves four components:<br />
<br />
# An existing deep model. While this can be any model, in theory, the authors leverage AlexNet in practice.<br />
# A bottleneck layer, used to reduce the dimensionality of the learned abstract feature space, directly after the existing network.<br />
# An MMD block, with the expressed intention of feature adaptation.<br />
# A residual block, with the expressed intention of classifier adaptation. <br />
<br />
This structure is then optimized against a loss function which combines the standard cross-entropy penalty with MMD and target entropy penalties, yielding the proposed Residual Transfer Network (RTN) structure.<br />
<br />
=== Feature Adaptation ===<br />
Feature adaptation refers to the process in which the features which are learned to represent the source domain are made applicable to the target domain. Broadly speaking a CNN works to generate abstract feature representations of the distribution that the inputs are sampled from. It has been found that using these deep features can reduce, but not remove, cross-domain distribution discrepancy, hence the need for feature adaptation. It is important to note that CNN's transfer from general to specific features as the network gets deeper. In this light, the discrepancy between the feature representation of the source and the target will grow through a deeper convolutional net. As such a technique for forcing these distributions to be similar is needed.<br />
<br />
In particular the authors of this paper impose a bottleneck layer (call it $fc_b$) which is included after the final convolutional layer of AlexNet. This dense layer is connected to an additional dense layer $fc_c$, (which will serve as the target classification layer). They then compute the tensor product between the activations of the layers, performing "lossless multi-layer feature fusion". That is for the source domain they define $z_i^s \overset{\underset{\mathrm{def}}{}}{=} x_i^{s,fc_b}\otimes x_i^{s,fc_c}$ and for the target domain, $z_i^t \overset{\underset{\mathrm{def}}{}}{=} x_i^{t,fc_b}\otimes x_i^{t,fc_c}$. The authors then employ feature adaptation by means of Maximum Mean Discrepancy, between the source and target domains, on these fusion features.<br />
<br />
[[File:RTN-MMD-Block.png|right|thumb|The Maximum Mean Discrepancy Block (MMD) included in the RTN. The outputs of $fc_b$ and $fc_c$ are fused through a tensor product, and then passed through the MMD penalty, ensuring distributional similarity.]]<br />
<br />
==== Maximum Mean Discrepancy ==== <br />
The Maximum Mean Discrepancy (MMD) is a Kernel method involes mapping to a Reproducing Kernel Hilbert Space (RKHS) [https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space]. Denote the RKHS $\mathcal{H}_K$ with a characteristic kernel $K$. We then define the '''mean embedding''' of a distribution $p$ in $\mathcal{H}_K$ to be the unique element $\mu_K(p)$ such that $\mathbf{E}_{x\sim p}f(x) = \langle f(x), \mu_K(p)\rangle_{\mathcal{H}_K}$ for all $f \in \mathcal{H}_K$. Now, if we take $\phi: \mathcal{X} \to \mathcal{H}_K$, then we can define the MMD between two distributions $p$ and $q$ as follows:<br />
<br />
<center><br />
<math display="block"><br />
d_k(p, q) \overset{\underset{\mathrm{def}}{}}{=} ||\mathbf{E}_{x\sim p}(\phi(x^s)) - \mathbf{E}_{x\sim q}(\phi(x^t))||_{\mathcal{H}_K}<br />
</math><br />
</center><br />
<br />
Effectively, the MMD will compute the self-similarity of $p$ and $q$, and subtract twice the cross-similarity between the distributions: $\widehat{\text{MMD}}^2 = \text{mean}(K_{pp}) + \text{mean}(K_{qq}) - 2\times\text{mean}(K_{pq})$. From here we can infer that $p$ and $q$ are equivalent distributions if and only if the $\text{MMD} = 0$. If we then wish to force two distributions to be similar, this becomes a minimization problem over the MMD.<br />
<br />
Two important notes:<br />
# The RKHS, and as such MMD, depend on the choice of the kernel;<br />
# Computing the MMD efficiently requires an unbiased estimate of the MMD (as outlined [https://arxiv.org/pdf/1502.02791.pdf]).<br />
<br />
==== MMD for Feature Adaptation in the RTN ====<br />
The authors wish to minimize the MMD between the fusion features outlined above derived from the source and target domains. Concretely this amounts to forcing the distribution of the abstract representation of the source domain $\mathcal{D}_s$ to be similar to the distribution of the abstract representation of the target domain $\mathcal{D}_t$. Performing this optimization over the fused features between the $fb_b$ and $fb_c$ forces each of those layers towards similar distributions.<br />
<br />
Practically this involves an additional penalty function given by the following:<br />
<br />
<center><br />
<math display="block"><br />
D_{\mathcal{L}}(\mathcal{D}_s, \mathcal{D}_t) = \sum_{i,j=1}^{n_s} \frac{k(z_i^s, z_j^s)}{n_s^2} + \sum_{i,j=1}^{n_t} \frac{k(z_i^t, z_j^t)}{n_t^2} + \sum_{i=1}^{n_s}\sum_{j=1}^{n_t} \frac{k(z_i^s, z_j^t)}{n_sn_t} <br />
</math><br />
</center><br />
<br />
Where the characteristic kernel $k(z, z')$ is the Gaussian kernel, defined on the vectorization of tensors, with bandwidth parameter $b$. That is: $k(z, z') = \exp(-||vec(z) - vec(z')||^2/b)$.<br />
<br />
=== Classifier Adaptation ===<br />
In traditional unsupervised domain adaptation there is a '''shared-classifier assumption''' which is made. In essence, if $f_s(x)$ represents the classifier on the source domain, and $f_t(x)$ represents the classifier on the target domain then this assumption simply states that $f_s = f_t$. While this may seem to be a reasonable assumption at first glance, it is problematic largely in that this is an assumption that is incredibly difficult to check. If it could be readily confirmed that the source and target classifiers could be shared, then the problem of domain adaptation would be largely trivialized. Instead, the authors here relax this assumption slightly. They postulate that instead of being equivalent, the source and target classifier differ by some perturbation function $\Delta f$. The general idea is that, by assuming $f_S(x) = f_T(x) + \Delta f(x)$, where $f_S$ and $f_T$ correspond to the source and target classifiers, pre-activation, and $\Delta f(x)$ is some residual function.<br />
<br />
The authors then suggest using residual blocks, as popularized by the ResNet framework [https://arxiv.org/pdf/1512.03385.pdf], to learn this residual function.<br />
<br />
[[File:Residual-Block-vs-DNN.png|thumb|left|A comparison of a standard Deep Neural Network block which is designed to fit a function H(x) compared to a residual block which fits H(x) as the sum of the input, x, and a learned residual function, F(X).]]<br />
==== Residual Networks Framework ==== <br />
A (Deep) Residual Network, as proposed initially in ResNet, employs residual blocks to assist in the learning process, and were a key component of being able to train extraordinarily deep networks. The Residual Network is comprised largely in the same manner as standard neural networks, with one key difference, namely the inclusion of residual blocks - sets of layers which aim to estimate a residual function in place of estimating the function itself. <br />
<br />
That is, if we wish to use a DNN to estimate some function $h(x)$, a residual block will decompose this to $h(x) = F(x) + x$. The layers are then used to learn $F(x)$, and after the layers which aim to learn this residual function, the input $x$ is recombined through element-wise addition, to form $h(x) = F(x) + x$. This was initially proposed as a manner to allow for deeper networks to be effectively trained, but has since used in novel contexts.<br />
<br />
==== Residual Blocks in the RTN ====<br />
[[File:RTN-Residual-Block.png|thumb|right|The Structure of the Residual Block in the RTN framework. The block relies on two additional dense layers following the target classifier in an attempt to learn the residual difference between the source and target classifiers.]] The authors leverage residual blocks for the purpose of classifier adaptation. Operating under the assumption that the source and target classifiers differ by an arbitrary perturbation function, $f(x)$, the authors add an additional set of densely connected layers which the source data will flow through. In particular, the authors take the $fc_c$ layer above as the desired target classifier. For the source data an additional set of layers ($fc-1$ and $fc-2$) are added following $fc_c$, which are connected as a residual block. The output of the classifier layer is then added back to the output of the residual block in order to form the source classifier.<br />
<br />
It is necessary to note that in this case the output from $fc_c$ passes the non-activated (i.e. pre-softmax activation) to the element-wise addition, the result of which is passed through the activation layer, yielding the source prediction. In the provided diagram, we have that $f_S(x)$ represents the non-activated output from the additive layer in the residual block; $f_T(x)$ represents the non-activated output from the target classifier; and $fc-1$/$fc-2$ are used to learn the perturbation function $\Delta f(x)$.<br />
<br />
==== Entropy Minimization ====<br />
In addition to the residual blocks, the authors make use of the '''entropy minimization principle''' [http://www.iro.umontreal.ca/~lisa/pointeurs/semi-supervised-entropy-nips2004.pdf] to further refine the classifier adaptation. In particular, by minimizing the entropy of the target classifier (or more correctly, the entropy of the class conditional distribution $f_j^t(x_i^t) = p(y_i^t = j \mid x_i^t; f_t)$), low-density separation between the classes is encouraged. '''Low-Density Separation''' is a concept used predominantly in semi-supervised learning, which in essence tries to draw class decision boundaries in regions where there are few data points (labeled or unlabeled). The above paper leverages an entropy regularization scheme to achieve the goal low-density separation goal; this is adopted here to the case of unsupervised domain adaptation.<br />
<br />
In practice this amounts to adding a further penalty based on the entropy of the class conditional distribution. In particular, if $H(\cdot)$ is defined to be the entropy function, such that $H(f_t(x_i^t)) = - \sum_{j=1}^c f_j^t(x_i^t)\log f_j^t(x_i^t)$, where $c$ is the number of classes and $f_j^t(x_i^t)$ represents the probability of predicting class $j$ for point $x_i^t$, then over the target domain $\mathcal{D}_t$ we define the entropy penalty to be:<br />
<br />
<center><br />
<math display="block"><br />
\frac{1}{n_t} \sum_{i=1}^{n_t} H(f_t(x_i^t))<br />
</math><br />
</center><br />
<br />
The combination of the residual learning and the entropy penalty, the authors hypothesize will enable effective classifier adaptation.<br />
<br />
=== Residual Transfer Network ===<br />
The combination of the MMD loss introduced in feature adaptation, the residual block introduced in classifier adaptation, and the application of the entropy minimization principle cumulates in the Residual Transfer Network proposed by the authors. The model will be optimized according to the following loss function, which combines the standard cross-entropy, MMD penalty, and entropy penalty:<br />
<br />
<center><br />
<math display="block"><br />
\min_{f_s = f_t + \Delta f} \underbrace{\left(\frac{1}{n_s} \sum_{i=1}^{n_s} L(f_s(x_i^s), y_i^s)\right)}_{\text{Typical Cross-Entropy}} + \underbrace{\frac{\gamma}{n_t}\left(\sum_{i=1}^{n_t} H(f_t(x_i^t)) \right)}_{\text{Target Entropy Minimization}} + \underbrace{\lambda\left(D_{\mathcal{L}}(\mathcal{D}_s, \mathcal{D}_t)\right)}_{\text{MMD Penalty}}<br />
</math><br />
</center><br />
<br />
Where we take $\gamma$ and $\lambda$ to be tradeoff parameters between the entropy penalty and the MMD penalty. As classifier adaptation proposed in this paper and feature adaptation studied in [5, 6] are tailored to adapt different layers of deep networks, they are expected to complement each other and to establish better performance.<br />
<br />
The full network, which is trained subject to the above optimization problem, thus takes on the following structure.<br />
<br />
[[File:rtn-full-paper-structure.png||center|alt=The Structure of the RTN]]<br />
<br />
== Experiments == <br />
<br />
=== Set-up ===<br />
The performance of RTN was jointly compared across two key data sets in the area of Unsupervised Domain Adaptation. Specifically, Office-31 (discussed in the introduction) and Office-Caltech (maintained by the same project group). Office-31 is comprised of images from 3 sources, Amazon ('''A'''), Webcam ('''W'''), and DSLR ('''D'''), of 31 different objects. Office-Caltech is derived by considering 10 classes common to both the Office-31 and the Caltech data sets, thus providing further adaptation possibilities. This provides 6 Transfer Tasks on the 31 classes of Office-31 ($\{(A,W), (A,D), (W,A), (W,D), (D,A), (D,W)\}$) and 12 Transfer Tasks on the 10 classes of Office-Caltech ($\{(A,W), (A,D), (A,C), (W,A), (W,D), (W,C), (D,A), (D,W), (D,C), (C,A), (C,W), (C,D)\}$).<br />
<br />
The authors then compare the results on the 18 different adaptation tasks against 6 other models. In order to determine the efficacy of the various contributions outlined in the paper they perform an ablation study, evaluating variants of the RTN. Specifically, they consider the RTN with only the MMD module ('''RTN (mmd)'''), the RTN with the MMD module and the entropy minimization ('''RTN (mmd+ent)'''), and the complete RTN ('''RTN (mmd+ent+res)'''). The experiments leverage all the labeled training data and compute accuracy across all unlabeled domain data. The parameters of the model (i.e. $\gamma$, and $\lambda$) are fixed based on a single validation point on the transfer task $\mathbf{A}\to\mathbf{W}$. These parameters are then maintained across all transfer tasks. <br />
<br />
As for specification details, the authors use mini-batch SGD, with momentum $0.9$, and with the learning rate adjusted based on $\eta_p = \frac{\eta_0}{(1 + \alpha p)^\beta}$, where $p$ indicates the portion of training completed (linear from $0$ to $1$), $\eta_0 = 0.01$, $\alpha = 10$ and $\beta = 0.75$, which was optimized for low error on the source. The MMD and entropy parameters, set as above, were maintained at $\lambda = 0.3$ and $\gamma - 0.3$.<br />
<br />
=== Results ===<br />
[[File:table-1-results.PNG|thumb|right|Results from the Office-31 Experiment]][[File:table-2-results.PNG|thumb|right|Results from the Office-Caltech Experiment]]<br />
In aggregate, the network outperformed all comparison methods, across all transfer tasks. Broadly speaking the network saw the largest increases in accuracy on the hard transfer tasks (for instance $\mathbf{A} \to \mathbf{C}$), where the source-domain discrepancy is large. The authors take this to mean that the proposed model learns "more adaptive classifiers and transferable features for safer domain adaptation." They further indicate that standard deep learning techniques (i.e. just AlexNet) perform similarly to standard shallow techniques (TCA and GFK). Deep-transfer methods which focus on feature adaptation perform significantly better than the standard methods. The proposed RTN, which adds in additional considerations for classifier adaptation, performs even better.<br />
<br />
In addition, the ablation study found a number of interesting results:<br />
# The RTN (mmd) outperforms DAN, which is founded on a similar method, but contains multiple MMD penalties (one for each layer instead of on a bottleneck), and is as such less computationally efficient;<br />
# The addition of the entropy penalty [RTN (mmd+ent)] provides significant marginal benefit over the previous RTN (mmd);<br />
# The full RTN [RTN (mmd+ent+res)] performs the best of all variants, by diminishing returns are seen over the addition of the entropy penalty.<br />
<br />
Overall the authors claim that the RTN (mmd+ent+res) is now regarded as state-of-the-art for unsupervised domain adaptation.<br />
<br />
=== Discussion ===<br />
[[File:t-sne-embeddings.png|thumb|left|t-SNE Embeddings Comparing the Performance of DAN and RTN]] <br />
[[File:mean-sd-layer-outputs.png|thumb|right|The Mean and Standard Deviations of the outputs from the Source Classifier, Target Classifier, and Residual Functions. As expected, the residual function provides a small, but non-zero, contribution.]] <br />
[[File:gamma-tradeoff.png|thumb|left|The accuracy of tests by varying the parameter $\gamma$. We first see an increase in accuracy up to an ideal point, before having the accuracy fall again.]]<br />
[[File:classifier-shift.png|thumb|right|The corresponding weights of the classifier layers, if trained on the labeled source and target data, exhibiting the differences which exist between the two classifiers in an ideal state. ]]<br />
<br />
==== Visualizing Predictions (Versus DAN) ====<br />
DAN uses a similar method for feature adaptation but neglects any attempt at classifier adaptation (i.e. it makes the shared-classifier assumption). In order to demonstrate that this leads to the worse performance, the authors provide images showing the t-SNE embeddings by DAN and RTN on the transfer task $\mathbf{A} \to \mathbf{W}$. The images show that the target categories are not well discriminated by the source classifier, suggesting a violation of the shared-classifier assumption. Conversely, the target classifier for the RTN exhibits better discrimination.<br />
<br />
==== Layer Responses and Classifier Shift ==== <br />
The authors further consider the mean and standard deviation of the outputs of $f_S(x)$, $f_T(x)$ and $\Delta f(x)$ to consider the relative contributions of the different components. As expected, $\Delta f(x)$ provides a small (though non-zero) contribution to the learned source classifier. This provides some merit to the idea of residual learning on the classifiers. <br />
<br />
In addition, the authors train classifiers on the source and target data, with labels present, and compare the realized weights. This is used to test how different the ideal weights are on separate classifiers. The results suggest that there is, in fact, a discrepancy between the classifiers, further motivating the use of tactics to avoid the shared-classifier assumption. <br />
<br />
==== Parameter Sensitivity ==== <br />
Lastly, the authors test the sensitivity of these results against the parameter $\gamma$. They run this test on $\mathbf{A}\to\mathbf{W}$ in addition to $\mathbf{C}\to\mathbf{W}$, varying the parameter from $0.01$ to $1.0$. They find that, on both tasks, the increase of the parameter initially improves accuracy, before seeing a drop-off.<br />
<br />
== Conclusion ==<br />
This paper presented a novel approach to unsupervised domain adaptation which relaxed assumptions made by previous models with regard to the shared nature of classifiers. Like previous models this proposed network leverages feature adaptation by matching the distributions of features across the domains. In addition, using a residual network and entropy minimization tactic, the target classifier is allowed to differ from the source classifier. In particular, this approach allows for easy integration into existing networks, and can be implemented with any standard deep learning software.<br />
<br />
For follow-up considerations, the authors propose looking for adaptations which may be useful in the semi-supervised domain adaptation problem.<br />
<br />
== Critique ==<br />
While the paper presents a clear approach, which empirically attains great results on the desired tasks, I question the benefit to the residual block that is employed. The results of the ablation study seem to suggest that the majority of the benefits can be derived from using the MMD and Entropy penalties. The residual block appears to add marginal, perhaps insignificant contributions to the outcome. Despite this, the use of MMD loss is not novel, and the entropy loss is less well documented, and less thoroughly explored. Perhaps a different set of ablations would have indicated that the three parts, indeed, are equally effective (and the diminishing returns stems from stacking the three methods), but as it is presented, I question the utility of the final structure versus a less complicated, less novel approach.<br />
<br />
==References==<br />
# https://en.wikipedia.org/wiki/Domain_adaptation<br />
# https://people.eecs.berkeley.edu/~jhoffman/domainadapt/<br />
# Glorot, Xavier, Antoine Bordes, and Yoshua Bengio. "Domain adaptation for large-scale sentiment classification: A deep learning approach." Proceedings of the 28th international conference on machine learning (ICML-11). 2011.<br />
# Tzeng, Eric, et al. "Deep domain confusion: Maximizing for domain invariance." arXiv preprint arXiv:1412.3474 (2014).<br />
# Long, Mingsheng, et al. "Learning transferable features with deep adaptation networks." International Conference on Machine Learning. 2015.<br />
# Ganin, Yaroslav, and Victor Lempitsky. "Unsupervised domain adaptation by backpropagation." International Conference on Machine Learning. 2015.<br />
# Tzeng, Eric, et al. "Simultaneous deep transfer across domains and tasks." Proceedings of the IEEE International Conference on Computer Vision. 2015.<br />
# He, Kaiming, et al. "Deep residual learning for image recognition." Proceedings of the IEEE conference on computer vision and pattern recognition. 2016.<br />
# Yang, Jun, Rong Yan, and Alexander G. Hauptmann. "Cross-domain video concept detection using adaptive svms." Proceedings of the 15th ACM international conference on Multimedia. ACM, 2007.<br />
# Duan, Lixin, et al. "Domain adaptation from multiple sources via auxiliary classifiers." Proceedings of the 26th Annual International Conference on Machine Learning. ACM, 2009.<br />
# Duan, Lixin, et al. "Visual event recognition in videos by learning from web data." IEEE Transactions on Pattern Analysis and Machine Intelligence 34.9 (2012): 1667-1680.<br />
# http://vision.stanford.edu/teaching/cs231b_spring1415/slides/alexnet_tugce_kyunghee.pdf<br />
# https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space<br />
#Long, Mingsheng, et al. "Learning transferable features with deep adaptation networks." International Conference on Machine Learning. 2015.<br />
#He, Kaiming, et al. "Deep residual learning for image recognition." Proceedings of the IEEE conference on computer vision and pattern recognition. 2016.<br />
# Grandvalet, Yves, and Yoshua Bengio. "Semi-supervised learning by entropy minimization." Advances in neural information processing systems. 2005.<br />
<br />
Expert review from the NIPS community can be found in https://media.nips.cc/nipsbooks/nipspapers/paper_files/nips29/reviews/99.html.<br />
<br />
Implementation Example: https://github.com/thuml/Xlearn</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Model-Agnostic_Meta-Learning_for_Fast_Adaptation_of_Deep_Networks&diff=30933Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks2017-11-20T22:47:51Z<p>H4lyu: /* Reinforcement Learning */</p>
<hr />
<div>='''Introduction & Background'''=<br />
Learning quickly is a hallmark of human intelligence, whether it involves recognizing objects from a few examples or quickly learning new skills after just minutes of experience. In this work, we propose a meta-learning algorithm that is general and model-agnostic, in the sense that it can be directly applied to any learning problem and model that is trained with a gradient descent procedure. Our focus is on deep neural network models, but we illustrate how our approach can easily handle different architectures and different problem settings, including classification, regression, and policy gradient reinforcement learning, with minimal modification. Unlike prior meta-learning methods that learn an update function or learning rule (Schmidhuber, 1987; Bengio et al., 1992; Andrychowicz et al., 2016; Ravi & Larochelle, 2017), this algorithm does not expand the number of learned parameters nor place constraints on the model architecture (e.g. by requiring a recurrent model (Santoro et al., 2016) or a Siamese network (Koch, 2015)), and it can be readily combined with fully connected, convolutional, or recurrent neural networks. It can also be used with a variety of loss functions, including differentiable supervised losses and nondifferentiable reinforcement learning objectives.<br />
<br />
The primary contribution of this work is a simple model and task-agnostic algorithm for meta-learning that trains a model’s parameters such that a small number of gradient updates will lead to fast learning on a new task. The paper shows the effectiveness of the proposed algorithm in different domains, including classification, regression, and reinforcement learning problems.<br />
<br />
='''Model-Agnostic Meta Learning (MAML)'''=<br />
The goal of the proposed model is rapid adaptation. This setting is usually formalized as few-shot learning.<br />
<br />
=== Problem set-up ===<br />
The goal of few-shot meta-learning is to train a model that can quickly adapt to a new task using only a few datapoints and training iterations. To do so. the model is trained during a meta-learning phase on a set of tasks, such that it can then be adapted to a new task using only a small number of parameter updates. In effect, the meta-learning problem treats entire tasks as training examples. <br />
<br />
Let us consider a model denoted by $f$, that maps the observation $\mathbf{x}$ into the output variable $a$. During meta-learning, the model is trained to be able to adapt to a large or infinite number of tasks. <br />
<br />
Let us consider a generic notion of task as below. Each task $\mathcal{T} = \{\mathcal{L}(\mathbf{x}_1.a_1,\mathbf{x}_2,a_2,..., \mathbf{x}_H,a_H), q(\mathbf{x}_1),q(\mathbf{x}_{t+1}|\mathbf{x}_t,a_t),H \}$, consists of a loss function $\mathcal{L}$, a distribution over initial observations $q(\mathbf{x}_1)$, a transition distribution $q(\mathbf{x}_{t+1}|\mathbf{x}_t)$, and an episode length $H$. In i.i.d. supervised learning problems,<br />
the length $H =1$. The model may generate samples of length $H$ by choosing an output at at each time $t$. The cost $\mathcal{L}$ provides a task-specific feedback, which is defined based on the nature of the problem. <br />
<br />
A distribution over tasks is denoted by $p(\mathcal{T})$. In the K-shot learning setting, the model is trained to learn a new task $\mathcal{T}_i$ drawn from $p(\mathcal{T})$ from only K samples drawn from $q_i$ and feedback $\mathcal{L}_{\mathcal{T}_i}$ generated by $\mathcal{T}_i$. During meta-training, a task $\mathcal{T}_i$ is sampled from $p(\mathcal{T})$, the model is trained with K samples and feedback from the corresponding loss LTi from Ti, and then tested on new samples from Ti. The model f is then improved by considering how the test error on new data from $q_i$ changes with respect to the parameters. In effect, the test error on sampled tasks $\mathcal{T}_i$ serves as the training error of the meta-learning process. At the end of meta-training, new tasks are sampled from $p(\mathcal{T})$, and meta-performance is measured by the model’s performance after learning from K samples.<br />
<br />
=== MAML Algorithm ===<br />
[[File:model.png|200px|right|thumb|Figure 1: Diagram of the MAML algorithm]]<br />
The paper proposes a method that can learn the parameters of any standard model via meta-learning in such a way as to prepare that model for fast adaptation. The intuition behind this approach is that some internal representations are more transferrable than others. Since the model will be fine-tuned using a gradient-based learning rule on a new task, we will aim to learn a model in such a way that this gradient-based learning rule can make rapid progress on new tasks drawn from $p(\mathcal{T})$, without overfitting. In effect, we will aim to find model parameters that are sensitive to changes in the task, such that small changes in the parameters will produce large improvements on the loss function of any task drawn from $p(\mathcal{T})$, see Fig 1.<br />
<br />
Note that there is no assumption about the form of the model. Only assumption is that it is parameterized by a vector of parameters $\theta$, and the loss is smooth so that the parameters can be leaned using gradient-based techniques. Formally lets assume that the model is denoted by $f_{\theta}$. When adapting<br />
to a new task $\mathcal{T}_i $, the model’s parameters $\theta$ become $\theta_i'$. In our method, the updated parameter vector $\theta_i'$ is computed using one or more gradient descent updates on task $\mathcal{T}_i $. For example, when using one gradient update:<br />
<br />
$$<br />
\theta_i ' = \theta - \alpha \nabla_{\theta \mathcal{L}_{\mathcal{T}_i}}(f_{\theta}).<br />
$$<br />
<br />
Here $\alpha$ is a the learning rate of each task and considered as a hyperparameter. They consider a single step of update for the rest of the paper, for the sake of the simplicity. <br />
<br />
The model parameters are trained by optimizing for the performance<br />
of $f_{\theta_i'}$ with respect to $\theta$ across tasks sampled from $p(\mathcal{T})$. More concretely, the meta-objective is as follows: <br />
<br />
$$<br />
\min_{\theta} \sum \limits_{\mathcal{T}_i \sim p(\mathcal{T})} \mathcal{L}_{\mathcal{T}_i} (f_{\theta_i'}) = \sum \limits_{\mathcal{T}_i \sim p(\mathcal{T})} \mathcal{L}_{\mathcal{T}_i} (f_{\theta - \alpha \nabla_{\theta \mathcal{L}_{\mathcal{T}_i}}(f_{\theta})})<br />
$$<br />
<br />
Note that the meta-optimization is performed over the model parameters $\theta$, whereas the objective is computed using the updated model parameters $\theta'$. The model aims to optimize the model parameters such that one or a small number of gradient step on a new task will produce maximally effective behavior on that task. <br />
<br />
Therefore the meta-learning across the tasks is performed via stochastic gradient descent (SGD), such that the model parameters $\theta$ are updated as:<br />
<br />
$$<br />
\theta \gets \theta - \beta \nabla_{\theta } \sum \limits_{\mathcal{T}_i \sim p(\mathcal{T})} \mathcal{L}_{\mathcal{T}_i} (f_{\theta_i'})<br />
$$<br />
where $\beta$ is the meta step size. Outline of the algorithm is shown in Algorithm 1. <br />
[[File:ershad_alg1.png|500px|center|thumb]]<br />
<br />
The MAML meta-gradient update involves a gradient through a gradient. Computationally, this requires an additional backward pass through f to compute Hessian-vector products, which is supported by standard deep learning libraries such as TensorFlow.<br />
<br />
='''Different Types of MAML'''=<br />
In this section the MAML algorithm is discussed for different supervised learning and reinforcement learning tasks. The differences between each of these tasks are in their loss function and the way the data is generated. In general, this method does not require additional model parameters nor using any additional meta-learner to learn the update of parameters. Compared to other approaches that tend to “learn to compare new examples in a learned metric space using e.g. Siamese networks or recurrence with attention mechanisms”, the proposed method can be generalized to any other problems including classification, regression and reinforcement learning. <br />
<br />
=== Supervised Regression and Classification ===<br />
Few-shot learning is well-studied in this field. For these two types of tasks the horizon $H$ is equal to 1, since the data points are generated i.i.d. <br />
<br />
Although any common classification and regression objectives can be used as the loss, the paper uses the following losses for these two tasks. <br />
<br />
Regression : For regression we use the mean-square error (MSE):<br />
<br />
$$<br />
\mathcal{L}_{\mathcal{T}_i} (f_{\theta}) = \sum \limits_{\mathbf{x}^{(j)}, \mathcal{y}^{(j)} \sim \mathcal{T}_i} \parallel f_{\theta} (\mathbf{x}^{(j)} - \mathbf{y}^{(j)})\parallel_2^2<br />
$$<br />
<br />
where $\mathbf{x}^{(j)}$ and $\mathbf{y}^{()j}$ are the input/output pair sampled from task $\mathcal{T}_i$. In K-shot regression tasks, K input/output pairs are provided for learning for each task. <br />
<br />
Classification: For classification we use the cross entropy loss:<br />
<br />
$$<br />
\mathcal{L}_{\mathcal{T}_i} (f_{\theta}) = \sum \limits_{\mathbf{x}^{(j)}, \mathcal{y}^{(j)} \sim \mathcal{T}_i} \mathbf{y}^{(j)} \log f_{\theta}(\mathbf{x}^{(j)}) + (1-\mathbf{y}^{(j)}) \log (1-f_{\theta}(\mathbf{x}^{(j)}))<br />
$$<br />
<br />
According to the conventional terminology, K-shot classification tasks use K input/output pairs from each class, for a total of $NK$ data points for N-way classification.<br />
<br />
Given a distribution over tasks, these loss functions can be directly inserted into the equations in the previous section to perform meta-learning, as detailed in Algorithm 2.<br />
[[File:ershad_alg2.png|500px|center|thumb]]<br />
<br />
=== Reinforcement Learning ===<br />
In reinforcement learning (RL), the goal of few-shot meta learning is to enable an agent to quickly acquire a policy for a new test task using only a small amount of experience in the test setting. A new task might involve achieving a new goal or succeeding on a previously trained goal in a new environment. For example an agent may learn how to navigate mazes very quickly so that, when faced with a new maze, it can determine how to reliably reach the exit with only a few samples.<br />
<br />
Each RL task $\mathcal{T}_i$ contains an initial state distribution $q_i(\mathbf{x}_1)$ and a transition distribution $q_i(\mathbf{x}_{t+1}|\mathbf{x}_t,a_t)$ , and the loss $\mathcal{L}_{\mathcal{T}_i}$ corresponds to the (negative) reward function $R$. The entire task is therefore a Markov decision process (MDP) with horizon H, where the learner is allowed to query a limited number of sample trajectories for few-shot learning. Any aspect of the MDP may change across tasks in $p(\mathcal{T})$. The model being learned, $f_{\theta}$, is a policy that maps from states $\mathbf{x}_t$ to a distribution over actions $a_t$ at each timestep $t \in \{1,...,H\}$. The loss for task $\mathcal{T}_i$ and model $f_{\theta}$ takes the form<br />
<br />
$$<br />
\mathcal{L}_{\mathcal{T}_i}(f_{\theta}) = -\mathbb{E}_{\mathbf{x}_t,a_t \sim f_{\theta},q_{\mathcal{T}_i}} \big [\sum_{t=1}^H R_i(\mathbf{x}_t,a_t)\big ]<br />
$$<br />
<br />
<br />
In K-shot reinforcement learning, K rollouts from $f_{\theta}$ and task $\mathcal{T}_i$, $(\mathbf{x}_1,a_1,...,\mathbf{x}_H)$, and the corresponding rewards $ R(\mathbf{x}_t,a_t)$, may be used for adaptation on a new task $\mathcal{T}_i$.<br />
<br />
Since the expected reward is generally not differentiable due to unknown dynamics, we use policy gradient methods to estimate the gradient both for the model gradient update(s) and the meta-optimization. Since policy gradients are an on-policy algorithm, each additional gradient step during the adaptation of $f_{\theta}$ requires new samples from the current policy $f_{\theta_i}$ . We detail the algorithm in Algorithm 3, which has the same structure as Algorithm 2 but also which requires sampling trajectories from the environment corresponding to task $\mathcal{T}_i$ in steps 5 and 8.<br />
[[File:ershad_alg3.png|500px|center|thumb]]<br />
<br />
='''Experiments'''=<br />
<br />
=== Regression ===<br />
We start with a simple regression problem that illustrates the basic principles of MAML. Each task involves regressing from the input to the output of a sine wave, where the amplitude and phase of the sinusoid are varied between tasks. Thus, $p(\mathcal{T})$ is continuous, and the input and output both have a dimensionality of 1. During training and testing, datapoints are sampled uniformly. The loss is the mean-squared error between the prediction and true value. The regressor is a neural network model with 2 hidden layers of size 40 with ReLU nonlinearities. When training with MAML, we use one gradient update with K = 10 examples with a fixed step size 0.01, and use Adam as the metaoptimizer [2]. The baselines are likewise trained with Adam. To evaluate performance, we finetune a single meta-learned model on varying numbers of K examples, and compare performance to two baselines: (a) pretraining on all of the tasks, which entails training a network to regress to random sinusoid functions and then, at test-time, fine-tuning with gradient descent on the K provided points, using an automatically tuned step size, and (b) an oracle which receives the true amplitude and phase as input.<br />
<br />
We evaluate performance by fine-tuning the model learned by MAML and the pretrained model on $K = \{ 5,10,20 \}$ datapoints. During fine-tuning, each gradient step is computed using the same $K$ datapoints. Results are shown in Fig 2.<br />
<br />
<br />
[[File:ershad_results1.png|500px|center|thumb|Figure 2: Few-shot adaptation for the simple regression task. Left: Note that MAML is able to estimate parts of the curve where there are no datapoints, indicating that the model has learned about the periodic structure of sine waves. Right: Fine-tuning of a model pretrained on the same distribution of tasks without MAML, with a tuned step size. Due to the often contradictory outputs on the pre-training tasks, this model is unable to recover a suitable representation and fails to extrapolate from the small number of test-time samples.]]<br />
<br />
=== Classification ===<br />
<br />
For classification evaluation, Omniglot and MiniImagenet datasets are used. The Omniglot dataset consists of 20 instances of 1623 characters from 50 different alphabets. <br />
<br />
The experiment involves fast learning of N-way classification with 1 or 5 shots. The problem of N-way classification is set up as follows: select N unseen classes, provide the model with K different instances of each of the N classes, and evaluate the model’s ability to classify new instances within the N classes. For Omniglot, 1200 characters are selected randomly for training, irrespective of alphabet, and use the remaining for testing. The Omniglot dataset is augmented with rotations by multiples of 90 degrees.<br />
<br />
The model follows the same architecture as the embedding function that has 4 modules with a 3-by-3 convolutions and 64 filters, followed by batch normalization, a ReLU nonlinearity, and 2-by-2 max-pooling. The Omniglot images are downsampled to 28-by-28, so the dimensionality of the last hidden layer is 64. The last layer is fed into a softmax. For Omniglot, strided convolutions is used instead of max-pooling. For MiniImagenet, 32 filters per layer are used to reduce overfitting. In order to also provide a fair comparison against memory-augmented neural networks [3] and to test the flexibility of MAML, the results for a non-convolutional network are also provided. <br />
<br />
[[File:ershad_results2.png|500px|center|thumb|Table 1: Few-shot classification on held-out Omniglot characters (top) and the MiniImagenet test set (bottom). MAML achieves results that are comparable to or outperform state-of-the-art convolutional and recurrent models. Siamese nets, matching nets, and the memory module approaches are all specific to classification, and are not directly applicable to regression or RL scenarios. The $\pm$ shows 95% confidence intervals over tasks. ]]<br />
<br />
=== Reinforcement Learning ===<br />
Several simulated continuous control environments are used for RL evaluation. In all of the domain, the MAML model is a neural network policy with two hidden layers of size 100, and ReLU activations. The gradient updates are computed using vanilla policy gradient and trust-region policy optimization (TRPO) is used as the meta-optimizer.<br />
<br />
In order to avoid computing third derivatives, finite differences are computed to <br />
compute the Hessian-vector products for TRPO. For both learning and meta-learning updates, we use the standard linear feature baseline proposed by [4], which is fitted separately at each iteration for each sampled task in the batch. <br />
<br />
Three baseline models for the comparison are: <br />
(a) pretraining one policy on all of the tasks and then fine-tuning<br />
(b) training a policy from randomly initialized weights<br />
(c) an oracle policy which receives the parameters of the task as input, which for the tasks below corresponds to a goal position, goal direction, or goal velocity for the agent. <br />
<br />
The baseline models of (a) and (b) are fine-tuned with gradient descent with a manually tuned step size.<br />
<br />
2D Navigation: In the first meta-RL experiment, the authors study a set of tasks where a point agent must move to different goal positions in 2D, randomly chosen for each task within a unit square. The observation is the current 2D position, and actions correspond to velocity commands clipped to be in the range [-0.1; 0.1]. The reward is the negative squared distance to the goal, and episodes terminate when the agent is within 0:01 of the goal or at the horizon ofH = 100. The policy was trained with MAML <br />
to maximize performance after 1 policy gradient update using 20 trajectories. They compare adaptation to a new task with up to 4 gradient updates, each with 40 samples. Results are shown in Fig. 3.<br />
<br />
[[File:ershad_results3.png|500px|center|thumb|Figure 3: Top: quantitative results from 2D navigation task, Bottom: qualitative comparison between model learned with MAML and with fine-tuning from a pretrained network ]]<br />
<br />
Locomotion. To study how well MAML can scale to more complex deep RL problems, we also study adaptation on high-dimensional locomotion tasks with the MuJoCo simulator [5]. The tasks require two simulated robots – a planar cheetah and a 3D quadruped (the “ant”) – to run in a particular direction or at a particular velocity. In the goal velocity experiments, the reward is the negative absolute value between the current velocity of the agent and a goal, which is chosen uniformly at random between 0 and 2 for the cheetah and between 0 and 3 for the ant. In the goal direction experiments, the reward is the magnitude of the velocity in either the forward or backward direction, chosen at random for each task in p(T ). The horizon is H = 200, with 20 rollouts per gradient step for all problems except the ant forward/backward task, which used 40 rollouts per step. The results in Figure 5 show that MAML learns a model that can quickly adapt its velocity and direction with even <br />
just a single gradient update, and continues to improve with more gradient steps. The results also show that, on these challenging tasks, the MAML initialization substantially outperforms random initialization and pretraining.<br />
[[File:ershad_results4.png|500px|center|thumb|Figure 4: Reinforcement learning results for the half-cheetah and ant locomotion tasks, with the tasks shown on the far right. ]]<br />
<br />
='''Conclusion'''=<br />
<br />
The paper introduced a meta-learning method based on learning easily adaptable model parameters through gradient descent. The approach has a number of benefits. It is simple and does not introduce any learned parameters for meta-learning. It can be combined with any model representation that is amenable to gradient-based training, and any differentiable objective, including classification, regression, and reinforcement learning. Lastly, since our method merely produces a weight initialization, adaptation can be performed with any amount of data and any number of gradient steps, though it demonstrates state-of-the-art results on classification with only one or five examples per class. The authors also show that the method can adapt an RL agent using policy gradients and a very modest amount of experience.<br />
<br />
='''References'''=<br />
# Schmidhuber, J¨urgen. Learning to control fast-weight memories: An alternative to dynamic recurrent networks. Neural Computation, 1992.<br />
<br />
# Lake, Brenden M, Salakhutdinov, Ruslan, Gross, Jason, and Tenenbaum, Joshua B. One shot learning of simple visual concepts. In Conference of the Cognitive Science Society (CogSci), 2011.<br />
<br />
# Santoro, Adam, Bartunov, Sergey, Botvinick, Matthew, Wierstra, Daan, and Lillicrap, Timothy. Meta-learning with memory-augmented neural networks. In International Conference on Machine Learning (ICML), 2016.<br />
<br />
# Duan, Yan, Chen, Xi, Houthooft, Rein, Schulman, John, and Abbeel, Pieter. Benchmarking deep reinforcement learning for continuous control. In International Conference on Machine Learning (ICML), 2016.<br />
<br />
# Todorov, Emanuel, Erez, Tom, and Tassa, Yuval. Mujoco: A physics engine for model-based control. In International Conference on Intelligent Robots and Systems (IROS), 2012.<br />
<br />
# Videos the learned policies can be found in https://sites.google.com/view/maml.<br />
<br />
Implementation Example: https://github.com/cbfinn/maml</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Model-Agnostic_Meta-Learning_for_Fast_Adaptation_of_Deep_Networks&diff=30932Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks2017-11-20T22:45:04Z<p>H4lyu: minor correction /* Reinforcement Learning */</p>
<hr />
<div>='''Introduction & Background'''=<br />
Learning quickly is a hallmark of human intelligence, whether it involves recognizing objects from a few examples or quickly learning new skills after just minutes of experience. In this work, we propose a meta-learning algorithm that is general and model-agnostic, in the sense that it can be directly applied to any learning problem and model that is trained with a gradient descent procedure. Our focus is on deep neural network models, but we illustrate how our approach can easily handle different architectures and different problem settings, including classification, regression, and policy gradient reinforcement learning, with minimal modification. Unlike prior meta-learning methods that learn an update function or learning rule (Schmidhuber, 1987; Bengio et al., 1992; Andrychowicz et al., 2016; Ravi & Larochelle, 2017), this algorithm does not expand the number of learned parameters nor place constraints on the model architecture (e.g. by requiring a recurrent model (Santoro et al., 2016) or a Siamese network (Koch, 2015)), and it can be readily combined with fully connected, convolutional, or recurrent neural networks. It can also be used with a variety of loss functions, including differentiable supervised losses and nondifferentiable reinforcement learning objectives.<br />
<br />
The primary contribution of this work is a simple model and task-agnostic algorithm for meta-learning that trains a model’s parameters such that a small number of gradient updates will lead to fast learning on a new task. The paper shows the effectiveness of the proposed algorithm in different domains, including classification, regression, and reinforcement learning problems.<br />
<br />
='''Model-Agnostic Meta Learning (MAML)'''=<br />
The goal of the proposed model is rapid adaptation. This setting is usually formalized as few-shot learning.<br />
<br />
=== Problem set-up ===<br />
The goal of few-shot meta-learning is to train a model that can quickly adapt to a new task using only a few datapoints and training iterations. To do so. the model is trained during a meta-learning phase on a set of tasks, such that it can then be adapted to a new task using only a small number of parameter updates. In effect, the meta-learning problem treats entire tasks as training examples. <br />
<br />
Let us consider a model denoted by $f$, that maps the observation $\mathbf{x}$ into the output variable $a$. During meta-learning, the model is trained to be able to adapt to a large or infinite number of tasks. <br />
<br />
Let us consider a generic notion of task as below. Each task $\mathcal{T} = \{\mathcal{L}(\mathbf{x}_1.a_1,\mathbf{x}_2,a_2,..., \mathbf{x}_H,a_H), q(\mathbf{x}_1),q(\mathbf{x}_{t+1}|\mathbf{x}_t,a_t),H \}$, consists of a loss function $\mathcal{L}$, a distribution over initial observations $q(\mathbf{x}_1)$, a transition distribution $q(\mathbf{x}_{t+1}|\mathbf{x}_t)$, and an episode length $H$. In i.i.d. supervised learning problems,<br />
the length $H =1$. The model may generate samples of length $H$ by choosing an output at at each time $t$. The cost $\mathcal{L}$ provides a task-specific feedback, which is defined based on the nature of the problem. <br />
<br />
A distribution over tasks is denoted by $p(\mathcal{T})$. In the K-shot learning setting, the model is trained to learn a new task $\mathcal{T}_i$ drawn from $p(\mathcal{T})$ from only K samples drawn from $q_i$ and feedback $\mathcal{L}_{\mathcal{T}_i}$ generated by $\mathcal{T}_i$. During meta-training, a task $\mathcal{T}_i$ is sampled from $p(\mathcal{T})$, the model is trained with K samples and feedback from the corresponding loss LTi from Ti, and then tested on new samples from Ti. The model f is then improved by considering how the test error on new data from $q_i$ changes with respect to the parameters. In effect, the test error on sampled tasks $\mathcal{T}_i$ serves as the training error of the meta-learning process. At the end of meta-training, new tasks are sampled from $p(\mathcal{T})$, and meta-performance is measured by the model’s performance after learning from K samples.<br />
<br />
=== MAML Algorithm ===<br />
[[File:model.png|200px|right|thumb|Figure 1: Diagram of the MAML algorithm]]<br />
The paper proposes a method that can learn the parameters of any standard model via meta-learning in such a way as to prepare that model for fast adaptation. The intuition behind this approach is that some internal representations are more transferrable than others. Since the model will be fine-tuned using a gradient-based learning rule on a new task, we will aim to learn a model in such a way that this gradient-based learning rule can make rapid progress on new tasks drawn from $p(\mathcal{T})$, without overfitting. In effect, we will aim to find model parameters that are sensitive to changes in the task, such that small changes in the parameters will produce large improvements on the loss function of any task drawn from $p(\mathcal{T})$, see Fig 1.<br />
<br />
Note that there is no assumption about the form of the model. Only assumption is that it is parameterized by a vector of parameters $\theta$, and the loss is smooth so that the parameters can be leaned using gradient-based techniques. Formally lets assume that the model is denoted by $f_{\theta}$. When adapting<br />
to a new task $\mathcal{T}_i $, the model’s parameters $\theta$ become $\theta_i'$. In our method, the updated parameter vector $\theta_i'$ is computed using one or more gradient descent updates on task $\mathcal{T}_i $. For example, when using one gradient update:<br />
<br />
$$<br />
\theta_i ' = \theta - \alpha \nabla_{\theta \mathcal{L}_{\mathcal{T}_i}}(f_{\theta}).<br />
$$<br />
<br />
Here $\alpha$ is a the learning rate of each task and considered as a hyperparameter. They consider a single step of update for the rest of the paper, for the sake of the simplicity. <br />
<br />
The model parameters are trained by optimizing for the performance<br />
of $f_{\theta_i'}$ with respect to $\theta$ across tasks sampled from $p(\mathcal{T})$. More concretely, the meta-objective is as follows: <br />
<br />
$$<br />
\min_{\theta} \sum \limits_{\mathcal{T}_i \sim p(\mathcal{T})} \mathcal{L}_{\mathcal{T}_i} (f_{\theta_i'}) = \sum \limits_{\mathcal{T}_i \sim p(\mathcal{T})} \mathcal{L}_{\mathcal{T}_i} (f_{\theta - \alpha \nabla_{\theta \mathcal{L}_{\mathcal{T}_i}}(f_{\theta})})<br />
$$<br />
<br />
Note that the meta-optimization is performed over the model parameters $\theta$, whereas the objective is computed using the updated model parameters $\theta'$. The model aims to optimize the model parameters such that one or a small number of gradient step on a new task will produce maximally effective behavior on that task. <br />
<br />
Therefore the meta-learning across the tasks is performed via stochastic gradient descent (SGD), such that the model parameters $\theta$ are updated as:<br />
<br />
$$<br />
\theta \gets \theta - \beta \nabla_{\theta } \sum \limits_{\mathcal{T}_i \sim p(\mathcal{T})} \mathcal{L}_{\mathcal{T}_i} (f_{\theta_i'})<br />
$$<br />
where $\beta$ is the meta step size. Outline of the algorithm is shown in Algorithm 1. <br />
[[File:ershad_alg1.png|500px|center|thumb]]<br />
<br />
The MAML meta-gradient update involves a gradient through a gradient. Computationally, this requires an additional backward pass through f to compute Hessian-vector products, which is supported by standard deep learning libraries such as TensorFlow.<br />
<br />
='''Different Types of MAML'''=<br />
In this section the MAML algorithm is discussed for different supervised learning and reinforcement learning tasks. The differences between each of these tasks are in their loss function and the way the data is generated. In general, this method does not require additional model parameters nor using any additional meta-learner to learn the update of parameters. Compared to other approaches that tend to “learn to compare new examples in a learned metric space using e.g. Siamese networks or recurrence with attention mechanisms”, the proposed method can be generalized to any other problems including classification, regression and reinforcement learning. <br />
<br />
=== Supervised Regression and Classification ===<br />
Few-shot learning is well-studied in this field. For these two types of tasks the horizon $H$ is equal to 1, since the data points are generated i.i.d. <br />
<br />
Although any common classification and regression objectives can be used as the loss, the paper uses the following losses for these two tasks. <br />
<br />
Regression : For regression we use the mean-square error (MSE):<br />
<br />
$$<br />
\mathcal{L}_{\mathcal{T}_i} (f_{\theta}) = \sum \limits_{\mathbf{x}^{(j)}, \mathcal{y}^{(j)} \sim \mathcal{T}_i} \parallel f_{\theta} (\mathbf{x}^{(j)} - \mathbf{y}^{(j)})\parallel_2^2<br />
$$<br />
<br />
where $\mathbf{x}^{(j)}$ and $\mathbf{y}^{()j}$ are the input/output pair sampled from task $\mathcal{T}_i$. In K-shot regression tasks, K input/output pairs are provided for learning for each task. <br />
<br />
Classification: For classification we use the cross entropy loss:<br />
<br />
$$<br />
\mathcal{L}_{\mathcal{T}_i} (f_{\theta}) = \sum \limits_{\mathbf{x}^{(j)}, \mathcal{y}^{(j)} \sim \mathcal{T}_i} \mathbf{y}^{(j)} \log f_{\theta}(\mathbf{x}^{(j)}) + (1-\mathbf{y}^{(j)}) \log (1-f_{\theta}(\mathbf{x}^{(j)}))<br />
$$<br />
<br />
According to the conventional terminology, K-shot classification tasks use K input/output pairs from each class, for a total of $NK$ data points for N-way classification.<br />
<br />
Given a distribution over tasks, these loss functions can be directly inserted into the equations in the previous section to perform meta-learning, as detailed in Algorithm 2.<br />
[[File:ershad_alg2.png|500px|center|thumb]]<br />
<br />
=== Reinforcement Learning ===<br />
In reinforcement learning (RL), the goal of few-shot meta learning is to enable an agent to quickly acquire a policy for a new test task using only a small amount of experience in the test setting. A new task might involve achieving a new goal or succeeding on a previously trained goal in a new environment. For example an agent may learn how to navigate mazes very quickly so that, when faced with a new maze, it can determine how to reliably reach the exit with only a few samples.<br />
<br />
Each RL task $\mathcal{T}_i$ contains an initial state distribution $q_i(\mathbf{x}_1)$ and a transition distribution $q_i(\mathbf{x}_{t+1}|\mathbf{x}_t,a_t)$ , and the loss $\mathcal{L}_{\mathcal{T}_i}$ corresponds to the (negative) reward function $R$. The entire task is therefore a Markov decision process (MDP) with horizon H, where the learner is allowed to query a limited number of sample trajectories for few-shot learning. Any aspect of the MDP may change across tasks in $p(\mathcal{T})$. The model being learned, $f_{\theta}$, is a policy that maps from states $\mathbf{x}_t$ to a distribution over actions $a_t$ at each timestep $t \in \{1,...,H\}$. The loss for task $\mathcal{T}_i$ and model $f_{\theta}$ takes the form<br />
<br />
$$<br />
\mathcal{L}_{\mathcal{T}_i}(f_{\theta}) = -\mathbb{E}_{\mathbf{x}_t,a_t \sim f_{\theta},q_{\mathcal{T}_i}} \big [\sum_{t=1}^H R_i(\mathbf{x}_t,a_t)\big ]<br />
$$<br />
<br />
<br />
In K-shot reinforcement learning, K rollouts from $f_{\theta}$ and task $\mathcal{T}_i$, $(\mathbf{x}_1,a_1,...,\mathbf{x}_H)$, and the corresponding rewards $ R(\mathbf{x}_t,a_t)$, may be used for adaptation on a new task $\mathcal{T}_i$.<br />
<br />
Since the expected reward is generally not differentiable due to unknown dynamics, we use policy gradient methods to estimate the gradient both for the model gradient update(s) and the meta-optimization. Since policy gradients are an on-policy algorithm, each additional gradient step during the adaptation of $f_{\theta}$ requires new samples from the current policy $f_{\theta_i}$ . We detail the algorithm in Algorithm 3.<br />
[[File:ershad_alg3.png|500px|center|thumb]]<br />
<br />
='''Experiments'''=<br />
<br />
=== Regression ===<br />
We start with a simple regression problem that illustrates the basic principles of MAML. Each task involves regressing from the input to the output of a sine wave, where the amplitude and phase of the sinusoid are varied between tasks. Thus, $p(\mathcal{T})$ is continuous, and the input and output both have a dimensionality of 1. During training and testing, datapoints are sampled uniformly. The loss is the mean-squared error between the prediction and true value. The regressor is a neural network model with 2 hidden layers of size 40 with ReLU nonlinearities. When training with MAML, we use one gradient update with K = 10 examples with a fixed step size 0.01, and use Adam as the metaoptimizer [2]. The baselines are likewise trained with Adam. To evaluate performance, we finetune a single meta-learned model on varying numbers of K examples, and compare performance to two baselines: (a) pretraining on all of the tasks, which entails training a network to regress to random sinusoid functions and then, at test-time, fine-tuning with gradient descent on the K provided points, using an automatically tuned step size, and (b) an oracle which receives the true amplitude and phase as input.<br />
<br />
We evaluate performance by fine-tuning the model learned by MAML and the pretrained model on $K = \{ 5,10,20 \}$ datapoints. During fine-tuning, each gradient step is computed using the same $K$ datapoints. Results are shown in Fig 2.<br />
<br />
<br />
[[File:ershad_results1.png|500px|center|thumb|Figure 2: Few-shot adaptation for the simple regression task. Left: Note that MAML is able to estimate parts of the curve where there are no datapoints, indicating that the model has learned about the periodic structure of sine waves. Right: Fine-tuning of a model pretrained on the same distribution of tasks without MAML, with a tuned step size. Due to the often contradictory outputs on the pre-training tasks, this model is unable to recover a suitable representation and fails to extrapolate from the small number of test-time samples.]]<br />
<br />
=== Classification ===<br />
<br />
For classification evaluation, Omniglot and MiniImagenet datasets are used. The Omniglot dataset consists of 20 instances of 1623 characters from 50 different alphabets. <br />
<br />
The experiment involves fast learning of N-way classification with 1 or 5 shots. The problem of N-way classification is set up as follows: select N unseen classes, provide the model with K different instances of each of the N classes, and evaluate the model’s ability to classify new instances within the N classes. For Omniglot, 1200 characters are selected randomly for training, irrespective of alphabet, and use the remaining for testing. The Omniglot dataset is augmented with rotations by multiples of 90 degrees.<br />
<br />
The model follows the same architecture as the embedding function that has 4 modules with a 3-by-3 convolutions and 64 filters, followed by batch normalization, a ReLU nonlinearity, and 2-by-2 max-pooling. The Omniglot images are downsampled to 28-by-28, so the dimensionality of the last hidden layer is 64. The last layer is fed into a softmax. For Omniglot, strided convolutions is used instead of max-pooling. For MiniImagenet, 32 filters per layer are used to reduce overfitting. In order to also provide a fair comparison against memory-augmented neural networks [3] and to test the flexibility of MAML, the results for a non-convolutional network are also provided. <br />
<br />
[[File:ershad_results2.png|500px|center|thumb|Table 1: Few-shot classification on held-out Omniglot characters (top) and the MiniImagenet test set (bottom). MAML achieves results that are comparable to or outperform state-of-the-art convolutional and recurrent models. Siamese nets, matching nets, and the memory module approaches are all specific to classification, and are not directly applicable to regression or RL scenarios. The $\pm$ shows 95% confidence intervals over tasks. ]]<br />
<br />
=== Reinforcement Learning ===<br />
Several simulated continuous control environments are used for RL evaluation. In all of the domain, the MAML model is a neural network policy with two hidden layers of size 100, and ReLU activations. The gradient updates are computed using vanilla policy gradient and trust-region policy optimization (TRPO) is used as the meta-optimizer.<br />
<br />
In order to avoid computing third derivatives, finite differences are computed to <br />
compute the Hessian-vector products for TRPO. For both learning and meta-learning updates, we use the standard linear feature baseline proposed by [4], which is fitted separately at each iteration for each sampled task in the batch. <br />
<br />
Three baseline models for the comparison are: <br />
(a) pretraining one policy on all of the tasks and then fine-tuning<br />
(b) training a policy from randomly initialized weights<br />
(c) an oracle policy which receives the parameters of the task as input, which for the tasks below corresponds to a goal position, goal direction, or goal velocity for the agent. <br />
<br />
The baseline models of (a) and (b) are fine-tuned with gradient descent with a manually tuned step size.<br />
<br />
2D Navigation: In the first meta-RL experiment, the authors study a set of tasks where a point agent must move to different goal positions in 2D, randomly chosen for each task within a unit square. The observation is the current 2D position, and actions correspond to velocity commands clipped to be in the range [-0.1; 0.1]. The reward is the negative squared distance to the goal, and episodes terminate when the agent is within 0:01 of the goal or at the horizon ofH = 100. The policy was trained with MAML <br />
to maximize performance after 1 policy gradient update using 20 trajectories. They compare adaptation to a new task with up to 4 gradient updates, each with 40 samples. Results are shown in Fig. 3.<br />
<br />
[[File:ershad_results3.png|500px|center|thumb|Figure 3: Top: quantitative results from 2D navigation task, Bottom: qualitative comparison between model learned with MAML and with fine-tuning from a pretrained network ]]<br />
<br />
Locomotion. To study how well MAML can scale to more complex deep RL problems, we also study adaptation on high-dimensional locomotion tasks with the MuJoCo simulator [5]. The tasks require two simulated robots – a planar cheetah and a 3D quadruped (the “ant”) – to run in a particular direction or at a particular velocity. In the goal velocity experiments, the reward is the negative absolute value between the current velocity of the agent and a goal, which is chosen uniformly at random between 0 and 2 for the cheetah and between 0 and 3 for the ant. In the goal direction experiments, the reward is the magnitude of the velocity in either the forward or backward direction, chosen at random for each task in p(T ). The horizon is H = 200, with 20 rollouts per gradient step for all problems except the ant forward/backward task, which used 40 rollouts per step. The results in Figure 5 show that MAML learns a model that can quickly adapt its velocity and direction with even <br />
just a single gradient update, and continues to improve with more gradient steps. The results also show that, on these challenging tasks, the MAML initialization substantially outperforms random initialization and pretraining.<br />
[[File:ershad_results4.png|500px|center|thumb|Figure 4: Reinforcement learning results for the half-cheetah and ant locomotion tasks, with the tasks shown on the far right. ]]<br />
<br />
='''Conclusion'''=<br />
<br />
The paper introduced a meta-learning method based on learning easily adaptable model parameters through gradient descent. The approach has a number of benefits. It is simple and does not introduce any learned parameters for meta-learning. It can be combined with any model representation that is amenable to gradient-based training, and any differentiable objective, including classification, regression, and reinforcement learning. Lastly, since our method merely produces a weight initialization, adaptation can be performed with any amount of data and any number of gradient steps, though it demonstrates state-of-the-art results on classification with only one or five examples per class. The authors also show that the method can adapt an RL agent using policy gradients and a very modest amount of experience.<br />
<br />
='''References'''=<br />
# Schmidhuber, J¨urgen. Learning to control fast-weight memories: An alternative to dynamic recurrent networks. Neural Computation, 1992.<br />
<br />
# Lake, Brenden M, Salakhutdinov, Ruslan, Gross, Jason, and Tenenbaum, Joshua B. One shot learning of simple visual concepts. In Conference of the Cognitive Science Society (CogSci), 2011.<br />
<br />
# Santoro, Adam, Bartunov, Sergey, Botvinick, Matthew, Wierstra, Daan, and Lillicrap, Timothy. Meta-learning with memory-augmented neural networks. In International Conference on Machine Learning (ICML), 2016.<br />
<br />
# Duan, Yan, Chen, Xi, Houthooft, Rein, Schulman, John, and Abbeel, Pieter. Benchmarking deep reinforcement learning for continuous control. In International Conference on Machine Learning (ICML), 2016.<br />
<br />
# Todorov, Emanuel, Erez, Tom, and Tassa, Yuval. Mujoco: A physics engine for model-based control. In International Conference on Intelligent Robots and Systems (IROS), 2012.<br />
<br />
# Videos the learned policies can be found in https://sites.google.com/view/maml.<br />
<br />
Implementation Example: https://github.com/cbfinn/maml</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Deep_Alternative_Neural_Network:_Exploring_Contexts_As_Early_As_Possible_For_Action_Recognition&diff=30881Deep Alternative Neural Network: Exploring Contexts As Early As Possible For Action Recognition2017-11-20T04:44:48Z<p>H4lyu: /* Alternative Layer */</p>
<hr />
<div>==Introduction==<br />
<br />
Action recognition deals with recognizing and classifying the actions or activities done by humans or other agents in a video clip. In action recognition, Contexts contribute semantic clues for action recognition in video(See Fig below[8]). Conventional Neural Networks [1,2,3] and their shifted version 3D CNNs [4,5,6] have been employed in action recognition but they identify and aggregate the contexts at later stages. <br />
[[File:ActionRecognition1.jpg|center|400px|border|context and action region]]<br />
<br />
The authors have come up with a strategy to identify contexts in the videos as early as possible and leverage their evolutions for action recognition. Contexts contribute semantic clues for action recognition in videos. The network themselves involve a lot of layers and the first layer typically being receptive fields (RF) output only extra local features. As we go deeper into the layers the Receptive Fields expand and we start getting the contexts. The authors identified that increasing the number of layers will only cause additional burden in terms of handling the parameters and contexts could be obtained even in the earlier stages. The authors also cite the papers [9,10] that relate the CNNs and the visual systems of our brain, one remarkable difference being the abundant recurrent connections in our brain compared to the forward connections in the CNNs. <br />
<br />
The main contributions in the paper can be summarized as follows: <br />
* A Deep Alternative Neural Network (DANN) is proposed for action recognition. <br />
* DANN consists of alternative volumetric convolutional and recurrent layers. <br />
* An adaptive method to determine the temporal size of the video clip <br />
* A volumetric pyramid pooling layer to resize the output before fully connected layers.<br />
<br />
===Related Work===<br />
There are already exists a very related paper ([11]) in the literature which proposed a similar alternation architecture. In particular the similarity between the authors work and the aforementioned paper is that they both propose alternating CNN-RNN architectures. This similarity between the two works was noted by Reviewer 1 in the NIPS review process.<br />
<br />
=== Optic Flow ===<br />
Optical flow or optic flow is the pattern of apparent motion of objects, surfaces, and edges in a visual scene caused by the relative motion between an observer and a scene.<br />
It can be used for affordance perception, the ability to discern possibilities for action within the environment.<br />
<br />
==Deep Alternative Neural Network:==<br />
===Adaptive Network Input===<br />
The input size of the video clip is generally determined empirically and various approaches have been taken in the past with a different number of frames. For instance, many previous papers suggested to used shorter intervals of between 1 to 16 frames. However, more recent work[9] recognized that human-based actions often “span tens or hundreds of frames” and longer intervals such as 60 frames will outperform the one with shorter interval. However, there’s still no systematic way of determining number of frames for input size of the network. This serves the motives for the authors of this paper to develop this adaptive method. Past research shows that motion energy intensity induced by human activity exhibits a regular periodicity. This signal can be approximately estimated by optical flow computation as shown in Figure 1, and is particularly suitable to address our temporal estimation due to: <br />
* the local minima and maxima landmarks probably correspond to characteristic gesture and motion <br />
* it is relatively robust to changes in camera viewpoint.<br />
<br />
The authors have come up with an adaptive method to automatically select the most discriminative video fragments using the density of optical flow energy which exhibits regular periodicity. According to Wikipedia, optical flow is the pattern of apparent motion of objects, surfaces, and edges in a visual scene caused by the relative motion between an observer and a scene, and optical flow methods try to calculate the motion between two image frames which are taken at different times. The optimal flow energy of an optical field $(v_{x},v_{y})$ is defined as follows <br />
<br />
:<math>e(I)=\underset{(x,y)\in\mathbb{P}}{\operatorname{\Sigma}} ||v_{x}(x,y),v_{y}(x,y)||_{2}</math><br />
<br />
Here, P is the pixel level set of selected interest points. They locate the local minima and maxima landmarks $\{t\}$ of $\epsilon = \{e(I_1),\dots,e(I_t)\}$ and for each two consecutive landmarks create a video fragment $s$ by extracting the frames $s = \{I_{t-1},\dots,I_t\}$.<br />
<br />
[[File:golfswing.png]]<br />
<br />
===Alternative Layer===<br />
This is a key layer consisting of a standard volumetric convolutional layer followed by a designed recurrent layer. Volumetric convolutional extracts features from local neighborhoods and a recurrent layer is applied to the output and it proceeds iteratively for T times. The input of a unit at position (x,y,z) in the jth feature map of the ith AL in time t is given by,<br />
<br />
:<math>u_{ij}^{xyz}(t) = u_{ij}^{xyz}(0) + f(w_{ij}^{r}u_{ij}^{xyz}(t-1)) + b_{ij} \\ <br />
u_{ij}^{xyz}(0) = f(w_{i-1}^{c}u_{(i-1)j}^{xyz}) <br />
</math><br />
<br />
U(0): feed forward output of volumetric convolutional layer. <br />
U(t-1) : recurrent input of previous time <br />
$w_{k}^{c}$ and $w_{k}^{r}$: vectorized feed-forward kernels and recurrent kernels respectively <br />
f: ReLU function<br />
<br />
Figure 3 depicts this structure:<br />
[[File:unfolded.PNG|1000px]]<br />
<br />
The recurrent connections in AL provide two advantages. First, they enable every unit to incorporate contexts in an arbitrarily large region in the current layer; Second, the recurrent connections increase the network depth while keeping the number of adjustable parameters constant by weight sharing.<br />
<br />
===Volumetric Pyramid Pooling Layer===<br />
<br />
[[File:Volumetric Pyramid Pooling Layer.png|thumb|550px|Figure 2: Volumetric Pyramid Pooling Layer]]<br />
The authors have replaced the last pooling layer with a volumetric pyramid pooling layer as we need fixed-length vectors for the fully connected layers and the AL accepts video clips of arbitrary sizes and produces outputs of variable sizes. The authors have used the max pooling to pool the responses of each kernel in each volumetric bin. The outputs are kM dimensional vectors where:<br />
<br />
M: number of bins <br />
<br />
K: Number of kernels in the last alternative layer.<br />
<br />
This layer structure allows not only for arbitrary-length videos, but also arbitrary aspect ratios and scales.<br />
<br />
==Overall Architecture== <br />
[[File:DANN Architecture.png|thumb|550px|Figure 3:DANN Architecture]]<br />
The following are the components of the DANN (as shown in Figure 3)<br />
* 6 Alternative layers with 64, 128, 256, 256, 512 and 512 kernel response maps <br />
* 5 ReLU and volumetric pooling layers <br />
* 1 volumetric pyramid pooling layer <br />
* 3 fully connected layers of size 2048 each <br />
* A softmax layer<br />
<br />
==Implementation details==<br />
The authors have used the Torch toolbox platform for Implementations of volumetric convolutions, recurrent layers and optimizations. They have used a technique called as random clipping for data augmentation, in which they select a point randomly from the input video of fixed size 80x80xt after determining the temporal size t. For training the network the authors have used SGD applied to mini-batches of size 30 with a negative log likelihood criterion. Training is done by minimizing the cross-entropy loss function using backpropagation through time algorithm (BPTT). During testing, they applied a video clip divided into 80x80xt clips with a stride of 4 frames followed by testing with 10 crops. Final score is the average of all clip-level scores and the crop scores.<br />
Data augmentation techniques such as the multi-scale cropping method have been evaluated due to the recent success in the state-of-the-art performance displayed by Very Deep Two-stream ConvNets. Going by intuition, the corner cropping strategy could provide better results ( based on trade-off degree) since the receptive fields can focus harder on the central regions of the video frames [7].<br />
<br />
==Evaluations==<br />
===Datasets:===<br />
* The datasets used in the evaluation are UCF101 and HMDB51 <br />
* UCF101 – 13K videos annotated into 101 classes <br />
* HMDB51 – 6.8K videos with 51 actions. <br />
* Three training and test splits are provided <br />
* Performance measured by mean classification accuracy across the splits. <br />
* UCF101 split – 9.5K videos; HMDB51 – 3.7K training videos.<br />
<br />
===Quantitative Results===<br />
The authors used three types of optical flows, viz., sparse, RGB and TVL1 and found that TVL1 is suitable as action recognition is more easy to learn from motion information compared to raw pixel values. The influence of data augmentation is also studied. The baseline being sliding window with 75% overlap, the authors observed that the random clipping and multi-scale clipping outperformed the baseline on the UCF101 split 1 dataset. The authors were able to prove that the adaptive temporal length was able to give a boost of 4.2% when compared with architectures that had fixed-size temporal length. Experiments were also conducted to see if the learnings done in one dataset could improve the accuracy of another dataset. Fine tuning HMDB51 from UCF101 boosted the performance from 56.4% to 62.5%. The authors also observed that increasing the AL layers improves the performance as larger contexts are being embedded into the DANN. The DANN achieved an overall accuracy of 65.9% and 91.6% on HMDB51 and UCF101 respectively.<br />
<br />
<br />
[[File:Performance Comparison of different input modalities.png]]<br />
<br />
===Qualitative Analysis===<br />
The authors have discussed the quality of the prediction in the video clips taking examples of two different scenes involving bowling and haircut. In the bowling scene, the adaptive temporal choice used by DANN could aggregate more reasonable semantic structures and hence it leveraged reasonable video clips as input. On the other hand, the performance on the haircut video clip was not up to the mark as the rich contexts provided by the DANN was not helpful in a setting with simple actions performed in a simple background.<br />
<br />
==Conclusion==<br />
* Deep alternative neural network is introduced for action recognition. <br />
* DANN consists of volumetric convolutional layer and a recurrent layer. <br />
* The authors have experimented with different datasets like HMDB51 and UCF101 with different scenarios and compared the * * performance of DANN with other approaches. <br />
* The spatial size is still chosen in an ad hoc manner and this can be an area of improvement. <br />
* There are prospects for studying action tube which is a more compact input.<br />
<br />
Github code: https://github.com/wangjinzhuo/DANN<br />
<br />
==References==<br />
<br />
[1] Andrej Karpathy, George Toderici, Sachin Shetty, Tommy Leung, Rahul Sukthankar, and Li FeiFei. Large-scale video classification with convolutional neural networks. In CVPR, pages 1725–1732, 2014 <br />
<br />
[2] Karen Simonyan and Andrew Zisserman. Two-stream convolutional networks for action recognition in videos. In NIPS, pages 568–576, 2014. <br />
<br />
[3]Limin Wang, Yu Qiao, and Xiaoou Tang. Action recognition with trajectory-pooled deepconvolutional descriptors. In CVPR, pages 4305–4314, 2015. <br />
<br />
[4] Shuiwang Ji, Wei Xu, Ming Yang, and Kai Yu. 3d convolutional neural networks for human action recognition. TPAMI, 35(1):221–231, 2013. <br />
<br />
[5] Du Tran, Lubomir Bourdev, Rob Fergus, Lorenzo Torresani, and Manohar Paluri. Learning spatiotemporal features with 3d convolutional networks. In ICCV, pages 4489–4497, 2015. <br />
<br />
[6]Gül Varol, Ivan Laptev, and Cordelia Schmid. Long-term temporal convolutions for action recognition. arXiv preprint arXiv:1604.04494, 2016. <br />
<br />
[7]Limin Wang, Yuanjun Xiong, Zhe Wang, Yu Qiao. Towards Good Practices for Very Deep Two-Stream ConvNets. arXiv preprint arXiv:1507.02159 , 2015. <br />
<br />
[8] IEEE International Symposium on Multimedia 2013 <br />
<br />
[9] Gül Varol, Ivan Laptev, and Cordelia Schmid. Long-term temporal convolutions for action<br />
recognition. arXiv preprint arXiv:1604.04494, 2016<br />
<br />
[10] https://en.wikipedia.org/wiki/Optical_flow<br />
<br />
[11] Delving Deeper into Convolutional Networks for Learning Video Representations Nicolas Ballas, Li Yao, Chris Pal, Aaron Courville, ICLR 2016 <br />
<br />
[36] Christopher Zach, Thomas Pock, and Horst Bischof. A duality based approach for realtime tv-l<br />
1 optical flow. In Pattern Recognition, pages 214–223. 2007.<br />
<br />
A list of expert reviews: http://media.nips.cc/nipsbooks/nipspapers/paper_files/nips29/reviews/480.html</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Universal_Style_Transfer_via_Feature_Transforms&diff=30878Universal Style Transfer via Feature Transforms2017-11-20T03:50:06Z<p>H4lyu: /* Methodology */</p>
<hr />
<div>=Introduction=<br />
When viewing an image, whether it is a photograph or a painting, two types of mutually exclusive data are present. First, there is the content of the image, such as a person in a portrait. However, the content does not uniquely define the image. Consider a case where multiple artists paint a portrait of an identical subject, the results would vary despite the content being invariant. The cause of the variance is rooted in the style of each particular artist. Therefore, style transfer between two images results in the content being unaffected but the style being copied. Style transfer is an important image editing task which enables the creation of new artistic works. Typically one image is termed the content/reference image, whose style is discarded. The other image is called the style image, whose style, but the not content is copied to the content image.<br />
<br />
Deep learning techniques have been shown to be effective methods for implementing style transfer. Previous methods have been successful but with several key limitations and often trade off between generalization, quality and efficiency. Either they are fast, but have very few styles that can be transferred or they can handle arbitrary styles but are no longer efficient. The presented paper establishes a compromise between these two extremes by using only whitening and coloring transforms (WCT) to transfer a style within a feedforward image reconstruction architecture. No training of the underlying deep network is required per style.<br />
<br />
=Related Work=<br />
Gatys et al. developed a new method for generating textures from sample images in 2015 [1] and extended their approach to style transfer by 2016 [2]. They proposed the use of a pre-trained convolutional neural network (CNN) to separate content and style of input images. Having proven successful, a number of improvements quickly developed, reducing computational time, increasing the diversity of transferrable styes, and improving the quality of the results. Central to these approaches and of the present paper is the use of a CNN.<br />
<br />
In 2017, Mechrez et al. [12] proposed an approach that takes as input a stylized image and makes it more photorealistic. Their approach relied on the Screened Poisson Equation, maintaining the fidelity of the stylized image while constraining the gradients to those of the original input image. The method they proposed was fast, simple, fully automatic and showed positive progress in making a stylized image photorealistic.<br />
<br />
Alternative attempts, by using a single network to transfer<br />
multiple styles, include models conditioned on binary selection units [13], a network that learns a set of new filters for every new style [15], and a novel conditional normalization layer that learns normalization parameters for each style [3]<br />
==How Content and Style are Extracted using CNNs==<br />
A CNN was chosen due to its ability to extract high level feature from images. These features can be interpreted in two ways. Within layer <math> l </math> there are <math> N_l </math> feature maps of size <math> M_l </math>. With a particular input image, the feature maps are given by <math> F_{i,j}^l </math> where <math> i </math> and <math> j </math> locate the map within the layer. Starting with a white noise image and an reference (content) image, the features can be transferred by minimizing<br />
<br />
<center><br />
<math> \mathcal{L}_{content} = \frac{1}{2} \sum_{i,j} \left( F_{i,j}^l - P_{i,j}^l \right)^2 </math><br />
</center><br />
<br />
where <math> P_{i,j} </math> denotes the feature map output caused by the white noise image. Therefore this loss function preserves the content of the reference image. The style is described using a Gram matrix given by<br />
<br />
<center><br />
<math><br />
G_{i,j}^l = \sum_k F_{i,k}^l F_{j,k}^l<br />
</math><br />
</center><br />
<br />
Gram matrix $G$ of a set of vectors $v_1,\dots,v_n$ is the matrix of all possible inner products whose entries are given by $G_{ij}=v_i^Tv_j$. The loss function that describes a difference in style between two images is equal to:<br />
<br />
<center><br />
<math><br />
\mathcal{L}_{style} = \frac{1}{4 N_l^2 M_l^2} \sum_{i,j} \left(G_{i,j}^l - A_{i,j}^l \right)^2<br />
</math><br />
</center><br />
<br />
where <math> A_{i,j}^l </math> and <math> G_{i,j}^l </math> are the Gram matrices of the generated image and style image respectively. Therefore three images are required, a style image, a content image and an initial white noise image. Iterative optimization is then used to add content from one image to the white noise image, and style from the other. An additional parameter is used to balance the ratio of these loss functions.<br />
<br />
The 19-layer ImageNet trained VGG network was chosen by Gatys et al. VGG-19 is still commonly used in more recent works as will be shown in the presented paper, although training datasets vary. Such CNNs are typically used in classification problems by finalizing their output through a series of full connected layers. For content and style extraction it is the convolutional layers that are required. The method of Gatys et al. is style independent, since the CNN does not need to be trained for each style image. However the process of iterative optimization to generate the output image is computational expensive.<br />
<br />
==Other Methods==<br />
Other methods avoid the inefficiency of iterative optimization by training a network/networks on a set of styles. The network then directly transfers the style from the style image to the content image without solving the iterative optimization problem. V. Dumoulin et al. trained a single network on $N$ styles [3]. This improved upon previous work where a network was required per style [4]. The stylized output image was generated by simply running a feedforward pass of the network on the content image. While efficiency is high, the method is no longer able to apply an arbitrary style without retraining.<br />
<br />
=Methodology=<br />
Li et al. have proposed a novel method for generating the stylized image. A CNN is still used as in Gatys et al. to extract content and style. However, the stylized image is not generated through iterative optimization or a feed-forward pass as required by previous methods. Instead, whitening and colour transforms are used.<br />
<br />
==Image Reconstruction==<br />
[[File:image_resconstruction.png|thumb|150px|right|alt=Training a single decoder.|Training a single decoder. X denotes the layer of the VGG encoder that the decoder receives as input.]]<br />
An auto-encoder network is used to first encode an input image into a set of feature maps, and then decode it back to an image as shown in the adjacent figure. The encoder network used is VGG-19. This network is reponsible for obtaining feature maps (similar to Gatys et al.). The output of each of the first five layers is then fed into a corresponding decoder network, which is a mirrored version of VGG-19. Each decoder network then decodes the feature maps of the $l$th layer producing an output image. A mechanism for transferring style will be implemented by manipulating the feature maps between the encoder and decoder networks.<br />
<br />
First, the auto-encoder network needs to be trained. The following loss function is used<br />
<br />
<center><br />
<math><br />
<br />
\mathcal{L} = || I_{output} - I_{input} ||_2^2 + \lambda || \Phi(I_{output}) - \Phi(I_{input})||_2^2<br />
<br />
</math><br />
</center><br />
<br />
where $I_{input}$ and $I_{output}$ are the input and output images of the auto-encoder. $\Phi$ is the VGG encoder. The first term of the loss is the pixel reconstruction loss, while the second term is feature loss. Recall from "Related Work" that the feature maps correspond to the content of the image. Therefore the second term can also be seen as penalising for content differences that arise due to the encoder network. The network was trained using the Microsoft COCO dataset. <br />
<br />
They use whitening and coloring transforms to directly transform the $f_c$ (VGG feature<br />
map of content image at a certain layer) to match the covariance matrix of $f_s$ (VGG feature<br />
map of style image). This process is consisted of two steps, i.e., whitening and coloring transform. Note that the decoder will reconstruct the original content image if $f_c$ is directly fed into it.<br />
<br />
==Whitening Transform==<br />
Whitening first requires that the covariance of the data is a diagonal matrix. This is done by solving for the covariance matrix's eigenvalues and eigenvector matrices. Whitening then forces the diagonal elements of the eigenvalue matrix to be the same. This is achieved for a feature map from VGG through the following steps.<br />
<br />
# The feature map $f_c$ is extracted from a layer of the encoder network after activation on the content image. This is the data to be whitened.<br />
# $f_c$ is centered by subtracting its mean vector $m_c$.<br />
# Then, the eigenvectors $E_c$ and eigenvalues $D_c$ are found for the covariance matrix of $f_c$. <br />
# The whitened feature map is then given by $\hat{f}_c = E_c D_c^{-1/2} E_c^T f_c$. <br />
<br />
Note that this is indeed finding the symmetric transformer matrix $A$ in $\hat{f}_c = A f_c$ such that the covariance matrix of $\hat{f}_c$ is identity matrix. If interested, the derivation of the whitening equation can be seen in [5]. Li et al. found that whitening removed styles from the image.<br />
<br />
==Colour Transform==<br />
However, whitening does not transfer style from the style image. It only uses feature maps from the content image. The colour transform uses both $\hat{f}_c$ from above and $f_s$, the feature map from the style image.<br />
<br />
# $f_s$ is centered by subtracting its mean vector $m_s$.<br />
# Then, the eigenvectors $E_s$ and eigenvalues $D_s$ are calculated for the covariance matrix of $f_s$.<br />
# The colour transform is given by $\hat{f}_{cs} = E_s D_s^{1/2} E_s^T \hat{f}_c$.<br />
# Recenter $\hat{f}_{cs}$ using $m_s$.<br />
<br />
Intuitively, colouring results in a correlation between the $\hat{f}_c$ and $f_s$ feature maps, or rather, $\hat{f}_{cs}$ is a linear transform of the original feature map $f_c$ which takes on the variance of $f_s$. This is where the style transfer takes place.<br />
<br />
==Content/Style Balance==<br />
Using just $\hat{f}_{cs}$ as the input to the decoder may create a result that is too extreme in style. To balance content and style a new parameter $\alpha$ is defined.<br />
<br />
<center><br />
<math><br />
<br />
\hat{f}_{cs} = \alpha \hat{f}_{cs} + (1 - \alpha) f_c<br />
<br />
</math><br />
</center><br />
<br />
Authors use $\alpha$ = 0.6 in the style transfer experiments.<br />
<br />
==Using Multiple Layers==<br />
It has been previously mentioned that multiple decoders were trained, one for each of the first five layers of the encoder network. Each layer of a CNN perceives features at different levels. Levels close to the input image will detect lower level local features such as edges. Those levels deeper into the network will detect more complex global features. The style transfer algorithm is applied at each of these levels, which yields the question as to which results, as shown below, to use.<br />
<br />
[[File:multilevel_features.png|thumb|700px|center|alt=Results of style transfer from each of the first five layers of the encoder network.|Results of style transfer from each of the first five layers of the encoder network.]]<br />
<br />
Ideally, the results of each layer should be used to build the final output image. This captures the entire range of features detected by the encoder network. First, one full pass of the network is performed. Then the stylised image from the deepest layer (Relu_5_1 in this case) is taken and used as the content image for another iteration of the algorithm, where then the next layer (Relu_4_1) is used as the output. These steps are repeated until the final image is produced from the shallowest layer. This process is summarised in the figure below.<br />
<br />
[[File:process_summary.png|thumb|700px|center|alt=Process summary of the multi-level stylization algorithm.|The content (C) and style (S) are fed to the VGG encoding network. The output image (I) after a whitening and colour transform (WCT) is taken from the deepest level's decoder. The process is iteratively repeated until the most shallow layer is reached.]]<br />
<br />
The authors note that the transformations must be applied first at the highest level (most abstract) layers, which capture complicated local structures and pass this transformed image to lower layers, which improve on details. They observe that reversing this order (lowest to highest) leads to images with low visual quality, as low-level information cannot be preserved after manipulating high level features.<br />
<br />
[[File:Universal_Style_Transfer_Coarse_to_Fine.JPG|thumb|700px|center|alt=(a)-(c) Output from intermediate layers. (d) Reversed transformation order.|(a)-(c) Output from intermediate layers. (d) Reversed transformation order.]]<br />
<br />
=Evaluation=<br />
The success of style transfer might appear hard to quantify as it relies on qualitative judgement. However, the extremes of transferring no style, or transferring only style can be considered as performing poorly. Consistent transfer of style throughout the entire image is another parameter of success. Ideally, the viewer can recognize the content of the image, while seeing it expressed in an alternative style. Quantitatively, the quality of the style transfer can be calculated by taking the covariance matrix difference $L_s$ between the resulting image and the original style. The results of the presented paper also need to be considered within the contexts of generality, efficiency and training requirements.<br />
<br />
==Style Transfer==<br />
A number of style transfer examples are presented relative to other works. <br />
<br />
[[File:transfer_results_label.jpg|thumb|700px|center|alt=Style transfer results of the presented paper.|A: See [6]. B: See [7]. C: See [8]. D: Gatys et al. iterative optimization, see [2]. E: This paper's results.]]<br />
<br />
Li et al. then obtained the average $L_s$ using 10 random content images across 40 style images. They had the lowest average $log(L_s)$ of all referenced works at 6.3. Next lowest was Gatys et al. [2] with $log(L_s) = 6.7$. It should be noted that while $L_s$ quantitatively calculates the success of the style transfer, results are still subject to the viewer's impression. Reviewing the transfer results, rows five and six for Gatys et al.'s method shows local minimization issues. However, their method still achieves a competitive $L_s$ score.<br />
<br />
==Transfer Efficiency==<br />
It was hypothesized by Li et al. that using WCT would enable faster run-times than [2] while still supporting arbitrary style transfer. For a 256x256 image, using a 12GB TITAN X, they achieved a transfer time of 1.5 seconds. Gatys et al.'s method [2] required 21.2 seconds. The pure feed-forward approaches [7], and [8] had times equal to or less than 0.2 seconds. [6] had a time comparable to the presented paper's method. However, [6,7,8] do not generalize well to multiple styles as training is required. Therefore this paper obtained a near 15x speed up for a style agnostic transfer algorithm when compared to leading previous work. The authors also note that WCT was done using the CPU. They intend to port WCT to the GPU and expect to see the computational time be further reduced.<br />
<br />
==Other Applications==<br />
Li et al.'s method can also be used for texture synthesis. This was the original work of Gatys et. al. before they applied their algorithm to style transfer problems. Texture synthesis takes a reference texture/image and creates new textures from it. With proper boundary conditions enforced these synthesized textures can be tileable. Alternatively, higher resolution textures can be generated. Texture synthesis has applications in areas such as computer graphics, allowing for large surfaces to be texture mapped.<br />
<br />
The content image is set as white noise, similar to how [2] initializes their output image. Then the reference texture/image is set as the style image. Since the content image is initially random white noise, then the features generated by the encoder of this image are also random. Li et al. state that this increases the diversity of the resulting output textures.<br />
<br />
[[File:texture_synthesis_label.jpg|thumb|700px|center|alt=Texture synthesis results.|A: Reference image/texture. B: Result from [8]. C: Result of present paper.]]<br />
<br />
Reviewing the examples from the above figure, it can be observed that the method from this paper repeats fewer local features from the image than a competing feed forward network method [8]. While the analysis is qualitative, the authors claim that their method produces "more visually pleasing results".<br />
<br />
=Conclusion=<br />
Only a couple years ago were CNNs first used to stylize images. Today, a host of improvements have been developed, optimizing the original work of Gatys et al. for a number of different situations. Using additional training per style image, computational efficiency and image quality can be increased. However, the trained network then depends on that specific style image, or in some cases such as in [3], a set of style images. Till now, limited work has taken place in improving Gatys et al.'s method for arbitrary style images. The authors of this paper developed and evaluated a novel method for arbitrary style transfer. Their method and Gatys et al.'s method share the use of a VGG-19 CNN as the initial processing step. However, the authors replaced iterative optimization with whitening and colour transforms, which can be applied in a single step. This yields a decrease in computational time while maintaining generality with respect to the style image. After their CNN auto-encoder is initially trained no further training is required. This allows their method to be style agnostic. Their method also performs favourably, in terms of image quality, when compared to other current work.<br />
<br />
=Critique=<br />
In the paper, the authors only experimented with layers of VGG19. Given that architectures such as ResNet and Xception perform better on image recognition tasks, it would be interesting to see how residual layers and/or Inception modules may be applied to the task of disentangling style and content and whether they would improve performance relative to the results presented in the current paper is the encoder used were to utilize layers from these alternative convolutional architectures. Additionally, it is worth exploring whether one can invent a probabilistic and/or generative version of the encoder-decoder architecture used in the paper. More precisely, is it possible to come up with something in the spirit of variational autoencoders, wherein we the bottleneck layer can be used to sample noise vectors, which can then be input into each of the decoder units to generate synthetic style and content images.<br />
Alternative attempts would also involve the study of generative adversarial networks with a perturbation threshold value. GANs can produce surreal images, where the underlying structure (content) is preserved ( in CNNs the filters learn the edges and surfaces and shape of the image), provided the Discriminator is trained for style classification ( training set consists of images pertaining the style that requires to be transferred).<br />
<br />
=Additional Results and Figures=<br />
Given in this section are the additional figures of universal style transform found in supplementary file. They are typically for larger image sizes and more variety of styles.<br />
#[[File:style-1.PNG]]<br />
#[[File:style-2.PNG]]<br />
#[[File:style-3.PNG]]<br />
<br />
=References=<br />
[1] L. A. Gatys, A. S. Ecker, and M. Bethge. Texture synthesis using convolutional neural networks. In NIPS, 2015.<br />
<br />
[2] L. A. Gatys, A. S. Ecker, and M. Bethge. Image style transfer using convolutional neural networks. In CVPR, 2016.<br />
<br />
[3] V. Dumoulin, J. Shlens, and M. Kudlur. A learned representation for artistic style. In ICLR, 2017.<br />
<br />
[4] J. Johnson, A. Alahi, and L. Fei-Fei. Perceptual losses for real-time style transfer and super-resolution. In ECCV, 2016<br />
<br />
[5] R. Picard. MAS 622J/1.126J: Pattern Recognition and Analysis, Lecture 4. http://courses.media.mit.edu/2010fall/mas622j/whiten.pdf<br />
<br />
[6] T. Q. Chen and M. Schmidt. Fast patch-based style transfer of arbitrary style. arXiv preprint arXiv:1612.04337, 2016.<br />
<br />
[7] X. Huang and S. Belongie. Arbitrary style transfer in real-time with adaptive instance normalization. arXiv preprint arXiv:1703.06868, 2017.<br />
<br />
[8] D. Ulyanov, V. Lebedev, A. Vedaldi, and V. Lempitsky. Texture networks: Feed-forward synthesis of textures and stylized images. In ICML, 2016.<br />
<br />
[9] Leon A. Gatys, Alexander S. Ecker, Matthias Bethge, A Neural Algorithm of Artistic Style, https://arxiv.org/abs/1508.06576<br />
<br />
[10] Karen Simonyan et al. Very Deep Convolutional Networks for Large-Scale Image Recognition<br />
<br />
[11] VGG Architectures - [http://www.robots.ox.ac.uk/~vgg/research/very_deep/| More Details]<br />
<br />
[12] Mechrez, R., Shechtman, E., & Zelnik-Manor, L. (2017). Photorealistic Style Transfer with Screened Poisson Equation. arXiv preprint arXiv:1709.09828.<br />
<br />
[13] Y. Li, C. Fang, J. Yang, Z. Wang, X. Lu, and M.-H. Yang. Diversified texture synthesis with feed-forward networks. In CVPR, 2017<br />
<br />
[14] D. Chen, L. Yuan, J. Liao, N. Yu, and G. Hua. Stylebank: An explicit representation for neural image style transfer. In CVPR, 2017<br />
<br />
Implementation Example: https://github.com/titu1994/Neural-Style-Transfer</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Modular_Multitask_Reinforcement_Learning_with_Policy_Sketches&diff=30877Modular Multitask Reinforcement Learning with Policy Sketches2017-11-20T03:26:37Z<p>H4lyu: /* Ablations */</p>
<hr />
<div>='''Introduction & Background'''=<br />
[[File:MRL0.png|border|right|400px]]<br />
This paper describes a framework for learning composable deep subpolicies in a multitask setting. These policies are guided only by abstract sketches which are representative of the high-level behavior in the environment. General reinforcement learning algorithms allow agents to solve tasks in complex environments. Vanilla policies find it difficult to deal with tasks featuring extremely delayed rewards. Most approaches often require in-depth supervision in the form of explicitly specified high-level actions, subgoals, or behavioral primitives. The proposed methodology is particularly suitable where rewards are difficult to engineer by hand. It is enough to tell the learner about the abstract policy structure, without indicating how high-level behaviors should try to use primitive percepts or actions.<br />
<br />
This paper explores a multitask reinforcement learning setting where the learner is presented with policy sketches. Policy sketches are defined as short, ungrounded, symbolic representations of a task. It describe its components, as shown in Figure 1. While symbols might be shared across different tasks ( the predicate "get wood" appears in sketches for both the tasks : "make planks" and "make sticks"). The learner is not shown or told anything about what these symbols mean, either in terms of observations or intermediate rewards.<br />
<br />
The agent learns from policy sketches by associating each high-level action with a parameterization of a low-level subpolicy. It jointly optimizes over concatenated task-specific policies by tying/sharing parameters across common subpolicies. They find that this architecture uses the high-level guidance provided by sketches to drastically accelerate learning of complex multi-stage behaviors. The experiments show that most benefits of learning from very detailed low-level supervision (e.g. from subgoal rewards) can also be obtained from fairly coarse high-level policy sketches. Most importantly, sketches are much easier to construct. They require no additions or modifications to the environment dynamics or reward function, and can be easily provided by non-experts (third party mechanical turk providers). This makes it possible to extend the benefits of hierarchical RL to challenging environments where it may not be possible to specify by hand the details of relevant subtasks. This paper shows that their approach drastically outperforms purely unsupervised methods that do not provide the learner with any task-specific guidance. The specific use of sketches to parameterize modular subpolicies makes better use of sketches than conditioning on them directly.<br />
<br />
The modular structure of this whole approach, which associates every high-level action symbol with a discrete subpolicy, naturally leads to a library of interpretable policy fragments which can be are easily recombined. The authors evaluate the approach in a variety of different data conditions: <br />
# Learning the full collection of tasks jointly via reinforcement learning <br />
# In a zero-shot setting where a policy sketch is available for a held-out task<br />
# In a adaptation setting, where sketches are hidden and the agent must learn to use and adapt a pretrained policy to reuse high-level actions in a new task.<br />
<br />
The code has been released at http://github.com/jacobandreas/psketch.<br />
<br />
='''Related Work'''=<br />
The approach in this paper is a specific case of the options framework developed by Sutton et al., 1999. In that work, options are introduced as "closed-loop policies for taking action over the period of time". They show that options enable temporally abstract information to be included in reinforcement learning algorithms, though it was published before the large-scale popularity of neural networks for reinforcement.<br />
<br />
Other authors have recently explored techniques for learning policies which require less prior knowledge of the environment than the method presented in this paper. For example, in Vezhnevets et al. (2016), the authors propose a RNN architecture to build "implicit plans" only through interacting with the environment as in the classic reinforcement learning problem formulation.<br />
<br />
One closely related line of work is the Hierarchical Abstract Machines (HAM) framework introduced by Parr & Russell, 1998 [11]. Like the approach which the Modular Multitask Reinforcement Learning with Policy Sketches uses, HAMs begin with a representation of a high-level policy as an automaton (or a more general computer program; Andre & Russell,<br />
2001 [7]; Marthi et al., 2004 [12]) and use reinforcement learning to fill in low-level details.<br />
<br />
='''Learning Modular Policies from Sketches'''=<br />
The paper considers a multitask reinforcement learning problem arising from a family of infinite-horizon discounted Markov decision processes in a shared environment. This environment is specified by a tuple $(S, A, P, \gamma )$, with <br />
* $S$ a set of states<br />
* $A$ a set of low-level actions <br />
* $P : S \times A \times S \to R$ a transition probability distribution<br />
* $\gamma$ a discount factor<br />
<br />
Each task $t \in T$ is then specified by a pair $(R_t, \rho_t)$, with $R_t : S \to R$ a task-specific reward function and $\rho_t: S \to R$, an initial distribution over states. For a fixed sequence ${(s_i, a_i)}$ of states and actions obtained from a rollout of a given policy, we will denote the empirical return starting in state $s_i$ as $q_i = \sum_{j=i+1}^\infty \gamma^{j-i-1}R(s_j)$. In addition to the components of a standard multitask RL problem, we assume that tasks are annotated with sketches $K_t$ , each consisting of a sequence $(b_{t1},b_{t2},...)$ of high-level symbolic labels drawn from a fixed vocabulary $B$.<br />
<br />
==Model==<br />
The authors exploit the structural information provided by sketches by constructing for each symbol ''b'' a corresponding subpolicy $\pi_b$. By sharing each subpolicy across all tasks annotated with the corresponding symbol, their approach naturally learns the tied/shared abstraction for the corresponding subtask.<br />
<br />
[[File:Algorithm_MRL2.png|center|frame|Pseudo Algorithms for Modular Multitask Reinforcement Learning with Policy Sketches]]<br />
<br />
At every timestep, a subpolicy selects either a low-level action $a \in A$ or a special STOP action. The augmented state space is denoted as $A^+ := A \cup \{STOP\}$. At a high level, this framework is agnostic to the implementation of subpolicies: any function that takes a representation of the current state onto a distribution over $A^+$ will work fine with the approach.<br />
<br />
In this paper, $\pi_b$ is represented as a neural network. These subpolicies may be viewed as options of the kind described by [2], with the key distinction that they have no initiation semantics, but are instead invokable everywhere, and have no explicit representation as a function from an initial state to a distribution over final states (instead this paper uses the STOP action to terminate).<br />
<br />
Given a fixed sketch $(b_1, b_2,....)$, a task-specific policy $\Pi_r$ is formed by concatenating its associated subpolicies in sequence. In particular, the high-level policy maintains a sub-policy index ''i'' (initially 0), and executes actions from $\pi_{b_i}$ until the STOP symbol is emitted, at which point control is passed to bi+1 . We may thus think of as inducing a Markov chain over the state space $S \times B$, with transitions:<br />
[[File:MRL1.png|center|border|]]<br />
<br />
Note that $\Pi_r$ is semi-Markov with respect to projection of the augmented state space $S \times B$ onto the underlying state space ''S''. The complete family of task-specific policies is denoted as $\Pi := \bigcup_r \{ \Pi_r \}$. Assume each $\pi_b$ be an arbitrary function of the current environment state parameterized by some weight vector $\theta_b$. The learning problem is to optimize over all $\theta_b$ to maximize expected discounted reward<br />
[[File:MRL2.png|center|border|]]<br />
across all tasks $t \in T$.<br />
<br />
==Policy Optimization==<br />
<br />
Here that optimization is accomplished through a simple decoupled actor–critic method. In a standard policy gradient approach, with a single policy $\pi$ with parameters $\theta$, the gradient steps are of the form:<br />
[[File:MRL3.png|center|border|]]<br />
<br />
where the baseline or “critic” c can be chosen independently of the future without introducing bias into the gradient. Recalling the previous definition of $q_i$ as the empirical return starting from $s_i$, this form of the gradient corresponds to a generalized advantage estimator with $\lambda = 1$. Here ''c'' achieves close to the optimal variance[6] when it is set exactly equal to the state-value function $V_{\pi} (s_i) = E_{\pi} q_i$ for the target policy $\pi$ starting in state $s_i$.<br />
[[File:MRL4.png|frame|]]<br />
<br />
In the case of generalizing to modular policies built by sequencing sub-policies the authors suggest to have one subpolicy per symbol but one critic per task. This is because subpolicies $\pi_b$ might participate in many compound policies $\Pi_r$, each associated with its own reward function $R_r$ . Thus individual subpolicies are not uniquely identified or differentiated with value functions. The actor–critic method is extended to allow decoupling of policies from value functions by allowing the critic to vary per-sample (per-task-and-timestep) based on the reward function with which that particular sample is associated. Noting that <br />
[[File:MRL5.png|center|border|]]<br />
i.e. the sum of gradients of expected rewards across all tasks in which $\pi_b$ participates, we have:<br />
[[File:MRL6.png|center|border|]]<br />
where each state-action pair $(s_{t_i}, a_{t_i})$ was selected by the subpolicy $\pi_b$ in the context of the task ''t''.<br />
<br />
Now minimization of the gradient variance requires that each $c_t$ actually depend on the task identity. (This follows immediately by applying the corresponding argument in [6] individually to each term in the sum over ''t'' in Equation 2.) Because the value function is itself unknown, an approximation must be estimated from data. Here it is allowed that these $c_t$ to be implemented with an arbitrary function approximator with parameters $\eta_t$ . This is trained to minimize a squared error criterion, with gradients given by<br />
[[File:MRL7.png|center|border|]]<br />
Alternative forms of the advantage estimator (e.g. the TD residual $R_t (s_i) + \gamma V_t(s_{i+1} - V_t(s_i))$ or any other member of the generalized advantage estimator family) can be used to substitute by simply maintaining one such estimator per task. Experiments show that conditioning on both the state and the task identity results in dramatic performance improvements, suggesting that the variance reduction given by this objective is important for efficient joint learning of modular policies.<br />
<br />
The complete algorithm for computing a single gradient step is given in Algorithm 1. (The outer training loop over these steps, which is driven by a curriculum learning procedure, is shown in Algorithm 2.) Note that this is an on-policy algorithm. In every step, the agent samples tasks from a task distribution provided by a curriculum (described in the following subsection). The current family of policies '''$\Pi$''' is used to perform rollouts for every sampled task, accumulating the resulting tuples of (states, low-level actions, high-level symbols, rewards, and task identities) into a dataset ''$D$''. Once ''$D$'' reaches a maximum size D, it is used to compute gradients with respect to both policy and critic parameters, and the parameter vectors are updated accordingly. The step sizes $\alpha$ and $\beta$ in Algorithm 1 can be chosen adaptively using any first-order method.<br />
<br />
==Curriculum Learning==<br />
<br />
For complex tasks, like the one depicted in Figure 3b, it is difficult for the agent to discover any states with positive reward until many subpolicy behaviors have already been learned. It is thus a better use of the learner’s time (and computational resources) to focus on “easy” tasks, where many rollouts will result in high reward from which relevant subpolicy behavior can be obtained. But there is a fundamental tradeoff involved here: if the learner spends a lot of its time on easy tasks before being told of the existence of harder ones, it may overfit and learn subpolicies that exhibit the desired structural properties or no longer generalize.<br />
<br />
To resolve these issues, a curriculum learning scheme is used that allows the model to smoothly scale up from easy tasks to more difficult ones without overfitting. Initially the model is presented with tasks associated with short sketches. Once average reward on all these tasks reaches a certain threshold, the length limit is incremented. It is assumed that rewards across tasks are normalized with maximum achievable reward $0 < q_i < 1$ . Let $Er_t$ denote the empirical estimate of the expected reward for the current policy on task T. Then at each timestep, tasks are sampled in proportion $1-Er_t$, which by assumption must be positive.<br />
<br />
Intuitively, the tasks that provide the strongest learning signal are those in which <br />
# The agent does not on average achieve reward close to the upper bound<br />
# Many episodes result in high reward.<br />
<br />
The expected reward component of the curriculum solves condition (1) by making sure that time is not spent on nearly solved tasks, while the length bound component of the curriculum addresses condition (2) by ensuring that tasks are not attempted until high-reward episodes are likely to be encountered. The experiments performed show that both components of this curriculum learning scheme improve the rate at which the model converges to a good policy.<br />
<br />
The complete curriculum-based training algorithm is written as Algorithm 2 above. Initially, the maximum sketch length $l_{max}$ is set to 1, and the curriculum initialized to sample length-1 tasks uniformly. For each setting of $l_{max}$, the algorithm uses the current collection of task policies to compute and apply the gradient step described in Algorithm 1. The rollouts obtained from the call to TRAIN-STEP can also be used to compute reward estimates $Er_t$ ; these estimates determine a new task distribution for the curriculum. The inner loop is repeated until the reward threshold $r_{good}$ is exceeded, at which point $l_{max}$ is incremented and the process repeated over a (now-expanded) collection of tasks.<br />
<br />
='''Experiments'''=<br />
[[File:MRL8.png|border|right|400px]]<br />
This paper considers three families of tasks: a 2-D Minecraft-inspired crafting game (Figure 3a), in which the agent must acquire particular resources by finding raw ingredients, combining them together in the correct order, and in some cases building intermediate tools that enable the agent to alter the environment itself; a 2-D maze navigation task that requires the agent to collect keys and open doors, and a 3-D locomotion task (Figure 3b) in which a quadrupedal robot must actuate its joints to traverse a narrow winding cliff.<br />
<br />
In all tasks, the agent receives a reward only after the final goal is accomplished. For the most challenging tasks, involving sequences of four or five high-level actions, a task-specific agent initially following a random policy essentially never discovers the reward signal, so these tasks cannot be solved without considering their hierarchical structure. These environments involve various kinds of challenging low-level control: agents must learn to avoid obstacles, interact with various kinds of objects, and relate fine-grained joint activation to high-level locomotion goals.<br />
<br />
==Implementation==<br />
In all of the experiments, each subpolicy is implemented as a neural network with ReLU nonlinearities and a hidden layer with 128 hidden units. Each critic is a linear function of the current state. Each subpolicy network receives as input a set of features describing the current state of the environment, and outputs a distribution over actions. The agent acts at every timestep by sampling from this distribution. The gradient steps given in lines 8 and 9 of Algorithm 1 are implemented using RMSPROP with a step size of 0.001 and gradient clipping to a unit norm. They take the batch size D in Algorithm 1 to be 2000, and set $\gamma$= 0.9 in both environments. For curriculum learning, the improvement threshold $r_{good}$ is 0.8.<br />
<br />
==Environments==<br />
<br />
The environment in Figure 3a is inspired by the popular game Minecraft, but is implemented in a discrete 2-D world. The agent interacts with objects in the environment by executing a special USE action when it faces them. Picking up raw materials initially scattered randomly around the environment adds to an inventory. Interacting with different crafting stations causes objects in the agent’s inventory to be combined or transformed. Each task in this game corresponds to some crafted object the agent must produce; the most complicated goals require the agent to also craft intermediate ingredients, and in some cases build tools (like a pickaxe and a bridge) to reach ingredients located in initially inaccessible regions of the world.<br />
<br />
The maze environment is very similar to “light world” described by [4]. The agent is placed in a discrete world consisting of a series of rooms, some of which are connected by doors. The agent needs to first pick up a key to open them. For our experiments, each task corresponds to a goal room that the agent must reach through a sequence of intermediate rooms. The agent senses the distance to keys, closed doors, and open doors in each direction. Sketches specify a particular sequence of directions for the agent to traverse between rooms to reach the goal. The sketch always corresponds to a viable traversal from the start to the goal position, but other (possibly shorter) traversals may also exist.<br />
<br />
The cliff environment (Figure 3b) proves the effectiveness of the approach in a high-dimensional continuous control environment where a quadrupedal robot [5] is placed on a variable-length winding path, and must navigate to the end without falling off. This is a challenging RL problem since the walker must learn the low-level walking skill before it can make any progress. The agent receives a small reward for making progress toward the goal, and a large positive reward for reaching the goal square, with a negative reward for falling off the path.<br />
<br />
==Multitask Learning==<br />
<br />
[[File:MRL9.png|border|center|800px]]<br />
The primary experimental question in this paper is whether the extra structure provided by policy sketches alone is enough to enable fast learning of coupled policies across tasks. The aim is to explore the differences between the approach described and relevant prior work that performs either unsupervised or weakly supervised multitask learning of hierarchical policy structure. Specifically,they compare their '''modular''' approach to:<br />
<br />
# Structured hierarchical reinforcement learners:<br />
#* the fully unsupervised '''option–critic''' algorithm of Bacon & Precup[1]<br />
#* a '''Q automaton''' that attempts to explicitly represent the Q function for each task / subtask combination (essentially a HAM [8] with a deep state abstraction function)<br />
# Alternative ways of incorporating sketch data into standard policy gradient methods:<br />
#* learning an '''independent''' policy for each task<br />
#* learning a '''joint policy''' across all tasks, conditioning directly on both environment features and a representation of the complete sketch<br />
<br />
The joint and independent models performed best when trained with the same curriculum described in Section 3.3, while the option–critic model performed best with a length–weighted curriculum that has access to all tasks from the beginning of training.<br />
<br />
Learning curves for baselines and the modular model are shown in Figure 4. It can be seen that in all environments, our approach substantially outperforms the baselines: it induces policies with substantially higher average reward and converges more quickly than the policy gradient baselines. It can further be seen in Figure 4c that after policies have been learned on simple tasks, the model is able to rapidly adapt to more complex ones, even when the longer tasks involve high-level actions not required for any of the short tasks.<br />
<br />
==Ablations==<br />
[[File:MRL10.png|border|right|400px]]<br />
In addition to the overall modular parameter tying structure induced by sketches, the other critical component of the training procedure is the decoupled critic and the curriculum. The next experiments investigate the extent to which these are necessary for good performance.<br />
<br />
To evaluate the the critic, consider three ablations: <br />
# Removing the dependence of the model on the environment state, in which case the baseline is a single scalar per task<br />
# Removing the dependence of the model on the task, in which case the baseline is a conventional generalised advantage estimator<br />
# Removing both, in which case the baseline is a single scalar, as in a vanilla policy gradient approach.<br />
<br />
Results are shown in Figure 5a. Introducing both state and task dependence into the baseline leads to faster convergence of the model: the approach with a constant baseline achieves less than half the overall performance of the full critic after 3 million episodes. Introducing task and state dependence independently improve this performance; combining them gives the best result.<br />
<br />
Two other experiments are also performed as Figure 5b: starting with short examples and moving to long ones, and sampling tasks in inverse proportion to their accumulated reward. It is shown that both components help; prioritization by both length and weight gives the best results.<br />
<br />
==Zero-shot and Adaptation Learning==<br />
[[File:MRL11.png|border|left|320px]]<br />
In the final experiments, the authors test the model’s ability to generalize beyond the standard training condition. Consider two tests of generalization: a zero-shot setting, in which the model is provided a sketch for the new task and must immediately achieve good performance, and a adaptation setting, in which no sketch is provided leaving the model to learn the form of a suitable sketch via interaction in the new task.They hold out two length-four tasks from the full inventory used in Section 4.3, and train on the remaining tasks. For zero-shot experiments, the concatenated policy is formed to describe the sketches of the held-out tasks, and repeatedly executing this policy (without learning) in order to obtain an estimate of its effectiveness. For adaptation experiments, consider ordinary RL over high-level actions B rather than low-level actions A, implementing the high-level learner with the same agent architecture as described in Section 3.1. Results are shown in Table 1. The held-out tasks are sufficiently challenging that the baselines are unable to obtain more than negligible reward: in particular, the joint model overfits to the training tasks and cannot generalize to new sketches, while the independent model cannot discover enough of a reward signal to learn in the adaptation setting. The modular model does comparatively well: individual subpolicies succeed in novel zero-shot configurations (suggesting that they have in fact discovered the behavior suggested by the semantics of the sketch) and provide a suitable basis for adaptive discovery of new high-level policies.<br />
<br />
='''Conclusion & Critique'''=<br />
The paper's contributions are:<br />
<br />
* A general paradigm for multitask, hierarchical, deep reinforcement learning guided by abstract sketches of task-specific policies.<br />
<br />
* A concrete recipe for learning from these sketches, built on a general family of modular deep policy representations and a multitask actor–critic training objective.<br />
<br />
They have described an approach for multitask learning of deep multitask policies guided by symbolic policy sketches. By associating each symbol appearing in a sketch with a modular neural sub policy, they have shown that it is possible to build agents that share behavior across tasks in order to achieve success in tasks with sparse and delayed rewards. This process induces an inventory of reusable and interpretable sub policies which can be employed for zero-shot generalization when further sketches are available, and hierarchical reinforcement learning when they are not.<br />
<br />
<br />
<br />
One critique of this approach could be that building of different neural networks for each sub tasks could lead to overtly complicated networks and is not in the spirit of building efficient structure.<br />
<br />
='''References'''=<br />
[1] Bacon, Pierre-Luc and Precup, Doina. The option-critic architecture. In NIPS Deep Reinforcement Learning Work-shop, 2015.<br />
<br />
[2] Sutton, Richard S, Precup, Doina, and Singh, Satinder. Be-tween MDPs and semi-MDPs: A framework for tempo-ral abstraction in reinforcement learning. Artificial intel-ligence, 112(1):181–211, 1999.<br />
<br />
[3] Stolle, Martin and Precup, Doina. Learning options in reinforcement learning. In International Symposium on Abstraction, Reformulation, and Approximation, pp. 212– 223. Springer, 2002.<br />
<br />
[4] Konidaris, George and Barto, Andrew G. Building portable options: Skill transfer in reinforcement learning. In IJ-CAI, volume 7, pp. 895–900, 2007.<br />
<br />
[5] Schulman, John, Moritz, Philipp, Levine, Sergey, Jordan, Michael, and Abbeel, Pieter. Trust region policy optimization. In International Conference on Machine Learning, 2015b.<br />
<br />
[6] Greensmith, Evan, Bartlett, Peter L, and Baxter, Jonathan. Variance reduction techniques for gradient estimates in reinforcement learning. Journal of Machine Learning Research, 5(Nov):1471–1530, 2004.<br />
<br />
[7] Andre, David and Russell, Stuart. Programmable reinforce-ment learning agents. In Advances in Neural Information Processing Systems, 2001.<br />
<br />
[8] Andre, David and Russell, Stuart. State abstraction for pro-grammable reinforcement learning agents. In Proceedings of the Meeting of the Association for the Advance-ment of Artificial Intelligence, 2002.<br />
<br />
[9] Author Jacob Andreas presenting the paper - https://www.youtube.com/watch?v=NRIcDEB64x8<br />
<br />
[10] Vezhnevets, A., Mnih, V., Osindero, S., Graves, A., Vinyals, O., & Agapiou, J. (2016). Strategic attentive writer for learning macro-actions. In Advances in Neural Information Processing Systems (pp. 3486-3494).<br />
<br />
[11] Parr, Ron and Russell, Stuart. Reinforcement learning with hierarchies of machines. In Advances in Neural Information Processing Systems, 1998.<br />
<br />
[12] Marthi, Bhaskara, Lantham, David, Guestrin, Carlos, and Russell, Stuart. Concurrent hierarchical reinforcement learning. In Proceedings of the Meeting of the Association for the Advancement of Artificial Intelligence, 2004.</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Modular_Multitask_Reinforcement_Learning_with_Policy_Sketches&diff=30876Modular Multitask Reinforcement Learning with Policy Sketches2017-11-20T03:17:45Z<p>H4lyu: Undo revision 30875 by H4lyu (talk)</p>
<hr />
<div>='''Introduction & Background'''=<br />
[[File:MRL0.png|border|right|400px]]<br />
This paper describes a framework for learning composable deep subpolicies in a multitask setting. These policies are guided only by abstract sketches which are representative of the high-level behavior in the environment. General reinforcement learning algorithms allow agents to solve tasks in complex environments. Vanilla policies find it difficult to deal with tasks featuring extremely delayed rewards. Most approaches often require in-depth supervision in the form of explicitly specified high-level actions, subgoals, or behavioral primitives. The proposed methodology is particularly suitable where rewards are difficult to engineer by hand. It is enough to tell the learner about the abstract policy structure, without indicating how high-level behaviors should try to use primitive percepts or actions.<br />
<br />
This paper explores a multitask reinforcement learning setting where the learner is presented with policy sketches. Policy sketches are defined as short, ungrounded, symbolic representations of a task. It describe its components, as shown in Figure 1. While symbols might be shared across different tasks ( the predicate "get wood" appears in sketches for both the tasks : "make planks" and "make sticks"). The learner is not shown or told anything about what these symbols mean, either in terms of observations or intermediate rewards.<br />
<br />
The agent learns from policy sketches by associating each high-level action with a parameterization of a low-level subpolicy. It jointly optimizes over concatenated task-specific policies by tying/sharing parameters across common subpolicies. They find that this architecture uses the high-level guidance provided by sketches to drastically accelerate learning of complex multi-stage behaviors. The experiments show that most benefits of learning from very detailed low-level supervision (e.g. from subgoal rewards) can also be obtained from fairly coarse high-level policy sketches. Most importantly, sketches are much easier to construct. They require no additions or modifications to the environment dynamics or reward function, and can be easily provided by non-experts (third party mechanical turk providers). This makes it possible to extend the benefits of hierarchical RL to challenging environments where it may not be possible to specify by hand the details of relevant subtasks. This paper shows that their approach drastically outperforms purely unsupervised methods that do not provide the learner with any task-specific guidance. The specific use of sketches to parameterize modular subpolicies makes better use of sketches than conditioning on them directly.<br />
<br />
The modular structure of this whole approach, which associates every high-level action symbol with a discrete subpolicy, naturally leads to a library of interpretable policy fragments which can be are easily recombined. The authors evaluate the approach in a variety of different data conditions: <br />
# Learning the full collection of tasks jointly via reinforcement learning <br />
# In a zero-shot setting where a policy sketch is available for a held-out task<br />
# In a adaptation setting, where sketches are hidden and the agent must learn to use and adapt a pretrained policy to reuse high-level actions in a new task.<br />
<br />
The code has been released at http://github.com/jacobandreas/psketch.<br />
<br />
='''Related Work'''=<br />
The approach in this paper is a specific case of the options framework developed by Sutton et al., 1999. In that work, options are introduced as "closed-loop policies for taking action over the period of time". They show that options enable temporally abstract information to be included in reinforcement learning algorithms, though it was published before the large-scale popularity of neural networks for reinforcement.<br />
<br />
Other authors have recently explored techniques for learning policies which require less prior knowledge of the environment than the method presented in this paper. For example, in Vezhnevets et al. (2016), the authors propose a RNN architecture to build "implicit plans" only through interacting with the environment as in the classic reinforcement learning problem formulation.<br />
<br />
One closely related line of work is the Hierarchical Abstract Machines (HAM) framework introduced by Parr & Russell, 1998 [11]. Like the approach which the Modular Multitask Reinforcement Learning with Policy Sketches uses, HAMs begin with a representation of a high-level policy as an automaton (or a more general computer program; Andre & Russell,<br />
2001 [7]; Marthi et al., 2004 [12]) and use reinforcement learning to fill in low-level details.<br />
<br />
='''Learning Modular Policies from Sketches'''=<br />
The paper considers a multitask reinforcement learning problem arising from a family of infinite-horizon discounted Markov decision processes in a shared environment. This environment is specified by a tuple $(S, A, P, \gamma )$, with <br />
* $S$ a set of states<br />
* $A$ a set of low-level actions <br />
* $P : S \times A \times S \to R$ a transition probability distribution<br />
* $\gamma$ a discount factor<br />
<br />
Each task $t \in T$ is then specified by a pair $(R_t, \rho_t)$, with $R_t : S \to R$ a task-specific reward function and $\rho_t: S \to R$, an initial distribution over states. For a fixed sequence ${(s_i, a_i)}$ of states and actions obtained from a rollout of a given policy, we will denote the empirical return starting in state $s_i$ as $q_i = \sum_{j=i+1}^\infty \gamma^{j-i-1}R(s_j)$. In addition to the components of a standard multitask RL problem, we assume that tasks are annotated with sketches $K_t$ , each consisting of a sequence $(b_{t1},b_{t2},...)$ of high-level symbolic labels drawn from a fixed vocabulary $B$.<br />
<br />
==Model==<br />
The authors exploit the structural information provided by sketches by constructing for each symbol ''b'' a corresponding subpolicy $\pi_b$. By sharing each subpolicy across all tasks annotated with the corresponding symbol, their approach naturally learns the tied/shared abstraction for the corresponding subtask.<br />
<br />
[[File:Algorithm_MRL2.png|center|frame|Pseudo Algorithms for Modular Multitask Reinforcement Learning with Policy Sketches]]<br />
<br />
At every timestep, a subpolicy selects either a low-level action $a \in A$ or a special STOP action. The augmented state space is denoted as $A^+ := A \cup \{STOP\}$. At a high level, this framework is agnostic to the implementation of subpolicies: any function that takes a representation of the current state onto a distribution over $A^+$ will work fine with the approach.<br />
<br />
In this paper, $\pi_b$ is represented as a neural network. These subpolicies may be viewed as options of the kind described by [2], with the key distinction that they have no initiation semantics, but are instead invokable everywhere, and have no explicit representation as a function from an initial state to a distribution over final states (instead this paper uses the STOP action to terminate).<br />
<br />
Given a fixed sketch $(b_1, b_2,....)$, a task-specific policy $\Pi_r$ is formed by concatenating its associated subpolicies in sequence. In particular, the high-level policy maintains a sub-policy index ''i'' (initially 0), and executes actions from $\pi_{b_i}$ until the STOP symbol is emitted, at which point control is passed to bi+1 . We may thus think of as inducing a Markov chain over the state space $S \times B$, with transitions:<br />
[[File:MRL1.png|center|border|]]<br />
<br />
Note that $\Pi_r$ is semi-Markov with respect to projection of the augmented state space $S \times B$ onto the underlying state space ''S''. The complete family of task-specific policies is denoted as $\Pi := \bigcup_r \{ \Pi_r \}$. Assume each $\pi_b$ be an arbitrary function of the current environment state parameterized by some weight vector $\theta_b$. The learning problem is to optimize over all $\theta_b$ to maximize expected discounted reward<br />
[[File:MRL2.png|center|border|]]<br />
across all tasks $t \in T$.<br />
<br />
==Policy Optimization==<br />
<br />
Here that optimization is accomplished through a simple decoupled actor–critic method. In a standard policy gradient approach, with a single policy $\pi$ with parameters $\theta$, the gradient steps are of the form:<br />
[[File:MRL3.png|center|border|]]<br />
<br />
where the baseline or “critic” c can be chosen independently of the future without introducing bias into the gradient. Recalling the previous definition of $q_i$ as the empirical return starting from $s_i$, this form of the gradient corresponds to a generalized advantage estimator with $\lambda = 1$. Here ''c'' achieves close to the optimal variance[6] when it is set exactly equal to the state-value function $V_{\pi} (s_i) = E_{\pi} q_i$ for the target policy $\pi$ starting in state $s_i$.<br />
[[File:MRL4.png|frame|]]<br />
<br />
In the case of generalizing to modular policies built by sequencing sub-policies the authors suggest to have one subpolicy per symbol but one critic per task. This is because subpolicies $\pi_b$ might participate in many compound policies $\Pi_r$, each associated with its own reward function $R_r$ . Thus individual subpolicies are not uniquely identified or differentiated with value functions. The actor–critic method is extended to allow decoupling of policies from value functions by allowing the critic to vary per-sample (per-task-and-timestep) based on the reward function with which that particular sample is associated. Noting that <br />
[[File:MRL5.png|center|border|]]<br />
i.e. the sum of gradients of expected rewards across all tasks in which $\pi_b$ participates, we have:<br />
[[File:MRL6.png|center|border|]]<br />
where each state-action pair $(s_{t_i}, a_{t_i})$ was selected by the subpolicy $\pi_b$ in the context of the task ''t''.<br />
<br />
Now minimization of the gradient variance requires that each $c_t$ actually depend on the task identity. (This follows immediately by applying the corresponding argument in [6] individually to each term in the sum over ''t'' in Equation 2.) Because the value function is itself unknown, an approximation must be estimated from data. Here it is allowed that these $c_t$ to be implemented with an arbitrary function approximator with parameters $\eta_t$ . This is trained to minimize a squared error criterion, with gradients given by<br />
[[File:MRL7.png|center|border|]]<br />
Alternative forms of the advantage estimator (e.g. the TD residual $R_t (s_i) + \gamma V_t(s_{i+1} - V_t(s_i))$ or any other member of the generalized advantage estimator family) can be used to substitute by simply maintaining one such estimator per task. Experiments show that conditioning on both the state and the task identity results in dramatic performance improvements, suggesting that the variance reduction given by this objective is important for efficient joint learning of modular policies.<br />
<br />
The complete algorithm for computing a single gradient step is given in Algorithm 1. (The outer training loop over these steps, which is driven by a curriculum learning procedure, is shown in Algorithm 2.) Note that this is an on-policy algorithm. In every step, the agent samples tasks from a task distribution provided by a curriculum (described in the following subsection). The current family of policies '''$\Pi$''' is used to perform rollouts for every sampled task, accumulating the resulting tuples of (states, low-level actions, high-level symbols, rewards, and task identities) into a dataset ''$D$''. Once ''$D$'' reaches a maximum size D, it is used to compute gradients with respect to both policy and critic parameters, and the parameter vectors are updated accordingly. The step sizes $\alpha$ and $\beta$ in Algorithm 1 can be chosen adaptively using any first-order method.<br />
<br />
==Curriculum Learning==<br />
<br />
For complex tasks, like the one depicted in Figure 3b, it is difficult for the agent to discover any states with positive reward until many subpolicy behaviors have already been learned. It is thus a better use of the learner’s time (and computational resources) to focus on “easy” tasks, where many rollouts will result in high reward from which relevant subpolicy behavior can be obtained. But there is a fundamental tradeoff involved here: if the learner spends a lot of its time on easy tasks before being told of the existence of harder ones, it may overfit and learn subpolicies that exhibit the desired structural properties or no longer generalize.<br />
<br />
To resolve these issues, a curriculum learning scheme is used that allows the model to smoothly scale up from easy tasks to more difficult ones without overfitting. Initially the model is presented with tasks associated with short sketches. Once average reward on all these tasks reaches a certain threshold, the length limit is incremented. It is assumed that rewards across tasks are normalized with maximum achievable reward $0 < q_i < 1$ . Let $Er_t$ denote the empirical estimate of the expected reward for the current policy on task T. Then at each timestep, tasks are sampled in proportion $1-Er_t$, which by assumption must be positive.<br />
<br />
Intuitively, the tasks that provide the strongest learning signal are those in which <br />
# The agent does not on average achieve reward close to the upper bound<br />
# Many episodes result in high reward.<br />
<br />
The expected reward component of the curriculum solves condition (1) by making sure that time is not spent on nearly solved tasks, while the length bound component of the curriculum addresses condition (2) by ensuring that tasks are not attempted until high-reward episodes are likely to be encountered. The experiments performed show that both components of this curriculum learning scheme improve the rate at which the model converges to a good policy.<br />
<br />
The complete curriculum-based training algorithm is written as Algorithm 2 above. Initially, the maximum sketch length $l_{max}$ is set to 1, and the curriculum initialized to sample length-1 tasks uniformly. For each setting of $l_{max}$, the algorithm uses the current collection of task policies to compute and apply the gradient step described in Algorithm 1. The rollouts obtained from the call to TRAIN-STEP can also be used to compute reward estimates $Er_t$ ; these estimates determine a new task distribution for the curriculum. The inner loop is repeated until the reward threshold $r_{good}$ is exceeded, at which point $l_{max}$ is incremented and the process repeated over a (now-expanded) collection of tasks.<br />
<br />
='''Experiments'''=<br />
[[File:MRL8.png|border|right|400px]]<br />
This paper considers three families of tasks: a 2-D Minecraft-inspired crafting game (Figure 3a), in which the agent must acquire particular resources by finding raw ingredients, combining them together in the correct order, and in some cases building intermediate tools that enable the agent to alter the environment itself; a 2-D maze navigation task that requires the agent to collect keys and open doors, and a 3-D locomotion task (Figure 3b) in which a quadrupedal robot must actuate its joints to traverse a narrow winding cliff.<br />
<br />
In all tasks, the agent receives a reward only after the final goal is accomplished. For the most challenging tasks, involving sequences of four or five high-level actions, a task-specific agent initially following a random policy essentially never discovers the reward signal, so these tasks cannot be solved without considering their hierarchical structure. These environments involve various kinds of challenging low-level control: agents must learn to avoid obstacles, interact with various kinds of objects, and relate fine-grained joint activation to high-level locomotion goals.<br />
<br />
==Implementation==<br />
In all of the experiments, each subpolicy is implemented as a neural network with ReLU nonlinearities and a hidden layer with 128 hidden units. Each critic is a linear function of the current state. Each subpolicy network receives as input a set of features describing the current state of the environment, and outputs a distribution over actions. The agent acts at every timestep by sampling from this distribution. The gradient steps given in lines 8 and 9 of Algorithm 1 are implemented using RMSPROP with a step size of 0.001 and gradient clipping to a unit norm. They take the batch size D in Algorithm 1 to be 2000, and set $\gamma$= 0.9 in both environments. For curriculum learning, the improvement threshold $r_{good}$ is 0.8.<br />
<br />
==Environments==<br />
<br />
The environment in Figure 3a is inspired by the popular game Minecraft, but is implemented in a discrete 2-D world. The agent interacts with objects in the environment by executing a special USE action when it faces them. Picking up raw materials initially scattered randomly around the environment adds to an inventory. Interacting with different crafting stations causes objects in the agent’s inventory to be combined or transformed. Each task in this game corresponds to some crafted object the agent must produce; the most complicated goals require the agent to also craft intermediate ingredients, and in some cases build tools (like a pickaxe and a bridge) to reach ingredients located in initially inaccessible regions of the world.<br />
<br />
The maze environment is very similar to “light world” described by [4]. The agent is placed in a discrete world consisting of a series of rooms, some of which are connected by doors. The agent needs to first pick up a key to open them. For our experiments, each task corresponds to a goal room that the agent must reach through a sequence of intermediate rooms. The agent senses the distance to keys, closed doors, and open doors in each direction. Sketches specify a particular sequence of directions for the agent to traverse between rooms to reach the goal. The sketch always corresponds to a viable traversal from the start to the goal position, but other (possibly shorter) traversals may also exist.<br />
<br />
The cliff environment (Figure 3b) proves the effectiveness of the approach in a high-dimensional continuous control environment where a quadrupedal robot [5] is placed on a variable-length winding path, and must navigate to the end without falling off. This is a challenging RL problem since the walker must learn the low-level walking skill before it can make any progress. The agent receives a small reward for making progress toward the goal, and a large positive reward for reaching the goal square, with a negative reward for falling off the path.<br />
<br />
==Multitask Learning==<br />
<br />
[[File:MRL9.png|border|center|800px]]<br />
The primary experimental question in this paper is whether the extra structure provided by policy sketches alone is enough to enable fast learning of coupled policies across tasks. The aim is to explore the differences between the approach described and relevant prior work that performs either unsupervised or weakly supervised multitask learning of hierarchical policy structure. Specifically,they compare their '''modular''' approach to:<br />
<br />
# Structured hierarchical reinforcement learners:<br />
#* the fully unsupervised '''option–critic''' algorithm of Bacon & Precup[1]<br />
#* a '''Q automaton''' that attempts to explicitly represent the Q function for each task / subtask combination (essentially a HAM [8] with a deep state abstraction function)<br />
# Alternative ways of incorporating sketch data into standard policy gradient methods:<br />
#* learning an '''independent''' policy for each task<br />
#* learning a '''joint policy''' across all tasks, conditioning directly on both environment features and a representation of the complete sketch<br />
<br />
The joint and independent models performed best when trained with the same curriculum described in Section 3.3, while the option–critic model performed best with a length–weighted curriculum that has access to all tasks from the beginning of training.<br />
<br />
Learning curves for baselines and the modular model are shown in Figure 4. It can be seen that in all environments, our approach substantially outperforms the baselines: it induces policies with substantially higher average reward and converges more quickly than the policy gradient baselines. It can further be seen in Figure 4c that after policies have been learned on simple tasks, the model is able to rapidly adapt to more complex ones, even when the longer tasks involve high-level actions not required for any of the short tasks.<br />
<br />
==Ablations==<br />
[[File:MRL10.png|border|right|400px]]<br />
In addition to the overall modular parameter tying structure induced by sketches, the other critical component of the training procedure is the decoupled critic and the curriculum. The next experiments investigate the extent to which these are necessary for good performance.<br />
<br />
To evaluate the the critic, consider three ablations: <br />
# Removing the dependence of the model on the environment state, in which case the baseline is a single scalar per task<br />
# Removing the dependence of the model on the task, in which case the baseline is a conventional generalised advantage estimator<br />
# Removing both, in which case the baseline is a single scalar, as in a vanilla policy gradient approach.<br />
<br />
Results are shown in Figure 5a. Introducing both state and task dependence into the baseline leads to faster convergence of the model: the approach with a constant baseline achieves less than half the overall performance of the full critic after 3 million episodes. Introducing task and state dependence independently improve this performance; combining them gives the best result.<br />
<br />
==Zero-shot and Adaptation Learning==<br />
[[File:MRL11.png|border|left|320px]]<br />
In the final experiments, the authors test the model’s ability to generalize beyond the standard training condition. Consider two tests of generalization: a zero-shot setting, in which the model is provided a sketch for the new task and must immediately achieve good performance, and a adaptation setting, in which no sketch is provided leaving the model to learn the form of a suitable sketch via interaction in the new task.They hold out two length-four tasks from the full inventory used in Section 4.3, and train on the remaining tasks. For zero-shot experiments, the concatenated policy is formed to describe the sketches of the held-out tasks, and repeatedly executing this policy (without learning) in order to obtain an estimate of its effectiveness. For adaptation experiments, consider ordinary RL over high-level actions B rather than low-level actions A, implementing the high-level learner with the same agent architecture as described in Section 3.1. Results are shown in Table 1. The held-out tasks are sufficiently challenging that the baselines are unable to obtain more than negligible reward: in particular, the joint model overfits to the training tasks and cannot generalize to new sketches, while the independent model cannot discover enough of a reward signal to learn in the adaptation setting. The modular model does comparatively well: individual subpolicies succeed in novel zero-shot configurations (suggesting that they have in fact discovered the behavior suggested by the semantics of the sketch) and provide a suitable basis for adaptive discovery of new high-level policies.<br />
<br />
='''Conclusion & Critique'''=<br />
The paper's contributions are:<br />
<br />
* A general paradigm for multitask, hierarchical, deep reinforcement learning guided by abstract sketches of task-specific policies.<br />
<br />
* A concrete recipe for learning from these sketches, built on a general family of modular deep policy representations and a multitask actor–critic training objective.<br />
<br />
They have described an approach for multitask learning of deep multitask policies guided by symbolic policy sketches. By associating each symbol appearing in a sketch with a modular neural sub policy, they have shown that it is possible to build agents that share behavior across tasks in order to achieve success in tasks with sparse and delayed rewards. This process induces an inventory of reusable and interpretable sub policies which can be employed for zero-shot generalization when further sketches are available, and hierarchical reinforcement learning when they are not.<br />
<br />
<br />
<br />
One critique of this approach could be that building of different neural networks for each sub tasks could lead to overtly complicated networks and is not in the spirit of building efficient structure.<br />
<br />
='''References'''=<br />
[1] Bacon, Pierre-Luc and Precup, Doina. The option-critic architecture. In NIPS Deep Reinforcement Learning Work-shop, 2015.<br />
<br />
[2] Sutton, Richard S, Precup, Doina, and Singh, Satinder. Be-tween MDPs and semi-MDPs: A framework for tempo-ral abstraction in reinforcement learning. Artificial intel-ligence, 112(1):181–211, 1999.<br />
<br />
[3] Stolle, Martin and Precup, Doina. Learning options in reinforcement learning. In International Symposium on Abstraction, Reformulation, and Approximation, pp. 212– 223. Springer, 2002.<br />
<br />
[4] Konidaris, George and Barto, Andrew G. Building portable options: Skill transfer in reinforcement learning. In IJ-CAI, volume 7, pp. 895–900, 2007.<br />
<br />
[5] Schulman, John, Moritz, Philipp, Levine, Sergey, Jordan, Michael, and Abbeel, Pieter. Trust region policy optimization. In International Conference on Machine Learning, 2015b.<br />
<br />
[6] Greensmith, Evan, Bartlett, Peter L, and Baxter, Jonathan. Variance reduction techniques for gradient estimates in reinforcement learning. Journal of Machine Learning Research, 5(Nov):1471–1530, 2004.<br />
<br />
[7] Andre, David and Russell, Stuart. Programmable reinforce-ment learning agents. In Advances in Neural Information Processing Systems, 2001.<br />
<br />
[8] Andre, David and Russell, Stuart. State abstraction for pro-grammable reinforcement learning agents. In Proceedings of the Meeting of the Association for the Advance-ment of Artificial Intelligence, 2002.<br />
<br />
[9] Author Jacob Andreas presenting the paper - https://www.youtube.com/watch?v=NRIcDEB64x8<br />
<br />
[10] Vezhnevets, A., Mnih, V., Osindero, S., Graves, A., Vinyals, O., & Agapiou, J. (2016). Strategic attentive writer for learning macro-actions. In Advances in Neural Information Processing Systems (pp. 3486-3494).<br />
<br />
[11] Parr, Ron and Russell, Stuart. Reinforcement learning with hierarchies of machines. In Advances in Neural Information Processing Systems, 1998.<br />
<br />
[12] Marthi, Bhaskara, Lantham, David, Guestrin, Carlos, and Russell, Stuart. Concurrent hierarchical reinforcement learning. In Proceedings of the Meeting of the Association for the Advancement of Artificial Intelligence, 2004.</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Modular_Multitask_Reinforcement_Learning_with_Policy_Sketches&diff=30875Modular Multitask Reinforcement Learning with Policy Sketches2017-11-20T03:12:04Z<p>H4lyu: /* Introduction & Background */</p>
<hr />
<div>='''Introduction & Background'''=<br />
[[File:MRL0.png|border|right|400px]]<br />
This paper describes a framework for learning composable deep subpolicies in a multitask setting. These policies are guided only by abstract sketches which are representative of the high-level behavior in the environment. General reinforcement learning algorithms allow agents to solve tasks in complex environments. Vanilla policies find it difficult to deal with tasks featuring extremely delayed rewards. Most approaches often require in-depth supervision in the form of explicitly specified high-level actions, subgoals, or behavioral primitives. The proposed methodology is particularly suitable where rewards are difficult to engineer by hand. It is enough to tell the learner about the abstract policy structure, without indicating how high-level behaviors should try to use primitive percepts or actions.<br />
<br />
This paper explores a multitask reinforcement learning setting where the learner is presented with policy sketches. Policy sketches are defined as short, ungrounded, symbolic representations of a task. It describe its components, as shown in Figure 1. While symbols might be shared across different tasks ( the predicate "get wood" appears in sketches for both the tasks : "make planks" and "make sticks"). The learner is not shown or told anything about what these symbols mean, either in terms of observations or intermediate rewards.<br />
<br />
The agent learns from policy sketches by associating each high-level action with a parameterization of a low-level subpolicy. It jointly optimizes over concatenated task-specific policies by tying/sharing parameters across common subpolicies. They find that this architecture uses the high-level guidance provided by sketches to drastically accelerate learning of complex multi-stage behaviors. The experiments show that most benefits of learning from very detailed low-level supervision (e.g. from subgoal rewards) can also be obtained from fairly coarse high-level policy sketches. Most importantly, sketches are much easier to construct. They require no additions or modifications to the environment dynamics or reward function, and can be easily provided by non-experts (third party mechanical turk providers). This makes it possible to extend the benefits of hierarchical RL to challenging environments where it may not be possible to specify by hand the details of relevant subtasks. This paper shows that their approach drastically outperforms purely unsupervised methods that do not provide the learner with any task-specific guidance. The specific use of sketches to parameterize modular subpolicies makes better use of sketches than conditioning on them directly. In short, the main contributions of this paper include:<br />
# A general paradigm for multitask, hierarchical, deep reinforcement learning guided by abstract sketches of task-speciﬁcpolicies<br />
# A concrete recipe for learning from these sketches, built on a general family of modular deep policy representations and a multitask actor–critic training objective.<br />
<br />
The modular structure of this whole approach, which associates every high-level action symbol with a discrete subpolicy, naturally leads to a library of interpretable policy fragments which can be are easily recombined. The authors evaluate the approach in a variety of different data conditions: <br />
# Learning the full collection of tasks jointly via reinforcement learning <br />
# In a zero-shot setting where a policy sketch is available for a held-out task<br />
# In a adaptation setting, where sketches are hidden and the agent must learn to use and adapt a pretrained policy to reuse high-level actions in a new task.<br />
<br />
The code has been released at http://github.com/jacobandreas/psketch.<br />
<br />
='''Related Work'''=<br />
The approach in this paper is a specific case of the options framework developed by Sutton et al., 1999. In that work, options are introduced as "closed-loop policies for taking action over the period of time". They show that options enable temporally abstract information to be included in reinforcement learning algorithms, though it was published before the large-scale popularity of neural networks for reinforcement.<br />
<br />
Other authors have recently explored techniques for learning policies which require less prior knowledge of the environment than the method presented in this paper. For example, in Vezhnevets et al. (2016), the authors propose a RNN architecture to build "implicit plans" only through interacting with the environment as in the classic reinforcement learning problem formulation.<br />
<br />
One closely related line of work is the Hierarchical Abstract Machines (HAM) framework introduced by Parr & Russell, 1998 [11]. Like the approach which the Modular Multitask Reinforcement Learning with Policy Sketches uses, HAMs begin with a representation of a high-level policy as an automaton (or a more general computer program; Andre & Russell,<br />
2001 [7]; Marthi et al., 2004 [12]) and use reinforcement learning to fill in low-level details.<br />
<br />
='''Learning Modular Policies from Sketches'''=<br />
The paper considers a multitask reinforcement learning problem arising from a family of infinite-horizon discounted Markov decision processes in a shared environment. This environment is specified by a tuple $(S, A, P, \gamma )$, with <br />
* $S$ a set of states<br />
* $A$ a set of low-level actions <br />
* $P : S \times A \times S \to R$ a transition probability distribution<br />
* $\gamma$ a discount factor<br />
<br />
Each task $t \in T$ is then specified by a pair $(R_t, \rho_t)$, with $R_t : S \to R$ a task-specific reward function and $\rho_t: S \to R$, an initial distribution over states. For a fixed sequence ${(s_i, a_i)}$ of states and actions obtained from a rollout of a given policy, we will denote the empirical return starting in state $s_i$ as $q_i = \sum_{j=i+1}^\infty \gamma^{j-i-1}R(s_j)$. In addition to the components of a standard multitask RL problem, we assume that tasks are annotated with sketches $K_t$ , each consisting of a sequence $(b_{t1},b_{t2},...)$ of high-level symbolic labels drawn from a fixed vocabulary $B$.<br />
<br />
==Model==<br />
The authors exploit the structural information provided by sketches by constructing for each symbol ''b'' a corresponding subpolicy $\pi_b$. By sharing each subpolicy across all tasks annotated with the corresponding symbol, their approach naturally learns the tied/shared abstraction for the corresponding subtask.<br />
<br />
[[File:Algorithm_MRL2.png|center|frame|Pseudo Algorithms for Modular Multitask Reinforcement Learning with Policy Sketches]]<br />
<br />
At every timestep, a subpolicy selects either a low-level action $a \in A$ or a special STOP action. The augmented state space is denoted as $A^+ := A \cup \{STOP\}$. At a high level, this framework is agnostic to the implementation of subpolicies: any function that takes a representation of the current state onto a distribution over $A^+$ will work fine with the approach.<br />
<br />
In this paper, $\pi_b$ is represented as a neural network. These subpolicies may be viewed as options of the kind described by [2], with the key distinction that they have no initiation semantics, but are instead invokable everywhere, and have no explicit representation as a function from an initial state to a distribution over final states (instead this paper uses the STOP action to terminate).<br />
<br />
Given a fixed sketch $(b_1, b_2,....)$, a task-specific policy $\Pi_r$ is formed by concatenating its associated subpolicies in sequence. In particular, the high-level policy maintains a sub-policy index ''i'' (initially 0), and executes actions from $\pi_{b_i}$ until the STOP symbol is emitted, at which point control is passed to bi+1 . We may thus think of as inducing a Markov chain over the state space $S \times B$, with transitions:<br />
[[File:MRL1.png|center|border|]]<br />
<br />
Note that $\Pi_r$ is semi-Markov with respect to projection of the augmented state space $S \times B$ onto the underlying state space ''S''. The complete family of task-specific policies is denoted as $\Pi := \bigcup_r \{ \Pi_r \}$. Assume each $\pi_b$ be an arbitrary function of the current environment state parameterized by some weight vector $\theta_b$. The learning problem is to optimize over all $\theta_b$ to maximize expected discounted reward<br />
[[File:MRL2.png|center|border|]]<br />
across all tasks $t \in T$.<br />
<br />
==Policy Optimization==<br />
<br />
Here that optimization is accomplished through a simple decoupled actor–critic method. In a standard policy gradient approach, with a single policy $\pi$ with parameters $\theta$, the gradient steps are of the form:<br />
[[File:MRL3.png|center|border|]]<br />
<br />
where the baseline or “critic” c can be chosen independently of the future without introducing bias into the gradient. Recalling the previous definition of $q_i$ as the empirical return starting from $s_i$, this form of the gradient corresponds to a generalized advantage estimator with $\lambda = 1$. Here ''c'' achieves close to the optimal variance[6] when it is set exactly equal to the state-value function $V_{\pi} (s_i) = E_{\pi} q_i$ for the target policy $\pi$ starting in state $s_i$.<br />
[[File:MRL4.png|frame|]]<br />
<br />
In the case of generalizing to modular policies built by sequencing sub-policies the authors suggest to have one subpolicy per symbol but one critic per task. This is because subpolicies $\pi_b$ might participate in many compound policies $\Pi_r$, each associated with its own reward function $R_r$ . Thus individual subpolicies are not uniquely identified or differentiated with value functions. The actor–critic method is extended to allow decoupling of policies from value functions by allowing the critic to vary per-sample (per-task-and-timestep) based on the reward function with which that particular sample is associated. Noting that <br />
[[File:MRL5.png|center|border|]]<br />
i.e. the sum of gradients of expected rewards across all tasks in which $\pi_b$ participates, we have:<br />
[[File:MRL6.png|center|border|]]<br />
where each state-action pair $(s_{t_i}, a_{t_i})$ was selected by the subpolicy $\pi_b$ in the context of the task ''t''.<br />
<br />
Now minimization of the gradient variance requires that each $c_t$ actually depend on the task identity. (This follows immediately by applying the corresponding argument in [6] individually to each term in the sum over ''t'' in Equation 2.) Because the value function is itself unknown, an approximation must be estimated from data. Here it is allowed that these $c_t$ to be implemented with an arbitrary function approximator with parameters $\eta_t$ . This is trained to minimize a squared error criterion, with gradients given by<br />
[[File:MRL7.png|center|border|]]<br />
Alternative forms of the advantage estimator (e.g. the TD residual $R_t (s_i) + \gamma V_t(s_{i+1} - V_t(s_i))$ or any other member of the generalized advantage estimator family) can be used to substitute by simply maintaining one such estimator per task. Experiments show that conditioning on both the state and the task identity results in dramatic performance improvements, suggesting that the variance reduction given by this objective is important for efficient joint learning of modular policies.<br />
<br />
The complete algorithm for computing a single gradient step is given in Algorithm 1. (The outer training loop over these steps, which is driven by a curriculum learning procedure, is shown in Algorithm 2.) Note that this is an on-policy algorithm. In every step, the agent samples tasks from a task distribution provided by a curriculum (described in the following subsection). The current family of policies '''$\Pi$''' is used to perform rollouts for every sampled task, accumulating the resulting tuples of (states, low-level actions, high-level symbols, rewards, and task identities) into a dataset ''$D$''. Once ''$D$'' reaches a maximum size D, it is used to compute gradients with respect to both policy and critic parameters, and the parameter vectors are updated accordingly. The step sizes $\alpha$ and $\beta$ in Algorithm 1 can be chosen adaptively using any first-order method.<br />
<br />
==Curriculum Learning==<br />
<br />
For complex tasks, like the one depicted in Figure 3b, it is difficult for the agent to discover any states with positive reward until many subpolicy behaviors have already been learned. It is thus a better use of the learner’s time (and computational resources) to focus on “easy” tasks, where many rollouts will result in high reward from which relevant subpolicy behavior can be obtained. But there is a fundamental tradeoff involved here: if the learner spends a lot of its time on easy tasks before being told of the existence of harder ones, it may overfit and learn subpolicies that exhibit the desired structural properties or no longer generalize.<br />
<br />
To resolve these issues, a curriculum learning scheme is used that allows the model to smoothly scale up from easy tasks to more difficult ones without overfitting. Initially the model is presented with tasks associated with short sketches. Once average reward on all these tasks reaches a certain threshold, the length limit is incremented. It is assumed that rewards across tasks are normalized with maximum achievable reward $0 < q_i < 1$ . Let $Er_t$ denote the empirical estimate of the expected reward for the current policy on task T. Then at each timestep, tasks are sampled in proportion $1-Er_t$, which by assumption must be positive.<br />
<br />
Intuitively, the tasks that provide the strongest learning signal are those in which <br />
# The agent does not on average achieve reward close to the upper bound<br />
# Many episodes result in high reward.<br />
<br />
The expected reward component of the curriculum solves condition (1) by making sure that time is not spent on nearly solved tasks, while the length bound component of the curriculum addresses condition (2) by ensuring that tasks are not attempted until high-reward episodes are likely to be encountered. The experiments performed show that both components of this curriculum learning scheme improve the rate at which the model converges to a good policy.<br />
<br />
The complete curriculum-based training algorithm is written as Algorithm 2 above. Initially, the maximum sketch length $l_{max}$ is set to 1, and the curriculum initialized to sample length-1 tasks uniformly. For each setting of $l_{max}$, the algorithm uses the current collection of task policies to compute and apply the gradient step described in Algorithm 1. The rollouts obtained from the call to TRAIN-STEP can also be used to compute reward estimates $Er_t$ ; these estimates determine a new task distribution for the curriculum. The inner loop is repeated until the reward threshold $r_{good}$ is exceeded, at which point $l_{max}$ is incremented and the process repeated over a (now-expanded) collection of tasks.<br />
<br />
='''Experiments'''=<br />
[[File:MRL8.png|border|right|400px]]<br />
This paper considers three families of tasks: a 2-D Minecraft-inspired crafting game (Figure 3a), in which the agent must acquire particular resources by finding raw ingredients, combining them together in the correct order, and in some cases building intermediate tools that enable the agent to alter the environment itself; a 2-D maze navigation task that requires the agent to collect keys and open doors, and a 3-D locomotion task (Figure 3b) in which a quadrupedal robot must actuate its joints to traverse a narrow winding cliff.<br />
<br />
In all tasks, the agent receives a reward only after the final goal is accomplished. For the most challenging tasks, involving sequences of four or five high-level actions, a task-specific agent initially following a random policy essentially never discovers the reward signal, so these tasks cannot be solved without considering their hierarchical structure. These environments involve various kinds of challenging low-level control: agents must learn to avoid obstacles, interact with various kinds of objects, and relate fine-grained joint activation to high-level locomotion goals.<br />
<br />
==Implementation==<br />
In all of the experiments, each subpolicy is implemented as a neural network with ReLU nonlinearities and a hidden layer with 128 hidden units. Each critic is a linear function of the current state. Each subpolicy network receives as input a set of features describing the current state of the environment, and outputs a distribution over actions. The agent acts at every timestep by sampling from this distribution. The gradient steps given in lines 8 and 9 of Algorithm 1 are implemented using RMSPROP with a step size of 0.001 and gradient clipping to a unit norm. They take the batch size D in Algorithm 1 to be 2000, and set $\gamma$= 0.9 in both environments. For curriculum learning, the improvement threshold $r_{good}$ is 0.8.<br />
<br />
==Environments==<br />
<br />
The environment in Figure 3a is inspired by the popular game Minecraft, but is implemented in a discrete 2-D world. The agent interacts with objects in the environment by executing a special USE action when it faces them. Picking up raw materials initially scattered randomly around the environment adds to an inventory. Interacting with different crafting stations causes objects in the agent’s inventory to be combined or transformed. Each task in this game corresponds to some crafted object the agent must produce; the most complicated goals require the agent to also craft intermediate ingredients, and in some cases build tools (like a pickaxe and a bridge) to reach ingredients located in initially inaccessible regions of the world.<br />
<br />
The maze environment is very similar to “light world” described by [4]. The agent is placed in a discrete world consisting of a series of rooms, some of which are connected by doors. The agent needs to first pick up a key to open them. For our experiments, each task corresponds to a goal room that the agent must reach through a sequence of intermediate rooms. The agent senses the distance to keys, closed doors, and open doors in each direction. Sketches specify a particular sequence of directions for the agent to traverse between rooms to reach the goal. The sketch always corresponds to a viable traversal from the start to the goal position, but other (possibly shorter) traversals may also exist.<br />
<br />
The cliff environment (Figure 3b) proves the effectiveness of the approach in a high-dimensional continuous control environment where a quadrupedal robot [5] is placed on a variable-length winding path, and must navigate to the end without falling off. This is a challenging RL problem since the walker must learn the low-level walking skill before it can make any progress. The agent receives a small reward for making progress toward the goal, and a large positive reward for reaching the goal square, with a negative reward for falling off the path.<br />
<br />
==Multitask Learning==<br />
<br />
[[File:MRL9.png|border|center|800px]]<br />
The primary experimental question in this paper is whether the extra structure provided by policy sketches alone is enough to enable fast learning of coupled policies across tasks. The aim is to explore the differences between the approach described and relevant prior work that performs either unsupervised or weakly supervised multitask learning of hierarchical policy structure. Specifically,they compare their '''modular''' approach to:<br />
<br />
# Structured hierarchical reinforcement learners:<br />
#* the fully unsupervised '''option–critic''' algorithm of Bacon & Precup[1]<br />
#* a '''Q automaton''' that attempts to explicitly represent the Q function for each task / subtask combination (essentially a HAM [8] with a deep state abstraction function)<br />
# Alternative ways of incorporating sketch data into standard policy gradient methods:<br />
#* learning an '''independent''' policy for each task<br />
#* learning a '''joint policy''' across all tasks, conditioning directly on both environment features and a representation of the complete sketch<br />
<br />
The joint and independent models performed best when trained with the same curriculum described in Section 3.3, while the option–critic model performed best with a length–weighted curriculum that has access to all tasks from the beginning of training.<br />
<br />
Learning curves for baselines and the modular model are shown in Figure 4. It can be seen that in all environments, our approach substantially outperforms the baselines: it induces policies with substantially higher average reward and converges more quickly than the policy gradient baselines. It can further be seen in Figure 4c that after policies have been learned on simple tasks, the model is able to rapidly adapt to more complex ones, even when the longer tasks involve high-level actions not required for any of the short tasks.<br />
<br />
==Ablations==<br />
[[File:MRL10.png|border|right|400px]]<br />
In addition to the overall modular parameter tying structure induced by sketches, the other critical component of the training procedure is the decoupled critic and the curriculum. The next experiments investigate the extent to which these are necessary for good performance.<br />
<br />
To evaluate the the critic, consider three ablations: <br />
# Removing the dependence of the model on the environment state, in which case the baseline is a single scalar per task<br />
# Removing the dependence of the model on the task, in which case the baseline is a conventional generalised advantage estimator<br />
# Removing both, in which case the baseline is a single scalar, as in a vanilla policy gradient approach.<br />
<br />
Results are shown in Figure 5a. Introducing both state and task dependence into the baseline leads to faster convergence of the model: the approach with a constant baseline achieves less than half the overall performance of the full critic after 3 million episodes. Introducing task and state dependence independently improve this performance; combining them gives the best result.<br />
<br />
==Zero-shot and Adaptation Learning==<br />
[[File:MRL11.png|border|left|320px]]<br />
In the final experiments, the authors test the model’s ability to generalize beyond the standard training condition. Consider two tests of generalization: a zero-shot setting, in which the model is provided a sketch for the new task and must immediately achieve good performance, and a adaptation setting, in which no sketch is provided leaving the model to learn the form of a suitable sketch via interaction in the new task.They hold out two length-four tasks from the full inventory used in Section 4.3, and train on the remaining tasks. For zero-shot experiments, the concatenated policy is formed to describe the sketches of the held-out tasks, and repeatedly executing this policy (without learning) in order to obtain an estimate of its effectiveness. For adaptation experiments, consider ordinary RL over high-level actions B rather than low-level actions A, implementing the high-level learner with the same agent architecture as described in Section 3.1. Results are shown in Table 1. The held-out tasks are sufficiently challenging that the baselines are unable to obtain more than negligible reward: in particular, the joint model overfits to the training tasks and cannot generalize to new sketches, while the independent model cannot discover enough of a reward signal to learn in the adaptation setting. The modular model does comparatively well: individual subpolicies succeed in novel zero-shot configurations (suggesting that they have in fact discovered the behavior suggested by the semantics of the sketch) and provide a suitable basis for adaptive discovery of new high-level policies.<br />
<br />
='''Conclusion & Critique'''=<br />
The paper's contributions are:<br />
<br />
* A general paradigm for multitask, hierarchical, deep reinforcement learning guided by abstract sketches of task-specific policies.<br />
<br />
* A concrete recipe for learning from these sketches, built on a general family of modular deep policy representations and a multitask actor–critic training objective.<br />
<br />
They have described an approach for multitask learning of deep multitask policies guided by symbolic policy sketches. By associating each symbol appearing in a sketch with a modular neural sub policy, they have shown that it is possible to build agents that share behavior across tasks in order to achieve success in tasks with sparse and delayed rewards. This process induces an inventory of reusable and interpretable sub policies which can be employed for zero-shot generalization when further sketches are available, and hierarchical reinforcement learning when they are not.<br />
<br />
<br />
<br />
One critique of this approach could be that building of different neural networks for each sub tasks could lead to overtly complicated networks and is not in the spirit of building efficient structure.<br />
<br />
='''References'''=<br />
[1] Bacon, Pierre-Luc and Precup, Doina. The option-critic architecture. In NIPS Deep Reinforcement Learning Work-shop, 2015.<br />
<br />
[2] Sutton, Richard S, Precup, Doina, and Singh, Satinder. Be-tween MDPs and semi-MDPs: A framework for tempo-ral abstraction in reinforcement learning. Artificial intel-ligence, 112(1):181–211, 1999.<br />
<br />
[3] Stolle, Martin and Precup, Doina. Learning options in reinforcement learning. In International Symposium on Abstraction, Reformulation, and Approximation, pp. 212– 223. Springer, 2002.<br />
<br />
[4] Konidaris, George and Barto, Andrew G. Building portable options: Skill transfer in reinforcement learning. In IJ-CAI, volume 7, pp. 895–900, 2007.<br />
<br />
[5] Schulman, John, Moritz, Philipp, Levine, Sergey, Jordan, Michael, and Abbeel, Pieter. Trust region policy optimization. In International Conference on Machine Learning, 2015b.<br />
<br />
[6] Greensmith, Evan, Bartlett, Peter L, and Baxter, Jonathan. Variance reduction techniques for gradient estimates in reinforcement learning. Journal of Machine Learning Research, 5(Nov):1471–1530, 2004.<br />
<br />
[7] Andre, David and Russell, Stuart. Programmable reinforce-ment learning agents. In Advances in Neural Information Processing Systems, 2001.<br />
<br />
[8] Andre, David and Russell, Stuart. State abstraction for pro-grammable reinforcement learning agents. In Proceedings of the Meeting of the Association for the Advance-ment of Artificial Intelligence, 2002.<br />
<br />
[9] Author Jacob Andreas presenting the paper - https://www.youtube.com/watch?v=NRIcDEB64x8<br />
<br />
[10] Vezhnevets, A., Mnih, V., Osindero, S., Graves, A., Vinyals, O., & Agapiou, J. (2016). Strategic attentive writer for learning macro-actions. In Advances in Neural Information Processing Systems (pp. 3486-3494).<br />
<br />
[11] Parr, Ron and Russell, Stuart. Reinforcement learning with hierarchies of machines. In Advances in Neural Information Processing Systems, 1998.<br />
<br />
[12] Marthi, Bhaskara, Lantham, David, Guestrin, Carlos, and Russell, Stuart. Concurrent hierarchical reinforcement learning. In Proceedings of the Meeting of the Association for the Advancement of Artificial Intelligence, 2004.</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=LightRNN:_Memory_and_Computation-Efficient_Recurrent_Neural_Networks&diff=30152LightRNN: Memory and Computation-Efficient Recurrent Neural Networks2017-11-13T23:30:16Z<p>H4lyu: /* Remarks */</p>
<hr />
<div>= Introduction =<br />
<br />
The study of natural language processing has been around for more than fifty years. It begins in 1950s which the specific field of natural language processing (NLP) is still embedded in the subject of linguistics (Hirschberg & Manning, 2015). After the emergence of strong computational power, computational linguistics began to evolve and gradually branch out to various applications in NLP, such as text classification, speech recognition and question answering (Brownlee, 2017). Computational linguistics or natural language processing is usually defined as “subfield of computer science concerned with using computational techniques to learn, understand, and produce human language content” (Hirschberg & Manning, 2015, p. 261). <br />
<br />
With the development in deep neural networks, one type of neural network, namely recurrent neural networks (RNN) have preformed significantly well in many natural language processing tasks. The reason is that nature of RNN takes into account the past inputs as well as the current input without resulting in vanishing or exploding gradient. More detail of how RNN works in the context of NLP will be discussed in the section of NLP using RNN. However, one limitation of RNN used in NLP is its enormous size of input vocabulary. This will result in a very complex RNN model with too many parameters to train and makes the training process both time and memory-consuming. This serves as the major motivation for this paper’s authors to develop a new technique utilized in RNN, which is particularly efficient at processing large size of vocabulary in many NLP tasks, namely LightRNN.<br />
<br />
= Motivations =<br />
<br />
In language modelling, researchers used to represent words by arbitrary codes, such as “Id143” is the code for “dog” (“Vector Representations of Words,” 2017). Such coding of words is completely random, and it loses the meaning of the words and (more importantly) connection with other words. Nowadays, one-hot representation of words is commonly used, in which a word is represented by a vector of numbers and the dimension of the vector is related to the size of the vocabulary. In RNN, all words in the vocabulary are coded using one-hot representation and then mapped to an embedding vector (Li, Qin, Yang, Hu, & Liu, 2016). Such embedding vector is “a continuous vector space where semantically similar words are mapped to nearby points” (“Vector Representations of Words” 2017, para. 6). Popular RNN structure used in NLP task is long short-term memory (LSTM). In order to predict the probability of the next word, the last hidden layer of the network needs to calculate the probability distribution over all other words in the vocabulary. Lastly, an activation function (commonly, softmax function) is used to select the next word with the highest probability. <br />
This method has 3 major limitations:<br />
<br />
1. Memory Constraint <br />
<br />
When input vocabulary contains enormous amount of unique words, which is very common in various NLP tasks, the size of model becomes very large. This means the number of trainable parameters is very big, which makes it difficult to fit such model on a regular GPU device.<br />
<br />
2. Computationally Heavy to Train<br />
<br />
As previously mentioned, probability distribution of all other words in the vocabulary needs to be computed to determine what predicted word it would be. When the size of the vocabulary is large, such calculation can be computationally heavy. <br />
<br />
3. Low Compressability<br />
<br />
Due to the memory and computation-consuming process of RNN applied in NLP tasks, mobile devices cannot usually handle such algorithm, which makes it undesirable and limits its usage.<br />
<br />
= LightRNN Structure =<br />
<br />
The authors of the paper proposed a new structure that effectively reduces the size of the model by arranging all words in the vocabulary into a word table, which is referred as “2-Component (2C) shared embedding for word representation”. Thus, a word is indexed by its location in such table, which in terms is characterized by the corresponding row and column components. Each row and column component are unique row vector and column vector respectively. By organizing each word in the vocabulary in this manner, multiple words can share the same row component or column component and it can reduce the number of trainable parameters significantly. <br />
The next question is how to construct such word table. More specifically, how to allocate each word in the vocabulary to different positions so that semantically similar words are in the same row or column. The authors proposed a bootstrap method to solve this problem. Essentially, we first randomly distribute words into the table. Then, we let the model “learn” better position of each word by minimizing training error. By repeating this process, each word can be allocated to a particular position within the table so that similar words share common row or column components. More details of those 2 parts of LightRNN structure will be discussed in the following sections.<br />
<br />
There are 2 major benefits of the proposed technique:<br />
<br />
1. Computationally efficient<br />
<br />
The name “LightRNN” is to illustrate the small model size and fast training speed. Because of these features of the new RNN architecture, it’s possible to launch such model onto regular GPU and other mobile devices. <br />
<br />
2. Higher scalability <br />
<br />
The authors briefly explained this algorithm is scalable because if parallel-computing is needed to train such model, the difficulty of combining smaller models is low. <br />
<br />
<br />
== Part I: 2-Component Shared Embedding ==<br />
<br />
The key aspect of LightRNN structure is its innovative method of word representation, namely 2-Component Shared Embedding. All words in the vocabulary is organized into a table with row components and column components. Each pair of element in a row component and a column component is corresponding to a unique word in the vocabulary. For instance, the <math>i^{th}</math> row and <math>j^{th}</math> column are the row and column indexes for <math>X_{ij}</math>. As shown in the following graph, <math>x_{1}</math> is corresponding to the words “January”. In 2C shared embedding table, it’s indexed by 2 elements: <math>x^{r}_{1}</math> and <math>x^{c}_{1}</math> where the subscript indicates which row component and column component this word belongs to. Ideally, words that share similar semantic features should be assigned to the same row or column. The shared embedding word table in Figure 1 serves as a good example: the word “one” and “January” are assigned to the same column, while the word “one” and “two” are allocated to the same row. <br />
<br />
[[File:2C shared embedding.png|700px|thumb|centre|Fig 1. 2-Component Shared Embedding for Word Representation]]<br />
<br />
The main advantage of using such word representation is it reduces the number of vector/element needed for input word embedding. For instance, if there are 25 unique words in the vocabulary, the number of vectors to represent all the words is 10, namely 5 row vectors/elements and 5 column vectors/elements. Therefore, the shared embedding word table is a 5 by 5 matrix. In general, the formula for calculating number of vector/element needed to represent |V| words is <math>2\sqrt{|V|}</math>. <br />
<br />
<br />
== Part II: How 2C Shared Embedding is Used in LightRNN ==<br />
<br />
After constructing such word representation table, those 2-component shared embedding matrices are fed into the recurrent neural network. The following Figure 2 demonstrates a portion of LightRNN structure (left) with comparison with the regular RNN (right). Compared to regular RNN where a single input <math>x_{t-1}</math> is fed into the network each time, 2 elements of a single input <math>x_{t-1}</math>: <math>x^{r}_{t-1}</math> and <math>x^{c}_{t-1}</math> are fed into LightRNN. <br />
<br />
[[File:LightRNN.PNG |700px|thumb|centre|Fig 1. LightRNN Structure & Regular RNN]]<br />
<br />
As mentioned before, the last hidden layer will produce the probabilities of <math>word_{t}</math>. Based on the diagram below, the following formulas are used:<br />
Let n = dimension/length of a row input vector/a column input vector, <math>X^{c}, X^{r} \in \mathbb{R}^{n \times \sqrt{|V|}}</math> denotes the input-embedding matrices: <br />
<center><br />
: row vector <math>x^{r}_{t-1} \in \mathbb{R}^n</math><br />
: column vector <math>x^{c}_{t-1} \in \mathbb{R}^n</math><br />
</center><br />
<br />
Let <math>h^{c}_{t-1}, h^{r}_{t-1} \in \mathbb{R}^m</math> denotes the two hidden layers where m = dimension of the hidden layer:<br />
<center><br />
: <math>h^{c}_{t-1} = f(W x_{t-1}^{c} + U h_{t-1}^{r} + b) </math><br />
: <math>h^{r}_{t} = f(W x_{t}^{r} + U h_{t-1}^{c} + b) </math><br />
</center><br />
where <math>W \in \mathbb{R}^m \times n</math>, <math>U \in \mathbb{R}^m \times n</math> & <math>b \in \mathbb{R}^m</math> and <math>f</math> is a nonlinear activation function<br />
<br />
The final step in LightRNN is to calculate <math>P_{r}(w_{t})</math> and <math>P_{c}(w_{t})</math> , which means the probability of a word w at time t, using the following formulas:<br />
<center><br />
: <math>P_{r}(w_t) = \frac{exp(h_{t-1}^{c} y_{r(w)}^{r})}{\sum\nolimits_{i \in S_r} exp(h_{t-1}^{c} y_{i}^{r}) }</math><br />
: <math>P_{c}(w_t) = \frac{exp(h_{t}^{r} y_{c(w)}^{c})}{\sum\nolimits_{i \in S_r} exp(h_{t}^{r} y_{i}^{c}) }</math><br />
: <math> P(w_t) = P_{r}(w_t) P_{c}(w_t) </math> <br />
</center><br />
where <br />
<center><br />
:<math> r(w) </math> = row index of word w <br />
:<math> c(w) </math> = column index of word w<br />
:<math> y_{i}^{r} \in \mathbb{R}^m </math> = i-th vector of <math> Y^r \in \mathbb{R}^{m \times \sqrt{|V|}}</math> <br />
:<math> y_{i}^{c} \in \mathbb{R}^m </math> = i-th vector of <math> Y^c \in \mathbb{R}^{m \times \sqrt{|V|}}</math><br />
:<math> S_r </math> = the set of rows of the word table<br />
:<math> S_c </math> = the set of columns of the word table<br />
</center><br />
<br />
Note that we don't see the t-th word before predicting it. So in the above diagram, given the input column vector <math>x^c_{t-1} </math> of the (t-1)-th word, we first infer the row probability <math>P_r(w_t)</math> of the t-th word, and then choose the index of the row the largest probability in <math>P_r(w_t)</math> to look up the next input row vector <math>x^r_{t} </math>. Similarly, we can infer the column probability <math>P_c(w_t)</math> of the t-th word. <br />
<br />
Essentially, in LightRNN, the prediction of the word at time t (<math> w_t </math>) based on word at time t-1 (<math> w_{t-1} </math>) is achieved by selecting the index <math> r </math> and <math> c </math> with the highest probabilities <math> P_{r}(w_t) </math>, <math> P_{c}(w_t) </math>. Then, the probability of each word is computed based on the multiplication of <math> P_{r}(w_t) </math> and <math> P_{c}(w_t) </math>.<br />
<br />
== Part III: Bootstrap for Word Allocation ==<br />
<br />
As mentioned before, the major innovative aspect of LightRNN is the development of 2-component shared embedding. Such structure can be used in building a recurrent neural network called LightRNN. However, how should such word table representation be constructed is the key part of building a successful LightRNN model. In this section, the procedures of constructing 2C shared embedding structure is explained. <br />
The fundamental idea is using bootstrap method by minimizing a loss function (namely, negative log-likelihood function). The detailed procedures are described as the following:<br />
<br />
Step 1: First, all words in a vocabulary are randomly assigned to individual position within the word table<br />
<br />
Step 2: Train LightRNN model based on word table produced in step 1 until certain criteria are met<br />
<br />
Step 3: By fixing the training results of input and output embedding matrices (W & U) from step 2, adjust the position of words by minimizing the loss function over all the words. Then, repeat from step 2<br />
<br />
The authors presented the overall loss function for word w moving to position [i, j] using a negative log-likelihood function (NLL) as the following:<br />
<center><br />
<math> NLL = \sum\limits_{i=1}^T -logP(w_t) = \sum\limits_{i=1}^T -log[P_{r}(w_t) P_{c}(w_t)] = \sum\limits_{i=1}^T -log[P_{r}(w_t)] – log[P_{c}(w_t)] = \sum\limits_{w=1}^{|V|} NLL_w </math><br />
</center><br />
where <math> NLL_w </math> is the negative log-likelihood of a word w. <br />
<br />
Since in 2-component shared embedding structure, a word (w) is represented by one row vector and one column vector, <math> NLL_w </math> can be rewritten as <math> l(w, r(w), c(w)) </math> where <math> r(w) </math> and <math> c(w) </math> are the position index of word w in the word table. Next, the authors defined 2 more terms to explain the meaning of <math> NLL_w </math>: <math> l_r(w,r(w)) </math> and <math> l_c(w,c(w)) </math>, namely the row component and column component of <math> l(w, r(w), c(w)) </math>. The above can be summarised by the following formulas: <br />
<center><br />
<math> NLL_w = \sum\limits_{t \in S_w} -logP(w_t) = l(w, r(w), c(w)) </math> <br><br />
<math> = \sum\limits_{t \in S_w} -logP_r(w_t) + \sum\limits_{t \in S_w} -logP_c(w_t) = l_r(w,r(w)) + l_c(w,c(w))</math> <br><br />
<math> = \sum\limits_{t \in S_w} -log (\frac{exp(h_{t-1}^{c} y_{i}^{r})}{\sum\nolimits_{k} exp(h_{t-1}^{c} y_{i}^{k})}) + \sum\limits_{t \in S_w} -log (\frac{exp(h_{t}^{r} y_{j}^{c})}{\sum\nolimits_{k} exp(h_{t}^{r} y_{k}^{c}) }) </math> <br> <br />
where <math> S_w </math> is the set of all possible positions within the word table <br />
</center><br />
In summary, the overall loss function for word w to move to position [i, j] is the sum of its row loss and column loss of moving to position [i, j]. Therefore, total loss of moving to position [i, j] <math> l(w, i, j) = l_r(w, i) + l_c(w, j)</math>. Thus, to update the table by reallocating each word, we are looking for position [i, j] for each word w that minimize the total loss function, mathematically written as for the following:<br />
<center><br />
<math> \min\limits_{a} \sum\limits_{w,i,j} l(w,i,j)a(w,i,j) </math> such that <br><br />
<math> \sum\limits_{(i,j)} a(w,i,j) = 1 \space \forall w \in V, \sum\limits_{(w)} a(w,i,j) = 1 \space \forall i \in S_r, j \in S_j</math> <br><br />
<math> a(w,i,j) \in {0,1}, \forall w \in V, i \in S_r, j \in S_j</math> <br><br />
where <math> a(w,i,j) =1 </math> indicates moving word w to position [i, j]<br />
</center><br />
<br />
= LightRNN Example =<br />
<br />
After describing the theoretical background of the LightRNN algorithm, the authors applied this method to 2 datasets (2013 ACL Workshop Morphological Language Dataset (ACLW) & One-Billion-Word Benchmark Dataset (BillonW)) and compared its performance with several other state-of-the-art RNN algorithms. The following table shows some summary statistics of those 2 datasets:<br />
<br />
[[File:Table1YH.PNG|700px|thumb|centre|Table 1. Summary Statistics of Datasets]]<br />
<br />
The goal of a probabilistic language model is either to compute the probability distribution of a sequence of given words (e.g. <math> P(W) = P(w_1, w_2, … , w_n)</math>) or to compute the probability of the next word given some previous words (e.g. <math> P(w_5 | w_1, w_2, w_3, w_4)</math>) (Jurafsky, 2017). In this paper, the evaluation matrix for the performance of LightRNN algorithm is perplexity <math> PPL </math> which is defined as the following: <br />
<center><br />
<math> PPL = exp(\frac{NLL}{T})</math> <br><br />
where T = number of tokens in the test set<br />
</center><br />
<br />
Based on the mathematical definition of PPL, a well-performed model will have a lower perplexity. <br />
The authors then trained “LSTM-based LightRNN using stochastic gradient descent with truncated backpropagation through time” (Li, Qin, Yang, Hu, & Liu, 2016). To begin with, the authors first used the ACLW French dataset to determine the size of embedding matrix. From the results shown in Table 2, larger embedding size corresponds to higher accuracy rate (expressed in terms of perplexity). Therefore, they adopted embedding size of 1000 to be used in LightRNN to analyze the ACLW datasets. <br />
<br />
[[File:Table2YH.PNG|700px|thumb|centre|Table 2. Testing PPL of LightRNN on ACLW-French dataset w.r.t. embedding size]]<br />
<br />
'''Advantage 1: small model size'''<br />
<br />
One of the major advantages of using LightRNN on NLP tasks is significantly reduced model size, which means fewer number of parameters to estimate. By comparing LightRNN with two other RNN algorithms and the baseline language model with Kneser-Ney smoothing. Those two RNN algorithms are: HSM which uses LSTM RNN algorithm with hierarchical softmax for word prediction; C-HSM which uses both hierarchical softmax and character-level convolutional filters for input embedding. From the results table shown below, we can see that LightRNN has the lowest perplexity while keeping the model size significantly smaller compared to the other three algorithms. <br />
<br />
[[File:Table5YH.PNG|700px|thumb|centre|Table 3. PPL Results in test set on ACLW datasets]]<br />
Italic results are the previous state-of-the-art. #P denotes the number of parameters. <br />
<br />
'''Advantage 2: high training efficiency'''<br />
<br />
Another advantage of LightRNN model is its shorter training time while maintaining same level of perplexity compared to other RNN algorithms. When comparing to both C-HSM and HSM (shown below in Table 4), LightRNN only takes half the runtime but achieve same level of perplexity when applied to both ACLW and BillionW datasets. In the last column of Table 3, the amount of time used for word table reconstruction is presented as the percentage of the total runtime. As we can see, the training time for word reallocation takes up only a very small proportion of the total runtime. However, the resulting reconstructed word table can be used as a valuable output, which is further explained in the next section. <br />
<br />
[[File:Table3YH.PNG|700px|thumb|centre|Table 4. Runtime comparisons in order to achieve the HSMs’ baseline PPL]]<br />
<br />
<br />
'''Advantage 3: semantically valid word allocation table'''<br />
<br />
As explained in the previous section, LightRNN uses a word allocation table that gets updated in every iteration of the algorithm. The optimal structure of the table should assign semantically similar words onto the same row or column in order to reduce the number of parameters to estimate. Below is a snapshot of the reconstructed word table used in LightRNN algorithm. Evidently, we can see in row 887, all URL addresses are grouped together and in row 872 all verbs in past tense are grouped together. As the authors explained in the paper, LightRNN doesn’t assume independence of each word but instead using a shared embedding table. In this way, it reduces the model size by utilizing common embedding elements of the table/matrix, and also uses such preprocessed data to improve the efficiency of this algorithm.<br />
<br />
[[File:Table6YH.PNG|700px|thumb|centre|Table 6. Sample Word Allocation Table]]<br />
<br />
= Remarks =<br />
In summary, the proposed method in this paper is mainly on developing a new way of using word embedding. Words with similar semantic meanings are embedded using similar vectors. Those vectors are then divided into row and column components where similar words are grouped together by having shared row and column components in the word representation table. This is indeed a dimensional reduction, i.e. use the row and column "semantic vectors" to approximate the coded word. Because of this structural change of input word embedding, RNN model needs to adapt by having both row and column components being fed into the network. However, the fundamental structure of RNN model does not changed. Therefore, personally I would say it’s a new word embedding technique rather than a new development in model construction. One major confusion I have when reading this paper is how those row and column components in the word allocation table are determined. From the paper itself, the authors didn’t explain how they are constructed. <br />
<br />
Such shared word embedding technique are prevalently used in NLP. For instance, in language translation, similar words from different languages are grouped together so that the machine can translate sentences from one language to another. In Socher et al. (2013a), English and Chinese words are embedded in the same space so that we can find similar English (Chinese) words for Chinese (English) words. (Zou, Socher, Cer, & Manning, 2013). Word2vec is also a commonly used technique for word embedding, which uses a two-layer neural network to transform text into numeric vectors where similar words will have similar numeric values. The key feature of word2vec is that semantically similar words (which is now represented by numeric vectors) can be grouped together (“Word2vec,” n.d.; Bengio, Ducharme, & Vincent, 2001; Bengio, Ducharme, Vincent, & Jauvin, 2003).<br />
<br />
An interesting area of further exploration proposed by the authors is an extension of this method to k-component shared embeddings where k>2. Words probably share similar semantic meanings on more than two dimensions, and this extension could reduce network size even further. However it could also further complicate the bootstrapping phase of training.<br />
<br />
= Reference =<br />
Bengio, Y, Ducharme, R., & Vincent, P. (2001). A Neural Probabilistic Language Model. In Journal of Machine Learning Research (Vol. 3, pp. 932–938). https://doi.org/10.1162/153244303322533223<br />
<br />
Bengio, Yoshua, Ducharme, R., Vincent, P., & Jauvin, C. (2003). A Neural Probabilistic Language Model. Journal of Machine Learning Research, 3(Feb), 1137–1155.<br />
<br />
Brownlee, J. (2017, September 20). 7 Applications of Deep Learning for Natural Language Processing. Retrieved October 27, 2017, from https://machinelearningmastery.com/applications-of-deep-learning-for-natural-language-processing/<br />
<br />
Hirschberg, J., & Manning, C. D. (2015). Advances in natural language processing. Science, 349(6245), 261–266. https://doi.org/10.1126/science.aaa8685<br />
<br />
Jurafsky, D. (2017, January). Language Modeling Introduction to N grams. Presented at the CS 124: From Languages to Information, Stanford University. Retrieved from https://web.stanford.edu/class/cs124/lec/languagemodeling.pdf<br />
<br />
Li, X., Qin, T., Yang, J., Hu, X., & Liu, T. (2016). LightRNN: Memory and Computation-Efficient Recurrent Neural Networks. Advances in Neural Information Processing Systems 29, 4385–4393.<br />
<br />
Recurrent Neural Networks. (n.d.). Retrieved October 8, 2017, from https://www.tensorflow.org/tutorials/recurrent<br />
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Vector Representations of Words. (2017, August 17). Retrieved October 8, 2017, from https://www.tensorflow.org/tutorials/word2vec<br />
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Word2vec. (n.d.). Retrieved October 26, 2017, from https://deeplearning4j.org/word2vec.html<br />
<br />
Zou, W. Y., Socher, R., Cer, D., & Manning, C. D. (2013). Bilingual word embeddings for phrase-based machine translation, 1393–1398.</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=%22Why_Should_I_Trust_You%3F%22:_Explaining_the_Predictions_of_Any_Classifier&diff=30147"Why Should I Trust You?": Explaining the Predictions of Any Classifier2017-11-13T23:04:06Z<p>H4lyu: /* Remarks and Critique */</p>
<hr />
<div>==Introduction==<br />
<br />
Understanding why machine learning models behave the way they do helps users in model selection, feature engineering, and finally trusting the model to deploy it. In many cases even though a non-interpretable model is more accurate on the validation data set than the interpretable one, an interpretable model is chosen. But restricting the models to just interpretable model isn't the best option. In this paper the authors argue for explaining machine learning predictions using model-agnostic approaches and propose LIME (Local Interpretable Model-Agnostic Explanations), a novel explanation technique that explains the predictions of any classifier in an interpretable and faithful manner, by learning an interpretable model locally around the prediction. They also propose a method called sub-modular optimization to explain models globally by selecting a representative individual predictions.<br />
<br />
The experiments are conducted mainly on text data and image data, where models used are random forest (RF) and Convolutional Neural Networks (CNN). The experiments highlight utility of explanations in deciding between the models, which model is to be trusted and which is not and also improving the model based on the explanations. The main contributions of this paper are summarized as follows.<br />
<br />
* LIME, an algorithm to explain any prediction of any model (classifier or regressor) by approximating locally with an interpretable model that is faithful to the original model locally. <br />
<br />
* SP-LIME, a method to select final set of representative samples using sub-modular optimization to show to the user that pretty much captures what the original model is doing globally. <br />
<br />
* A detailed set of experiments with simulated subjects to prove the usefulness of LIME and SP-LIME. <br />
<br />
[[File:LIME.jpg|thumb|550px|Figure 1: Explaining individual predictions. A model predicts that a patient has the flu, and LIME highlights the symptoms in the patient’s history that led to the prediction. Sneeze and headache are portrayed as contributing to the “flu” prediction, while “no fatigue” is evidence against it. With these, a doctor can make an informed decision about whether to trust the model’s prediction.]]<br />
<br />
== Need for Explanations ==<br />
<br />
Prediction explanation basically means answering question such as "What triggers the model to make such a prediction?", and an answer to that would be to say, these are the features with certain range of values in your input sample that are contributing to the prediction the most. For example, it can be words in a document for text classification or patches of pixels in an image for image classification. And these explanations need to make sense to the user and hence has to be simple. Figure 1 illustrates an explanation procedure. In this case, an explanation is a small weighted list symptoms that either contribute to the prediction (in green) or are evidence against it (in red). It is very clear that the life of a doctor is much easier in terms of making a decision with the help of a model if intelligible explanations are provided. This is similar to the topic of inference versus prediction. Some people tend to think of these two terms are the same. However, statistically, their meanings are very different. Inference is given some datasets, we would like to infer how the output/response is generated based on a sequence of explanatory variables. For instance, we would like to know/infer how education level affect people’s income level. Prediction , on the other hand is by using a given dataset, we fit/train a model that will correctly predict the outcome of a new observation. <br />
<br />
Even with highest accuracy on validation dataset, we sometimes can't judge how the model is going to behave on other dataset. There are many reasons why this can happen. For example, Data leakage where some of the features used for training are heavily correlated with the target value in both training and validation data that might result in great train and validation results but will be of no use if used on all together a new dataset. But if explanations such as the one in Figure 1 are given, it becomes easy to fix the fault by removing the heavily correlated feature from the data and train. This is how you convert an untrustworthy model to a trustworthy one. Another problem is dataset shift, where train and test data come from different distributions and providing explanations for predictions will help user to take measures to make model generalize better.<br />
<br />
Figure 2 shows how explanations help in selecting between the models in addition to accuracy measures. In this case, by looking at the explanations it is easy to say that model with higher accuracy is in-fact the worst. Further, many a times, there can be difference between what we want the model to optimize and what the model is actually optimizing which might result into model not behaving as well as expected (For example in binary classification using logistic regression, all we are concerned is with accuracy, but the model is trying to reduce the log-loss due to the learning algorithms limitations). While we may not be able to quantitatively measure what difference has that made, but we still have some intuitions as to how the model has to behave with respect to some of its features, and if explanations are given for model predictions, we will be able to ascertain if those explanations make any sense. Also explaining models being model agnostic helps as we then compare different class of models in making a choice.<br />
<br />
[[File:explanation_example.png|thumb|550px|Figure 2: Explaining individual predictions of competing classifiers trying to determine if a document is about “Christianity” or “Atheism”. The bar chart represents the importance given to the most relevant words, also highlighted in the text. Color indicates which class the word contributes to (green for “Christianity”, magenta for “Atheism”).]]<br />
<br />
===Desired Characteristics for Explainers===<br />
<br />
Explainers have to be simple models so that humans are able to comprehend what the model is doing. For example, even with linear models, if the number of features are too many, it becomes hard for humans to comprehend overwhelming set of features. In case of models trained with word-embedding as features, we can't give these word-embeddings as explanations, instead they need to something different than these features.<br />
<br />
Another requirement of a good explainer is that it has to be locally faithful to the original model, i.e, it must behave similar to the model in the vicinity of instance being predicted. And these local explanations should aid in forming global explanations.<br />
<br />
==Local Interpretable Model-Agnostic Explanations (LIME)==<br />
<br />
The overall goal of LIME is to basically identify an interpretable model over the interpretable representation (features) that is locally faithful to the classifier. For example in case of text classification, a possible interpretable representation of data would be to use bag of words (one-hot encodings) instead of the word embeddings. Similarly for image classification, it may be a binary vector (one-hot encoding) indicating the “presence” or “absence” of a contiguous patch of pixels (a super-pixel or a segment), while the classifier may represent the image as a tensor with three color channels per pixel. Let $x ∈ R^d$ be the original representation of an instance being explained, and $x' ∈ \{0, 1\}^{d'}$ be a binary vector for its interpretable representation.<br />
<br />
Formally, we define an explanation as a model $g ∈ G$, where $G$ is a class of potentially interpretable models, such as linear models, decision trees, or falling rule lists [3]. The domain of $g$ is $\{0, 1\}^{d'}$, i.e. $g$ acts over absence/presence of the interpretable components. Let $Ω(g)$ be a measure of complexity (like a regularizer term) of the explanation $g ∈ G$. For example, for decision trees $Ω(g)$ may be the depth of the tree, for linear models, $Ω(g)$ may be the number of non-zero weights. Let the model being explained be denoted $f : R^d → R$. In classification, $f(x)$ is the probability that $x$ belongs to a certain class. We further use $π_x(z)$ as a proximity measure between an instance $z$ to $x$, so as to define locality around $x$. Finally, let $L(f, g, π_x)$ be a measure of error between $g$ and $f$ in the locality defined by $π_x$. Our task is to minimize $L(f, g, π_x)$ while having $Ω(g)$ be low enough to be interpretable by humans. The explanation produced by LIME is obtained by the following:<br />
<br />
:<math>\xi(x) = \underset{g\in\mathbb{G}}{\operatorname{argmin}}\, \mathcal{L}(f,g,\pi_x) + \Omega(g)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, Eq. (1)</math><br />
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Even though $G$ can be any class of interpretable models, in this paper only sparse linear models are considered.<br />
<br />
=== Sampling for Local Exploration ===<br />
<br />
In order to learn the local behavior of $f$ using $g$ as the interpretable inputs vary, $L(f, g, π_x)$ is approximated by drawing samples, weighted by $π_x$. We sample instances around $x'$ by setting few features of $x'$ to 0, uniformly at random (where the number of such features set to 0 is also uniformly sampled). Given a perturbed sample $z' ∈ \{0, 1\}^{d'}$ (which contains a fraction of the features of $x'$ set to 0), we recover the sample in the original representation $z ∈ R^d$ and obtain the prediction $f(z)$ from the original classifier, and is used as a label for the explanation model. Given this dataset $Z$ of perturbed samples with the associated labels, we optimize Eq. (1) to get an explanation $ξ(x)$. The primary intuition behind LIME is presented in Figure 3, where we sample instances both in the vicinity of $x$ (which have a high weight due to $π_x$) and far away from $x$ (low weight from $π_x$). Even though the original model may be too complex to explain globally, LIME presents an explanation that is locally faithful (linear in this case), where the locality is captured by weight factor, $π_x$. A concrete example of this process is presented in the next section.<br />
<br />
[[File:decision_boundary.png|thumb|Figure 3: Toy example to present intuition for LIME. The black-box model’s complex decision function $f$ (unknown to LIME) is represented by the blue/pink background, which cannot be approximated well by a linear model. The bold red cross is the instance being explained. LIME samples instances, gets predictions using $f$, and weighs them by the proximity to the instance being explained (represented here by size). The dashed line is the learned explanation that is locally (but not globally) faithful.]]<br />
<br />
===Sparse Linear Explanations===<br />
<br />
For the rest of this paper, we let $G$ be the class of linear models, such that $g(z') = w_g · z'$ . We use the locally weighted square loss as $L$, as defined in Eq. (2), where we let $π_x(z) = exp(−D(x, z)^2 /σ^2 )$ be an exponential kernel defined on some distance function $D$ (e.g. cosine distance for text, L2 distance for images) with width $σ$ resulting in higher sample weights for perturbed samples that are in the vicinity of sample being explained and lower weights for samples that are far away.<br />
<br />
:<math>\mathcal{L}(f,g,\pi_x) = \underset{z,z'\in\mathbb{Z}}{\operatorname{\Sigma}} \pi_x(z)\,(f(z) - g(z'))\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, Eq. (2)</math><br />
<br />
For text classification, the features used for explanation models are bag of words, and we set a limit on the number of words used for explanation, $K$. i.e. $Ω(g) = ∞1[||w_g||_0 > K]$. We use the same $Ω$ for image classification, using “super-pixels” (computed using any standard image segmentation algorithm, eg. quickshift() function in skimage python library [5]) instead of words, such that the interpretable representation of an image is a binary vector where 1 indicates the original super-pixel and 0 indicates a grayed out super-pixel. This particular choice of $Ω$ makes directly solving Eq. (1) intractable, but we approximate it by first selecting $K$ features with Lasso (using the regularization path [4]) (scikit lars_path method [6]) and then learning the weights via least squares (together a procedure we call K-LASSO in Algorithm 1). . <br />
<br />
There are some limitations to class of locally interpretable models $G$. It can happen that no $g \in G$ is powerful enough to be locally faithful to the original model, i.e, $g$ has a high bias w.r.t $f$ locally. This may be because the underlying model is quite complex even locally. Another issue could be, the features used for building explanations are not representative of the inputs or factors that the underlying model relies on. For example, a model that predicts sepia-toned images to be retro cannot be explained by presence of absence of super pixels.<br />
<br />
<br />
[[File:algorithm1.png|thumb|550px]]<br />
<br />
==Examples==<br />
<br />
===Example 1: Text classification with SVMs===<br />
<br />
In Figure 2 (RHS), Here LIME explains an SVM classifier with RBF kernel trained on "20 newsgroup dataset" to classify documents into "Atheism" and "Christianity". Even though it achieves 94% accuracy on validation dataset, the explanation for an instance show that features that are used the most in predictions are very arbitrary such as words like "Posting", "Host" and "Re". Even if these stop words or headers are removed, the classier still considers the proper names of people writing the post to be important features, which clearly doesn't make sense. Hence we can conclude by looking at the explanations that the dataset has serious issues and when a classifier is trained on it, it wont be able to generalize well. Hence suitable steps needed to be taken to train a trustworthy classifier.<br />
<br />
===Example 2: Deep networks for images===<br />
<br />
In this example we use sparse linear explanations to explain Google’s pre-trained Inception neural network [7] which is an image classifier. Here features used for giving explanations are super-pixels (segments) which intuitively makes sense as we humans detect an object in an image based on certain parts of the object. Figure 4a show an arbitrary image that we want to explain. Figures 4b, 4c, 4d show the super-pixels that contribute the most while predicting the image to be one of the top 3 classes. Here we set max feature count for explainer, $K = 10$. It is quite intuitive from Figure 4b in particular that why acoustic guitar was predicted to be electric based on the fretboard that resembles acoustic guitar. So even though the prediction is incorrect, it still is not unreasonable at all.<br />
<br />
[[File:inception_example.png|thumb|center|550px]]<br />
<br />
==Submodular Pick for Explaining Models==<br />
<br />
Submodular pick provides some sort of a global understanding of the model by methodically picking a set on non-redundant sample explanations. This method is still model agnostic and complimentary to calculating validation accuracy in machine learning problem. If we choose a large number of instances to explain, the user may not be able to go through each of them and verify what is wrong or right with the model. We say humans have a budget $B$, that is the number of instance explanations that they are willing to examine. So given a set of $X$ instances, the task of sub-modular optimization is to pick a maximum of $B$ instances that effectively capture the essence of what the model is doing in a non-redundant way.<br />
<br />
Here we have a explanations for $n$ instances and we construct an $n × d'$ matrix $W$, each row of which represent local importances of features used for each instance explanations. In case of using sparse linear models as explanations, for an instance $x_i$ and explanation $g_i = ξ(x_i)$, we set $W_{ij} = |w_{gij}|$. Further, for each feature (column) $j$ in $W$, let $I_j$ denote the global importance of that component in the explanation space. Intuitively, $I$ represents global importance of each feature. In Figure 5, we show a toy example $W$, with $n = d' = 5$, where $W$ is binary (for simplicity). Since feature $f2$ is used in most of the explanations, $I$ should have higher value for feature $f2$ than $f1$, i.e. $I_2 > I_1$. Formally for the text classifiers, we set $I_j = \sqrt{\sum_{i=1}^nW_{ij}}$ . For images, $I$ must represent something that can be compared across the super-pixels (segments) in different images, such as color histograms or other features of super-pixels; But this is left as a future work for now.<br />
<br />
While picking the samples, we must be careful in not picking up samples that explain the same thing. So we want minimum set of samples that cover maximum number of features as possible. In Figure 5, after the second row is picked, the third row is redundant, as the user has already seen features $f2$ and $f3$ - while the last row gives the user completely new features. Hence selecting the second and last row results in giving the maximum coverage of features. Eq. (3) formalizes this process, where we define coverage as the set function $c$ that, given $W$ and $I$, computes the total importance of the features that at least appear in one instance in a set $V$ .<br />
<br />
:<math>c(V, W, I) = {{\sum_{j = 1}^{d'}}}\mathbb{1}_{[\exists i\in V:W_{ij}\gt 0]}I_j\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, Eq. (3)</math><br />
<br />
The pick problem, defined in Eq. (4), consists of finding the set $V, |V| ≤ B$ that achieves highest coverage.<br />
<br />
:<math>Pick(W, I) = \underset{V,|V|\leq B}{\operatorname{argmax}}\, c(V,W, I)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, Eq. (4)</math><br />
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The problem in Eq. (4) is maximizing a weighted coverage function, and is NP-hard [10]. Let $c(V\cup\{i\}, W, I)−c(V, W, I)$ be the marginal coverage gain of adding an instance $i$ to a set $V$ . Due to sub-modularity, a greedy algorithm that iteratively adds the instance with the highest marginal coverage gain to the solution offers a constant-factor approximation guarantee of $1−1/e$ to the optimum [8]. This approximation process is outlined in Algorithm 2 and is called sub-modular pick.<br />
<br />
[[File:algorithm2.png|thumb|550px]] <br />
[[File:toy_example.png|thumb|Figure 5: Toy example W. Rows represent instances (documents) and columns represent features (words). Feature f2 (dotted blue) has the highest importance. Rows 2 and 5 (in red) would be selected by the pick procedure, covering all but feature f1.]]<br />
<br />
==Simulated User Experiments==<br />
<br />
This section highlights simulated user experiments conducted to evaluate how explanations help the user in different tasks. In particular, following questions are addressed: (1) Are the explanations faithful to the model, that is are the explainers correctly highlighting the important features that the model is selecting at least locally? (2) Can the explanations help users in ascertaining trust in predictions that is to say if the predictions make sense or not and (3) Are the explanations useful for evaluating the model as a whole and thereby aid in finally selecting a model. Code and data for experiments are available at https://github.com/marcotcr/lime-experiments.<br />
<br />
===Experiment Setup===<br />
<br />
2 sentiment analysis datasets (books and DVDs, 2000 instances each) where the task is to classify product reviews as positive or negative [9] are used in the experiment. Decision trees (DT), logistic regression with L2 regularization (LR), nearest neighbors (NN), and SVMs with RBF kernel, all using bag of words as features are trained separately. There is also a random forest (with 1000 trees) trained over average word2vec embeddings (RF). Unless otherwise noted, default methods of scikit are used. Dataset is divided into train and test in the ratio of 4:1. To explain individual predictions, LIME is compared with parzen [10], a method that approximates the black box classifier globally with Parzen windows, and explains individual predictions by taking the gradient of the prediction probability function. For parzen, $K$ features are selected as explanations such that they have the highest absolute gradients. Hyper-parameters for LIME and parzen are set using cross-validations and number of perturbed samples for a single explanation, $N$ is set to 15000. There is also a comparison made against a greedy procedure in which features that contribute the most to the predicted class are greedily removed until prediction changes or we reach a maximum of $K$ features that are removed, and these $K$ or less features are finally selected to explain the model. Also also a random procedure that randomly picks K features as an explanation is also used for comparison. For all the experiments below $K$ is set to 10. For experiments where the pick procedure applies, it's either a random pick (RP) or sub-modular pick (SP). We refer to pick-explainer combinations by adding RP or SP as a prefix.<br />
<br />
[[File:recall1.png|thumb|Figure 6: Recall [11] on truly important features for two interpretable classifiers on the books dataset.]] [[File:recall2.png|thumb|Figure 7: Recall on truly important features for two interpretable classifiers on the DVDs dataset.]]<br />
<br />
===Are explanations faithful to the model?===<br />
<br />
Here we evaluate faithfulness of explanations on classifiers that are by themselves explicable such as sparse logistic regression or decision trees. So we train 2 models, a sparse logistc regressor and a decision tree such that maximum number of features they use 10 so we now know what the important features (gold features) really are for these 2 models. For each prediction on the test set, we generate explanations and compute the fraction of these gold features that are recovered by the explanations (recall). We report this recall averaged over all the test instances in Figures 6 and 7. From the bar graphs we can see that the greedy approach is comparable to parzen on logistic regression, but is substantially worse on decision trees since changing a single feature at a time often does not have an effect on the prediction. The overall recall by parzen is low, likely due to the difficulty in approximating the original high-dimensional classifier. LIME consistently provides > 90% recall for both classifiers on both datasets, and this demonstrates that LIME explanations are more faithful to the model than any other explanations.<br />
<br />
===Should I trust this prediction?===<br />
<br />
"""In order to simulate trust in individual predictions, we first randomly select 25% of the features to be “untrustworthy”, and assume that the users can identify and would not want to trust these features (such as the headers in 20 newsgroups, leaked data, etc). We thus develop oracle “trustworthiness” by labeling test set predictions from a black box classifier as “untrustworthy” if the prediction changes when untrustworthy features are removed from the instance, and “trustworthy” otherwise. In order to simulate users, we assume that users deem predictions untrustworthy from LIME and parzen explanations if the prediction from the linear approximation changes when all untrustworthy features that appear in the explanations are removed (the simulated human “discounts” the effect of untrustworthy features). For greedy and random, the prediction is mistrusted if any untrustworthy features are present in the explanation, since these methods do not provide a notion of the contribution of each feature to the prediction. Thus for each test set prediction, we can evaluate whether the simulated user trusts it using each explanation method, and compare it to the trustworthiness oracle. <br />
<br />
Using this setup, we report the score, F1 (average of precision and recall) on the trustworthy predictions for each explanation method, averaged over 100 runs, in Table 1. The results indicate that LIME dominates others (all results are significant at p = 0.01) on both datasets, and for all of the black box models. The other methods either achieve a lower recall (i.e. they mistrust predictions more than they should) or lower precision (i.e. they trust too many predictions), while LIME maintains both high precision and high recall. Even though we artificially select which features are untrustworthy, these results indicate that LIME is helpful in assessing trust in individual predictions.<br />
<br />
[[File:table1.png|thumb|Table 1: Average F1 of trustworthiness for different explainers on a collection of classifiers and datasets.]][[File:choose_classifier.png|thumb|Figure 8: Choosing between two classifiers, as the number of instances shown to a simulated user is varied. Averages and standard errors from 800 runs.]]<br />
<br />
===Can I trust this model?===<br />
<br />
In the final simulated user experiment, we consider a case where user has to pick between 2 competing models with similar accuracy on validation dataset. For this purpose, we add 10 artificially “noisy” features. Specifically, on training and validation sets (80/20 split of the original training data), each artificial feature appears in 10% of the examples in one class, and 20% of the other, while on the test instances, each artificial feature appears in 10% of the examples in each class. This recreates the situation where the models use not only features that are informative in the real world, but also ones that introduce spurious correlations. We create pairs of competing classifiers by repeatedly training pairs of random forests with 30 trees until their validation accuracy is within 0.1% of each other, but their test accuracy differs by at least 5%. Thus, it is not possible to identify the better classifier (the one with higher test accuracy) from the accuracy on the validation data. <br />
<br />
The goal of this experiment is to evaluate whether a user can identify the better classifier based on the explanations of $B$ instances from the validation set. The simulated human marks the set of artificial features that appear in the $B$ explanations as untrustworthy, following which we evaluate how many total predictions in the validation set should be trusted (as in the previous section, treating only marked features as untrustworthy). Then, we select the classifier with fewer untrustworthy predictions, and compare this choice to the classifier with higher held-out test set accuracy. <br />
<br />
We present the accuracy of picking the correct classifier as $B$ varies, averaged over 800 runs, in Figure 8. We omit SP-parzen and RP-parzen from the figure since they did not produce useful explanations, performing only slightly better than random. LIME is consistently better than greedy, irrespective of the pick method. Further, combining sub-modular pick with LIME outperforms all other methods, in particular it is much better than RP-LIME when only a few examples are shown to the users. These results demonstrate that the trust assessments provided by SP-selected LIME explanations are good indicators of generalization."""<br />
<br />
==Conclusion and Future Work==<br />
<br />
This paper successfully argues for importance of explaining the predictions to make the model more trustworthy to the user. It proposes LIME, a comprehensive approach to faithfully explain predictions of any model in the simplest way desired. Detailed experiments demonstrating how explanations helped users pick models or discard models are provided. As a future consideration, authors want to repeat the experiments with decision trees being the interpretable model instead of sparse linear models. Authors also want to come up with an approach to perform the pick step for images. They also want to look into an automated way of selecting hyper-parameters used in LIME and SP-LIME such as K, N etc. and finally consider running the code on GPU to create explanations in realtime.<br />
<br />
==Remarks and Critique==<br />
<br />
This paper is by far the best paper I have read on model interpretability. The paper is full of new ideas and the authors clearly explain the approaches used with reasons as to why they picked it and also highlight both weaknesses and strong points of the approaches. Even though some of the tricks used are a little ambiguous on paper such as K-Lasso, super-pixels, but going through the code on GitHub, it becomes more clear. The code is also very well documented and easy to install and run to reproduce the results published. I tried running it on CIFAR-10 dataset and found it to be useful in understanding my model better.<br />
<br />
There are few minor details that are unanswered in the paper.<br />
<br />
* To interpret prediction for an image classifier, authors use super-pixels (segments) as features, but they don't give any reason as to why they picked this approach. One of the reasons I can think of for this to make sense is that convolutional neural networks learn shapes from layer to layer that become more complete moving from top to lower layers,. So in a way you can say, the classifier classifies on the basis of set of groups of pixels close to each other (as in the case of segments) than just the pixels that are spread apart far away from each other.<br />
<br />
* The authors don't say how locally faithful the interpretable model should be to the classifier with constraints on maximum number of features to be used, i.e, what is the sort of error (mean squared error in case of linear interpretable models) we are content with?<br />
<br />
* It seems that LIME can easily extended to regression cases and is not just limited to classification tasks, but the paper doesn't discuss anything in this regard.<br />
<br />
This paper is also comparable to the DeepLIFT method, which provides a global numerical recovery of sample manifold. Both methods rely heavily on the reference point chosen. LIME has the advantage of matching the behaviour of the model with interpretable explanations, while DeepLIFT focuses on the model structure and yields overall assessment of the model by manually feeding several typical reference inputs into the method.<br />
<br />
==References==<br />
[1] Marco Tulio Ribeiro, Sameer Singh, Carlos Guestrin, Model-Agnostic Interpretability of Machine Learning, presented at 2016 ICML Workshop on Human Interpretability in Machine Learning (WHI 2016), New York, NY<br />
<br />
[2] Marco Tulio Ribeiro, Sameer Singh, Carlos Guestrin, “Why Should I Trust You?” Explaining the Predictions of Any Classifier, KDD 2016 San Francisco, CA, USA<br />
<br />
[3] F. Wang and C. Rudin. Falling rule lists. In Artificial Intelligence and Statistics (AISTATS), 2015.<br />
<br />
[4] B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani. Least angle regression. Annals of Statistics, 32:407–499, 2004.<br />
<br />
[5] http://scikit-image.org/docs/dev/api/skimage.segmentation.html#skimage.segmentation.quickshift<br />
<br />
[6] http://scikit-learn.org/stable/modules/generated/sklearn.linear_model.lars_path.html<br />
<br />
[7] C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich. Going deeper with convolutions. In Computer Vision and Pattern Recognition (CVPR), 2015.<br />
<br />
[8] A. Krause and D. Golovin. Submodular function maximization. In Tractability: Practical Approaches to Hard Problems. Cambridge University Press, February 2014.<br />
<br />
[9] J. Blitzer, M. Dredze, and F. Pereira. Biographies, bollywood, boom-boxes and blenders: Domain adaptation for sentiment classification. In Association for Computational Linguistics (ACL), 2007.<br />
<br />
[10] D. Baehrens, T. Schroeter, S. Harmeling, M. Kawanabe, K. Hansen, and K.-R. Muller. How to explain individual classification decisions. Journal of Machine Learning Research, 11, 2010.<br />
<br />
[11] https://en.wikipedia.org/wiki/Precision_and_recall</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Teaching_Machines_to_Describe_Images_via_Natural_Language_Feedback&diff=30125STAT946F17/ Teaching Machines to Describe Images via Natural Language Feedback2017-11-13T15:49:40Z<p>H4lyu: /* Related Works */</p>
<hr />
<div>= Introduction = <br />
In the era of Artificial Intelligence, one should ideally be able to educate the robot about its mistakes,<br />
possibly without needing to dig into the underlying software. Reinforcement learning has become a standard way of training artificial agents that interact with an environment. Several works explored the idea of incorporating humans in the learning process, in order to help the reinforcement learning agent to learn faster. In most cases, the guidance comes in the form of a simple numerical (or “good”/“bad”) reward. In this work, natural language is used as a way to guide an RL agent. The author argues that a sentence provides a much stronger learning signal than a numeric reward in that we can easily point to where the mistakes occur and suggest how to correct them. <br />
<br />
Here the goal is to allow a non-expert human teacher to give feedback to an RL agent in the form of natural language, just as one would to a learning child. The author has focused on the problem of image captioning in which the quality of the output can easily be judged by non-experts.<br />
<br />
= Related Works =<br />
Several works incorporate human feedback to help an RL agent learn faster.<br />
#Thomaz et al. (2006) exploits humans in the loop to teach an agent to cook in a virtual kitchen. The users watch the agent learn and may intervene at any time to give a scalar reward. Reward shaping (Ng et al., 1999) is used to incorporate this information in the Markov Decision Process (MDP).<br />
#Judah et al. (2010) iterates between “practice”, during which the agent interacts with the real environment, and a critique session where a human labels any subset of the chosen actions as good or bad.<br />
#Griffith et al. (2013) proposes policy shaping which incorporates right/wrong feedback by utilizing it as direct policy labels. <br />
<br />
Above approaches mostly assume that humans provide a numeric reward. A few attempts have been made to advise an RL agent using language.<br />
# Maclin et al. (1994) translated advice to a short program which was then implemented as a neural network. The units in this network represent Boolean concepts, which recognize whether the observed state satisfies the constraints given by the program. In such a case, the advice network will encourage the policy to take the suggested action.<br />
# Weston et al. (2016) incorporates human feedback to improve a text-based question answering agent.<br />
# Kaplan et al. (2017) exploits textual advice to improve training time of the A3C algorithm in playing an Atari game.<br />
<br />
The authors propose Phrase-based Image Captioning Model which is similar to most image captioning models except that it exploits attention and linguistic information. Several recent approaches trained the captioning model with policy gradients in order to directly optimize for the<br />
desired performance metrics. This work follows the same line. <br />
<br />
There is also similar efforts on dialogue based visual representation learning and conversation modeling. These models aim to mimic human-to-human conversations while in this work the human converses with and guides an artificial learning agent.<br />
<br />
= Methodology =<br />
The framework consists of a new phrase-based captioning model trained with Policy Gradients that incorporates natural language feedback provided by a human teacher. The phrase-based captioning model allows natural guidance by a non-expert.<br />
=== Phrase-based Image Captioning ===<br />
The captioning model uses a hierarchical recurrent neural network (RNN). The model is composed of a two-level LSTM, a phrase RNN at the top level, and a word RNN that generates a sequence of words for each phrase. One can think of the phrase RNN as providing a “topic” at each time step, which instructs the word RNN what to talk about. The structure of the model is explained through the following figure.<br />
[[File:modelham.png|center|500px]]<br />
<br />
<br />
A convolutional neural network is used in order to extract a set of feature vectors $a = (a_1, \dots, a_n)$, with $a_j$ a feature in location j in the input image. These feature vectors are given to the attention layer. There are also two more inputs to the attention layer, current hidden state of the phrase-RNN and output of the label unit. The label unit predicts one out of four possible phrase labels, i.e., a noun (NP), preposition (PP), verb (VP), and conjunction phrase (CP). This information could be useful for the attention layer. For example, when we have a NP the model may look at objects in the image, while for VP it may focus on more global information. Computations can be expressed with the following equations:<br />
<br />
$$<br />
\begin{align*}<br />
\small{\textrm{hidden state of the phrase-RNN at time step t}} \leftarrow h_t &= f_{phrase}(h_{t-1}, l_{t-1}, c_{t-1}, e_{t-1}) \\<br />
\small{\text{output of the label unit}} \leftarrow l_t &= softmax(f_{phrase-label}(h_t)) \\<br />
\small{\text{output of the attention layer}} \leftarrow c_t &= f_{att}(h_t, l_t, a)<br />
\end{align*}<br />
$$<br />
<br />
After deciding about phrases, the outputs of phrase-RNN go to another LSTM to produce words for each phrase. $w_{t,i}$ denotes the i-th word output of the word-RNN in the t-th phrase. There is an additional <EOP> token in word-RNN’s vocabulary, which signals the end-of-phrase. Furthermore, $h_{t,i}$ denotes the i-th hidden state of the word-RNN for the t-th phrase. <br />
$$<br />
h_{t,i} = f_{word}(h_{t,i-1}, c_t, w_{t,i}) \\<br />
w_{t,i} = f_{out}(h_{t,i}, c_t, w_{t,i-1}) \\ <br />
e_t = f_{word-phrase}(w_{t,1}, \dots ,w_{t,n})<br />
$$<br />
<br />
Note that $e_t$ encodes the generated phrase via simple mean-pooling over the words, which provides additional word-level context to the next phrase.<br />
<br />
=== Crowd-sourcing Human Feedback ===<br />
The authors have created a web interface that allows to collect feedback information. Two rounds of annotation are designed. In the first round, the annotator is shown a captioned image and is asked to assess the quality of the caption, by choosing between: perfect, acceptable, grammar mistakes only, minor or major errors. They ask the annotators to choose minor and major error if the<br />
caption contained errors in semantics. They advise them to choose minor for small errors such as wrong or missing attributes or awkward prepositions, and go with major errors whenever any object or action naming is wrong. <br />
<br />
For the next (more detailed, and thus more costly) round of annotation, They only select captions which are not marked as either perfect or acceptable in the first round. Since these captions contain errors, the new annotator is required to provide detailed feedback about the mistakes. Annotators are asked to:<br />
#Choose the type of required correction (something “ should be replaced”, “is missing”, or “should be deleted”) <br />
#Write feedback in natural language (annotators are asked to describe a single mistake at a time)<br />
#Mark the type of mistake (whether the mistake corresponds to an error in object, action, attribute, preposition, counting, or grammar)<br />
#Highlight the word/phrase that contains the mistake <br />
#Correct the chosen word/phrase<br />
#Evaluate the quality of the caption after correction (it could be bad even after one round of correction)<br />
<br />
<br />
[[File:ham1.png|660px]]<br />
[[File:crowd.png|530px]]<br />
<br />
In above figures, the plots on the left show the statics of the evaluations before and after one round of correction task, and the right figure depicts the interface and an example of caption correction. The authors acknowledge the costliness of the second round of annotation.<br />
<br />
=== Feedback Network ===<br />
<br />
The collected feedback provides strong supervisory signal which can be used in the RL framework. In particular, the authors design a neural network (feedback network or FBN) which will provide additional reward based on the feedback sentence.<br />
<br />
RL training will require us to generate samples (captions) from the model. Thus, during training, the sampled captions for each training image will differ from the reference maximum likelihood estimation (MLE) caption for which the feedback is provided. The goal of the feedback network is to read a newly sampled caption, and judge the correctness of each phrase conditioned on the feedback. This network performs the following computations:<br />
[[File:fbn.JPG|600px|right]]<br />
<br />
<br />
$$<br />
h_t^{caption} = f_{sent}(h_{t-1}^{caption}, \omega_t^c) \\<br />
h_t^{feedback} = f_{sent}(h_{t-1}^{feedback}, \omega_t^f) \\<br />
q_i = f_{phrase}(\omega_{i,1}^c, \omega_{i,2}^c, \dots, \omega_{i,N}^c) \\ <br />
o_i = f_{fbn}(h_T^{caption}, h_T^{feedback }, q_i, m) \\<br />
$$<br />
<br />
Here, $\omega_t^c$ and $\omega_t^f$ denote the one-hot encoding of words in the sampled caption and feedback sentence for the t-th phrase, respectively. FBN encodes both the caption and feedback using an LSTM ($f_{sent}$), performs mean pooling ($f_{phrase}$) over the words in a phrase to represent the phrase i with $q_i$, and passes this information through a 3-layer MLP ($f_{fbn}$). The MLP accepts additional information about the mistake type (e.g., wrong object/action) encoded<br />
as a one hot vector m.<br />
<br />
=== Policy Gradient Optimization using Natural Language Feedback ===<br />
<br />
One can think of an caption decoder as an agent following a parameterized policy $p_\theta$ that selects an action at each time step. An “action” in our case requires choosing a word from the vocabulary (for the word RNN), or a phrase label (for the phrase RNN). The objective for learning the parameters of the model is the expected reward received when completing the caption $w^s = (w^s_1, \dots ,w^s_T)$. Here, $w_t^s$ is the word sampled from the model at time step t.<br />
<br />
$$<br />
L(\theta) = -\mathop{{}\mathbb{E}}_{\omega^s \sim p_\theta}[r(w^s)] <br />
$$<br />
To optimize this objective, we need the gradient:<br />
$$<br />
\nabla_\theta L(\theta) = - \mathop{{}\mathbb{E}}_{\omega^s \sim p_\theta}[r(w^s)\nabla_\theta \log p_\theta(w^s)]<br />
$$<br />
Which is estimated using a single Monte-Carlo sample:<br />
$$<br />
\nabla_\theta L(\theta) \approx - r(w^s)\nabla_\theta \log p_\theta(w^s)<br />
$$<br />
Then a baseline $b = r(\hat \omega)$ is used. A baseline does not change the expected gradient but can drastically reduce the variance.<br />
$$<br />
\hat{\omega}_t = argmax \ p(\omega_t|h_t) \\<br />
\nabla_\theta L(\theta) \approx - (r(\omega^s) - r(\hat{\omega}))\nabla_\theta \log p_\theta(\omega^s)<br />
$$<br />
'''Reward:''' A sentence reward is defined as a weighted sum of the BLEU scores (more information about BLUE score [http://www1.cs.columbia.edu/nlp/sgd/bleu.pdf here]).<br />
<br />
$$<br />
r(\omega^s) = \beta \sum_i \lambda_i \cdot BLEU_i(\omega^s, ref)<br />
$$<br />
<br />
As reference captions to compute the reward, the authors either use the reference captions generated by a snapshot of the model which were evaluated as not having minor and major errors, or ground-truth captions. In addition, they weigh the reward by the caption quality as provided by the annotators (e.g. $\beta = 1$ for perfect and $\beta = 0.8$ for acceptable). They further incorporate the reward provided by the feedback network:<br />
$$<br />
r(\omega_t^p) = r(\omega^s) + \lambda_f f_{fbn}(\omega^s, feedback, \omega_t^p)<br />
$$<br />
Where $\omega^p_t$ denotes the sequence of words in the t-th phrase. Note that FBN produces a classification of each phrase. This can be converted into reward, by assigning<br />
correct to 1, wrong to -1, and 0 to not relevant. So the final gradient takes the following form:<br />
$$<br />
\nabla_\theta L(\theta) = - \sum_{p=1}^{P}(r(\omega^p) - r(\hat{\omega}^p))\nabla_\theta \log p_\theta(\omega^p)<br />
$$<br />
<br />
= Experimental Results =<br />
The authors used MS-COCO dataset. They used 82K images for training, 2K for validation, and 4K for testing. To collect feedback, they randomly chose 7K images from the training set, as well as all 2K images from validation. In addition, they use a word vocabulary size of 23,115.<br />
<br />
=== Phrase-based captioning model ===<br />
The authors analyze different instantiations of their phrase-based captioning in the following table. To sanity check, they compare it to a flat approach (word-RNN only). Overall, their model performs slightly worse (0.66 points). However, the main strength of their model is that it allows a more natural integration with feedback.<br />
<br />
[[File:table2.JPG|center]]<br />
<br />
=== Feedback network ===<br />
The authors use 9000 images to collect feedback; 5150 of them are evaluated as containing errors. Finally, they use 4174 images for the second round of annotation. They randomly select 9/10 of them to serve as a training set for feedback network, and 1/10 of them to be test set. The model achieves the highest accuracy of 74.66% when they provide it with the kind of mistake the reference caption had (e.g. an object, action, etc). This is not particularly surprising as it requires the most additional information to train the model and the most time to compile the dataset for.<br />
<br />
=== RL with Natural Language Feedback ===<br />
The following table report the performance for several instantiations of the RL models. All models have been pre-trained using cross-entropy loss (MLE) on the full MS-COCO training set. For the next rounds of training, all the models are trained only on the 9K images.<br />
<br />
The authors define “C” captions as all captions that were corrected by the annotators and were not evaluated as containing minor or major error, and ground-truth captions for the rest of the images. For “A”, they use all captions (including captions which were evaluated as correct) that did not have minor or major errors, and GT for the rest. A detailed break-down of these captions is reported in in the following table. The authors test their model in two separate cases:<br />
<br />
*They first test a model using the standard cross-entropy loss, but which now also has access to the corrected captions in addition to the 5GT captions. This model (MLEC) is able to improve over the original MLE model by 1.4 points. They then test the RL model by optimizing the metric wrt the 5GT captions. This brings an additional point, achieving 2.4 over the MLE model. Next, the RL agent is given access to 3GT captions, the “C" captions and feedback sentences. They show that this model outperforms the no-feedback baseline by 0.5 points. If the RL agent has access to 4GT captions and feedback descriptions, a total of 1.1 points over the baseline RL model and 3.5 over the MLE model will be achieved. <br />
<br />
*They also test a more realistic scenario, in which the models have access to either a single GT caption, “C" (or “A”), and feedback. This mimics a scenario in which the human teacher observes the agent and either gives feedback about the agent’s mistakes, or, if the agent’s caption is completely wrong, the teacher writes a new caption. Interestingly, RL when provided with the corrected captions performs better than when given GT captions. Overall, their model outperforms the base RL (no feedback) by 1.2 points.<br />
<br />
[[File:table3.PNG|center]]<br />
<br />
These experiments make an important point. Instead of giving the RL agent a completely new target (caption), a better strategy is to “teach” the agent about the mistakes it is doing and suggest a correction. This is not very difficult to understand intuitively - informing the agent of its error indeed conveys more information than teaching it a completely correct answer, because the latter forces the network to "train" its memory from a sample which is, at least seemingly, insulated from its prior memory.<br />
<br />
= Conclusion =<br />
In this paper, a human teacher is enabled to provide feedback to the learning agent in the form of natural language. The authors focused on the problem of image captioning. They proposed a hierarchical phrase-based RNN as the captioning model, which allowed natural integration with human feedback.<br />
They also crowd-sourced feedback, and showed how to incorporate it in policy gradient optimization.<br />
<br />
= References=<br />
[1] Huan Ling and Sanja Fidler. Teaching Machines to Describe Images via Natural Language Feedback. In arXiv:1706.00130, 2017.<br />
<br />
[2] Shane Griffith, Kaushik Subramanian, Jonathan Scholz, Charles L. Isbell, and Andrea Lockerd Thomaz. Policy shaping: Integrating human feedback with reinforcement learning. In NIPS, 2013. <br />
<br />
[3] K. Judah, S. Roy, A. Fern, and T. Dietterich. Reinforcement learning via practice and critique advice. In AAAI, 2010. <br />
<br />
[4] A. Thomaz and C. Breazeal. Reinforcement learning with human teachers: Evidence of feedback and guidance. In AAAI, 2006. <br />
<br />
[5] Richard Maclin and Jude W. Shavlik. Incorporating advice into agents that learn from reinforcements. In National Conference on Artificial Intelligence, pages 694–699, 1994. <br />
<br />
[6] Jason Weston. Dialog-based language learning. In arXiv:1604.06045, 2016. <br />
<br />
[7] Russell Kaplan, Christopher Sauer, and Alexander Sosa. Beating atari with natural language guided reinforcement learning. In arXiv:1704.05539, 2017.<br />
<br />
[8] Andrew Y. Ng, Daishi Harada, and Stuart J. Russell. Policy invariance under reward transformations: Theory and application to reward shaping. In ICML, pages 278–287, 1999.</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Teaching_Machines_to_Describe_Images_via_Natural_Language_Feedback&diff=30124STAT946F17/ Teaching Machines to Describe Images via Natural Language Feedback2017-11-13T15:48:29Z<p>H4lyu: /* RL with Natural Language Feedback */</p>
<hr />
<div>= Introduction = <br />
In the era of Artificial Intelligence, one should ideally be able to educate the robot about its mistakes,<br />
possibly without needing to dig into the underlying software. Reinforcement learning has become a standard way of training artificial agents that interact with an environment. Several works explored the idea of incorporating humans in the learning process, in order to help the reinforcement learning agent to learn faster. In most cases, the guidance comes in the form of a simple numerical (or “good”/“bad”) reward. In this work, natural language is used as a way to guide an RL agent. The author argues that a sentence provides a much stronger learning signal than a numeric reward in that we can easily point to where the mistakes occur and suggest how to correct them. <br />
<br />
Here the goal is to allow a non-expert human teacher to give feedback to an RL agent in the form of natural language, just as one would to a learning child. The author has focused on the problem of image captioning in which the quality of the output can easily be judged by non-experts.<br />
<br />
= Related Works =<br />
Several works incorporate human feedback to help an RL agent learn faster.<br />
#Thomaz et al. [2006] exploits humans in the loop to teach an agent to cook in a virtual kitchen. The users watch the agent learn and may intervene at any time to give a scalar reward. Reward shaping (Ng et al. [1999]) is used to incorporate this information in the Markov Decision Process (MDP).<br />
#Judah et al. [2010] iterates between “practice”, during which the agent interacts with the real environment, and a critique session where a human labels any subset of the chosen actions as good or bad.<br />
#Griffith et al. [2013] proposes policy shaping which incorporates right/wrong feedback by utilizing it as direct policy labels. <br />
<br />
Above approaches mostly assume that humans provide a numeric reward. A few attempts have been made to advise an RL agent using language.<br />
# Maclin et al. [1994] translated advice to a short program which was then implemented as a neural network. The units in this network represent Boolean concepts, which recognize whether the observed state satisfies the constraints given by the program. In such a case, the advice network will encourage the policy to take the suggested action.<br />
# Weston et al. [2016] incorporates human feedback to improve a text-based question answering agent.<br />
# Kaplan et al. [2017] exploits textual advice to improve training time of the A3C algorithm in playing an Atari game.<br />
<br />
The authors propose Phrase-based Image Captioning Model which is similar to most image captioning models except that it exploits attention and linguistic information. Several recent approaches trained the captioning model with policy gradients in order to directly optimize for the<br />
desired performance metrics. This work follows the same line. <br />
<br />
There is also similar efforts on dialogue based visual representation learning and conversation modeling. These models aim to mimic human-to-human conversations while in this work the human converses with and guides an artificial learning agent.<br />
<br />
= Methodology =<br />
The framework consists of a new phrase-based captioning model trained with Policy Gradients that incorporates natural language feedback provided by a human teacher. The phrase-based captioning model allows natural guidance by a non-expert.<br />
=== Phrase-based Image Captioning ===<br />
The captioning model uses a hierarchical recurrent neural network (RNN). The model is composed of a two-level LSTM, a phrase RNN at the top level, and a word RNN that generates a sequence of words for each phrase. One can think of the phrase RNN as providing a “topic” at each time step, which instructs the word RNN what to talk about. The structure of the model is explained through the following figure.<br />
[[File:modelham.png|center|500px]]<br />
<br />
<br />
A convolutional neural network is used in order to extract a set of feature vectors $a = (a_1, \dots, a_n)$, with $a_j$ a feature in location j in the input image. These feature vectors are given to the attention layer. There are also two more inputs to the attention layer, current hidden state of the phrase-RNN and output of the label unit. The label unit predicts one out of four possible phrase labels, i.e., a noun (NP), preposition (PP), verb (VP), and conjunction phrase (CP). This information could be useful for the attention layer. For example, when we have a NP the model may look at objects in the image, while for VP it may focus on more global information. Computations can be expressed with the following equations:<br />
<br />
$$<br />
\begin{align*}<br />
\small{\textrm{hidden state of the phrase-RNN at time step t}} \leftarrow h_t &= f_{phrase}(h_{t-1}, l_{t-1}, c_{t-1}, e_{t-1}) \\<br />
\small{\text{output of the label unit}} \leftarrow l_t &= softmax(f_{phrase-label}(h_t)) \\<br />
\small{\text{output of the attention layer}} \leftarrow c_t &= f_{att}(h_t, l_t, a)<br />
\end{align*}<br />
$$<br />
<br />
After deciding about phrases, the outputs of phrase-RNN go to another LSTM to produce words for each phrase. $w_{t,i}$ denotes the i-th word output of the word-RNN in the t-th phrase. There is an additional <EOP> token in word-RNN’s vocabulary, which signals the end-of-phrase. Furthermore, $h_{t,i}$ denotes the i-th hidden state of the word-RNN for the t-th phrase. <br />
$$<br />
h_{t,i} = f_{word}(h_{t,i-1}, c_t, w_{t,i}) \\<br />
w_{t,i} = f_{out}(h_{t,i}, c_t, w_{t,i-1}) \\ <br />
e_t = f_{word-phrase}(w_{t,1}, \dots ,w_{t,n})<br />
$$<br />
<br />
Note that $e_t$ encodes the generated phrase via simple mean-pooling over the words, which provides additional word-level context to the next phrase.<br />
<br />
=== Crowd-sourcing Human Feedback ===<br />
The authors have created a web interface that allows to collect feedback information. Two rounds of annotation are designed. In the first round, the annotator is shown a captioned image and is asked to assess the quality of the caption, by choosing between: perfect, acceptable, grammar mistakes only, minor or major errors. They ask the annotators to choose minor and major error if the<br />
caption contained errors in semantics. They advise them to choose minor for small errors such as wrong or missing attributes or awkward prepositions, and go with major errors whenever any object or action naming is wrong. <br />
<br />
For the next (more detailed, and thus more costly) round of annotation, They only select captions which are not marked as either perfect or acceptable in the first round. Since these captions contain errors, the new annotator is required to provide detailed feedback about the mistakes. Annotators are asked to:<br />
#Choose the type of required correction (something “ should be replaced”, “is missing”, or “should be deleted”) <br />
#Write feedback in natural language (annotators are asked to describe a single mistake at a time)<br />
#Mark the type of mistake (whether the mistake corresponds to an error in object, action, attribute, preposition, counting, or grammar)<br />
#Highlight the word/phrase that contains the mistake <br />
#Correct the chosen word/phrase<br />
#Evaluate the quality of the caption after correction (it could be bad even after one round of correction)<br />
<br />
<br />
[[File:ham1.png|660px]]<br />
[[File:crowd.png|530px]]<br />
<br />
In above figures, the plots on the left show the statics of the evaluations before and after one round of correction task, and the right figure depicts the interface and an example of caption correction. The authors acknowledge the costliness of the second round of annotation.<br />
<br />
=== Feedback Network ===<br />
<br />
The collected feedback provides strong supervisory signal which can be used in the RL framework. In particular, the authors design a neural network (feedback network or FBN) which will provide additional reward based on the feedback sentence.<br />
<br />
RL training will require us to generate samples (captions) from the model. Thus, during training, the sampled captions for each training image will differ from the reference maximum likelihood estimation (MLE) caption for which the feedback is provided. The goal of the feedback network is to read a newly sampled caption, and judge the correctness of each phrase conditioned on the feedback. This network performs the following computations:<br />
[[File:fbn.JPG|600px|right]]<br />
<br />
<br />
$$<br />
h_t^{caption} = f_{sent}(h_{t-1}^{caption}, \omega_t^c) \\<br />
h_t^{feedback} = f_{sent}(h_{t-1}^{feedback}, \omega_t^f) \\<br />
q_i = f_{phrase}(\omega_{i,1}^c, \omega_{i,2}^c, \dots, \omega_{i,N}^c) \\ <br />
o_i = f_{fbn}(h_T^{caption}, h_T^{feedback }, q_i, m) \\<br />
$$<br />
<br />
Here, $\omega_t^c$ and $\omega_t^f$ denote the one-hot encoding of words in the sampled caption and feedback sentence for the t-th phrase, respectively. FBN encodes both the caption and feedback using an LSTM ($f_{sent}$), performs mean pooling ($f_{phrase}$) over the words in a phrase to represent the phrase i with $q_i$, and passes this information through a 3-layer MLP ($f_{fbn}$). The MLP accepts additional information about the mistake type (e.g., wrong object/action) encoded<br />
as a one hot vector m.<br />
<br />
=== Policy Gradient Optimization using Natural Language Feedback ===<br />
<br />
One can think of an caption decoder as an agent following a parameterized policy $p_\theta$ that selects an action at each time step. An “action” in our case requires choosing a word from the vocabulary (for the word RNN), or a phrase label (for the phrase RNN). The objective for learning the parameters of the model is the expected reward received when completing the caption $w^s = (w^s_1, \dots ,w^s_T)$. Here, $w_t^s$ is the word sampled from the model at time step t.<br />
<br />
$$<br />
L(\theta) = -\mathop{{}\mathbb{E}}_{\omega^s \sim p_\theta}[r(w^s)] <br />
$$<br />
To optimize this objective, we need the gradient:<br />
$$<br />
\nabla_\theta L(\theta) = - \mathop{{}\mathbb{E}}_{\omega^s \sim p_\theta}[r(w^s)\nabla_\theta \log p_\theta(w^s)]<br />
$$<br />
Which is estimated using a single Monte-Carlo sample:<br />
$$<br />
\nabla_\theta L(\theta) \approx - r(w^s)\nabla_\theta \log p_\theta(w^s)<br />
$$<br />
Then a baseline $b = r(\hat \omega)$ is used. A baseline does not change the expected gradient but can drastically reduce the variance.<br />
$$<br />
\hat{\omega}_t = argmax \ p(\omega_t|h_t) \\<br />
\nabla_\theta L(\theta) \approx - (r(\omega^s) - r(\hat{\omega}))\nabla_\theta \log p_\theta(\omega^s)<br />
$$<br />
'''Reward:''' A sentence reward is defined as a weighted sum of the BLEU scores (more information about BLUE score [http://www1.cs.columbia.edu/nlp/sgd/bleu.pdf here]).<br />
<br />
$$<br />
r(\omega^s) = \beta \sum_i \lambda_i \cdot BLEU_i(\omega^s, ref)<br />
$$<br />
<br />
As reference captions to compute the reward, the authors either use the reference captions generated by a snapshot of the model which were evaluated as not having minor and major errors, or ground-truth captions. In addition, they weigh the reward by the caption quality as provided by the annotators (e.g. $\beta = 1$ for perfect and $\beta = 0.8$ for acceptable). They further incorporate the reward provided by the feedback network:<br />
$$<br />
r(\omega_t^p) = r(\omega^s) + \lambda_f f_{fbn}(\omega^s, feedback, \omega_t^p)<br />
$$<br />
Where $\omega^p_t$ denotes the sequence of words in the t-th phrase. Note that FBN produces a classification of each phrase. This can be converted into reward, by assigning<br />
correct to 1, wrong to -1, and 0 to not relevant. So the final gradient takes the following form:<br />
$$<br />
\nabla_\theta L(\theta) = - \sum_{p=1}^{P}(r(\omega^p) - r(\hat{\omega}^p))\nabla_\theta \log p_\theta(\omega^p)<br />
$$<br />
<br />
= Experimental Results =<br />
The authors used MS-COCO dataset. They used 82K images for training, 2K for validation, and 4K for testing. To collect feedback, they randomly chose 7K images from the training set, as well as all 2K images from validation. In addition, they use a word vocabulary size of 23,115.<br />
<br />
=== Phrase-based captioning model ===<br />
The authors analyze different instantiations of their phrase-based captioning in the following table. To sanity check, they compare it to a flat approach (word-RNN only). Overall, their model performs slightly worse (0.66 points). However, the main strength of their model is that it allows a more natural integration with feedback.<br />
<br />
[[File:table2.JPG|center]]<br />
<br />
=== Feedback network ===<br />
The authors use 9000 images to collect feedback; 5150 of them are evaluated as containing errors. Finally, they use 4174 images for the second round of annotation. They randomly select 9/10 of them to serve as a training set for feedback network, and 1/10 of them to be test set. The model achieves the highest accuracy of 74.66% when they provide it with the kind of mistake the reference caption had (e.g. an object, action, etc). This is not particularly surprising as it requires the most additional information to train the model and the most time to compile the dataset for.<br />
<br />
=== RL with Natural Language Feedback ===<br />
The following table report the performance for several instantiations of the RL models. All models have been pre-trained using cross-entropy loss (MLE) on the full MS-COCO training set. For the next rounds of training, all the models are trained only on the 9K images.<br />
<br />
The authors define “C” captions as all captions that were corrected by the annotators and were not evaluated as containing minor or major error, and ground-truth captions for the rest of the images. For “A”, they use all captions (including captions which were evaluated as correct) that did not have minor or major errors, and GT for the rest. A detailed break-down of these captions is reported in in the following table. The authors test their model in two separate cases:<br />
<br />
*They first test a model using the standard cross-entropy loss, but which now also has access to the corrected captions in addition to the 5GT captions. This model (MLEC) is able to improve over the original MLE model by 1.4 points. They then test the RL model by optimizing the metric wrt the 5GT captions. This brings an additional point, achieving 2.4 over the MLE model. Next, the RL agent is given access to 3GT captions, the “C" captions and feedback sentences. They show that this model outperforms the no-feedback baseline by 0.5 points. If the RL agent has access to 4GT captions and feedback descriptions, a total of 1.1 points over the baseline RL model and 3.5 over the MLE model will be achieved. <br />
<br />
*They also test a more realistic scenario, in which the models have access to either a single GT caption, “C" (or “A”), and feedback. This mimics a scenario in which the human teacher observes the agent and either gives feedback about the agent’s mistakes, or, if the agent’s caption is completely wrong, the teacher writes a new caption. Interestingly, RL when provided with the corrected captions performs better than when given GT captions. Overall, their model outperforms the base RL (no feedback) by 1.2 points.<br />
<br />
[[File:table3.PNG|center]]<br />
<br />
These experiments make an important point. Instead of giving the RL agent a completely new target (caption), a better strategy is to “teach” the agent about the mistakes it is doing and suggest a correction. This is not very difficult to understand intuitively - informing the agent of its error indeed conveys more information than teaching it a completely correct answer, because the latter forces the network to "train" its memory from a sample which is, at least seemingly, insulated from its prior memory.<br />
<br />
= Conclusion =<br />
In this paper, a human teacher is enabled to provide feedback to the learning agent in the form of natural language. The authors focused on the problem of image captioning. They proposed a hierarchical phrase-based RNN as the captioning model, which allowed natural integration with human feedback.<br />
They also crowd-sourced feedback, and showed how to incorporate it in policy gradient optimization.<br />
<br />
= References=<br />
[1] Huan Ling and Sanja Fidler. Teaching Machines to Describe Images via Natural Language Feedback. In arXiv:1706.00130, 2017.<br />
<br />
[2] Shane Griffith, Kaushik Subramanian, Jonathan Scholz, Charles L. Isbell, and Andrea Lockerd Thomaz. Policy shaping: Integrating human feedback with reinforcement learning. In NIPS, 2013. <br />
<br />
[3] K. Judah, S. Roy, A. Fern, and T. Dietterich. Reinforcement learning via practice and critique advice. In AAAI, 2010. <br />
<br />
[4] A. Thomaz and C. Breazeal. Reinforcement learning with human teachers: Evidence of feedback and guidance. In AAAI, 2006. <br />
<br />
[5] Richard Maclin and Jude W. Shavlik. Incorporating advice into agents that learn from reinforcements. In National Conference on Artificial Intelligence, pages 694–699, 1994. <br />
<br />
[6] Jason Weston. Dialog-based language learning. In arXiv:1604.06045, 2016. <br />
<br />
[7] Russell Kaplan, Christopher Sauer, and Alexander Sosa. Beating atari with natural language guided reinforcement learning. In arXiv:1704.05539, 2017.<br />
<br />
[8] Andrew Y. Ng, Daishi Harada, and Stuart J. Russell. Policy invariance under reward transformations: Theory and application to reward shaping. In ICML, pages 278–287, 1999.</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Imagination-Augmented_Agents_for_Deep_Reinforcement_Learning&diff=30022Imagination-Augmented Agents for Deep Reinforcement Learning2017-11-12T23:21:49Z<p>H4lyu: /* Sokoban */</p>
<hr />
<div>=Introduction=<br />
An interesting research area in reinforcement learning is developing intelligent agents for playing video games. Before the introduction of deep learning, video game agents were commonly coded based on Monte-Carlo Tree Search of pre-set rules. Recent research has shown deep reinforcement learning to be very successful at playing video games like Atari 2600. To be specific, the method (Figure 1) is called Deep Q-Learning (DQN) which learns the optimal actions based on current observations (raw pixels) [[#Reference|[Mnih et al., (2015)]]]. However, there are some complex games where DQN fails to learn: some games need to solve a sub-problem without explicit reward or contain irreversible domains, where actions can be catastrophic. A typical example of these games is [https://en.wikipedia.org/wiki/Sokoban Sokoban]. Even as humans are playing the game, planning and inference are needed. This kind of game raises challenges to RL.<br />
<br />
[[File:DQN.png|800px|center|thumb|Figure 1: Deep Q-Learning Architecture]]<br />
<br />
In Reinforcement Learning, the algorithms can be divided into two categories: '''model-free''' algorithm and '''model-based''' algorithm. The model-based reinforcement learning tries to infer environment to gain the reward while model-free reinforcement learning does not use the environment to learn the action that results in the best reward. DQN, mentioned above(Figure 1), is a model-free method. It takes raw pixels as input and maps them to values or actions. As a drawback, large amounts of training data is required. In addition, the policies are not generalized to new tasks in the same environment. A model-based method is trying to build a model for the environment. By querying the model, agents can avoid irreversible, poor decisions. As an approximation of the environment, it can enable better generalization across states. However, this method only shows success in limited settings, where an exact transition model is given or in simple domains. In complex environments, model-based methods suffer from model errors from function approximation. These errors compound during planning, causing poor agent performance. Currently, there is no model-based method that is robust against imperfections.<br />
<br />
In this paper, the authors introduce a novel deep reinforcement learning architecture called Imagination-Augmented Agents(I2As). Literally, this method enables agents to learn to interpret predictions from a learned environment model to construct implicit plans. It is a combination of model-free and model-based aspects. The advantage of this method is that it learns in an end-to-end way to extract information from model simulations without making any assumptions about the structure or the perfections of the environment model.<br />
As shown in the results, this method outperforms DQN in the games: Sokoban, and MniniPacman. In addition, the experiments all show that I2A is able to successfully use imperfect models.<br />
<br />
=Motivation=<br />
Although the structure of this method is complex, the motivation is intuitive: since the agent suffers from irreversible decisions, attempts in simulated states may be helpful.<br />
To improve the expensive search space in traditional MCTS methods, adding decision from policy network can reduce search steps. In order to keep context information, rollout results are encoded by an LSTM encoder. The final output is combining the result from model-free network and model-based network.<br />
<br />
=Related Work=<br />
There are some works that try to apply deep learning to model-based reinforcement learning. The popular approach is to learn a neural network from the environment and apply the network in classical planning algorithms. These works can not handle the mismatch between the learned model and the ground truth. Liu et al.(2017) use context information from trajectories, but in terms of imitation learning.<br />
<br />
To deal with imperfect models, Deisenroth and Rasmussen(2011) try to capture model uncertainty by applying high-computational Gaussian Process models. In order to develop such a policy search method, the authors of this paper used analytic gradients of an approximation to the expected return for indirect policy search. This means by learning a probabilistic dynamics model and explicitly incorporating model uncertainty into long-term planning, this policy search method can cope with very little data and facilitates learning from scratch in only a few trials. <br />
<br />
Similar ideas can be found in a study by Hamrick et al.(2017): they present a neural network that queries expert models, but just focus on meta-control for continuous contextual bandit problems. Pascanu et al.(2017) extend this work by focusing on explicit planning in sequential environments.<br />
<br />
=Approach=<br />
The summary of the architecture of I2A can be seen in Figure 2.<br />
[[File:i2a.png|800px|center|thumb|Figure 2: The Architecture of I2A]]<br />
The observation $O_t$ (Figure 2 right) is fed into two paths, the model-free path is just common DQN which predicts the best action given $O_t$, whereas the model-based path performs a rollout strategy, the aggregator combines the $n$ rollout encoded outputs($n$ equals to the number of actions in the action space), and forwards the results to next layer. Together they are used to generate a policy function $\pi$ to output an action. In each rollout operation, the imagination core is used to predict the future state and reward.<br />
<br />
===Imagination Core===<br />
The imagination core(Figure 2 left) is the key role in the model-based path. It consists of two parts: environment model and rollout policy. The former is an approximation of the environment and the latter is used to simulate imagined trajectories, which are interpreted by a neural network and provided as additional context to a policy network.<br />
<br />
====environment model====<br />
In order to augment agents with imagination, the method relies on environment models that, given current information, can be queried to make predictions about the future. In this work, the environment model is built based on action-conditional next-step predictors, which receive input contains current observation and current action, and predict the next observation and the next reward(Figure 3).<br />
[[File:environment model.png|800px|center|thumb|Figure 3: Environment Model]]<br />
<br />
The authors can either pretrain the environment model before embedding it (with frozen weights) within the I2A architecture, or jointly train it with the agent by adding $l_{model}$ to the total loss as an auxiliary loss. In practice, they found that pre-training the environment model led to faster<br />
runtime of the I2A architecture, so they adopted this strategy.<br />
<br />
====rollout policy====<br />
The rollout process is regarded as the simulated trajectories. In this work, the rollout is performed for each possible action in the environment. <br />
<br />
A rollout policy $\hat \pi$ is a function that takes current observation $O$ and outputs an action $a$ that potentially leads to maximal reward. In this architecture, the rollout policy can be a DQN network. In the experiment, the rollout pocily$\hat \pi$ is broadcasted and shared. After experiments on the types of rollout policies(random, pre-trained), the authors found the efficient strategy is to distill the policy into a model-free policy, which consists in creating a smaill model-free network $\hat \pi(O_t)$, and adding to the total loss a cross entropy auxiliary loss between the imagination-augmented policy $\pi(O_t)$ as computed on the current observation, and the policy $\hat \pi(O_t)$ as computed on the same observation.<br />
<br />
$$<br />
l_{dist} (\pi, \hat \pi)(O_t) = \lambda_{dist} \sum_a \pi(a|O_t)log(\hat \pi(a|O_t))<br />
$$<br />
<br />
Together as the imagination core, these two parts produces $n$ trajectories $\hat \tau_1,...,\hat \tau_n$. Each imagined trajectory $\hat \tau$ is a sequence of features $(\hat f_{t+1},...,\hat f_{t+\tau})$, where $t$ is the current time, $\tau$ the length of rollout, and $\hat f_{t+i}$ the output of the environment model(the predicted observation and reward). In order to guarantee success in imperfections, the architecture does not assume the learned model to be perfect. The output will not only depend on the predicted reward.<br />
<br />
===Trajectories Encoder===<br />
From the intuition to keep the sequence information in the trajectories, the architecture uses a rollout encoder $\varepsilon$ that processes the imagined rollout as a whole and learns to interpret it(Figure 2 middle). Each trajectory is encoded as a rollout embedding $e_i=\varepsilon(\hat \tau_i)$. Then, the aggregator $A$ combines the rollout embedding s into a single imagination code $c_{ia}=A(e_1,...,e_n)$ by simply concatenating all the summaries.<br />
In the experiments, the encoder is an LSTM that takes the predicted output from environment model as the input. One observation is that the order of the sequence $\hat f_{t+1}$ to $\hat f_{t+\tau}$ makes relatively little impact on the performance. The encodes mimics the Bellman type backup operations in DQN.<br />
<br />
===Model-Free Path===<br />
The model-free path contains a network that only takes the current observation as input that generates the potential optimal action. This network can be same as the one in imagination core.<br />
<br />
<br />
In conclusion, the I2A learns to combine information for two paths, and without the model-based path, I2A simply reduce to a standard model-free network(such as A3C, more explanations [https://medium.com/emergent-future/simple-reinforcement-learning-with-tensorflow-part-8-asynchronous-actor-critic-agents-a3c-c88f72a5e9f2 here]). The imperfect approximation results in a rollout policy with higher entropy, potentially striking a balance between exploration and exploitation.<br />
<br />
=Experiment=<br />
These following experiments were tested in Sokoban and MiniPacman games. All results are averages taken from top three agents. These agents were trained over 32 to 64 workers, and the network was optimized by RMSprop.<br />
As the pre-training strategy, the training data of I2A was pre-generated from trajectories of a partially trained standard model-free agent, the data is also taken into account for the budget. The total number of frames that were needed in pre-training is counted in the later process. Meanwhile, the authors show that the environment model can be reused to solve multiple tasks in the same environment.<br />
<br />
In the game Sokoban, the environment is a 10 x 10 grid world. All agents were trained directly on raw pixels(image size 80 x 80 with 3 channels). To make sure the network is not just simply "memorize" all states, the game procedurally generates a new level each episode. Out of 40 million levels generated, less than 0.7% were repeated. Therefore, a good agent should solve the unseen level as well.<br />
<br />
The reward settings for reinforcement learning algorithms are as follows:<br />
* Every time step, a penalty of -0.1 is applied to the agent.(encourage agents to finish levels faster)<br />
* Whenever the agent pushes a box on target, it receives a reward of +1.(encourage agents to push boxes onto targets)<br />
* Whenever the agent pushes a box off target, it receives a penalty of -1.(avoid artificial reward loop that would be induced by repeatedly pushing a box off and on target)<br />
* Finishing the level gives the agent a reward of +10 and the level terminates.(strongly reward solving a level)<br />
<br />
To show the advantage of I2A, the authors set a model-free standard architecture as a baseline. The architecture is a multi-layer convolutional neural network (CNN), taking the current observation $O_t$ as input, followed by a fully connected (FC) hidden layer. This FC layer feeds into two heads: into an FC layer with one output per action computing the policy logits $\log \pi(a_t|O_t, \theta)$; and into another FC layer with a single output that computes the value function $V(O_t; \theta_v)$.<br />
* for MiniPacman: the CNN has two layers, both with 3x3 kernels, 16 output channels and strides 1 and 2; the following FC layer has 256 units<br />
* for Sokoban: the CNN has three layers with kernel sizes 8x8, 4x4, 3x3, strides of 4, 2, 1 and number of output channels 32, 64, 64; the following FC has 512 units<br />
<br />
===Sokoban===<br />
<br />
The environment model for Sokoban is shown in figure 4<br />
[[File:sokoban_em.png|400px|center|thumb|Figure 4: The Sokoban environment model]]<br />
<br />
Besides, to demonstrate the influence of larger architecture in I2A, the authors set a copy-model agent that uses the same architecture of I2A but the environment model is replaced by identical map. This agent is regarded as an I2A agent without imagination.<br />
<br />
[[File:sokoban_result.png|800px|center|thumb|Figure 5: Sokoban learning curves. Left: training curves of I2A and baselines. Right: I2A training curves for various values of imagination depth]]<br />
The results are shown in Figure 4(left). I2A agents can solve much more levels compared to common DQN. Also, it far outperforms the copy-model version, suggesting that the environment model is crucial. The authors also trained an I2A where the environment model was predicting no rewards, only observations. This also performed worse. However, after much longer training (3e9 steps), these agents did recover the performance of the original I2A, which was never the case for the baseline agent even with that many steps. Hence, reward prediction is very helpful but not absolutely necessary in this task, and imagined observations alone are informative enough to obtain high performance on Sokoban. Note this is in contrast to many classical planning and model-based reinforcement learning methods, which often rely on reward prediction.<br />
<br />
====Length of Rollout====<br />
A further experiment was investigating how the length of individual rollouts affects performance. The authors performed a parameter searching. Figure 5(right) shows the influence of the rollout length. The strategy using 3 rollout steps improves the speed of learning and improves the performance significantly than 1 step, and 5 is the optimal number. This implies rollout can be very helpful and informative. This rollout enables the agent to learn moves it cannot recover from.<br />
<br />
[[File:sokoban_noisy.png|800px|center|thumb|Figure 6: Experiments with a noisy environment model Left: each row shows an example 5-step rollout after conditioning on an environment observation. Errors accumulate and lead to various artifacts, including missing or duplicate sprites. Right: comparison of Monte-Carlo (MC) search and I2A when using either the accurate or the noisy model for rollouts.]]<br />
<br />
====Imperfections====<br />
To demonstrate I2A can handle less reliable predictions, the authors set experiment where the I2A used a poor environment model(smaller number of parameters), where the error may accumulate across the rollout(Figure 6 left). As shown in figure 6(right), even with relatively poor environment model, the performance of I2A is stable, unlike traditional Monte-Carlo search, which explicitly estimates the value of each action from rollouts, rather than learning an arbitrary encoding of the rollouts. An interesting result is that a rollout length 5 no longer outperforms a length of 3, which matches our common sense. The authors suggest that it is learning a rollout encoder that enables I2As to deal with imperfect model predictions.<br />
<br />
====Perfections====<br />
As I2A shows the robustness towards environment models, the authors tested an I2A agent with a nearly perfect environment model, and the results are in Table 1 and Table 2. Traditional Mento-Carlo Tree Search is tested as the baseline. From the table, although it is able to solve many levels, the search steps are very huge. On the contrary, I2A with the nearly perfect model can achieve the same fraction with much fewer steps.<br />
<br />
====Generalization====<br />
Lastly, the authors probe the generalization capabilities of I2As, beyond handling random level layouts in Sokoban. The agents were trained on levels with 4 boxes. Table 2 shows the performance of I2A when such an agent was tested on levels with different numbers of boxes, and that of the standard model-free agent for comparison. It turns out that I2As generalizes well; at 7 boxes, the I2A agent is still able to solve more than half of the levels, nearly as many as the standard agent on 4 boxes.<br />
[[File:i2a_table.png|800px|center|thumb]]<br />
<br />
===MiniPacman===<br />
MiniPacman is a game modified from the classical game PacMan. In the game(Figure 8, left), the player explores a maze that contains food while being chased by ghosts. The maze also contains power pills; when eaten, for a fixed number of steps, the player moves faster, and the ghosts run away and can be eaten. These dynamics are common to all tasks. Each task is defined by a vector $w \in R^5$, associating a reward to each of the following five events: moving, eating food, eating a power pill, eating a ghost, and being eaten by a ghost.<br />
The goal of this part is the attempt that tries to apply the same I2A model to different tasks. The five tasks are described as follows:<br />
* Regular: level is cleared when all the food is eaten;<br />
* Avoid: level is cleared after 128 steps;<br />
* Hunt: level is cleared when all ghosts are eaten or after 80 steps.<br />
* Ambush: level is cleared when all ghosts are eaten or after 80 steps.<br />
* Rush: level is cleared when all power pills are eaten.<br />
<br />
[[File:minipacman_reward.png|800px|center|thumb|Table 3: the reward settings in different tasks]]<br />
<br />
Different from the task in Sokoban, in order to capture long-range dependencies across pixels, the authors also made use of a layer that is called pool-and-inject, which applies global max-pooling over each feature map and broadcasts the resulting values as feature maps of the same size and concatenates the result to the input. Pool-and-inject layers are therefore size-preserving layers which communicate the max-value of each layer globally to the next convolutional layer. The environment model for MiniPacman is shown in Figure 7.<br />
<br />
[[File:minipacman_model.png|800px|center|thumb|Figure 7: The MiniPacman environment model]]<br />
<br />
To illustrate the benefits of model-based methods in this multi-task setting, the authors trained a single environment model to predict both observations (frames) and events, where the environment model is effectively shared across all tasks. Results in Figure 7(right) illustrates the benefit of the I2A architecture, outperforming the standard agent in all tasks. Note that for tasks 4 & 5, the rewards are particularly sparse, and the anticipation of ghost dynamics is especially important. The I2A agent can leverage its environment and reward model to explore the environment much more effectively.<br />
<br />
[[File:minipacman.png|800px|center|thumb|Figure 8: Minipacman environment Left: Two frames from a minipacman game: the player is green, dangerous ghosts red, food dark blue, empty corridors black, power pills in cyan. After eating a power pill (right frame), the player can eat the 4 weak ghosts (yellow). Right: Performance after 300 million environment steps for different agents and all tasks. Note I2A clearly outperforms the other two agents on all tasks with sparse rewards.]]<br />
<br />
=Conclusion=<br />
In this paper, the authors applied recent success in CNN and reinforcement learning and raised a novel approach, which is a combination of model-free and model-based methods, called Imagination-augmented RL. Unlike classical model-based RL and planning methods, I2A is able to successfully use imperfect models to support model-free desicions. This approach outperforms model-free baselines in Sokoban and MiniPacman games. As experiments suggest, this method is able to successfully use imperfect models to interpret future states and rewards. The imagination core part is essential in irreversible domains, where actions can have catastrophic outcomes. Compared to traditional Mente-Carlo search methods, the search space in I2A only grows linearly with the extension of the length of rollouts whereas I2As require far fewer function calls. This work may significantly broaden the applicability of model-based RL concepts and ideas.<br />
<br />
=Insight=<br />
This is a paper with very interesting ideas. However, it seems that the work is really hard to reproduce for an individual researcher. Since the architecture works as a whole, it is very difficult to debug each single part. Meanwhile, the training process is kind of long with up to 1e9 steps, which is also a huge requirement for computing resources.<br />
<br />
In terms of the architecture itself, the design the CNN for the tasks seems to be very empirical. The authors did not include the reasons or rules for this part. Yet why authors applied residual connection in this shadow network is unknown. According to the paper, even the CNN network is quite simple, some details in LSTM encoder are omitted. Therefore, the backpropagation process is not so clear across the whole model.<br />
<br />
Back to the settings of environment model, the authors used pre-trained model instead of the jointly training way. Would it be hard to train both models simultaneously?<br />
<br />
Lastly, the authors raised a new layer as Pool-and-inject layer, the motivation and plausibility are not so clear. It would be better if the authors can compare it with common pooling layer.<br />
<br />
In spite of some missing details, this is a solid work with a novel idea and many tricks. In addition, the settings of the experiment are quite inspiring where we can learn from.<br />
<br />
=Reference=<br />
# Buesing, L., Badia, A.P., Battaglia, P.W., Guez, A., Heess, N., Li, Y., Pascanu, R., Racanière, S., Reichert, D.P., Rezende, D.J., Silver, D., Vinyals, O., Weber, T., & Wierstra, D. (2017). Imagination-Augmented Agents for Deep Reinforcement Learning. CoRR, abs/1707.06203.<br />
# YuXuan Liu, Abhishek Gupta, Pieter Abbeel, and Sergey Levine. Imitation from observation: Learning to imitate behaviors from raw video via context translation. arXiv preprint arXiv:1707.03374, 2017.<br />
# Marc Deisenroth and Carl E Rasmussen. Pilco: A model-based and data-efficient approach to policy search. In Proceedings of the 28th International Conference on machine learning (ICML-11), pages 465–472, 2011.<br />
# Jessica B. Hamrick, Andy J. Ballard, Razvan Pascanu, Oriol Vinyals, Nicolas Heess, and Peter W. Battaglia. Metacontrol for adaptive imagination-based optimization. In Proceedings of the 5th International Conference on Learning Representations (ICLR 2017), 2017.<br />
# Razvan Pascanu, Yujia Li, Oriol Vinyals, Nicolas Heess, David Reichert, Theophane Weber, Sebastien Racaniere, Lars Buesing, Daan Wierstra, and Peter Battaglia. Learning model-based planning from scratch. arXiv preprint, 2017.<br />
# Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A. A., Veness, J., Bellemare, M. G., ... & Petersen, S. (2015). Human-level control through deep reinforcement learning. Nature, 518(7540), 529-533.</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Imagination-Augmented_Agents_for_Deep_Reinforcement_Learning&diff=30017Imagination-Augmented Agents for Deep Reinforcement Learning2017-11-12T22:13:08Z<p>H4lyu: /* Conclusion */</p>
<hr />
<div>=Introduction=<br />
An interesting research area in reinforcement learning is developing intelligent agents for playing video games. Before the introduction of deep learning, video game agents were commonly coded based on Monte-Carlo Tree Search of pre-set rules. Recent research has shown deep reinforcement learning to be very successful at playing video games like Atari 2600. To be specific, the method (Figure 1) is called Deep Q-Learning (DQN) which learns the optimal actions based on current observations (raw pixels) [[#Reference|[Mnih et al., (2015)]]]. However, there are some complex games where DQN fails to learn: some games need to solve a sub-problem without explicit reward or contain irreversible domains, where actions can be catastrophic. A typical example of these games is [https://en.wikipedia.org/wiki/Sokoban Sokoban]. Even as humans are playing the game, planning and inference are needed. This kind of game raises challenges to RL.<br />
<br />
[[File:DQN.png|800px|center|thumb|Figure 1: Deep Q-Learning Architecture]]<br />
<br />
In Reinforcement Learning, the algorithms can be divided into two categories: '''model-free''' algorithm and '''model-based''' algorithm. The model-based reinforcement learning tries to infer environment to gain the reward while model-free reinforcement learning does not use the environment to learn the action that results in the best reward. DQN, mentioned above(Figure 1), is a model-free method. It takes raw pixels as input and maps them to values or actions. As a drawback, large amounts of training data is required. In addition, the policies are not generalized to new tasks in the same environment. A model-based method is trying to build a model for the environment. By querying the model, agents can avoid irreversible, poor decisions. As an approximation of the environment, it can enable better generalization across states. However, this method only shows success in limited settings, where an exact transition model is given or in simple domains. In complex environments, model-based methods suffer from model errors from function approximation. These errors compound during planning, causing poor agent performance. Currently, there is no model-based method that is robust against imperfections.<br />
<br />
In this paper, the authors introduce a novel deep reinforcement learning architecture called Imagination-Augmented Agents(I2As). Literally, this method enables agents to learn to interpret predictions from a learned environment model to construct implicit plans. It is a combination of model-free and model-based aspects. The advantage of this method is that it learns in an end-to-end way to extract information from model simulations without making any assumptions about the structure or the perfections of the environment model.<br />
As shown in the results, this method outperforms DQN in the games: Sokoban, and MniniPacman. In addition, the experiments all show that I2A is able to successfully use imperfect models.<br />
<br />
=Motivation=<br />
Although the structure of this method is complex, the motivation is intuitive: since the agent suffers from irreversible decisions, attempts in simulated states may be helpful.<br />
To improve the expensive search space in traditional MCTS methods, adding decision from policy network can reduce search steps. In order to keep context information, rollout results are encoded by an LSTM encoder. The final output is combining the result from model-free network and model-based network.<br />
<br />
=Related Work=<br />
There are some works that try to apply deep learning to model-based reinforcement learning. The popular approach is to learn a neural network from the environment and apply the network in classical planning algorithms. These works can not handle the mismatch between the learned model and the ground truth. Liu et al.(2017) use context information from trajectories, but in terms of imitation learning.<br />
<br />
To deal with imperfect models, Deisenroth and Rasmussen(2011) try to capture model uncertainty by applying high-computational Gaussian Process models. In order to develop such a policy search method, the authors of this paper used analytic gradients of an approximation to the expected return for indirect policy search. This means by learning a probabilistic dynamics model and explicitly incorporating model uncertainty into long-term planning, this policy search method can cope with very little data and facilitates learning from scratch in only a few trials. <br />
<br />
Similar ideas can be found in a study by Hamrick et al.(2017): they present a neural network that queries expert models, but just focus on meta-control for continuous contextual bandit problems. Pascanu et al.(2017) extend this work by focusing on explicit planning in sequential environments.<br />
<br />
=Approach=<br />
The summary of the architecture of I2A can be seen in Figure 2.<br />
[[File:i2a.png|800px|center|thumb|Figure 2: The Architecture of I2A]]<br />
The observation $O_t$ (Figure 2 right) is fed into two paths, the model-free path is just common DQN which predicts the best action given $O_t$, whereas the model-based path performs a rollout strategy, the aggregator combines the $n$ rollout encoded outputs($n$ equals to the number of actions in the action space), and forwards the results to next layer. Together they are used to generate a policy function $\pi$ to output an action. In each rollout operation, the imagination core is used to predict the future state and reward.<br />
<br />
===Imagination Core===<br />
The imagination core(Figure 2 left) is the key role in the model-based path. It consists of two parts: environment model and rollout policy. The former is an approximation of the environment and the latter is used to simulate imagined trajectories, which are interpreted by a neural network and provided as additional context to a policy network.<br />
<br />
====environment model====<br />
In order to augment agents with imagination, the method relies on environment models that, given current information, can be queried to make predictions about the future. In this work, the environment model is built based on action-conditional next-step predictors, which receive input contains current observation and current action, and predict the next observation and the next reward(Figure 3).<br />
[[File:environment model.png|800px|center|thumb|Figure 3: Environment Model]]<br />
<br />
The authors can either pretrain the environment model before embedding it (with frozen weights) within the I2A architecture, or jointly train it with the agent by adding $l_{model}$ to the total loss as an auxiliary loss. In practice, they found that pre-training the environment model led to faster<br />
runtime of the I2A architecture, so they adopted this strategy.<br />
<br />
====rollout policy====<br />
The rollout process is regarded as the simulated trajectories. In this work, the rollout is performed for each possible action in the environment. <br />
<br />
A rollout policy $\hat \pi$ is a function that takes current observation $O$ and outputs an action $a$ that potentially leads to maximal reward. In this architecture, the rollout policy can be a DQN network. In the experiment, the rollout pocily$\hat \pi$ is broadcasted and shared. After experiments on the types of rollout policies(random, pre-trained), the authors found the efficient strategy is to distill the policy into a model-free policy, which consists in creating a smaill model-free network $\hat \pi(O_t)$, and adding to the total loss a cross entropy auxiliary loss between the imagination-augmented policy $\pi(O_t)$ as computed on the current observation, and the policy $\hat \pi(O_t)$ as computed on the same observation.<br />
<br />
$$<br />
l_{dist} (\pi, \hat \pi)(O_t) = \lambda_{dist} \sum_a \pi(a|O_t)log(\hat \pi(a|O_t))<br />
$$<br />
<br />
Together as the imagination core, these two parts produces $n$ trajectories $\hat \tau_1,...,\hat \tau_n$. Each imagined trajectory $\hat \tau$ is a sequence of features $(\hat f_{t+1},...,\hat f_{t+\tau})$, where $t$ is the current time, $\tau$ the length of rollout, and $\hat f_{t+i}$ the output of the environment model(the predicted observation and reward). In order to guarantee success in imperfections, the architecture does not assume the learned model to be perfect. The output will not only depend on the predicted reward.<br />
<br />
===Trajectories Encoder===<br />
From the intuition to keep the sequence information in the trajectories, the architecture uses a rollout encoder $\varepsilon$ that processes the imagined rollout as a whole and learns to interpret it(Figure 2 middle). Each trajectory is encoded as a rollout embedding $e_i=\varepsilon(\hat \tau_i)$. Then, the aggregator $A$ combines the rollout embedding s into a single imagination code $c_{ia}=A(e_1,...,e_n)$ by simply concatenating all the summaries.<br />
In the experiments, the encoder is an LSTM that takes the predicted output from environment model as the input. One observation is that the order of the sequence $\hat f_{t+1}$ to $\hat f_{t+\tau}$ makes relatively little impact on the performance. The encodes mimics the Bellman type backup operations in DQN.<br />
<br />
===Model-Free Path===<br />
The model-free path contains a network that only takes the current observation as input that generates the potential optimal action. This network can be same as the one in imagination core.<br />
<br />
<br />
In conclusion, the I2A learns to combine information for two paths, and without the model-based path, I2A simply reduce to a standard model-free network(such as A3C, more explanations [https://medium.com/emergent-future/simple-reinforcement-learning-with-tensorflow-part-8-asynchronous-actor-critic-agents-a3c-c88f72a5e9f2 here]). The imperfect approximation results in a rollout policy with higher entropy, potentially striking a balance between exploration and exploitation.<br />
<br />
=Experiment=<br />
These following experiments were tested in Sokoban and MiniPacman games. All results are averages taken from top three agents. These agents were trained over 32 to 64 workers, and the network was optimized by RMSprop.<br />
As the pre-training strategy, the training data of I2A was pre-generated from trajectories of a partially trained standard model-free agent, the data is also taken into account for the budget. The total number of frames that were needed in pre-training is counted in the later process. Meanwhile, the authors show that the environment model can be reused to solve multiple tasks in the same environment.<br />
<br />
In the game Sokoban, the environment is a 10 x 10 grid world. All agents were trained directly on raw pixels(image size 80 x 80 with 3 channels). To make sure the network is not just simply "memorize" all states, the game procedurally generates a new level each episode. Out of 40 million levels generated, less than 0.7% were repeated. Therefore, a good agent should solve the unseen level as well.<br />
<br />
The reward settings for reinforcement learning algorithms are as follows:<br />
* Every time step, a penalty of -0.1 is applied to the agent.(encourage agents to finish levels faster)<br />
* Whenever the agent pushes a box on target, it receives a reward of +1.(encourage agents to push boxes onto targets)<br />
* Whenever the agent pushes a box off target, it receives a penalty of -1.(avoid artificial reward loop that would be induced by repeatedly pushing a box off and on target)<br />
* Finishing the level gives the agent a reward of +10 and the level terminates.(strongly reward solving a level)<br />
<br />
To show the advantage of I2A, the authors set a model-free standard architecture as a baseline. The architecture is a multi-layer convolutional neural network (CNN), taking the current observation $O_t$ as input, followed by a fully connected (FC) hidden layer. This FC layer feeds into two heads: into an FC layer with one output per action computing the policy logits $\log \pi(a_t|O_t, \theta)$; and into another FC layer with a single output that computes the value function $V(O_t; \theta_v)$.<br />
* for MiniPacman: the CNN has two layers, both with 3x3 kernels, 16 output channels and strides 1 and 2; the following FC layer has 256 units<br />
* for Sokoban: the CNN has three layers with kernel sizes 8x8, 4x4, 3x3, strides of 4, 2, 1 and number of output channels 32, 64, 64; the following FC has 512 units<br />
<br />
===Sokoban===<br />
<br />
The environment model for Sokoban is shown in figure 4<br />
[[File:sokoban_em.png|400px|center|thumb|Figure 4: The Sokoban environment model]]<br />
<br />
Besides, to demonstrate the influence of larger architecture in I2A, the authors set a copy-model agent that uses the same architecture of I2A but the environment model is replaced by identical map. This agent is regarded as an I2A agent without imagination.<br />
<br />
[[File:sokoban_result.png|800px|center|thumb|Figure 5: Sokoban learning curves. Left: training curves of I2A and baselines. Right: I2A training curves for various values of imagination depth]]<br />
The results are shown in Figure 4(left). I2A agents can solve much more levels compared to common DQN. Also, it far outperforms the copy-model. The authors also trained an I2A where the environment model was predicting no rewards, only observations. This also performed worse. However, after much longer training (3e9 steps), these agents did recover the performance of the original I2A, which was never the case for the baseline agent even with that many steps. Hence, reward prediction is helpful but not absolutely necessary in this task, and imagined observations alone are informative enough to obtain high performance on Sokoban. Note this is in contrast to many classical planning and model-based reinforcement learning methods, which often rely on reward prediction.<br />
<br />
====Length of Rollout====<br />
A further experiment was investigating how the length of individual rollouts affects performance. The authors performed a parameter searching. Figure 5(right) shows the influence of the rollout length. The strategy using 3 rollout steps improves the speed of learning and improves the performance significantly than 1 step, and 5 is the optimal number. This implies rollout can be very helpful and informative. This rollout enables the agent to learn moves it cannot recover from.<br />
<br />
[[File:sokoban_noisy.png|800px|center|thumb|Figure 6: Experiments with a noisy environment model Left: each row shows an example 5-step rollout after conditioning on an environment observation. Errors accumulate and lead to various artifacts, including missing or duplicate sprites. Right: comparison of Monte-Carlo (MC) search and I2A when using either the accurate or the noisy model for rollouts.]]<br />
<br />
====Imperfections====<br />
To demonstrate I2A can handle less reliable predictions, the authors set experiment where the I2A used a poor environment model(smaller number of parameters), where the error may accumulate across the rollout(Figure 6 left). As shown in figure 6(right), even with relatively poor environment model, the performance of I2A is stable, unlike traditional Monte-Carlo search, which explicitly estimates the value of each action from rollouts, rather than learning an arbitrary encoding of the rollouts. An interesting result is that a rollout length 5 no longer outperforms a length of 3, which matches our common sense. The authors suggest that it is learning a rollout encoder that enables I2As to deal with imperfect model predictions.<br />
<br />
====Perfections====<br />
As I2A shows the robustness towards environment models, the authors tested an I2A agent with a nearly perfect environment model, and the results are in Table 1 and Table 2. Traditional Mento-Carlo Tree Search is tested as the baseline. From the table, although it is able to solve many levels, the search steps are very huge. On the contrary, I2A with the nearly perfect model can achieve the same fraction with much fewer steps.<br />
<br />
====Generalization====<br />
Lastly, the authors probe the generalization capabilities of I2As, beyond handling random level layouts in Sokoban. The agents were trained on levels with 4 boxes. Table 2 shows the performance of I2A when such an agent was tested on levels with different numbers of boxes, and that of the standard model-free agent for comparison. It turns out that I2As generalizes well; at 7 boxes, the I2A agent is still able to solve more than half of the levels, nearly as many as the standard agent on 4 boxes.<br />
[[File:i2a_table.png|800px|center|thumb]]<br />
<br />
<br />
===MiniPacman===<br />
MiniPacman is a game modified from the classical game PacMan. In the game(Figure 8, left), the player explores a maze that contains food while being chased by ghosts. The maze also contains power pills; when eaten, for a fixed number of steps, the player moves faster, and the ghosts run away and can be eaten. These dynamics are common to all tasks. Each task is defined by a vector $w \in R^5$, associating a reward to each of the following five events: moving, eating food, eating a power pill, eating a ghost, and being eaten by a ghost.<br />
The goal of this part is the attempt that tries to apply the same I2A model to different tasks. The five tasks are described as follows:<br />
* Regular: level is cleared when all the food is eaten;<br />
* Avoid: level is cleared after 128 steps;<br />
* Hunt: level is cleared when all ghosts are eaten or after 80 steps.<br />
* Ambush: level is cleared when all ghosts are eaten or after 80 steps.<br />
* Rush: level is cleared when all power pills are eaten.<br />
<br />
[[File:minipacman_reward.png|800px|center|thumb|Table 3: the reward settings in different tasks]]<br />
<br />
Different from the task in Sokoban, in order to capture long-range dependencies across pixels, the authors also made use of a layer that is called pool-and-inject, which applies global max-pooling over each feature map and broadcasts the resulting values as feature maps of the same size and concatenates the result to the input. Pool-and-inject layers are therefore size-preserving layers which communicate the max-value of each layer globally to the next convolutional layer. The environment model for MiniPacman is shown in Figure 7.<br />
<br />
[[File:minipacman_model.png|800px|center|thumb|Figure 7: The MiniPacman environment model]]<br />
<br />
To illustrate the benefits of model-based methods in this multi-task setting, the authors trained a single environment model to predict both observations (frames) and events, where the environment model is effectively shared across all tasks. Results in Figure 7(right) illustrates the benefit of the I2A architecture, outperforming the standard agent in all tasks. Note that for tasks 4 & 5, the rewards are particularly sparse, and the anticipation of ghost dynamics is especially important. The I2A agent can leverage its environment and reward model to explore the environment much more effectively.<br />
<br />
[[File:minipacman.png|800px|center|thumb|Figure 8: Minipacman environment Left: Two frames from a minipacman game: the player is green, dangerous ghosts red, food dark blue, empty corridors black, power pills in cyan. After eating a power pill (right frame), the player can eat the 4 weak ghosts (yellow). Right: Performance after 300 million environment steps for different agents and all tasks. Note I2A clearly outperforms the other two agents on all tasks with sparse rewards.]]<br />
<br />
=Conclusion=<br />
In this paper, the authors applied recent success in CNN and reinforcement learning and raised a novel approach, which is a combination of model-free and model-based methods, called Imagination-augmented RL. Unlike classical model-based RL and planning methods, I2A is able to successfully use imperfect models to support model-free desicions. This approach outperforms model-free baselines in Sokoban and MiniPacman games. As experiments suggest, this method is able to successfully use imperfect models to interpret future states and rewards. The imagination core part is essential in irreversible domains, where actions can have catastrophic outcomes. Compared to traditional Mente-Carlo search methods, the search space in I2A only grows linearly with the extension of the length of rollouts whereas I2As require far fewer function calls. This work may significantly broaden the applicability of model-based RL concepts and ideas.<br />
<br />
=Insight=<br />
This is a paper with very interesting ideas. However, it seems that the work is really hard to reproduce for an individual researcher. Since the architecture works as a whole, it is very difficult to debug each single part. Meanwhile, the training process is kind of long with up to 1e9 steps, which is also a huge requirement for computing resources.<br />
<br />
In terms of the architecture itself, the design the CNN for the tasks seems to be very empirical. The authors did not include the reasons or rules for this part. Yet why authors applied residual connection in this shadow network is unknown. According to the paper, even the CNN network is quite simple, some details in LSTM encoder are omitted. Therefore, the backpropagation process is not so clear across the whole model.<br />
<br />
Back to the settings of environment model, the authors used pre-trained model instead of the jointly training way. Would it be hard to train both models simultaneously?<br />
<br />
Lastly, the authors raised a new layer as Pool-and-inject layer, the motivation and plausibility are not so clear. It would be better if the authors can compare it with common pooling layer.<br />
<br />
In spite of some missing details, this is a solid work with a novel idea and many tricks. In addition, the settings of the experiment are quite inspiring where we can learn from.<br />
<br />
=Reference=<br />
# Buesing, L., Badia, A.P., Battaglia, P.W., Guez, A., Heess, N., Li, Y., Pascanu, R., Racanière, S., Reichert, D.P., Rezende, D.J., Silver, D., Vinyals, O., Weber, T., & Wierstra, D. (2017). Imagination-Augmented Agents for Deep Reinforcement Learning. CoRR, abs/1707.06203.<br />
# YuXuan Liu, Abhishek Gupta, Pieter Abbeel, and Sergey Levine. Imitation from observation: Learning to imitate behaviors from raw video via context translation. arXiv preprint arXiv:1707.03374, 2017.<br />
# Marc Deisenroth and Carl E Rasmussen. Pilco: A model-based and data-efficient approach to policy search. In Proceedings of the 28th International Conference on machine learning (ICML-11), pages 465–472, 2011.<br />
# Jessica B. Hamrick, Andy J. Ballard, Razvan Pascanu, Oriol Vinyals, Nicolas Heess, and Peter W. Battaglia. Metacontrol for adaptive imagination-based optimization. In Proceedings of the 5th International Conference on Learning Representations (ICLR 2017), 2017.<br />
# Razvan Pascanu, Yujia Li, Oriol Vinyals, Nicolas Heess, David Reichert, Theophane Weber, Sebastien Racaniere, Lars Buesing, Daan Wierstra, and Peter Battaglia. Learning model-based planning from scratch. arXiv preprint, 2017.<br />
# Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A. A., Veness, J., Bellemare, M. G., ... & Petersen, S. (2015). Human-level control through deep reinforcement learning. Nature, 518(7540), 529-533.</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Dialog-based_Language_Learning&diff=30013Dialog-based Language Learning2017-11-12T21:47:54Z<p>H4lyu: /* Dialog-based Supervision tasks */</p>
<hr />
<div>This page is a summary for NIPS 2016 paper <i>Dialog-based Language Learning</i> [1].<br />
==Introduction==<br />
One of the ways humans learn language, especially second language or language learning by students, is by communication and getting its feedback. However, most existing research in Natural Language Understanding has focused on supervised learning from fixed training sets of labeled data. This kind of supervision is not realistic of how humans learn, where language is both learned by, and used for, communication. When humans act in dialogs (i.e., make speech utterances) the feedback from other human’s responses contain very rich information. This is perhaps most pronounced in a student/teacher scenario where the teacher provides positive feedback for successful communication and corrections for unsuccessful ones. <br />
<br />
This paper is about dialog-based language learning, where supervision is given naturally and implicitly in the response of the dialog partner during the conversation. This paper is a step towards the ultimate goal of being able to develop an intelligent dialog agent that can learn while conducting conversations. Specifically this paper explores whether we can train machine learning models to learn from dialog.<br />
<br />
===Contributions of this paper===<br />
*Introduce a set of tasks that model natural feedback from a teacher and hence assess the feasibility of dialog-based language learning. <br />
*Evaluated some baseline models on this data and compared them to standard supervised learning. <br />
*Introduced a novel forward prediction model, whereby the learner tries to predict the teacher’s replies to its actions, which yields promising results, even with no reward signal at all<br />
<br />
==Background on Memory Networks==<br />
<br/><br />
[[File:DB F1.png|center|700px]]<br />
<br/><br />
A memory network combines learning strategies from the machine learning literature with a memory component that can be read and written to.<br />
<br />
The high-level view of a memory network is as follows:<br />
*There is a memory, $m$, an indexed array of objects (e.g. vectors or arrays of strings).<br />
*An input feature map $I$, which converts the incoming input to the internal feature representation<br />
*A generalization component $G$ which updates old memories given the new input. <br />
*An output feature map $O$, which produces a new output in the feature representation space given the new input and the current memory state.<br />
*A response component $R$ which converts the output into the response format desired – for example, a textual response or an action.<br />
<br />
$I$, $G$, $O$ and $R$ can all potentially be learned components and make use of any ideas from the existing machine learning literature.<br />
<br />
In question answering systems for example, the components may be instantiated as follows:<br />
*$I$ can make use of standard pre-processing such as parsing, coreference, and entity resolution. It could also encode the input into an internal feature representation by converting from text to a sparse or dense feature vector.<br />
*The simplest form of $G$ is to introduce a function $H$ which maps the internal feature representation produced by I to an individual memory slot, and just updates the memory at $H(I(x))$.<br />
*$O$ Reads from memory and performs inference to deduce the set of relevant memories needed to perform a good response.<br />
*$R$ would produce the actual wording of the question answer based on the memories found by $O$. For example, $R$ could be an RNN conditioned on the output of $O$<br />
<br />
When the components $I$,$G$,$O$, & $R$ are neural networks, the authors describe the resulting system as a <b>Memory Neural Network (MemNN)</b>. They build a MemNN for QA (question answering) problems and compare it to RNNs (Recurrent Neural Network) and LSTMs (Long Short Term Memory RNNs) and find that it gives superior performance.<br />
<br />
Extensions of this method include the Key-Value Memory Network model [16] which have been especially effective for utilizing KBs to answer questions. Key-value paired memories are a generalization of the way context (e.g. knowledge bases or documents to be read) are stored in memory. The lookup (addressing) stage is based on the key memory while the reading stage (giving the returned result) uses the value memory. This gives both (i) greater flexibility for the practitioner to encode prior knowledge about their task; and (ii) more effective power in the model via nontrivial transforms between key and value. An important property of the model is that the entire model can be trained with key-value transforms while still using standard back-propagation via stochastic gradient descent.<br />
<br />
[[File:DB_F2.png|center|800px]]<br />
<br />
==Related Work==<br />
<br />
'''Usefulness of feedback in language learning:''' Social interaction and natural infant directed conversations are shown to useful for language learning[2]. Several studies[3][4][5][6] have shown that feedback is especially useful in second language learning and learning by students.<br />
<br />
'''Supervised learning from dialogs using neural models:''' Neural networks has been used for response generation that can be trained end to end on large quantities of unstructured Twitter conversations[7]. However this does not incorporate feedback from dialog partner during real time conversation<br />
<br />
'''Reinforcement learning:''' Reinforcement learning works on dialogs[8][9], often consider reward as the feedback model rather than exploiting the dialog feedback per se. To be more specific, the reinforcement learning utilizes the system of rewards or what the authors from paper [8] called “trail-and-error”. The learning agent (in this case the language-learning agent) interacts with the dynamic environment (in this case through active dialog) and it receives feedback in the form of positive or negative rewards. By setting the objective function as maximizing the rewards, the model can be trained without explicit y responses. The reason why such algorithm is not particularly efficient in training a dialog-based language learning model is that there’s no explicit/fixed threshold of a positive or negative reward. One possible way to measure such action is define what a successful completion of a dialog should be and use that as the objective function. <br />
<br />
'''Forward prediction models:''' Although forward prediction models, have been used in other applications like learning eye-tracking[10], controlling robot arms[11] and vehicles[12], it has not been used for dialog.<br />
<br />
==Dialog-based Supervision tasks==<br />
For testing their models, the authors chose two datasets (i) the single supporting fact problem from the bAbI datasets [13] which consists of short stories from a simulated world followed by questions; and (ii) the MovieQA dataset [14] which is a large-scale dataset (∼ 100k questions over ∼ 75k entities) based on questions with answers in the open movie database (OMDb)<br />
<br />
However, since these datasets were not designed to model the supervision from dialogs, the authors modified them to create 10 supervision task types on these datasets(Fig 3).<br />
<br />
[[File:DB F3.png|center|700px]]<br />
<br/><br />
<br />
*'''Task 1: Imitating an Expert Student''': The dialogs take place between a teacher and an expert student who gives semantically coherent answers. Hence, the task is for the learner to imitate that expert student, and become an expert themselves <br />
<br />
*'''Task 2: Positive and Negative Feedback:''' When the learner answers a question the teacher then replies with either positive or negative feedback. In the experiments the subsequent responses are variants of “No, that’s incorrect” or “Yes, that’s right”. In the datasets there are 6 templates for positive feedback and 6 templates for negative feedback, e.g. ”Sorry, that’s not it.”, ”Wrong”, etc. To distinguish the notion of positive from negative, an additional external reward signal that is not part of the text<br />
<br />
*'''Task 3: Answers Supplied by Teacher:''' The teacher gives positive and negative feedback as in Task 2, however when the learner’s answer is incorrect, the teacher also responds with the correction. For example if “where is Mary?” is answered with the incorrect answer “bedroom” the teacher responds “No, the answer is kitchen”’<br />
<br />
*'''Task 4: Hints Supplied by Teacher:''' The corrections provided by the teacher do not provide the exact answer as in Task 3, but only a useful hint. This setting is meant to mimic the real life occurrence of being provided only partial information about what you did wrong.<br />
<br />
*'''Task 5: Supporting Facts Supplied by Teacher:''' Another way of providing partial supervision for an incorrect answer is explored. Here, the teacher gives a reason (explanation) why the answer is wrong by referring to a known fact that supports the true answer that the incorrect answer may contradict. <br />
<br />
*'''Task 6: Partial Feedback:''' External rewards are only given some of (50% of) the time for correct answers, the setting is otherwise identical to Task 3. This attempts to mimic the realistic situation of some learning being more closely supervised (a teacher rewarding you for getting some answers right) whereas other dialogs have less supervision (no external rewards). The task attempts to assess the impact of such partial supervision.<br />
<br />
*'''Task 7: No Feedback:''' External rewards are not given at all, only text, but is otherwise identical to Tasks 3 and 6. This task explores whether it is actually possible to learn how to answer at all in such a setting.<br />
<br />
*'''Task 8: Imitation and Feedback Mixture:''' Combines Tasks 1 and 2. The goal is to see if a learner can learn successfully from both forms of supervision at once. This mimics a child both observing pairs of experts talking (Task 1) while also trying to talk (Task 2).<br />
<br />
*'''Task 9: Asking For Corrections:''' The learner will ask questions to the teacher about what it has done wrong. Task 9 tests one of the most simple instances, where asking “Can you help me?” when wrong obtains from the teacher the correct answer.<br />
<br />
*'''Task 10: Asking for Supporting Facts:''' A less direct form of supervision for the learner after asking for help is to receive a hint rather than the correct answer, such as “A relevant fact is John moved to the bathroom” when asking “Can you help me?”. This is thus related to the supervision in Task 5 except the learner must request help<br />
<br />
[[File:F4.png|center|700px]]<br />
<br/><br />
<br />
For each task a fixed policy is considered for performing actions (answering questions) which gets questions correct with probability $π_{acc}$ (i.e. the chance of getting the red text correct in Figs. 3 and 4). We thus can compare different learning algorithms for each task over different values of $π_{acc}$ (0.5, 0.1 and 0.01). In all cases a training, validation and test set is provided. For the bAbI dataset this consists of 1000, 100 and 1000 questions respectively per task, and for movieQA there are ∼ 96k, ∼ 10k and ∼ 10k respectively. MovieQA also includes a knowledge base (KB) of ∼ 85k facts from OMDB(Open Movie Database), the memory network model we employ uses inverted index retrieval based on the question to form relevant memories from this set. Note that since policy is fixed, our model is supervised learning rather than reinforcement learning.<br />
<br />
==Learning models==<br />
This work evaluates four possible learning strategies for each of the 10 tasks: imitation learning, reward-based imitation, forward prediction, and a combination of reward-based imitation and forward prediction<br />
<br />
All of these approaches are evaluated with the same model architecture: an end-to-end memory network (MemN2N) [15], which has been used as a baseline model for exploring differing modes of learning.<br />
<br />
[[File:F5.png|center|700px]]<br />
<br/><br />
The input is the last utterance of the dialog, $x$, as well as a set of memories (context) (<math> c_1</math>, . . . , <math> c_n</math> ) which can encode both short-term memory, e.g. recent previous utterances and replies, and long-term memories, e.g. facts that could be useful for answering questions. The context inputs <math> c_i</math> are converted into vectors <math> m_i</math> via embeddings and are stored in the memory. The goal is to produce an output $\hat{a}$ by processing the input $x$ and using that to address and read from the memory, $m$, possibly multiple times, in order to form a coherent reply. In the figure the memory is read twice, which is termed multiple “hops” of attention. <br />
<br />
In the first step, the input $x$ is embedded using a matrix $A$ of size $d$ × $V$ where $d$ is the embedding dimension and $V$ is the size of the vocabulary, giving $q$ = $A$$x$, where the input $x$ is as a bag-of words vector. Each memory <math> c_i</math> is embedded using the same matrix, giving $m_i$ = $A$$c_i$ . The output of addressing and then reading from memory in the first hop is: <br />
<br />
[[File:eq1.png|center|400px]]<br />
<br />
Here, $p^{1}$ is a probability vector over the memories, and is a measure of how much the input and the memories match. The goal is to select memories relevant to the last utterance $x$, i.e. the most relevant have large values of $p^{1}_i$ . The output memory representation $o_1$ is then constructed using the weighted sum of memories, i.e. weighted by $p^{1}$ . The memory output is then added to the original input, <math> c_1</math> = <math> R_1</math>(<math> o_1</math> + $q$), to form the new state of the controller, where <math> R_1</math> is a $d$ × $d$ rotation matrix . The attention over the memory can then be repeated using <math> u_1</math> instead of $q$ as the addressing vector, yielding: <br />
<br />
[[File:eq2.png|center|400px]]<br />
<br />
The controller state is updated again with <math> u_2</math> = <math> R_2</math>(<math> o_2</math> + <math> u_1</math>), where <math> R_2</math> is another $d$ × $d$ matrix to be learnt. In a two-hop model the final output is then defined as: <br />
<br />
[[File:eq3.png|center|400px]]<br />
<br />
where there are $C$ candidate answers in $y$. In our experiments $C$ is the set of actions that occur in the training set for the bAbI tasks, and for MovieQA it is the set of words retrieved from the KB.<br />
<br />
==Training strategies==<br />
*'''1. Imitation Learning'''<br />
This approach involves simply imitating one of the speakers in observed dialogs. Examples arrive as $(x, c, a)$ triples, where $a$ is (assumed to be) a good response to the last utterance $x$ given context $c$. Here, the whole memory network model defined above is trained using stochastic gradient descent by minimizing a standard cross-entropy loss between $\hat{a}$ and the label $a$<br />
<br />
*'''2. Reward-based Imitation''' <br />
If some actions are poor choices, then one does not want to repeat them, that is we shouldn’t treat them as a supervised objective. Here, positive reward is only obtained immediately after (some of) the correct actions, or else is zero. Only apply imitation learning on the rewarded actions. The rest of the actions are simply discarded from the training set. <br />
<br />
*'''3. Forward Prediction''' <br />
The aim is, given an utterance $x$ from speaker 1 and an answer $a$ by speaker 2 (i.e., the learner), to predict $\bar{x}$, the response to the answer from speaker 1. That is, in general to predict the changed state of the world after action $a$, which in this case involves the new utterance $\bar{x}$.<br />
<br />
[[File:F6.png|center|700px]]<br />
<br/><br />
As shown in Figure (b), this is achieved by chopping off the final output from the original network of Fig (a) and replace it with some additional layers that compute the forward prediction. The first part of the network remains exactly the same and only has access to input x and context c, just as before. The computation up to $u_2$ = $R_2$($o_2$ + $u_1$) is thus exactly the same as before. <br />
<br />
Then perform another “hop” of attention but over the candidate answers rather than the memories. The information of which action (candidate) was actually selected in the dialog (i.e. which one is a) is also incorporated which is crucial. After this “hop”, the resulting state of the controller is then used to do the forward prediction.<br />
<br />
Concretely, we compute: <br />
<br />
[[File:eq4.png|center|550px]]<br />
<br />
where $β^{*}$ is a d-dimensional vector, that is also learnt, that represents in the output $o_3$ the action that was actually selected. The degree to which $β^{*}$ is represented in $o_3$ is determined by whether the answer a given in the dialog is correct, and how accurately the model can identify the true correct response (i.e. how much probability is placed on the correct response). After obtaining $o_3$, the forward prediction is then computed as: <br />
<br />
[[File:eq5.png|center|500px]]<br />
<br />
where $u_3$ = $R_3$($o_3$ + $u_2$). That is, it computes the scores of the possible responses to the answer a over $C^{¯}$ possible candidates.<br />
<br />
<br />
*'''4. Reward-based Imitation + Forward Prediction'''<br />
As the reward-based imitation learning uses the architecture of Fig (a), and forward prediction uses the same architecture but with the additional layers of Fig (b), we can learn jointly with both strategies. This is a powerful combination as it makes use of reward signal when available and the dialog feedback when the reward signal is not available.<br />
<br />
==Experiments==<br />
<br />
Experiments were conducted on the two test datasets - babi and MovieQA. For each task, a fixed policy is considered for performing actions (answering questions) which gets questions correct with probability $π_{acc}$. This helps to compare the different training strategies described earlier over each task for different values of $π_{acc}$. Hyperparameters for all methods are optimized on the validation sets.<br />
<br/><br/><br />
[[File:DB F7.png|center|800px]]<br />
<br/><br />
The following results are observed by the authors:<br />
*Imitation learning, ignoring rewards, is a poor learning strategy when imitating inaccurate answers, e.g. for $\pi_{acc}$ < 0.5. For imitating an expert however (Task 1) it is hard to beat. <br />
*Reward-based imitation (RBI) performs better when rewards are available, particularly in Table 1, but also degrades when they are too sparse e.g. for πacc = 0.01.<br />
*Forward prediction (FP) is more robust and has stable performance at different levels of $\pi_{acc}$. However as it only predicts answers implicitly and does not make use of rewards, it is outperformed by RBI on several tasks, notably Tasks 1 and 8 (because it cannot do supervised learning) and Task 2 (because it does not take advantage of positive rewards).<br />
*FP makes use of dialog feedback in Tasks 3-5 whereas RBI does not. This explains why FP does better with useful feedback (Tasks 3-5) than without (Task 2), whereas RBI cannot.<br />
*Supplying full answers (Task 3) is more useful than hints (Task 4) but hints still help FP more than just yes/no answers without extra information (Task 2).<br />
*When positive feedback is sometimes missing (Task 6) RBI suffers especially in Table 1. FP does not as it does not use this feedback.<br />
*One of the most surprising results of our experiments is that FP performs well overall, given that it does not use feedback, which we will attempt to explain subsequently. This is particularly evident on Task 7 (no feedback) where RBI has no hope of succeeding as it has no positive examples. FP on the other hand learns adequately.<br />
*Tasks 9 and 10 are harder for FP as the question is not immediately before the feedback.<br />
*Combining RBI and FP ameliorates the failings of each, yielding the best overall results<br />
*An interesting thing is that FP learns the relationship of words, e.g. "right" and "correct", without any rewards.<br />
<br />
==Future work==<br />
* Any reply in a dialog can be seen as feedback, and should be useful for learning. Evaluate if forward prediction, and the other approaches in this paper, work there too. <br />
* Develop further evaluation methodologies to test how the models presented here work in more complex settings where actions that are made lead to long-term changes in the environment and delayed rewards, i.e. extending to the reinforcement learning setting, and to full language generation. <br />
* How dialog-based feedback could also be used as a medium to learn non-dialog based skills, e.g. natural language dialog for completing visual or physical tasks.<br />
<br />
==References==<br />
# Jason Weston. Dialog-based Language Learning. NIPS, 2016.<br />
# P. K. Kuhl. Early language acquisition: cracking the speech code. Nature reviews neuroscience, 5(11): 831–843, 2004.<br />
# M. A. Bassiri. Interactional feedback and the impact of attitude and motivation on noticing l2 form. English Language and Literature Studies, 1(2):61, 2011.<br />
# R. Higgins, P. Hartley, and A. Skelton. The conscientious consumer: Reconsidering the role of assessment feedback in student learning. Studies in higher education, 27(1):53–64, 2002.<br />
# A. S. Latham. Learning through feedback. Educational Leadership, 54(8):86–87, 1997.<br />
# M. G. Werts, M. Wolery, A. Holcombe, and D. L. Gast. Instructive feedback: Review of parameters and effects. Journal of Behavioral Education, 5(1):55–75, 1995.<br />
# A. Sordoni, M. Galley, M. Auli, C. Brockett, Y. Ji, M. Mitchell, J.-Y. Nie, J. Gao, and B. Dolan. A neural network approach to context-sensitive generation of conversational responses. Proceedings of NAACL, 2015.<br />
# V. Rieser and O. Lemon. Reinforcement learning for adaptive dialogue systems: a data-driven methodology for dialogue management and natural language generation. Springer Science & Business Media, 2011.<br />
# J. Schatzmann, K. Weilhammer, M. Stuttle, and S. Young. A survey of statistical user simulation techniques for reinforcement-learning of dialogue management strategies. The knowledge engineering review, 21(02):97–126, 2006.<br />
# J. Schmidhuber and R. Huber. Learning to generate artificial fovea trajectories for target detection. International Journal of Neural Systems, 2(01n02):125–134, 1991.<br />
# I. Lenz, R. Knepper, and A. Saxena. Deepmpc: Learning deep latent features for model predictive control. In Robotics Science and Systems (RSS), 2015.<br />
# G. Wayne and L. Abbott. Hierarchical control using networks trained with higher-level forward models. Neural computation, 2014.<br />
# B. C. Stadie, S. Levine, and P. Abbeel. Incentivizing exploration in reinforcement learning with deep predictive models. arXiv preprint arXiv:1507.00814, 2015.<br />
# J. Clarke, D. Goldwasser, M.-W. Chang, and D. Roth. Driving semantic parsing from the world’s response. In Proceedings of the fourteenth conference on computational natural language learning, pages 18–27. Association for Computational Linguistics, 2010.<br />
# J. Schatzmann, K. Weilhammer, M. Stuttle, and S. Young. A survey of statistical user simulation techniques for reinforcement-learning of dialogue management strategies. The knowledge engineering review, 21(02):97–126, 2006.<br />
# Miller, Alexander, et al. "Key-value memory networks for directly reading documents." arXiv preprint arXiv:1606.03126 (2016).</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Dialog-based_Language_Learning&diff=30012Dialog-based Language Learning2017-11-12T21:42:02Z<p>H4lyu: /* Experiments */</p>
<hr />
<div>This page is a summary for NIPS 2016 paper <i>Dialog-based Language Learning</i> [1].<br />
==Introduction==<br />
One of the ways humans learn language, especially second language or language learning by students, is by communication and getting its feedback. However, most existing research in Natural Language Understanding has focused on supervised learning from fixed training sets of labeled data. This kind of supervision is not realistic of how humans learn, where language is both learned by, and used for, communication. When humans act in dialogs (i.e., make speech utterances) the feedback from other human’s responses contain very rich information. This is perhaps most pronounced in a student/teacher scenario where the teacher provides positive feedback for successful communication and corrections for unsuccessful ones. <br />
<br />
This paper is about dialog-based language learning, where supervision is given naturally and implicitly in the response of the dialog partner during the conversation. This paper is a step towards the ultimate goal of being able to develop an intelligent dialog agent that can learn while conducting conversations. Specifically this paper explores whether we can train machine learning models to learn from dialog.<br />
<br />
===Contributions of this paper===<br />
*Introduce a set of tasks that model natural feedback from a teacher and hence assess the feasibility of dialog-based language learning. <br />
*Evaluated some baseline models on this data and compared them to standard supervised learning. <br />
*Introduced a novel forward prediction model, whereby the learner tries to predict the teacher’s replies to its actions, which yields promising results, even with no reward signal at all<br />
<br />
==Background on Memory Networks==<br />
<br/><br />
[[File:DB F1.png|center|700px]]<br />
<br/><br />
A memory network combines learning strategies from the machine learning literature with a memory component that can be read and written to.<br />
<br />
The high-level view of a memory network is as follows:<br />
*There is a memory, $m$, an indexed array of objects (e.g. vectors or arrays of strings).<br />
*An input feature map $I$, which converts the incoming input to the internal feature representation<br />
*A generalization component $G$ which updates old memories given the new input. <br />
*An output feature map $O$, which produces a new output in the feature representation space given the new input and the current memory state.<br />
*A response component $R$ which converts the output into the response format desired – for example, a textual response or an action.<br />
<br />
$I$, $G$, $O$ and $R$ can all potentially be learned components and make use of any ideas from the existing machine learning literature.<br />
<br />
In question answering systems for example, the components may be instantiated as follows:<br />
*$I$ can make use of standard pre-processing such as parsing, coreference, and entity resolution. It could also encode the input into an internal feature representation by converting from text to a sparse or dense feature vector.<br />
*The simplest form of $G$ is to introduce a function $H$ which maps the internal feature representation produced by I to an individual memory slot, and just updates the memory at $H(I(x))$.<br />
*$O$ Reads from memory and performs inference to deduce the set of relevant memories needed to perform a good response.<br />
*$R$ would produce the actual wording of the question answer based on the memories found by $O$. For example, $R$ could be an RNN conditioned on the output of $O$<br />
<br />
When the components $I$,$G$,$O$, & $R$ are neural networks, the authors describe the resulting system as a <b>Memory Neural Network (MemNN)</b>. They build a MemNN for QA (question answering) problems and compare it to RNNs (Recurrent Neural Network) and LSTMs (Long Short Term Memory RNNs) and find that it gives superior performance.<br />
<br />
Extensions of this method include the Key-Value Memory Network model [16] which have been especially effective for utilizing KBs to answer questions. Key-value paired memories are a generalization of the way context (e.g. knowledge bases or documents to be read) are stored in memory. The lookup (addressing) stage is based on the key memory while the reading stage (giving the returned result) uses the value memory. This gives both (i) greater flexibility for the practitioner to encode prior knowledge about their task; and (ii) more effective power in the model via nontrivial transforms between key and value. An important property of the model is that the entire model can be trained with key-value transforms while still using standard back-propagation via stochastic gradient descent.<br />
<br />
[[File:DB_F2.png|center|800px]]<br />
<br />
==Related Work==<br />
<br />
'''Usefulness of feedback in language learning:''' Social interaction and natural infant directed conversations are shown to useful for language learning[2]. Several studies[3][4][5][6] have shown that feedback is especially useful in second language learning and learning by students.<br />
<br />
'''Supervised learning from dialogs using neural models:''' Neural networks has been used for response generation that can be trained end to end on large quantities of unstructured Twitter conversations[7]. However this does not incorporate feedback from dialog partner during real time conversation<br />
<br />
'''Reinforcement learning:''' Reinforcement learning works on dialogs[8][9], often consider reward as the feedback model rather than exploiting the dialog feedback per se. To be more specific, the reinforcement learning utilizes the system of rewards or what the authors from paper [8] called “trail-and-error”. The learning agent (in this case the language-learning agent) interacts with the dynamic environment (in this case through active dialog) and it receives feedback in the form of positive or negative rewards. By setting the objective function as maximizing the rewards, the model can be trained without explicit y responses. The reason why such algorithm is not particularly efficient in training a dialog-based language learning model is that there’s no explicit/fixed threshold of a positive or negative reward. One possible way to measure such action is define what a successful completion of a dialog should be and use that as the objective function. <br />
<br />
'''Forward prediction models:''' Although forward prediction models, have been used in other applications like learning eye-tracking[10], controlling robot arms[11] and vehicles[12], it has not been used for dialog.<br />
<br />
==Dialog-based Supervision tasks==<br />
For testing their models, the authors chose two datasets (i) the single supporting fact problem from the bAbI datasets [13] which consists of short stories from a simulated world followed by questions; and (ii) the MovieQA dataset [14] which is a large-scale dataset (∼ 100k questions over ∼ 75k entities) based on questions with answers in the open movie database (OMDb)<br />
<br />
However, since these datasets were not designed to model the supervision from dialogs, the authors modified them to create 10 supervision task types on these datasets(Fig 3).<br />
<br />
[[File:DB F3.png|center|700px]]<br />
<br/><br />
<br />
*'''Task 1: Imitating an Expert Student''': The dialogs take place between a teacher and an expert student who gives semantically coherent answers. Hence, the task is for the learner to imitate that expert student, and become an expert themselves <br />
<br />
*'''Task 2: Positive and Negative Feedback:''' When the learner answers a question the teacher then replies with either positive or negative feedback. In the experiments the subsequent responses are variants of “No, that’s incorrect” or “Yes, that’s right”. In the datasets there are 6 templates for positive feedback and 6 templates for negative feedback, e.g. ”Sorry, that’s not it.”, ”Wrong”, etc. To distinguish the notion of positive from negative, an additional external reward signal that is not part of the text<br />
<br />
*'''Task 3: Answers Supplied by Teacher:''' The teacher gives positive and negative feedback as in Task 2, however when the learner’s answer is incorrect, the teacher also responds with the correction. For example if “where is Mary?” is answered with the incorrect answer “bedroom” the teacher responds “No, the answer is kitchen”’<br />
<br />
*'''Task 4: Hints Supplied by Teacher:''' The corrections provided by the teacher do not provide the exact answer as in Task 3, but only a useful hint. This setting is meant to mimic the real life occurrence of being provided only partial information about what you did wrong.<br />
<br />
*'''Task 5: Supporting Facts Supplied by Teacher:''' Another way of providing partial supervision for an incorrect answer is explored. Here, the teacher gives a reason (explanation) why the answer is wrong by referring to a known fact that supports the true answer that the incorrect answer may contradict. <br />
<br />
*'''Task 6: Partial Feedback:''' External rewards are only given some of (50% of) the time for correct answers, the setting is otherwise identical to Task 3. This attempts to mimic the realistic situation of some learning being more closely supervised (a teacher rewarding you for getting some answers right) whereas other dialogs have less supervision (no external rewards). The task attempts to assess the impact of such partial supervision.<br />
<br />
*'''Task 7: No Feedback:''' External rewards are not given at all, only text, but is otherwise identical to Tasks 3 and 6. This task explores whether it is actually possible to learn how to answer at all in such a setting.<br />
<br />
*'''Task 8: Imitation and Feedback Mixture:''' Combines Tasks 1 and 2. The goal is to see if a learner can learn successfully from both forms of supervision at once. This mimics a child both observing pairs of experts talking (Task 1) while also trying to talk (Task 2).<br />
<br />
*'''Task 9: Asking For Corrections:''' The learner will ask questions to the teacher about what it has done wrong. Task 9 tests one of the most simple instances, where asking “Can you help me?” when wrong obtains from the teacher the correct answer.<br />
<br />
*'''Task 10: Asking for Supporting Facts:''' A less direct form of supervision for the learner after asking for help is to receive a hint rather than the correct answer, such as “A relevant fact is John moved to the bathroom” when asking “Can you help me?”. This is thus related to the supervision in Task 5 except the learner must request help<br />
<br />
[[File:F4.png|center|700px]]<br />
<br/><br />
<br />
For each task a fixed policy is considered for performing actions (answering questions) which gets questions correct with probability $π_{acc}$ (i.e. the chance of getting the red text correct in Figs. 3 and 4). We thus can compare different learning algorithms for each task over different values of $π_{acc}$ (0.5, 0.1 and 0.01). In all cases a training, validation and test set is provided. For the bAbI dataset this consists of 1000, 100 and 1000 questions respectively per task, and for movieQA there are ∼ 96k, ∼ 10k and ∼ 10k respectively. MovieQA also includes a knowledge base (KB) of ∼ 85k facts from OMDB(Open Movie Database), the memory network model we employ uses inverted index retrieval based on the question to form relevant memories from this set.<br />
<br />
==Learning models==<br />
This work evaluates four possible learning strategies for each of the 10 tasks: imitation learning, reward-based imitation, forward prediction, and a combination of reward-based imitation and forward prediction<br />
<br />
All of these approaches are evaluated with the same model architecture: an end-to-end memory network (MemN2N) [15], which has been used as a baseline model for exploring differing modes of learning.<br />
<br />
[[File:F5.png|center|700px]]<br />
<br/><br />
The input is the last utterance of the dialog, $x$, as well as a set of memories (context) (<math> c_1</math>, . . . , <math> c_n</math> ) which can encode both short-term memory, e.g. recent previous utterances and replies, and long-term memories, e.g. facts that could be useful for answering questions. The context inputs <math> c_i</math> are converted into vectors <math> m_i</math> via embeddings and are stored in the memory. The goal is to produce an output $\hat{a}$ by processing the input $x$ and using that to address and read from the memory, $m$, possibly multiple times, in order to form a coherent reply. In the figure the memory is read twice, which is termed multiple “hops” of attention. <br />
<br />
In the first step, the input $x$ is embedded using a matrix $A$ of size $d$ × $V$ where $d$ is the embedding dimension and $V$ is the size of the vocabulary, giving $q$ = $A$$x$, where the input $x$ is as a bag-of words vector. Each memory <math> c_i</math> is embedded using the same matrix, giving $m_i$ = $A$$c_i$ . The output of addressing and then reading from memory in the first hop is: <br />
<br />
[[File:eq1.png|center|400px]]<br />
<br />
Here, $p^{1}$ is a probability vector over the memories, and is a measure of how much the input and the memories match. The goal is to select memories relevant to the last utterance $x$, i.e. the most relevant have large values of $p^{1}_i$ . The output memory representation $o_1$ is then constructed using the weighted sum of memories, i.e. weighted by $p^{1}$ . The memory output is then added to the original input, <math> c_1</math> = <math> R_1</math>(<math> o_1</math> + $q$), to form the new state of the controller, where <math> R_1</math> is a $d$ × $d$ rotation matrix . The attention over the memory can then be repeated using <math> u_1</math> instead of $q$ as the addressing vector, yielding: <br />
<br />
[[File:eq2.png|center|400px]]<br />
<br />
The controller state is updated again with <math> u_2</math> = <math> R_2</math>(<math> o_2</math> + <math> u_1</math>), where <math> R_2</math> is another $d$ × $d$ matrix to be learnt. In a two-hop model the final output is then defined as: <br />
<br />
[[File:eq3.png|center|400px]]<br />
<br />
where there are $C$ candidate answers in $y$. In our experiments $C$ is the set of actions that occur in the training set for the bAbI tasks, and for MovieQA it is the set of words retrieved from the KB.<br />
<br />
==Training strategies==<br />
*'''1. Imitation Learning'''<br />
This approach involves simply imitating one of the speakers in observed dialogs. Examples arrive as $(x, c, a)$ triples, where $a$ is (assumed to be) a good response to the last utterance $x$ given context $c$. Here, the whole memory network model defined above is trained using stochastic gradient descent by minimizing a standard cross-entropy loss between $\hat{a}$ and the label $a$<br />
<br />
*'''2. Reward-based Imitation''' <br />
If some actions are poor choices, then one does not want to repeat them, that is we shouldn’t treat them as a supervised objective. Here, positive reward is only obtained immediately after (some of) the correct actions, or else is zero. Only apply imitation learning on the rewarded actions. The rest of the actions are simply discarded from the training set. <br />
<br />
*'''3. Forward Prediction''' <br />
The aim is, given an utterance $x$ from speaker 1 and an answer $a$ by speaker 2 (i.e., the learner), to predict $\bar{x}$, the response to the answer from speaker 1. That is, in general to predict the changed state of the world after action $a$, which in this case involves the new utterance $\bar{x}$.<br />
<br />
[[File:F6.png|center|700px]]<br />
<br/><br />
As shown in Figure (b), this is achieved by chopping off the final output from the original network of Fig (a) and replace it with some additional layers that compute the forward prediction. The first part of the network remains exactly the same and only has access to input x and context c, just as before. The computation up to $u_2$ = $R_2$($o_2$ + $u_1$) is thus exactly the same as before. <br />
<br />
Then perform another “hop” of attention but over the candidate answers rather than the memories. The information of which action (candidate) was actually selected in the dialog (i.e. which one is a) is also incorporated which is crucial. After this “hop”, the resulting state of the controller is then used to do the forward prediction.<br />
<br />
Concretely, we compute: <br />
<br />
[[File:eq4.png|center|550px]]<br />
<br />
where $β^{*}$ is a d-dimensional vector, that is also learnt, that represents in the output $o_3$ the action that was actually selected. The degree to which $β^{*}$ is represented in $o_3$ is determined by whether the answer a given in the dialog is correct, and how accurately the model can identify the true correct response (i.e. how much probability is placed on the correct response). After obtaining $o_3$, the forward prediction is then computed as: <br />
<br />
[[File:eq5.png|center|500px]]<br />
<br />
where $u_3$ = $R_3$($o_3$ + $u_2$). That is, it computes the scores of the possible responses to the answer a over $C^{¯}$ possible candidates.<br />
<br />
<br />
*'''4. Reward-based Imitation + Forward Prediction'''<br />
As the reward-based imitation learning uses the architecture of Fig (a), and forward prediction uses the same architecture but with the additional layers of Fig (b), we can learn jointly with both strategies. This is a powerful combination as it makes use of reward signal when available and the dialog feedback when the reward signal is not available.<br />
<br />
==Experiments==<br />
<br />
Experiments were conducted on the two test datasets - babi and MovieQA. For each task, a fixed policy is considered for performing actions (answering questions) which gets questions correct with probability $π_{acc}$. This helps to compare the different training strategies described earlier over each task for different values of $π_{acc}$. Hyperparameters for all methods are optimized on the validation sets.<br />
<br/><br/><br />
[[File:DB F7.png|center|800px]]<br />
<br/><br />
The following results are observed by the authors:<br />
*Imitation learning, ignoring rewards, is a poor learning strategy when imitating inaccurate answers, e.g. for $\pi_{acc}$ < 0.5. For imitating an expert however (Task 1) it is hard to beat. <br />
*Reward-based imitation (RBI) performs better when rewards are available, particularly in Table 1, but also degrades when they are too sparse e.g. for πacc = 0.01.<br />
*Forward prediction (FP) is more robust and has stable performance at different levels of $\pi_{acc}$. However as it only predicts answers implicitly and does not make use of rewards, it is outperformed by RBI on several tasks, notably Tasks 1 and 8 (because it cannot do supervised learning) and Task 2 (because it does not take advantage of positive rewards).<br />
*FP makes use of dialog feedback in Tasks 3-5 whereas RBI does not. This explains why FP does better with useful feedback (Tasks 3-5) than without (Task 2), whereas RBI cannot.<br />
*Supplying full answers (Task 3) is more useful than hints (Task 4) but hints still help FP more than just yes/no answers without extra information (Task 2).<br />
*When positive feedback is sometimes missing (Task 6) RBI suffers especially in Table 1. FP does not as it does not use this feedback.<br />
*One of the most surprising results of our experiments is that FP performs well overall, given that it does not use feedback, which we will attempt to explain subsequently. This is particularly evident on Task 7 (no feedback) where RBI has no hope of succeeding as it has no positive examples. FP on the other hand learns adequately.<br />
*Tasks 9 and 10 are harder for FP as the question is not immediately before the feedback.<br />
*Combining RBI and FP ameliorates the failings of each, yielding the best overall results<br />
*An interesting thing is that FP learns the relationship of words, e.g. "right" and "correct", without any rewards.<br />
<br />
==Future work==<br />
* Any reply in a dialog can be seen as feedback, and should be useful for learning. Evaluate if forward prediction, and the other approaches in this paper, work there too. <br />
* Develop further evaluation methodologies to test how the models presented here work in more complex settings where actions that are made lead to long-term changes in the environment and delayed rewards, i.e. extending to the reinforcement learning setting, and to full language generation. <br />
* How dialog-based feedback could also be used as a medium to learn non-dialog based skills, e.g. natural language dialog for completing visual or physical tasks.<br />
<br />
==References==<br />
# Jason Weston. Dialog-based Language Learning. NIPS, 2016.<br />
# P. K. Kuhl. Early language acquisition: cracking the speech code. Nature reviews neuroscience, 5(11): 831–843, 2004.<br />
# M. A. Bassiri. Interactional feedback and the impact of attitude and motivation on noticing l2 form. English Language and Literature Studies, 1(2):61, 2011.<br />
# R. Higgins, P. Hartley, and A. Skelton. The conscientious consumer: Reconsidering the role of assessment feedback in student learning. Studies in higher education, 27(1):53–64, 2002.<br />
# A. S. Latham. Learning through feedback. Educational Leadership, 54(8):86–87, 1997.<br />
# M. G. Werts, M. Wolery, A. Holcombe, and D. L. Gast. Instructive feedback: Review of parameters and effects. Journal of Behavioral Education, 5(1):55–75, 1995.<br />
# A. Sordoni, M. Galley, M. Auli, C. Brockett, Y. Ji, M. Mitchell, J.-Y. Nie, J. Gao, and B. Dolan. A neural network approach to context-sensitive generation of conversational responses. Proceedings of NAACL, 2015.<br />
# V. Rieser and O. Lemon. Reinforcement learning for adaptive dialogue systems: a data-driven methodology for dialogue management and natural language generation. Springer Science & Business Media, 2011.<br />
# J. Schatzmann, K. Weilhammer, M. Stuttle, and S. Young. A survey of statistical user simulation techniques for reinforcement-learning of dialogue management strategies. The knowledge engineering review, 21(02):97–126, 2006.<br />
# J. Schmidhuber and R. Huber. Learning to generate artificial fovea trajectories for target detection. International Journal of Neural Systems, 2(01n02):125–134, 1991.<br />
# I. Lenz, R. Knepper, and A. Saxena. Deepmpc: Learning deep latent features for model predictive control. In Robotics Science and Systems (RSS), 2015.<br />
# G. Wayne and L. Abbott. Hierarchical control using networks trained with higher-level forward models. Neural computation, 2014.<br />
# B. C. Stadie, S. Levine, and P. Abbeel. Incentivizing exploration in reinforcement learning with deep predictive models. arXiv preprint arXiv:1507.00814, 2015.<br />
# J. Clarke, D. Goldwasser, M.-W. Chang, and D. Roth. Driving semantic parsing from the world’s response. In Proceedings of the fourteenth conference on computational natural language learning, pages 18–27. Association for Computational Linguistics, 2010.<br />
# J. Schatzmann, K. Weilhammer, M. Stuttle, and S. Young. A survey of statistical user simulation techniques for reinforcement-learning of dialogue management strategies. The knowledge engineering review, 21(02):97–126, 2006.<br />
# Miller, Alexander, et al. "Key-value memory networks for directly reading documents." arXiv preprint arXiv:1606.03126 (2016).</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Coupled_GAN&diff=30006STAT946F17/ Coupled GAN2017-11-12T20:03:33Z<p>H4lyu: /* Coupled GAN (CoGAN) Framework and Learning */</p>
<hr />
<div><br />
This is a summary of NIPS 2016 paper [1].<br />
== Introduction==<br />
<br />
Generative models attempt to characterize and estimate the underlying probability distribution of the data (typically images) and in doing so generate samples from the aforementioned learned distribution. Moment-matching generative networks, Variational auto-encoders, and Generative Adversarial Networks (GANs) are some of the most popular (and recent) class of techniques in this burgeoning literature on generative models. The authors of the paper we are reviewing focus on proposing an extension to the class of GANs.<br />
<br />
The novelty of the proposed Coupled GAN (CoGAN) method lies in extending the GAN procedure (described in the next section) to the multi-domain setting. That is, the CoGAN methodology attempts to learn the (underlying) joint probability distribution of multi-domain images as a natural extension from the marginal setting associated with the vanilla GAN framework. This is inspired by the idea that deep neural networks learn a hierarchical feature representation. Given the dense and active literature on generative models, generating images in multiple domains is far from groundbreaking. Related works revolve around multi-modal deep learning ([2],[3]), semi-coupled dictionary learning ([4]), joint embedding space learning ([5]), cross-domain image generation ([6],[7]) to name a few. Thus, the novelty of the authors' contributions to this field comes from two key differentiating points. Firstly, this was (one of) the first papers to endeavor to generate multi-domain images with the GAN framework. Secondly, and perhaps more significantly, the authors proposed to learn the underlying joint distribution without requiring the presence of tuples of corresponding images in the training set. Only sets of images drawn from the (marginal) distributions of the separate domains is sufficient. As per the authors' claim, constructing tuples of corresponding images to train from is challenging and a potential bottle-neck for multi-domain image generation. One way around this bottleneck is thus to use their proposed CoGAN methodology. More details of how the authors achieve joint-distribution learning will be provided in the Coupled GAN section below.<br />
<br />
== Generative Adversarial Networks==<br />
<br />
A typical GAN framework consists of a generative model and a discriminative model. The generative model, which often is a de-convolutional network, takes as input a random ''latent'' vector (typically uniform or Gaussian) and synthesizes novel images resembling the real images (training set). The discriminative model, often a convolutional network, on the other hand, tries to distinguish between the fake synthesized images and the real images. The idea then is to let the two component models of the GAN framework "compete" with each other in the form of a min-max two player game. <br />
<br />
To further clarify and fix this idea, we introduce the mathematical setup of GANs following the notation used by the authors of this paper for sake of consistency. Let us define the following in our setup:<br />
<br />
:<math> \mathbf{x}-</math> natural image drawn from underlying distribution <math> p_X</math>,<br />
:<math> \mathbf{z} \sim U[-1,1]^d-</math> a latent random vector,<br />
:$g-$ generative model, $f-$ discriminative model.<br />
<br />
Ideally we are aiming for the system of these two ''adversarial'' networks to behave as:<br />
:Generator: $g(\mathbf{z})$ outputs an image with same support as $\mathbf{x}$. The probability density of the images output by $g$ can be denoted by $p_G$,<br />
:Discriminator: $f(\mathbf{x})=1$ if $\mathbf{x} \sim p_X$ and $f(\mathbf{x})=0$ if $\mathbf{x} \sim p_G$.<br />
<br />
To train such a system of networks given our goal (i.e., $p_G \rightarrow p_X$) we must treat such a framework as the following minimax two player game:<br />
<br />
$\displaystyle \max_{g}$<br />
$\min\limits_{f} V(g,f) = \mathop{\mathbb{E}}_{\mathbf{x} \sim p_X}[-\log(f(\mathbf{x}))] + \mathop{\mathbb{E}}_{\mathbf{z} \sim p_{Z}(\mathbf{z})}[-\log(1-f(g(\mathbf{z})))] $.<br />
<br />
See [8], the seminal paper on this topic, for more information.<br />
<br />
== Coupled Generative Adversarial Networks==<br />
<br />
The overarching goal of this framework is to learn a joint distribution of multi-domain images from data. That is, a density value is assigned to each joint occurrence of images in different domains. Examples of such pair of images in different domains include images of a particular scene with different modalities (color and depth) or images of the same face but with different facial attributes. <br />
<br />
To this end, the CoGAN setup consists of a pair of GANs, denoted as $GAN_1$ and $GAN_2$. Each GAN is tasked with synthesizing images in one domain. A naive training of such a system will result in learning the product of the two marginal distributions i.e., independence. However, by forcing the two GANs to share weights, the authors were able to demonstrate that they could in ''some sense'' learn the joint distribution of images. We will now describe the details of the generator and discriminator components of the setup and conclude this section with a summary of CoGAN learning algorithm.<br />
<br />
===Generator Models===<br />
<br />
Suppose $\mathbf{x_1} \sim p_{X_1}$ and $\mathbf{x_2} \sim p_{X_2}$ denote the natural images being drawn from the two marginal distributions of <br />
domain 1 and domain 2. Further, let $g_1$ be the generator of $GAN_1$ and $g_2$ be the generator of $GAN_2$. Both these generators take as input the latent vector $\mathbf{z}$ as defined in the previous section as input and out images in their specific domains. For completeness, denote the distributions of $g_1(\mathbf{z})$ and $g_2(\mathbf{z})$ as $p_{G_1}$ and $p_{G_2}$ respectively. We can characterize these two generator models as multi-layer perceptrons in the following way:<br />
<br />
\begin{align*}<br />
g_1(\mathbf{z})=g_1^{(m_1)}(g_1^{(m_1 -1)}(\dots g_1^{(2)}(g_1^{(1)}(\mathbf{z})))), \quad g_2(\mathbf{z})=g_2^{(m_2)}(g_2^{(m_2-1)}(\dots g_2^{(2)}(g_2^{(1)}(\mathbf{z})))),<br />
\end{align*}<br />
where $g_1^{(i)}$ and $g_2^{(i)}$ are the $i^{th}$ layers of $g_1$ and $g_2$ which respectively have a total of $m_1$ and $m_2$ layers each. Note $m_1$ need not be the same as $m_2$.<br />
<br />
As the generator networks can be thought of as an inverse of the prototypical convolutional networks (just as an example), the layers of these generator networks gradually decodes information from high-level abstract concepts to low-level details(last few layers). Taking this idea as the blueprint for the inner-workings of generator networks, the author's hypothesize that corresponding images in two domains share the same high-level semantics but with differing lower-level details. To put this hypothesis to practice, they forced the first $k$ layers of $g_1$ and $g_2$ to have identical structures and share the same weights. That is, $\mathbf{\theta}_{g_1^{(i)}}=\mathbf{\theta}_{g_2^{(i)}}$ for $i=1,\dots,k$ where $\mathbf{\theta}_{g_1^{(i)}}$ and $\mathbf{\theta}_{g_1^{(i)}}$ represents the parameters of the layers $g_1^{(i)}$ and $g_2^{(i)}$ respectively. Hence the two generator networks share the starting $k$ layers of the deep network and have different last layers to decode the differing material details in each domain.<br />
<br />
===Discriminative Models===<br />
<br />
Suppose $f_1$ and $f_2$ are the respective discriminative models of the two GANs. These models can be characterized by <br />
\begin{align*}<br />
f_1(\mathbf{x}_1)=f_1^{(n_1)}(f_1^{(n_1 -1)}(\dots f_1^{(2)}(f_1^{(1)}(\mathbf{x}_1)))), \quad f_2(\mathbf{x}_2)=f_2^{(n_2)}(f_2^{(n_2-1)}(\dots f_2^{(2)}(f_2^{(1)}(\mathbf{x}_1)))),<br />
\end{align*}<br />
where $f_1^{(i)}$ and $f_2^{(i)}$ are the $i^{th}$ layers of $f_1$ and $f_2$ which respectively have a total of $n_1$ and $n_2$ layers each. Note $n_1$ need not be the same as $n_2$. In contrast to generator models, the first layers of $f_1$ and $f_2$ extract the lower level details where the last layers extract the abstract higher level details. To reflect the prior hypothesis of shared higher level semantics between corresponding images, we can force $f_1$ and $f_2$ to now share the weights for last $l$ layers. That is, $\mathbf{\theta}_{f_1^{(n_1-i)}}=\mathbf{\theta}_{f_2^{(n_2-i)}}$ for $i=0,\dots,l-1$ where $\mathbf{\theta}_{f_1^{(i)}}$ and $\mathbf{\theta}_{f_1^{(i)}}$ represents the parameters of the layers $f_1^{(i)}$ and $f_2^{(i)}$ respectively. Unlike in the generative models, weight sharing in the discriminative models is not essential to estimating the joint distribution of images, however it is beneficial by reducing the total number of parameters in the network.<br />
<br />
===Coupled GAN (CoGAN) Framework and Learning===<br />
The following figure taken from the paper summarizes the system of models described in the previous subsections. <br />
<center><br />
[[File:CoGAN-1.PNG]]<br />
</center><br />
The CoGAN framework can be expressed as the following constrained min-max game<br />
<br />
\begin{align*}<br />
\max\limits_{g_1,g_2} \min\limits_{f_1, f_2} V(f_1,f_2,g_1,g_2)\quad \text{subject to} \ \mathbf{\theta}_{g_1^{(i)}}=\mathbf{\theta}_{g_2^{(i)}}, i=1,\dots k, \quad \mathbf{\theta}_{f_1^{(n_1-j)}}=\mathbf{\theta}_{f_2^{(n_2-j)}}, j=1,\dots,l-1, <br />
\end{align*}<br />
where the value function V is characterized as <br />
\begin{align*}<br />
\mathop{\mathbb{E}}_{\mathbf{x}_1 \sim p_{X_1}}[-\log(f_1(\mathbf{x_1}))] + \mathop{\mathbb{E}}_{\mathbf{z} \sim p_{Z}(\mathbf{z})}[-\log(1-f_1(g_1(\mathbf{z})))]+\mathop{\mathbb{E}}_{\mathbf{x}_2 \sim p_{X_2}}[-\log(f_2(\mathbf{\mathbf{x}_2}))] + \mathop{\mathbb{E}}_{\mathbf{z} \sim p_{Z}(\mathbf{z})}[-\log(1-f_2(g_2(\mathbf{z})))]. <br />
\end{align*}<br />
<br />
For the purposes of storytelling, we can describe this game to have two teams with two players each. The generative models are in the same team and collaborate with each other to synthesize a pair of images in two different domains with goal of fooling the discriminative models. The discriminative models than with collaboration try to differentiate between images drawn from the training data in their respective domains and the images generated by the respective generative models. The training algorithm for the CoGAN that was used is described in the following figure.<br />
<center><br />
[[File:CoGAN-2.PNG]]<br />
</center><br />
<br />
'''Important Remarks:''' <br />
<br />
CoGAN learning requires training samples drawn from the marginal distributions, $p_{X_1}$ and $p_{X_2}$ . It does not rely on samples drawn from the joint distribution, $p_{X_1,X_2}$ , where corresponding supervision would be available. Here, the main contribution is in showing that with just samples drawn separately from the marginal distributions, CoGAN can learn a joint distribution of images in the two domains. Both weight-sharing constraint and adversarial training are essential for enabling this capability. <br />
<br />
Unlike autoencoder learning which encourages the generated image pair to be identical to the target pair , the adversarial training only encourages the generated pair of images to be individually resembling the images in the respective domains, and ignore the correlation between them. Shared parameters, on the other hand, contribute to matching the correlation: the neurons responsible for decoding high-level semantics can be shared to produce highly correlated image pairs.<br />
<br />
==Experiments==<br />
<br />
To begin with, note that the authors do not use corresponding images in the training set in accordance with the goal of ''learning'' the joint distribution of multi-domain images without correspondence supervision. As at the time the paper was written, there were no existing approached with identical prerogatives (i.e., training with no correspondence supervision), they compared CoGAN with conditional GAN (see [10]) for more details on conditional GAN).<br />
<br />
===MNIST Dataset===<br />
<br />
The MNIST training set was experimented with two tasks:<br />
<br />
# Task A: Learning a joint distribution of a digit and its edge image. <br />
# Task B: Learning a joint distribution of a digit and its negative image. <br />
<br />
For the generative models, the authors used convolutional networks with 5 identical layers. They varied the number of shared layers as part of their experimental setup. The two discriminative models were a version of the LeNet ([9]). The results of the CoGAN generation scheme are displayed in the figure below.<br />
<br />
<center><br />
[[File:CoGAN-3.PNG]]<br />
</center><br />
As you can see from the figure above, the CoGAN system was able to generate pairs of corresponding images without explicitly training with correspondence supervision. This was naturally due to sharing weights in lower levels used for decoding high-level semantics. Without sharing these weights, the CoGAN would just output a pair of unrelated images in the two domains.<br />
<br />
To investigate the effects of weight sharing in the generator/discriminator models used for both tasks, the authors varied the number of shared levels. To quantify the performance of the generator, the image generated by $GAN_1$ (domain 1) was transformed to the 2nd domain using the same method used to generate training images in the 2nd domain. Then this transformed image was compared with the image generated by $GAN_2$. Naturally, if the joint distribution was learned completely, these two images would be identical. With that goal in mind, the authors used pixel agreement ratios for 10000 images as the evaluation metric. In particular, 5 trails with different weight initializations were used and an average pixel agreement ratio was taken. The results depicting the relationship between the average pixel agreement ratio and a number of shared layers are summarized in the figure below. <br />
<center><br />
[[File:CoGAN-4.PNG]]<br />
</center><br />
The results naturally offered some corroboration to our intuitions. The number of shared layers in the generator models, the higher the pixel agreement ratios. Interestingly the number of shared layers in the discriminative model does not seem to affect the pixel agreement ratios. Note this is a pretty naive and toy example as we by nature of the evaluation criteria have a deterministic way of generating an image in the 2nd domain.<br />
<br />
Finally, for this example, the authors compared the CoGAN framework with the conditional GAN model. For the conditional GAN, the generative and discriminative models were identical to those used for the CoGAN results. The conditional GAN additionally took a binary variable (the conditioning variable) as input. When the binary variable was 0, the conditional GAN synthesized an image in domain 1 and when it was 1 an image in domain 2. Naturally, for a fair comparison, the training set did not already contain corresponding pairs of images. The experiments were conducted for the two tasks described above and the pixel agreement ratio (PAR) was the evaluation criteria. For Task A, the CoGAN resulted in a PAR of 0.952 in comparison with 0.909 for the conditional GAN. For Task B, the CoGAN resulted in a PAR of 0.967 compared with a PAR of 0.778 for the conditional GAN. The results are not particularly eye opening as the CoGAN was more specifically designed for purpose of learning the joint distribution of multi-domain images whereas these tasks are just a very niche application for the conditional GAN. Nevertheless, for Task B, the results look promising.<br />
<br />
=== CelebFaces Attributes Dataset===<br />
<br />
For this experiment, the authors trained the CoGAN, using the CelebFaces Attributes Dataset, to generate pairs of faces with an attribute (domain 1) and without the attribute (domain 2). CelebFaces Attributes Dataset (CelebA) is a large-scale face attributes dataset with more than 200K celebrity images, each with 40 attribute annotations. The images in this dataset cover large pose variations and background clutter. CelebA has large diversities, large quantities, and rich annotations, including 10,177 number of identities, 202,599 number of face images, and 5 landmark locations, 40 binary attributes annotations per image.<br />
Convolutional networks with 7 layers for both the generative and discriminative models were used. The dataset contains a large variety of poses and background clutter. Attributes can include blonde/non-blonde hair, smiling/ not smiling, or with/without sunglasses for example. The resulting synthesized pair of images are shown in the figure below realized as spectrum traveling from one point to another (resembles changing faces).<br />
<br />
<center><br />
[[File:CoGAN-5.PNG]]<br />
</center><br />
<br />
=== Color and Depth Images===<br />
For this experiment the authors used two sources: the RGBD dataset and the NYU dataset. The two image domains (color-domain 1 and depth-domain 2) under consideration are the same for both the datasets. As usual, no corresponding images were fed into the training of the CoGAN framework. The resulting rendering of pairs of color and depth images for both the datasets are depicted in the figure below. <br />
<br />
<center><br />
[[File:CoGAN-6.PNG]]<br />
</center><br />
<br />
As is evident from the images, through the sharing of layers during training, the CoGAN was able to learn the appearance-depth correspondence.<br />
<br />
== Applications==<br />
<br />
===Unsupervised Domain Adaptation (UDA)===<br />
<br />
UDA involves adapting a classifier trained in one domain to conduct a classification task in a new domain which only contains ''unlabeled'' training data which disqualifies re-training of the classifier in this new domain. Some prior work in the field includes subspace learning ([11],[12]) and deep discriminative network learning ([13],[14]). The authors in the paper experimented with the MNIST and USPS datasets to showcase the applicability of the CoGAN framework for the UDA problem. A similar network architecture as was used for the MNIST experiment was employed for this application. The MNIST and USPS datasets has been denoted as $D_1$ and $D_2$ respectively in the paper. In accordance with the problem specification, no label information was used from $D_2$. <br />
<br />
The CoGAN is trained by jointly solving the classification problem in the MNIST domain using the labels provided in $D_1$ and the CoGAN learning problem which uses images in both $D_1$ and $D_2$. This training process produces two classifiers. That is, $c_1(x_1)=c(f_1^{(3)}(f_1^{(2)}(f_1^{(1)}(x_1))))$ for MNIST and $c_2(x_2)=c(f_2^{(3)}(f_2^{(2)}(f_2^{(1)}(x_2))))$ and USPS. Note $c()$ here denotes the softmax layer which is added on top of the other layers in the respective discriminative networks of the GANs. Further due to weight sharing the last two layers of the discriminative model would be identical and have the same weights. The classifier $c_2$ is then used for digit classification in the USPS dataset. The author reported a 91.2% average accuracy when classifying the USPS dataset. The mirror problem of classifying the MNIST dataset without labels using the fully characterized USPS dataset achieved an average accuracy of 89.1%. These results appear to significantly outperform (prior top classification accuracy lies roughly around 60-65%) what the authors ''claim'' to be the state of the art methods in the UDA literature. In particular the state of the art was noted to be described in [20].<br />
<br />
===Cross-Domain Image Transformation===<br />
<br />
Let $\mathbf{x}_1$ be a image in the 1st domain. The goal then is to find a corresponding image $\mathbf{x}_2$ in 2nd domain such that the joint probability density $p(\mathbf{x}_1,\mathbf{x}_2)$ is maximized. Given the two generators $g_1$ and $g_2$, one can achieve the cross-domain transformation by first finding the latent random vector that generates the input image $\mathbf{x}_1$ in the 1st domain. This amounts to the optimization: $\mathbf{z}^{*}=argmin_{\mathbf{z}}L(g_1(\mathbf{z}),\mathbf{x}_1)$, where $L$ is the loss function measuring the difference/distance between the two images. After finding $z^*$, one can apply $g_2$ to obtain the transformed image, $x_2 = g_2(z^*)$. Some very preliminary results were provided by the authors but there is nothing noteworthy to warrant discussion. The figure depicting their results is provided below for sake of completeness.<br />
[[File:CoGAN-7.PNG]]<br />
<br />
== Discussion and Summary==<br />
The main contributions of the paper can be summarized as:<br />
<br />
# A CoGAN framework for learning a joint distribution of multi-domain images was proposed. <br />
# The training is achieved by a simple weight sharing scheme for the generative and discriminative networks in the absence of any correspondence supervision in the training set. This can be construed as learning the joint distribution by using samples from the marginal distribution of images. <br />
# The experiments with digits, faces, and color/depth images provided some corroboration that the CoGAN system could synthesize corresponding pairs of images. <br />
# An application of the CoGAN framework for the problem of Unsupervised Domain Adaptation (UDA) was introduced. The preliminary results appear to be extremely promising for the task of adapting digit classifiers from MNIST to USPS data and vice-versa. <br />
#An application for the task of cross-domain image transformation was hypothesized with some very basic proof of concept results provided. <br />
# The setup naturally lends to more than the two domain setting focused on in the paper for experimental purposes. <br />
<br />
While the summary provided above adopted an objective filter, the following list enumerates the major ''subjective'' critical review points for this paper:<br />
# It appears the authors took components of various well-established techniques in the literature and produced the CoGAN framework. Weight-sharing is a well documented idea as was correspondance/ multi-modal learning along with the GAN problem formation and training. However, when the components are put together in this way, they form a modest and timely novel contribution to the literature of generative networks.<br />
# With such prominent preliminary results for the problem of UDA, the authors could have provided some additional details of their training procedure (slightly unclear) and additional experiments under the UDA umbrella to fortify what appears to be a ''ground breaking'' result when compared with state of the art methods. <br />
# The cross-domain image transformation application example was almost an after thought. More details could have been provided in the supplementary file if pressed for space or perhaps just merely relegated to a follow-up paper/work.<br />
<br />
== References and Supplementary Resources==<br />
:[1] Liu, Ming-Yu, and Oncel Tuzel. "Coupled generative adversarial networks." Advances in neural information processing systems. 2016.<br />
:[2] Srivastava, Nitish, and Ruslan R. Salakhutdinov. "Multimodal learning with deep boltzmann machines." Advances in neural information processing systems. 2012.<br />
:[3] Ngiam, Jiquan, et al. "Multimodal deep learning." Proceedings of the 28th international conference on machine learning (ICML-11). 2011.<br />
:[4] Wang, Shenlong, et al. "Semi-coupled dictionary learning with applications to image super-resolution and photo-sketch synthesis." Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference on. IEEE, 2012.<br />
:[5] Kiros, Ryan, Ruslan Salakhutdinov, and Richard S. Zemel. "Unifying visual-semantic embeddings with multimodal neural language models." arXiv preprint arXiv:1411.2539 (2014).<br />
:[6] Yim, Junho, et al. "Rotating your face using multi-task deep neural network." Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2015.<br />
:[7] Reed, Scott E., et al. "Deep visual analogy-making." Advances in neural information processing systems. 2015. <br />
:[8] Goodfellow, Ian, et al. "Generative adversarial nets." Advances in neural information processing systems. 2014.<br />
:[9] LeCun, Yann, et al. "Gradient-based learning applied to document recognition." Proceedings of the IEEE 86.11 (1998): 2278-2324.<br />
:[10] Mirza, Mehdi, and Simon Osindero. "Conditional generative adversarial nets." arXiv preprint arXiv:1411.1784 (2014).<br />
:[11] Long, Mingsheng, et al. "Transfer feature learning with joint distribution adaptation." Proceedings of the IEEE international conference on computer vision. 2013.<br />
:[12] Fernando, Basura, Tatiana Tommasi, and Tinne Tuytelaars. "Joint cross-domain classification and subspace learning for unsupervised adaptation." Pattern Recognition Letters 65 (2015): 60-66. <br />
:[13] Tzeng, Eric, et al. "Deep domain confusion: Maximizing for domain invariance." arXiv preprint arXiv:1412.3474 (2014).<br />
:[14] Rozantsev, Artem, Mathieu Salzmann, and Pascal Fua. "Beyond sharing weights for deep domain adaptation." arXiv preprint arXiv:1603.06432 (2016).<br />
:[15] http://mmlab.ie.cuhk.edu.hk/projects/CelebA.html<br />
<br />
Implementation Example: https://github.com/mingyuliutw/cogan</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/Cognitive_Psychology_For_Deep_Neural_Networks:_A_Shape_Bias_Case_Study&diff=29754STAT946F17/Cognitive Psychology For Deep Neural Networks: A Shape Bias Case Study2017-11-09T00:06:44Z<p>H4lyu: /* Inductive Biases & Probe Data */</p>
<hr />
<div>= Introduction =<br />
<br />
The recent burgeon on the use of Deep Neural Networks (DNNs) have resulted in giant leaps of accuracy in prediction. They are also being used to solve a variety of complex tasks which earlier methodologies have struggled to excel in.<br />
<br />
While it is all good to see incredibly high accuracy as a result of the use of DNN, we must begin to question why they perform so well. It has become an interesting field of study to actually represent the features/feature maps or interpret the meaning of the learnt values in a DNN's hidden layers. Currently we treat models of DNNs as black boxes which we practically tune the tweakable parameters like number of layers, number of units in each layer, number & size of feature maps(in case of CNN) etc. The opacity created by the lack of an intuitive representation of the internal learnt parameters of DNNs hinders both basic research as well as its application to real world problems.<br />
<br />
Recent pushes have aimed to better understand DNNs: tailor-made loss functions and architectures produce more interpretable features (Higgins et al., 2016; Raposo et al., 2017) while output-behavior analyses unveil previously opaque operations of these networks (Karpathy et al., 2015). Parallel to this work, neuroscience-inspired methods such as activation visualization (Li et al., 2015), ablation analysis (Zeiler & Fergus, 2014) and activation maximization (Yosinski et al., 2015) have also been applied<br />
<br />
This paper aims to provide another methodology to attempt to decipher & better understand how DNNs solve a particular task. This methodology was inspired by psychological concepts to test whether the DNN's were able to make accurate predictions with biases similar to that the human mind makes.<br />
<br />
Research in developmental psychology shows that when learning new words, humans tend to assign the same name to similarly shaped items rather than to items with similar color, texture, or size. This bias/knowledge tend to be forged into the brains of humans and humans then take this forward to easily associate these shapes with new objects they have not seen before.<br />
<br />
The authors of this paper try to simulate if DNNs behave similarly in one-shot learning applications. They attempt to prove that when the models of state-of-the-art DNNs are used to learn objects from images, they exhibit a stronger shape bias than a color bias. To emulate the human brain, they use the parameters of pre-trained DNN models and use this to perform one-shot learning on a new data set with different labels.<br />
<br />
= Background =<br />
== One Shot Learning ==<br />
One-shot learning is an object categorization problem in computer vision. Whereas most machine learning based object categorization algorithms require training on hundreds or thousands of images and very large datasets, one-shot learning aims to learn information about object categories from one, or only a few, training images.<br />
<br />
The one-shot word learning task is to label a novel data example $\hat{x}$ (e.g. a novel probe image) with a novel class label $\hat{y}$ (e.g. a new word) after only a single example.<br />
<br />
More specifically, given a support set $S = {(x_i, y_i) , i \in [1, k]}$, of images $x_i$, and their associated labels $y_i$, and an unlabeled probe image $\hat{x}$,<br />
the one-shot learning task is to identify the true label of the probe image, $\hat{y}$, from the support set labels $ {y_i , i \in [1, k]} $:<br />
<br />
<br />
$\displaystyle \hat{y} = arg \max_{y}$ $P(y | \hat{x}, S)$<br />
<br />
<br />
We assume that the image labels $y_i$ are represented using a one-hot encoding and that $P(y|\hat{x}, S)$ is parameterised by a DNN, allowing us to leverage the ability of deep networks to learn powerful representations.<br />
<br />
== Inception Networks ==<br />
<br />
A probe image $\hat{x}$ is given the label of the nearest neighbour from the<br />
support set:<br />
<br />
$\hat{y} = y$<br />
<br />
$(x, y) = \displaystyle arg \min_{(x_i,y_i) \in S} d(h(x_i), h(\hat{x})) $<br />
<br />
where d is a distance function.<br />
<br />
The function h is parameterized by Inception – one of the best performing ImageNet classification models. Specifically, h returns features from the last layer (the softmax input) of a pre-trained Inception classifier. With these features as input and cosine distance as the distance function, the classifier in achieves 87.6% accuracy on one-shot classification on the ImageNet dataset (Vinyals et al., 2016). We call the Inception classifier together with the nearest-neighbor component the Inception Baseline (IB) model.<br />
<br />
<br />
== Matching Networks ==<br />
<br />
MNs (Vinyals et al.,2016) are neural network architectures with state-of-the-art one shot learning performance on ImageNet (93.2% one-shot labelling accuracy).<br />
MNs are trained to assign label $\hat{y}$ to probe image $\hat{x}$ using an attention mechanism a acting on image embeddings stored in the support set S:<br />
\begin{align*}<br />
a(\hat{x},x_i)=\frac{e^{d(f(\hat{x},S), g(x_i,S))}}{\sum_{j}e^{d(f(\hat{x},S), g(x_j,S))}},<br />
\end{align*}<br />
<br />
<br />
where d is a cosine distance and where f and g provide context-dependent embeddings of $\hat{x}$ and $x_i$ (with contextS). The embedding $g(x_i, S)$ is a bi-directional LSTM (Hochreiter & Schmidhuber, 1997) with the support set S provided as an input sequence. The embedding $f(\hat{x}, S)$ is an LSTM with a read-attention mechanism operating over the entire embedded support set. The input to the LSTM is given by the penultimate layer features of a pre-trained deep convolutional network, specifically Inception.<br />
<br />
To train MNs we proceed as follows:<br />
<br />
=== Training MN ===<br />
* Step 1: At each step of training, the model is given a small support set of images and associated labels. In addition to the support set, the model is fed an unlabeled probe image $\hat{x}$<br />
<br />
* Step 2: The model parameters are then updated to improve classification accuracy of the probe image $\hat{x}$ given the support set. Parameters are updated using stochastic gradient descent with a learning rate of 0.1<br />
<br />
* Step 3: After each update, the labels ${(y_i, i \in [1, k]}$ in the training set are randomly re-assigned to new image classes (the label indices are randomly permuted,<br />
but the image labels are not changed). This is a critical step. It prevents MNs from learning a consistent mapping between a category and a label. Usually, in classification, this is what we want, but in one-shot learning we want to train our model for classification after viewing a single in-class example from the support set. <br />
<br />
The objective function used is:<br />
<br />
\begin{align*}<br />
L=E_{C\sim T}\biggr[E_{S \sim C, B \sim C}\bigr[\sum_{(x,y)\in B}\log P(y|x,S)\bigr]\biggr]<br />
\end{align*}<br />
<br />
<br />
where T is the set of all possible labelings of our classes, S is a support set sampled with a class labeling C ~ T and B is a batch of probe images and labels, also with the same randomly chosen class labeling as the support set.<br />
<br />
== Cognitive Biases ==<br />
Cognitive bias is a concept from developmental psychology which attempts to explain how children can extract meanings of words with very few examples, similar to the concept of one-shot learning discussed above. The theory, as explained by the authors, is that humans form biases that allow them to eliminate many potential hypotheses about word meaning where the amount of data available is insufficient for this purpose. These include:<br />
* Whole object bias<br />
* Taxonomic bias<br />
* Mutual exclusivity bias<br />
* Shape bias<br />
A more complete list of cognitive biases is given by [[#References|(Bloom, 2000)]]. The bias the authors investigate in this paper is the shape bias, which denotes a tendency to assign the same name to similarly shaped items rather than to items with similar color, texture, or size.<br />
<br />
= Methodology =<br />
== Inductive Biases & Probe Data ==<br />
<br />
Inductive biases are those criteria which are artificially selected or learnt by the network as a classifying/distinguishing property.<br />
It has been observed that the biases that DNNs learnt are complex composite features. We, as researchers can take advantage of the fact that DNNs learnt complex distinguishing features by constructing probe data sets which particularly target on exposing a particular bias that a DNN might have. <br />
<br />
* Step 1: Take a known composite feature which we suspect the DNNs are biased against. It is possible that the feature is not numerical but intuitive for human researchers to understand.<br />
* Step 2: Train the target model with an appropriate dataset.<br />
* Step 3: Transfer Learning: Use the pre-trained model with a new data set which is curated to contain data to prove/disprove the existence of the bias<br />
* Step 4: Model/Decide on a function which quantifies the bias under study<br />
* Step 5: Measure the bias with the bias function<br />
<br />
Then it is possible to deduce whether the DNN uses the feature to solve the task by the value of bias function.<br />
<br />
== Data Sets Used ==<br />
<br />
* Training Set: ImageNet<br />
* Test Set:<br />
** The Cognitive Psychology Probe Data (CogPsyc data) that is used consists of 150 images of objects. The images are arranged in triples consisting of a probe image, a shape-match image (that matches the probe in colour but not shape), and a color-match image (that matches the probe in shape but not colour). In the dataset there are 10 triples, each shown on 5 different backgrounds, giving a total of 50 triples. [[File:CogPsy.PNG|center|350px]]<br />
** A real-world dataset consisting of 90 images of objects (30 triples) collected using Google Image Search. The images are arranged in triples consisting of a probe, a shape-match and a colour-match.<br />
<br />
= Experiments =<br />
== Evaluation Criteria ==<br />
<br />
* For a given probe image $\hat{x}$, we loaded the shape-match image $x_s$ and corresponding label $y_s$, along with the colour-match image $x_c$ and corresponding label $y_c$ into memory, as the support set $S = \{(x_s, ys), (x_c, y_c)\}$<br />
* Calculate $\hat{y}$<br />
* The model assigns either $y_c$ or $y_s$ to the probe image.<br />
* To estimate the shape bias Bs, calculate the proportion of shape labels assigned to the probe: $B_s = E(\delta(\hat{y} - y_s))$ <br />
where E is an expectation across probe images and $\delta$ is the Dirac delta function.<br />
<br />
== Experiment 1: Shape bias statistics in Inception Baseline: ==<br />
* Shape bias of IB to be $B_s = 0.68$. Similarly, the shape bias of IB using our real-world dataset was $B_s = 0.97$. Together, these results strongly suggest that IB trained on ImageNet has a stronger bias towards shape than colour<br />
<br />
== Experiment 2: Shape bias statistics in Matching Network: ==<br />
* They found that MNs have a shape of bias $B_s = 0.7$ using the CogPsyc dataset and a bias of $Bs = 1$ using the real-world dataset. Once again, these results suggest that MNs trained seeding from Inception using ImageNet has a stronger bias towards shape than colour.<br />
<br />
== Experiment 3: Shape bias statistics between and across models: ==<br />
<br />
The authors extended the shape bias analysis to calculate the shape bias in a population of IB models and in a population of MN models with different random initialization<br />
<br />
=== Dependence on the initialization of parameters: ===<br />
<br />
[[File:3.1.PNG|right|250px]] A strong variability was observed when variation in the initial values of the parameters. For the CogPsyc dataset, the average shape bias was $B_s = 0.628$ with standard deviation $\sigma B_s = 0.049$ at the end of training and for the real-world dataset the average shape bias was $B_s = 0:958$ with $\sigma B_s = 0.037$.<br />
<br />
=== Dependence of shape bias on model performance: ===<br />
<br />
For the CogPsych dataset, the correlation between bias and classification accuracy was $\rho = 0.15$, and for the real world dataset, correlation between bias and classification accuracy was $\rho = -0.06$. This would be evident since the accuracy of the models remained nearly constant when the initialization parameters varied whereas the shape bias tended to vary a lot, hence highlighting the lack of correlation amongst them. <br />
<br />
=== Emergence of shape bias during training: ===<br />
The shape bias spiked to a large value very early. <br />
<br />
=== Variation of shape bias within models & across models: ===<br />
With different initialization parameters, the shape bias varied a lot within IB during training while the shape bias did not fluctuate during the training of MN. It was found that the MN inherits the shape bias of the IB which seeded its embeddings and thereafter, the shape bias remained constant throughout training. It is important to note that the output of the penultimate layer of the Inception was not fine tuned when it was pipelined to the MN. This was to ensure that the MN properties were independent of the IB model properties. [[File:3.3.PNG|center|250px]] [[File:3.4.PNG|center|250px]]<br />
<br />
= Learnings, Inferences & Implications =<br />
* Both the Inception Baseline and the Matching Network exhibit strong shape bias when trained on ImageNet. Researchers who use Inception & MN DNNs can now use this fact as a consideration for their application while using pre-trained models for new datasets. If it is known before hand that the new data set is strongly classifiable through a color bias, then they would either want to defer using the pre-trained models or explore methods to decrease/remove the strong shape bias.<br />
<br />
* There exists a high variability in the shape bias with the variation in the initialization parameters. This is an important finding since it uncovers the fact that the same architecture which exhibit similar accuracy in predictions can display a variety of shape bias just with different initialization parameters. Researchers can explore methods of tuning the random initialization such that the models start out with a low shape bias without compromising the accuracy of the model.<br />
<br />
* MNs inherit the shape bias which is seeded to it by the Inception Network's input embedding. This is also another fact which researchers & practitioners should be careful about. When using cascaded or pipelined heterogeneous architectures, the models downstream tend to inherit/become/are fed with the properties/biases of the models upstream. This may be desirable or undesirable according to the application, but it is important to be aware of its presence.<br />
<br />
* The biases under consideration are the property of the collection of the architecture, the dataset and the optimization procedure. Hence in order to increase or decrease the effect of a particular bias, one or more of the mentioned factors must be adjusted/tuned/changed.<br />
<br />
* The fact that a high shape bias emerged in the early epochs with less variability in further epochs can be thought of analogous to the biases that humans develop at an infancy which gets fortified as they age.<br />
<br />
=== Modeling human word learning === <br />
The authors note how there have been several previous attempts to model human word learning in the field of cognitive science. A major shortcoming of these works is that none of the models are capable of one-shot word learning when presented with real-world images. Recognizing this and given the success of MNs in this work, the authors propose MNs as a computational-level account of one-shot word learning. <br />
In the paper, it is discussed how shape bias increases dramatically during early training of the model. This agrees with what is observed in many psychological studies: older children show bias more than younger children. <br />
<br />
= Conclusion, Future Work and Open questions =<br />
<br />
* Just as cognitive psychology exposes the shape bias observed in this experiment, we should try to uncover other biases as well using multiple approaches<br />
* Study the underlying mechanisms which cause biases such as shape bias in DNNs<br />
* Research into various methods of probing and creating probe data sets which can be used to test architectures for various biases<br />
* Exploration into a research field called Artificial Cognitive Psychology which focuses on probing how DNN architectures can be understood further using known behaviors of the human brain<br />
<br />
= References =<br />
<br />
* Ritter, Samuel & G. T. Barrett, David & Santoro, Adam & M. Botvinick, Matt. (2017). Cognitive Psychology for Deep Neural Networks: A Shape Bias Case Study<br />
<br />
* Vinyals, Oriol, Blundell, Charles, Lillicrap, Timothy, Kavukcuoglu, Koray, and Wierstra, Daan. Matching networks for one shot learning. arXiv preprint arXiv:1606.04080, 2016.<br />
<br />
* Bloom, P. (2000). How children learn the meanings of words. The MIT Press.<br />
<br />
* https://www.slideshare.net/KazukiFujikawa/matching-networks-for-one-shot-learning-71257100<br />
<br />
* https://deepmind.com/blog/cognitive-psychology/<br />
<br />
* https://hacktilldawn.com/2016/09/25/inception-modules-explained-and-implemented/</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=FeUdal_Networks_for_Hierarchical_Reinforcement_Learning&diff=29744FeUdal Networks for Hierarchical Reinforcement Learning2017-11-08T23:36:41Z<p>H4lyu: /* ATARI action repeat transfer */</p>
<hr />
<div>= Introduction =<br />
<br />
Even though deep reinforcement learning has been hugely successful in a variety of domains, it has not been able to succeed in environments which have sparsely spaced reward signals and encounters a major challenge of long-term credit assignment, where the agent is not able to attribute a reward to an action taken several timesteps back. <br />
<br />
This paper proposes a hierarchical reinforcement learning architecture (HRL), called FeUdal Networks (FuN), which has been inspired from Feudal Reinforcement Learning (FRL)[3]. It is a fully-differentiable neural network with two levels of hierarchy – a Manager module at the top level and a Worker module below. The Manager sets abstract goals, which are learned, at a lower temporal resolution in a latent state-space. The Worker operates at a higher temporal resolution and produces primitive actions at every tick of the environment, motivated to follow the goals received from Manager, by an intrinsic reward.<br />
<br />
The key contributions of the authors in this paper are: (1) A consistent, end-to-end differentiable FRL inspired HRL. (2) A novel, approximate transition policy gradient update for training the Manager (3) The use of goals that are directional rather than absolute in nature. (4) Dilated LSTM – a novel RNN design for the Manager that allows gradients to flow through large hops in time.<br />
<br />
The experiments conducted on several tasks which involve sparse rewards show that FuN significantly outperforms a strong baseline agent on tasks that involve long-term credit assignment and memorization.<br />
<br />
= Related Work =<br />
<br />
Several hierarchical reinforcement learning models were proposed to solve this problem. The options framework [4] considers the problem with a two-level hierarchy, with options being typically learned using sub-goals and ‘pseudo-rewards’ that are provided explicitly. Whereas, the option-critic architecture[1] uses the policy gradient theorem for learning options in an end-to-end fashion. A problem with learning options end-to-end is that they tend to a trivial solution where: (i) only one option is active, which solves the whole task; (ii) a policy-over-options changes options at every step, micro-managing the behavior. The authors state that the option-critic architecture is the only other end-to-end trainable system with sub-policies. A key difference between the authors' approach and the options framework is that the top level (manager) produces a meaningful and explicit goal for the bottom level (worker) to achieve.<br />
<br />
Non-hierarchical deep RL (non-HRL) methods using auxiliary losses and rewards such as pseudo count for exploration[2] have significantly improved results by stimulating agents to explore new parts of the state space. The UNREAL agent[9] is another non-HRL method that showed a strong improvement using unsupervised auxiliary tasks.<br />
<br />
= Model =<br />
<br />
[[File:feudal_network_model_diagram.png|frame]]<br />
<br />
A high-level explanation of the model is as follows: <br />
<br />
The Manager computes a latent state representation <math>s_t</math> and outputs a goal vector <math>g_t</math> . The Worker outputs actions based on the environment observation, its own state, and the Manager’s goal. A perceptual module computes intermediate representation, <math>z_t</math> of the environment observation <math>x_t</math>, and is shared as input by both Manager and Worker. The Manager’s goals <math>g_t</math> are trained using an approximate transition policy gradient. The Worker is then trained via intrinsic reward which stimulates it to output actions that will achieve the goals set by the Manager.<br />
<br />
<center><br />
[[File:model_definition.png|500px]]<br />
</center><br />
<br />
Manager and Worker are recurrent networks (<math>{h^M}</math> and <math>{h^W}</math> being their internal states). <math>\phi</math> is a linear transform that maps a goal <math>g_t</math> into an embedding vector <math>w_t \in {R^k}</math> , which is then combined with matrix <math>U_t</math> (Worker's output) via a matrix-vector product to produce policy <math>\pi</math> – vector of probabilities over primitive actions. The projection <math>\phi</math> is linear, with no biases, and is learnt with gradients coming from the Worker’s actions.Since <math>\phi</math> has no biases it can never produce a constant non-zero vector – which is the only way the setup could ignore the Manager’s input. This makes sure that the goal output by the Manager always influences the final policy.<br />
<br />
===Learning===<br />
The learning considers a standard reinforcement learning setup where the goal of the agent is to maximize the discounted return <math>R_t = \sum_{k=0}^{&infin;} \gamma^k r_{t+k+1}</math>; where <math>\gamma \in [0,1]; r_t</math> is the reward from environment for action at timestep, <math>t</math>. The agent's behavior is defined by its action-selection policy, <math>\pi</math>.<br />
<br />
Since FuN is fully differentiable, the authors could have trained it end-to-end using a policy gradient algorithm operating on the actions taken by the Worker such the outputs <math>g</math> of the Manager would be trained by gradients coming from the Worker. This, however, would deprive Manager’s goals <math>g</math> of any semantic meaning, making them just internal latent variables of the model. So instead, Manager is independently trained to predict advantageous directions (transitions) in state space and to intrinsically reward the Worker to follow these directions.<br />
<br />
Update rule for manager:<br />
<br />
<br />
<center><br />
<math>\nabla g_t = A_t^M \nabla_\theta d_{cos}(s_{t+c} - s_t, g_t(\theta))</math><br />
</center><br />
<br />
<br />
In above equation, <math>d_{cos}(\alpha, \beta) = \alpha^T \beta/(|\alpha||\beta|)</math> is the cosine similarity between two vectors and <math>A_t^M = R_t - V_t^M(x_t,\theta)</math> is the Manager’s advantage function, computed using a value function estimate <math>V_t^M(x_t,\theta)</math> from the internal critic. Here c is an event horizon for the Manager to optimize its direction on. It must be treated as a hyperparameter of the model. It controls the temporal resolution of the Manager.<br />
<br />
The intrinsic reward that encourages the Worker to follow the goals are defined as:<br />
<br />
<br />
<center><br />
<math>r_t^I = 1/c \sum_{i=1}^c d_{cos}(s_t - s_{t-i}, g_{t-i})</math> <br />
</center><br />
<br />
<br />
Compared to FRL[3], which advocated concealing the reward from lower levels of the hierarchy, the Worker in FuN network is trained using an advantage actor-critic[5] to maximise a weighted sum <math>R_t + &alpha; R_t^I</math> , where <math>&alpha;</math> is a hyper-parameter that regulates the influence of the intrinsic reward:<br />
<br />
<br />
<center><br />
<math>\nabla {\pi}_t = A_t^D \nabla_\theta log \pi (a_t|x_t;\theta)</math><br />
</center><br />
<br />
<br />
The Advantage function <math>A_t^D = (R_t + \alpha R_t^I - V_t^D(x_t;\theta))</math> is calculated using an internal critic, which estimates the value functions for both rewards.<br />
<br />
===Transition Policy Gradient===<br />
The update rule for the Manager given above is a novel form of policy gradient with respect to a ''model'' of the Worker’s behavior. The Worker can follow a complex trajectory but it is not necessarily required to learn from these samples. If the trajectories can be predicted, by modeling the transitions, then the policy gradient of the predicted transition can be followed instead of the Worker's actual path. FuN assumes a particular form for the transition model: that the direction in state-space, <math>s_{t+c} − s_t</math>, follows a von Mises-Fisher distribution.<br />
<br />
=Architecture=<br />
The perceptual module <math>f^{percept}</math> is a convolutional network (CNN) followed by a fully connected layer. Each convolutional and fully-connected layer is followed by a rectifier non-linearity. <math>f_{Mspace}</math>, which is another fully connected layer followed by a rectifier non-linearity, is used to compute the state space, which the Manager uses to formulate goals. The Worker’s recurrent network <math>f^{Wrnn}</math> is a standard LSTM[6].<br />
<br />
<br />
The Manager uses a novel architecture called a dilated LSTM (dLSTM), which operates at lower temporal resolution than the data stream. It is similar to dilated convolutional networks[7] and clockwork RNN. For a dilation radius r, the network is composed of r separate groups of sub-states or ‘cores’, denoted by <math>h = \{\hat{h}^i\}_{i=1}^r</math>. At time <math>t</math>, the network is governed by the following equations: <math>\hat{h}_t^{t\%r},g_t = LSTM(s_t, \hat{h}_{t-1}^{t\%r};\theta^{LSTM})</math> where % denotes the modulo operation and allows us to indicate which group of cores is currently being updated. At each time step, only the corresponding part of the state is updated and the output is pooled across the previous c outputs. This allows the r groups of cores inside the dLSTM to preserve the memories for long periods, yet the dLSTM as a whole is still able to process and learn from every input experience and is also able to update its output at every step.<br />
<br />
=Experiments=<br />
The baseline the authors are using is a recurrent LSTM[6] network on top of a representation learned by a CNN. The A3C method[5] is used for all reinforcement learning experiments. Backpropagation through time (BPTT)[8] is run after K forward passes of a network or if a terminal signal is received. For each method, 100 experiments were run. A training epoch is defined as one million observations.<br />
<br />
==Montezuma’s Revenge==<br />
Montezuma’s revenge is a prime example of an environment with sparse rewards. FuN starts learning much earlier and achieves much higher scores. It takes > 300 epochs for LSTM to reach the score 400, which corresponds to solving the first room (take the key, open a door). FuN solves the first room in less than 200 epochs and immediately moves on to explore further, eventually visiting several other rooms and scoring up to 2600 points.<br />
<br />
<center><br />
[[File:feudal_figure2.png|900px]]<br />
</center><br />
<br />
==ATARI==<br />
The experiment was run on a diverse set of ATARI games, some of which involve long-term credit assignment and some which are more reactive. Enduro stands out as all the LSTM agents completely fail at it. Frostbite is a hard game that requires both long-term credit assignment and good exploration. The best-performing frostbite agent is FuN with 0.95 Manager discount, which outperforms the rest by a factor of 7. The other results can be seen in the figure.<br />
<br />
<center><br />
[[File:feudal_figure4.png|900px]]<br />
</center><br />
<br />
==Comparing the option-critic architecture==<br />
FuN network was run on the same games as Option-Critic (Asterix, Ms. Pacman, Seaquest, and Zaxxon) and after 200 epochs it achieves a similar score on Seaquest, doubles it on Ms. Pacman, more than triples it on Zaxxon and gets more than 20x improvement on Asterix.<br />
<br />
<center><br />
[[File:feudal_figure7.png]]<br />
</center><br />
<br />
==Memory in Labyrinth==<br />
DeepMind Lab (Beattie et al., 2016) is a first-person 3D game platform extended from OpenArena. The games on which the experiments were run on include a Water maze, T-maze, and Non-match (which is a visual memorization task). FuN consistently outperforms the LSTM baseline – it learns faster and also reaches a higher final reward. Interestingly, the LSTM agent doesn’t appear to use its memory for water maze task at all, always circling the maze at the roughly the same radius.<br />
<br />
<center><br />
[[File:feudal_figure5.png|800px]]<br />
[[File:feudal_figure6.png|800px]]<br />
</center><br />
<br />
==Ablative Analysis==<br />
Empirical evaluation of the main contributions of this paper:<br />
<br />
===Transition policy gradient===<br />
Experiments were run on modified FuN networks in which: 1) the Managers output g is trained with gradients coming directly from the Worker and no intrinsic reward is used, 2) g is learned using a standard<br />
policy gradient approach with the Manager emitting the mean of a Gaussian distribution from which goals are sampled, 3) a variant of FuN in which g specifies absolute, rather than relative/directional, goals and 4) a purely feudal version of FuN – in which the Worker is trained from the intrinsic reward alone. The experiments (Figure 8) reveal that, although alternatives do work to some degree their performance is significantly inferior.<br />
<br />
<center><br />
[[File:feudal_figure8.png|900px]]<br />
</center><br />
<br />
===Temporal resolution ablations===<br />
To test the effectiveness of the dilation LSTM, FuN was compared with two baselines 1) the Manager uses a vanilla LSTM with no dilation; 2) FuN with Manager’s prediction horizon c = 1. The non-dilated LSTM fails catastrophically, most likely overwhelmed by the recurrent gradient. Reducing the horizon c to 1 did hurt the performance, although not that much, which means that even at high temporal resolution Manager captures certain properties of the underlying MDP.<br />
<br />
<center><br />
[[File:feudal_figure10.png|900px]]<br />
</center><br />
<br />
===Intrinsic motivation weight===<br />
Evaluates the effect of weight <math>&alpha;</math> which regulates the relative weight of intrinsic reward. Figure below shows scatter plots of agents final score vs α hyper-parameter where there is a clear improvement in score for high <math>\alpha</math> in some games.<br />
<br />
<center><br />
[[File:feudal_figure11.png|900px]]<br />
</center><br />
<br />
===Dilate LSTM agent baseline===<br />
For this experiment, just the dLSTM is used in an agent on top of a CNN, without the rest of FuN structures. Figure below plots the learning curves for FuN, LSTM, and dLSTM agents. dLSTM generally underperforms both LSTM and FuN.<br />
<br />
<center><br />
[[File:feudal_figure12.png|900px]]<br />
</center><br />
<br />
===ATARI action repeat transfer===<br />
This experiment is to demonstrate the advantage of FeUdal Network, i.e. separating policy and primitive operations. It also implies that the transition policy can be transferred between agents with different embodiment, for example, across agents with different action repeat on ATARI. The figure below shows the corresponding learning curves. The transferred FuN agent (green curve) significantly outperforms every other method.<br />
<br />
<center><br />
[[File:feudal_figure9.png|900px]]<br />
</center><br />
<br />
=Conclusion=<br />
FuN currently holds state-of-the-art score in the Atari game, Montezuma's revenge among HRL methods. It is a novel approach to hierarchical reinforcement learning which separates the goal setting behavior from the generation of action primitives. This creates a natural hierarchy that is stable and the experiments clearly demonstrate that the FeUdal network makes long-term credit assignment and memorization more tractable.<br />
<br />
Deeper hierarchies by setting goals at multiple time scales is an avenue for further research. The modular structure looks promising for transfer and multitask learning as well.<br />
<br />
An implementation of this paper can be found on : https://github.com/dmakian/feudal_networks<br />
<br />
=References=<br />
#Bacon, Pierre-Luc, Precup, Doina, and Harb, Jean. The option-critic architecture. In AAAI, 2017.<br />
#Bellemare, Marc, Srinivasan, Sriram, Ostrovski, Georg, Schaul, Tom, Saxton, David, and Munos, Remi. Unifying count-based exploration and intrinsic motivation.In NIPS, 2016a.<br />
#Dayan, Peter and Hinton, Geoffrey E. Feudal reinforcement learning. In NIPS. Morgan Kaufmann Publishers,1993.<br />
#Sutton, Richard S, Precup, Doina, and Singh, Satinder. Between mdps and semi-mdps: A framework for temporal abstraction in reinforcement learning. Artificial intelligence, 1999.<br />
#Mnih, Volodymyr, Badia, Adria Puigdomenech, Mirza,Mehdi, Graves, Alex, Lillicrap, Timothy P, Harley, Tim,Silver, David, and Kavukcuoglu, Koray. Asynchronousmethods for deep reinforcement learning. ICML, 2016.<br />
#Hochreiter, Sepp and Schmidhuber, Jürgen. Long short-term memory. Neural computation, 1997.<br />
#Yu, Fisher and Koltun, Vladlen. Multi-scale context aggregation by dilated convolutions. ICLR, 2016.<br />
#Mozer, Michael C. A focused back-propagation algorithm for temporal pattern recognition. Complex systems, 1989.<br />
#Jaderberg, Max, Mnih, Volodymyr, Czarnecki, Wojciech Marian, Schaul, Tom, Leibo, Joel Z, Silver,David, and Kavukcuoglu, Koray. Reinforcement learning with unsupervised auxiliary tasks. arXiv preprint arXiv:1611.05397, 2016.<br />
#A. S. Vezhnevets, S. Osindero, T. Schaul, N. Heess, M. Jaderberg, D. Silver, and K. Kavukcuoglu. Feudal networks for hierarchical reinforcement learning. arXiv preprint arXiv:1703.01161, 2017.<br />
# https://www.quora.com/What-is-hierachical-reinforcement-learning<br />
# Tutorial for Hierarchial Reinforcement Learning: https://www.youtube.com/watch?v=K5MlmO0UJtI<br />
# Videos of FUN agent playing various Atari games can be found in supplementary file accessed through: http://proceedings.mlr.press/v70/vezhnevets17a.html</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=FeUdal_Networks_for_Hierarchical_Reinforcement_Learning&diff=29736FeUdal Networks for Hierarchical Reinforcement Learning2017-11-08T23:13:57Z<p>H4lyu: /* Related Work */</p>
<hr />
<div>= Introduction =<br />
<br />
Even though deep reinforcement learning has been hugely successful in a variety of domains, it has not been able to succeed in environments which have sparsely spaced reward signals and encounters the major challenge of long-term credit assignment, where the agent is not able to attribute a reward to an action taken several timesteps back. <br />
<br />
This paper proposes a hierarchical reinforcement learning architecture (HRL), called FeUdal Networks (FuN), which has been inspired from Feudal Reinforcement Learning (FRL)[3]. It is a fully-differentiable neural network with two levels of hierarchy – a Manager module at the top level and a Worker module below. The Manager sets abstract goals, which are learned, at a lower temporal resolution in a latent state-space. The Worker operates at a higher temporal resolution and produces primitive actions at every tick of the environment, motivated to follow the goals received from Manager, by an intrinsic reward.<br />
<br />
The key contributions of the authors in this paper are: (1) A consistent, end-to-end differentiable FRL inspired HRL. (2) A novel, approximate transition policy gradient update for training the Manager (3) The use of goals that are directional rather than absolute in nature. (4) Dilated LSTM – a novel RNN design for the Manager that allows gradients to flow through large hops in time.<br />
<br />
The experiments conducted on several tasks which involve sparse rewards show that FuN significantly outperforms a strong baseline agent on tasks that involve long-term credit assignment and memorization.<br />
<br />
= Related Work =<br />
<br />
Several hierarchical reinforcement learning models were proposed to solve this problem. The options framework [4] considers the problem with a two-level hierarchy, with options being typically learned using sub-goals and ‘pseudo-rewards’ that are provided explicitly. Whereas, the option-critic architecture[1] uses the policy gradient theorem for learning options in an end-to-end fashion. A problem with learning options end-to-end is that they tend to a trivial solution where: (i) only one option is active, which solves the whole task; (ii) a policy-over-options changes options at every step, micro-managing the behavior. The authors state that the option-critic architecture is the only other end-to-end trainable system with sub-policies. A key difference between the authors' approach and the options framework is that the top level (manager) produces a meaningful and explicit goal for the bottom level (worker) to achieve.<br />
<br />
Non-hierarchical deep RL (non-HRL) methods using auxiliary losses and rewards such as pseudo count for exploration[2] have significantly improved results by stimulating agents to explore new parts of the state space. The UNREAL agent[9] is another non-HRL method that showed a strong improvement using unsupervised auxiliary tasks.<br />
<br />
= Model =<br />
<br />
[[File:feudal_network_model_diagram.png|frame]]<br />
<br />
A high-level explanation of the model is as follows: <br />
<br />
The Manager computes a latent state representation <math>s_t</math> and outputs a goal vector <math>g_t</math> . The Worker outputs actions based on the environment observation, its own state, and the Manager’s goal. A perceptual module computes intermediate representation, <math>z_t</math> of the environment observation <math>x_t</math>, and is shared as input by both Manager and Worker. The Manager’s goals <math>g_t</math> are trained using an approximate transition policy gradient. The Worker is then trained via intrinsic reward which stimulates it to output actions that will achieve the goals set by the Manager.<br />
<br />
<center><br />
[[File:model_definition.png|500px]]<br />
</center><br />
<br />
Manager and Worker are recurrent networks (<math>{h^M}</math> and <math>{h^W}</math> being their internal states). <math>\phi</math> is a linear transform that maps a goal <math>g_t</math> into an embedding vector <math>w_t \in {R^k}</math> , which is then combined with matrix <math>U_t</math> (Worker's output) via a matrix-vector product to produce policy <math>\pi</math> – vector of probabilities over primitive actions. The projection <math>\phi</math> is linear, with no biases, and is learnt with gradients coming from the Worker’s actions.Since <math>\phi</math> has no biases it can never produce a constant non-zero vector – which is the only way the setup could ignore the Manager’s input. This makes sure that the goal output by the Manager always influences the final policy.<br />
<br />
===Learning===<br />
The learning considers a standard reinforcement learning setup where the goal of the agent is to maximize the discounted return <math>R_t = \sum_{k=0}^{&infin;} \gamma^k r_{t+k+1}</math>; where <math>\gamma \in [0,1]; r_t</math> is the reward from environment for action at timestep, <math>t</math>. The agent's behavior is defined by its action-selection policy, <math>\pi</math>.<br />
<br />
Since FuN is fully differentiable, the authors could have trained it end-to-end using a policy gradient algorithm operating on the actions taken by the Worker such the outputs <math>g</math> of the Manager would be trained by gradients coming from the Worker. This, however, would deprive Manager’s goals <math>g</math> of any semantic meaning, making them just internal latent variables of the model. So instead, Manager is independently trained to predict advantageous directions (transitions) in state space and to intrinsically reward the Worker to follow these directions.<br />
<br />
Update rule for manager:<br />
<br />
<br />
<center><br />
<math>\nabla g_t = A_t^M \nabla_\theta d_{cos}(s_{t+c} - s_t, g_t(\theta))</math><br />
</center><br />
<br />
<br />
In above equation, <math>d_{cos}(\alpha, \beta) = \alpha^T \beta/(|\alpha||\beta|)</math> is the cosine similarity between two vectors and <math>A_t^M = R_t - V_t^M(x_t,\theta)</math> is the Manager’s advantage function, computed using a value function estimate <math>V_t^M(x_t,\theta)</math> from the internal critic. Here c is an event horizon for the Manager to optimize its direction on. It must be treated as a hyperparameter of the model. It controls the temporal resolution of the Manager.<br />
<br />
The intrinsic reward that encourages the Worker to follow the goals are defined as:<br />
<br />
<br />
<center><br />
<math>r_t^I = 1/c \sum_{i=1}^c d_{cos}(s_t - s_{t-i}, g_{t-i})</math> <br />
</center><br />
<br />
<br />
Compared to FRL[3], which advocated concealing the reward from lower levels of the hierarchy, the Worker in FuN network is trained using an advantage actor-critic[5] to maximise a weighted sum <math>R_t + &alpha; R_t^I</math> , where <math>&alpha;</math> is a hyper-parameter that regulates the influence of the intrinsic reward:<br />
<br />
<br />
<center><br />
<math>\nabla {\pi}_t = A_t^D \nabla_\theta log \pi (a_t|x_t;\theta)</math><br />
</center><br />
<br />
<br />
The Advantage function <math>A_t^D = (R_t + \alpha R_t^I - V_t^D(x_t;\theta))</math> is calculated using an internal critic, which estimates the value functions for both rewards.<br />
<br />
===Transition Policy Gradient===<br />
The update rule for the Manager given above is a novel form of policy gradient with respect to a ''model'' of the Worker’s behavior. The Worker can follow a complex trajectory but it is not necessarily required to learn from these samples. If the trajectories can be predicted, by modeling the transitions, then the policy gradient of the predicted transition can be followed instead of the Worker's actual path. FuN assumes a particular form for the transition model: that the direction in state-space, <math>s_{t+c} − s_t</math>, follows a von Mises-Fisher distribution.<br />
<br />
=Architecture=<br />
The perceptual module <math>f^{percept}</math> is a convolutional network (CNN) followed by a fully connected layer. Each convolutional and fully-connected layer is followed by a rectifier non-linearity. <math>f_{Mspace}</math>, which is another fully connected layer followed by a rectifier non-linearity, is used to compute the state space, which the Manager uses to formulate goals. The Worker’s recurrent network <math>f^{Wrnn}</math> is a standard LSTM[6].<br />
<br />
<br />
The Manager uses a novel architecture called a dilated LSTM (dLSTM), which operates at lower temporal resolution than the data stream. It is similar to dilated convolutional networks[7] and clockwork RNN. For a dilation radius r, the network is composed of r separate groups of sub-states or ‘cores’, denoted by <math>h = \{\hat{h}^i\}_{i=1}^r</math>. At time <math>t</math>, the network is governed by the following equations: <math>\hat{h}_t^{t\%r},g_t = LSTM(s_t, \hat{h}_{t-1}^{t\%r};\theta^{LSTM})</math> where % denotes the modulo operation and allows us to indicate which group of cores is currently being updated. At each time step, only the corresponding part of the state is updated and the output is pooled across the previous c outputs. This allows the r groups of cores inside the dLSTM to preserve the memories for long periods, yet the dLSTM as a whole is still able to process and learn from every input experience and is also able to update its output at every step.<br />
<br />
=Experiments=<br />
The baseline the authors are using is a recurrent LSTM[6] network on top of a representation learned by a CNN. The A3C method[5] is used for all reinforcement learning experiments. Backpropagation through time (BPTT)[8] is run after K forward passes of a network or if a terminal signal is received. For each method, 100 experiments were run. A training epoch is defined as one million observations.<br />
<br />
==Montezuma’s Revenge==<br />
Montezuma’s revenge is a prime example of an environment with sparse rewards. FuN starts learning much earlier and achieves much higher scores. It takes > 300 epochs for LSTM to reach the score 400, which corresponds to solving the first room (take the key, open a door). FuN solves the first room in less than 200 epochs and immediately moves on to explore further, eventually visiting several other rooms and scoring up to 2600 points.<br />
<br />
<center><br />
[[File:feudal_figure2.png|900px]]<br />
</center><br />
<br />
==ATARI==<br />
The experiment was run on a diverse set of ATARI games, some of which involve long-term credit assignment and some which are more reactive. Enduro stands out as all the LSTM agents completely fail at it. Frostbite is a hard game that requires both long-term credit assignment and good exploration. The best-performing frostbite agent is FuN with 0.95 Manager discount, which outperforms the rest by a factor of 7. The other results can be seen in the figure.<br />
<br />
<center><br />
[[File:feudal_figure4.png|900px]]<br />
</center><br />
<br />
==Comparing the option-critic architecture==<br />
FuN network was run on the same games as Option-Critic (Asterix, Ms. Pacman, Seaquest, and Zaxxon) and after 200 epochs it achieves a similar score on Seaquest, doubles it on Ms. Pacman, more than triples it on Zaxxon and gets more than 20x improvement on Asterix.<br />
<br />
<center><br />
[[File:feudal_figure7.png]]<br />
</center><br />
<br />
==Memory in Labyrinth==<br />
DeepMind Lab (Beattie et al., 2016) is a first-person 3D game platform extended from OpenArena. The games on which the experiments were run on include a Water maze, T-maze, and Non-match (which is a visual memorization task). FuN consistently outperforms the LSTM baseline – it learns faster and also reaches a higher final reward. Interestingly, the LSTM agent doesn’t appear to use its memory for water maze task at all, always circling the maze at the roughly the same radius.<br />
<br />
<center><br />
[[File:feudal_figure5.png|800px]]<br />
[[File:feudal_figure6.png|800px]]<br />
</center><br />
<br />
==Ablative Analysis==<br />
Empirical evaluation of the main contributions of this paper:<br />
<br />
===Transition policy gradient===<br />
Experiments were run on modified FuN networks in which: 1) the Managers output g is trained with gradients coming directly from the Worker and no intrinsic reward is used, 2) g is learned using a standard<br />
policy gradient approach with the Manager emitting the mean of a Gaussian distribution from which goals are sampled, 3) a variant of FuN in which g specifies absolute, rather than relative/directional, goals and 4) a purely feudal version of FuN – in which the Worker is trained from the intrinsic reward alone. The experiments (Figure 8) reveal that, although alternatives do work to some degree their performance is significantly inferior.<br />
<br />
<center><br />
[[File:feudal_figure8.png|900px]]<br />
</center><br />
<br />
===Temporal resolution ablations===<br />
To test the effectiveness of the dilation LSTM, FuN was compared with two baselines 1) the Manager uses a vanilla LSTM with no dilation; 2) FuN with Manager’s prediction horizon c = 1. The non-dilated LSTM fails catastrophically, most likely overwhelmed by the recurrent gradient. Reducing the horizon c to 1 did hurt the performance, although not that much, which means that even at high temporal resolution Manager captures certain properties of the underlying MDP.<br />
<br />
<center><br />
[[File:feudal_figure10.png|900px]]<br />
</center><br />
<br />
===Intrinsic motivation weight===<br />
Evaluates the effect of weight <math>&alpha;</math> which regulates the relative weight of intrinsic reward. Figure below shows scatter plots of agents final score vs α hyper-parameter where there is a clear improvement in score for high <math>\alpha</math> in some games.<br />
<br />
<center><br />
[[File:feudal_figure11.png|900px]]<br />
</center><br />
<br />
===Dilate LSTM agent baseline===<br />
For this experiment, just the dLSTM is used in an agent on top of a CNN, without the rest of FuN structures. Figure below plots the learning curves for FuN, LSTM, and dLSTM agents. dLSTM generally underperforms both LSTM and FuN.<br />
<br />
<center><br />
[[File:feudal_figure12.png|900px]]<br />
</center><br />
<br />
===ATARI action repeat transfer===<br />
This experiment is to demonstrate that the transition policy can be transferred between agents with different embodiment, for example, across agents with different action repeat on ATARI. The figure below shows the corresponding learning curves. The transferred FuN agent (green curve) significantly outperforms every other method.<br />
<br />
<center><br />
[[File:feudal_figure9.png|900px]]<br />
</center><br />
<br />
=Conclusion=<br />
FuN currently holds state-of-the-art score in the Atari game, Montezuma's revenge among HRL methods. It is a novel approach to hierarchical reinforcement learning which separates the goal setting behavior from the generation of action primitives. This creates a natural hierarchy that is stable and the experiments clearly demonstrate that the FeUdal network makes long-term credit assignment and memorization more tractable.<br />
<br />
Deeper hierarchies by setting goals at multiple time scales is an avenue for further research. The modular structure looks promising for transfer and multitask learning as well.<br />
<br />
An implementation of this paper can be found on : https://github.com/dmakian/feudal_networks<br />
<br />
=References=<br />
#Bacon, Pierre-Luc, Precup, Doina, and Harb, Jean. The option-critic architecture. In AAAI, 2017.<br />
#Bellemare, Marc, Srinivasan, Sriram, Ostrovski, Georg, Schaul, Tom, Saxton, David, and Munos, Remi. Unifying count-based exploration and intrinsic motivation.In NIPS, 2016a.<br />
#Dayan, Peter and Hinton, Geoffrey E. Feudal reinforcement learning. In NIPS. Morgan Kaufmann Publishers,1993.<br />
#Sutton, Richard S, Precup, Doina, and Singh, Satinder. Between mdps and semi-mdps: A framework for temporal abstraction in reinforcement learning. Artificial intelligence, 1999.<br />
#Mnih, Volodymyr, Badia, Adria Puigdomenech, Mirza,Mehdi, Graves, Alex, Lillicrap, Timothy P, Harley, Tim,Silver, David, and Kavukcuoglu, Koray. Asynchronousmethods for deep reinforcement learning. ICML, 2016.<br />
#Hochreiter, Sepp and Schmidhuber, Jürgen. Long short-term memory. Neural computation, 1997.<br />
#Yu, Fisher and Koltun, Vladlen. Multi-scale context aggregation by dilated convolutions. ICLR, 2016.<br />
#Mozer, Michael C. A focused back-propagation algorithm for temporal pattern recognition. Complex systems, 1989.<br />
#Jaderberg, Max, Mnih, Volodymyr, Czarnecki, Wojciech Marian, Schaul, Tom, Leibo, Joel Z, Silver,David, and Kavukcuoglu, Koray. Reinforcement learning with unsupervised auxiliary tasks. arXiv preprint arXiv:1611.05397, 2016.<br />
#A. S. Vezhnevets, S. Osindero, T. Schaul, N. Heess, M. Jaderberg, D. Silver, and K. Kavukcuoglu. Feudal networks for hierarchical reinforcement learning. arXiv preprint arXiv:1703.01161, 2017.<br />
# https://www.quora.com/What-is-hierachical-reinforcement-learning<br />
# Tutorial for Hierarchial Reinforcement Learning: https://www.youtube.com/watch?v=K5MlmO0UJtI<br />
# Videos of FUN agent playing various Atari games can be found in supplementary file accessed through: http://proceedings.mlr.press/v70/vezhnevets17a.html</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Learning_the_Number_of_Neurons_in_Deep_Networks&diff=29734Learning the Number of Neurons in Deep Networks2017-11-08T23:05:10Z<p>H4lyu: /* Critique */</p>
<hr />
<div>='''Introduction'''=<br />
<br />
Due to the availability of large-scale datasets and powerful computation, '''Deep Learning''' has made huge breakthroughs in many areas, like Language Models and Computer Vision. In deep neural networks, we need to determine the number of layers and the number of neurons in each layer, i.e, we need to determine the number of parameters, or complexity of the model. Typically, this is determined by errors manually. Currently, this is mostly achieved by manually tuning these hyper-parameters using validation data or building very deep networks. However, building a very deep model is still challenging, especially for very large datasets, which leads to high cost on memory and reduction in speed.<br />
<br />
In this paper, we used an approach to automatically select the number of neurons in each layer when we learn the network. Our approach introduces a '''group sparsity regularizer''' on the parameters of the network, and each group acts on the parameters of one neuron, rather than trains an initial network as as pre-processing step(training shallow or thin networks to mimic the behaviour of deep ones [Hinton et al., 2014, Romero et al., 2015]). We set those useless parameters to zero, which cancels out the effects of a particular neuron. Therefore, our approach does not need to learn a redundant network successfully and then reduce its parameters, instead, it learns the number of relevant neurons in each layer and the parameters of those neurons simultaneously.<br />
<br />
In the experiments on several image recognition datasets, we showed the effectiveness of our approach, which reduces the number of parameters by up to 80% compared to the complete model, and has no recognition accuracy loss at the same time. Actually, our approach even yields more effective and faster networks, and occupies less memory.<br />
<br />
='''Related Work'''=<br />
<br />
The recent researches tend to build very deep networks. Building very deep networks means we need to learn more parameters, which leads to a significant cost on the memory of the equipment as well as the speed. Even though automatic model selection has developed in the past years by constructive and destructive approaches, there are some drawbacks. For '''constructive method''', it starts a super shallow architecture, and then adds additional parameters [Bello, 1992]. A similar work that adds new layers to the initial shallow networks was successfully employed [Simonyan and Zisserman, 2014] at the process of learning. However, we know shallow networks have fewer parameters, so that it can not handle the non-linearities as effectively as the deep networks [Montufar et al., 2014], so shallow networks may easily get stuck by the bad optima. Therefore, the drawback of this method is that these networks may produce poor initializations for the later processes. The authors make this claim without ever providing any evidence for it. For '''destructive method''', it starts by a deep network to reduce a significant number of redundant parameters [Denil et al., 2013, Cheng et al., 2015] while keeping its behaviour unchanged. Even though this technique has shown removing the redundant parameters [LeCun et al., 1990, Hassibi et al., 1993] or the neurons [Mozer and Smolensky, 1988, Ji et al., 1990, Reed, 1993] has little influence on the output, it requires the analysis of each parameter and neuron by network Hessian, which is very computationally expensive for large architectures. The main motivation of these works was to build a more compact network.<br />
<br />
Particularly, building a compact network is a research focus for '''Convolutional Neural Networks'''(CNNs). Some works has proposed to decompose the filters of a pre-trained network into low-rank filters, which reduces the number of parameters [Jaderberg et al., 2014b, Denton et al., 2014, Gong et al., 2014]. The issue of this proposal is that we need to successfully train an initial deep network, since it acts as as post-processing step. [Weigend et al., 1991] and [Collins and Kohl, 2014] used direct training to develop regularizers that eliminate some of the parameters of the network. The problem is that the number of layers and neurons each layer is determined manually. A very similar work using the group lasso method for CNN was previously done in [Liu et al., 2015]. The big-picture idea appears to be very similar but they differ in details of methodology.<br />
<br />
='''Model Training and Model Selection'''=<br />
<br />
In general, a deep network has L layers containing linear operations on their inputs, intertwined with activation functions. The activation function we generally use is '''Rectified Linear Units(RELU) or sigmoids'''. Suppose each layer l has $N_{l}$ neurons, and each of them has parameters $\Theta=(\theta_{l})_{1\leqslant{l}\leqslant{L}}$, where $\theta_{l}=({\theta^n _{l}})_{1\leqslant{n}\leqslant{N_{l}}}$ and where $\theta^n _{l}=[w_{l}^{n},b_{l}^{n}]$ is consisted of linear component $w_{l}^{n}$ and linear bias $b_{l}^{n}$. Given an input $x$, under the linear, on-linear and pooling operations, we obtain the output $\hat{y}=f(x,\theta)$, where $f(*)$ encodes the succession of linear, non-linear and pooling operations.<br />
<br />
At the step of training, we have N input-output pairs ${(x_{i},y_{i})}_{1\leqslant{i}\leqslant{N}}$, and the loss function is given by $\ell(y_{i},f(x_{i},\Theta))$, which compares the predicted output with the ground-truth output. Generally, we choose logistic loss for classification and the square loss for regression. Therefore, learning the parameters of the network is equivalent to solving the optimization of the following:<br />
$$\displaystyle \min_{\Theta}\frac{1}{N}\sum_{i=1}^{N}\ell(y_{i},f(x_{i},\Theta))+\gamma(\Theta),$$ where $\gamma(\Theta)$ represents a regularizer on the network parameters. Our choice for the regularizer can be $\ell_{2}$-norm(i.e, weight decay) or $\ell_{1}$-norm. $\ell_{2}$-norm usually favours small parameter values, and $\ell_{1}$-norm can only delete those irrelevant parameters, but not the neurons. The goal in this paper is to automatically determine the number of neurons of each layer, but neither of the above techniques achieve this goal. Here, we make use of the '''group sparsity''' [Yuan and Lin., 2007] (starting from an overcomplete network and canceling the influence of some neurons). The regularizer, therefore, can be written as $$\gamma(\Theta)=\sum_{l=1}^{L}\beta_{l}\sqrt{P_{l}}\sum_{n=1}^{N_{l}}||\theta_{l}^{n}||_{2},$$ where $P_{l}$ means the size of the vector that includes the parameters of each neuron in layer $l$, and $\beta_{l}$ balances the influence of the penalty. In practice, we found the most effective way to select $\beta$ is a relatively small one for the first few layers, and a larger weight for the remaining layers. The reason we choose a small weight is that it can prevent deleting too much neurons in the first few layers, so that we have enough information for learning the remaining parameters. The original premise of this paper seemed to suggest a new method that was different from both the constructive and destructive methods described above. However, this approach of starting with an overcomplete network and training with group sparsity appears to be no different from destructive methods. The main contribution here is then the regularization function to act on entire neurons, which is in fairness an interesting approach.<br />
<br />
The group sparsity helps us effectively remove some of the neurons, and also standard regularizers on the individual parameters are effective for the generalization purpose [Bartlett, 19996, Krogh and Hertz, 1992, Theodoridis, 2015, Collins and Kohli, 2014]. By this idea, we introduce '''sparse group Lasso''', which considers a more generalised penalty that merges L1 norm in Lasso with the group lasso (i.e. "two-norm"). This leads to the production of a penalty which specifies solutions that are sparse enough both at an individual and group feature levels [1]. It specifies that the regularizer can be written as $$\gamma(\Theta)=\sum_{l=1}^{L}((1-\alpha)\beta_{l}\sqrt{P_{l}}\sum_{n=1}^{N_{l}}||\theta_{l}^{n}||_{2}+\alpha\beta_{l}||\theta_{l}||_{1},$$ where $\alpha\in[0,1]$. We find that if $\alpha=0$, then we have the group sparsity regularizer. In practice, we use both $\alpha=0$ and $\alpha=0.5$ in the experiments.<br />
<br />
This reminds me of the relationships among Lasso regression, Ridge regression and Elastic Net regression. In lasso regression, the penalized residual sum of squares is composed of the regular residual sum of squared plus a L1 regularizer. In ridge regression, its penalized residual sum of squares is composed of the regular residual sum of squared plus a L2 regularizer. Finally, an elastic net regression is a combination of lasso regularizer and ridge regularizer, where its objective function is to optimize parameters by including both L1 and L2 norms. <br />
<br />
To find the optimization, in this paper we use proximal gradient descent [Parikh and Boyed, 2014]. This approach iteratively takes a gradient step of size t with respect to the loss. The following is the algorithm for it: <br />
<br />
We define proximal operator of f as $$prox_{f}(v)=\displaystyle \min_{x}(\frac{1}{2t}||x-v||_{2}^{2}+f(x))$$ <br />
<br />
<br />
Suppose we want to minimize $f(x)+g(x)$, and the proximal gradient method is given by $$x^{(k+1)}=prox_{t^{k}g}(x^{k}-t^{k}\nabla{f}(x^{k})), k=1,2,3...$$ <br />
<br />
Therefore, we can update our parameter by the above method as $$\tilde{\theta}_{l}^{n}=\displaystyle \min_{\theta_{l}^{n}}\frac{1}{2t}||\theta_{l}^{n}-\hat{\theta}_{l}^{n}||_{2}^{2}+\gamma(\Theta),$$<br />
where $\hat{\theta}_{l}^{n}$ is the solution obtained from the general loss gradient. By the derivative of [Simon et al., 2013], we have a closed-form solution for this problem: <br />
$$\tilde{\theta}_{l}^{n}=(1-\frac{t(1-\alpha)\beta_{l}\sqrt{P_{l}}}{||S(\hat{\theta}_{l}^{n},t\alpha\beta_{l})||_{2})})_{+}S(\hat{\theta}_{l}^{n},t\alpha\beta_{l}),$$<br />
where + refers to taking the maximum between the argument and 0, and $S(*)$ is $$S(a,b)=sign(a)(|a|-b)_{+}$$<br />
In practice, we use stochastic gradient descent and work with mini-batches, and then update the variables of all the groups according to the closed-form of $\tilde{\theta}_{l}^{n}$. When the learning steps terminate, we remove the neurons whose parameters have gone to zero.<br />
<br />
='''Experiment'''=<br />
<br />
==='''Set Up'''===<br />
<br />
They use two large-scale image classification datasets, '''ImageNet''' [Russakovsky et al., 2015] and '''Places2-401''' [Zhou et al., 2015]. They also conducted additional experiments on the '''ICDAR''' character recognition dataset of [Jaderberg et al., 2014a]. <br />
<br />
For ImageNet, they used the subset which contains 1000 categories, with 1.2 million training images and 50000 validation images. For Places2-401, it has more than 10 million images with 401 unique scene categories. 5000 to 30000 images are comprised into per category. Both architectures of these two datasets are based on the VGG-B network(BNet) [Simonyan and Zisserman, 2014] and on DecomposeMe8($Dec_{8}$) [ALvarez and Petersson, 2016]. BNet has 10 convolutional layers followed by 3 fully-connected layers. In the experiment, they remove the first 2 fully-connected layers, which we call $BNet^{C}$. $Dec_{8}$ contains 16 convolutional layers with 1D kernels, which can model 8 2D convolutional layers. Both models were trained for a total of 55 epochs with 12000 batches per epoch and a batch size of 48 and 180 for BNet and $Dec_{8}$, respectively. The learning rate was initialized by 0.01 and then multiplied by 0.1. They set $\beta_{l}$=0.102 for the first three layers and $\beta_{l}$=0.255 for the remaining ones.<br />
<br />
For ICDAR dataset, it consists of 185639 training and 5198 test data split into 36 categories. The architecture here starts 6 1D convolutional layers with max-pooling, rather than 3 convolutional layers with a maxout layer [Goodfellow et al., 2013] after each convolution, followed by one fully-connected layer. They call their architecture as Dec3. The model was trained for a total of 45 epochs with a batch size of 256 and 1000 iterations per epoch. The learning rate was initialized by 0.1 and multiplied by 0.1 in the second, seventh and fifteenth epochs. They set $\beta_{l}$=5.1 for the first layer and $\beta_{l}$=10.2 for the remaining ones.<br />
<br />
==='''Results'''===<br />
<br />
[[File:imageNet.png]]<br />
<br />
The above table show the accuracy comparisons between the original architectures and ours. For $Dec_{8}$ on the ImageNet dataset, we evaluated two additional models: $Dec_{8}-640$ with 640 neurons per layer and $Dec_{8}-768$ with 768 neurons per layer. $Dec_{8}-640_{SGL}$ means the sparse group Lasso regularizer with $\alpha=0.5$ and $Dec_{8}-640_{GS}$ represents the group sparsity regularizer. Note that all our architectures yield an improvement over the original network except $Dec_{8}-768$. For instance, Ours-$Bnet_{GS}^{C}$ increases the performance of 1.6% compared to $BNet^{C}$. <br />
<br />
[[File:44.png]]<br />
<br />
[[File:2.png]]<br />
<br />
The above figures reports the reduced percentage of neurons/parameters with our approach for $BNet^{C}$ and $Dec_{8}$. For example, in the first figure, our approach reduces the number of neurons by over 12% and the number of parameters by around 14%, while improving the generalization ability of 1.6%(as indicated by accuracy gap). The left image in the first figure also shows that reduction in number of neurons is spread all the layers with the largest difference in the L10. For $Dec_{8}$, in the second figure, we find when we increase the number of neurons in each layer, the benefits of our approach become more significant. For instance, $Dec_{8}-640$ with group sparsity regularizer reduces the number of neurons by 10%, and of parameters by 12.48%. The left image in the second figure also shows that reduction in number of neurons is spread all the layers. <br />
<br />
[[File:ICDA.png]]<br />
<br />
Finally, the above figure indicates the experiment results for ICDAR dataset. Here, we used the $Dec_{3}$ architecture, where the last two layers initially contain 512 neurons. The accuracy rate for $MaxPllo_{2Dneurons}$ is 83.8%, and accuracy rate for $Dec_{3}$ is 89.3%, which means 1D filters perform better than a network with 2D kernels. Our model on this dataset reduces 38.64% of neurons and totally up to 80% of the number of parameters with group sparsity regularizer.<br />
<br />
All the above results evidence that our algorithm effectively removes the number of parameters and increases the model accuracy. Our algorithm of automatic model selection effectively performs on the classification task.<br />
<br />
='''Analysis on Testing'''=<br />
<br />
Our algorithm does not remove neurons during the training time, however, we remove those neurons after training, which yields a smaller network at test time. This improvement not only reduces the number of parameters of the network, but also decreases the computational memory cost and increases the speed. <br />
<br />
[[File:table2.png]]<br />
<br />
The above table reports the runtime, memory, as well as the percentage of reduced parameters after removing the zeroed-out neurons. The BNet and $Dec_{8}$ were tested on the dataset of ImageNet, while $Dec_{3-GS}$ was tested on the dataset of ICDAR. From the table, we find that all the models for the ImageNet and ICDAR have speeded up the runtime, for example, $Dec_{8}-768_{GS}$ on ImageNet data speeds up the runtime nearly 16% at the batch size of 8, and $Dec_{3}$ on ICDAR data speeds up nearly 50% at natch size of 16. For the percentage of parameters reduced, we find BNet, $Dec_{8}-640_{GS}$ and $Dec_{8}-768_{GS}$ have reduced 12.06%, 26.51%, and 46.73% respectively. More significantly, for $Dec_{3-GS}$, it reduces 82.35% of the parameters. All of these changes show the benefits at the testing time.<br />
<br />
='''Conclusion'''=<br />
<br />
In this paper, we have introduced an approach that relies on group sparsity regularizer. This approach automatically determines the number of neurons in each layer of a deep network. From the experiments, we found our approach not only reduces the number of parameters in our model, but also saves the computation memory and increases the speed at test time. However, the limitation of our approach is that the number of layers in the network remains fixed.<br />
<br />
='''Critique'''=<br />
The authors of the paper state that "...we assume that the parameters of each neuron in layer $l$ are grouped in a vector of size $P_{l}$ and where $\lambda_{l}$ sets the influence of the penalty. Note that, in the general case, this weight can be different for each layer $l$. In practice, however, we found most effective to have<br />
two different weights: a relatively small one for the first few layers, and a larger weight for the<br />
remaining ones. This effectively prevents killing too many neurons in the first few layers, and thus<br />
retains enough information for the remaining ones." However, the authors fail to present any guidance as to what gets counted as "the first few layers" and what the relative sizes for the two weights should be even after we have chosen the "first few layers". Indeed, such choice seems to be an unaccounted component of tuning the model but this receives scant attention in the current paper. Several numerical comparisons should be carried out to allow further discussion on this question.<br />
<br />
The experiments could have included better baseline models to compare against. For example, how do we know the original model was not overly complex to begin with? It might have been a good idea for the authors to compare their group sparse lasso method against the naive method of (blindly) reducing the number of neurons in each layer by 10-20% just for a very preliminary check. Also, the author could have applied the theory of ridge and Lasso regression to analyze the effect of the regularization mathematically.<br />
<br />
A rather reliable method of experimentation to compare the performance and accuracy has been left out. The authors have not stated any comparisons of this method with the Dropout method [Srivastava,2014], which is similar in terms of the physical effects on the network.<br />
<br />
='''References'''=<br />
<br />
P. L. Bartlett. For valid generalization the size of the weights is more important than the size of the network. In NIPS, 1996.<br />
<br />
M. G. Bello. Enhanced training algorithms, and integrated training/architecture selection for multilayer perceptron networks. IEEE Transactions on Neural Networks, 3(6):864–875, Nov 1992.<br />
<br />
Yu Cheng, Felix X. Yu, Rogério Schmidt Feris, Sanjiv Kumar, Alok N. Choudhary, and Shih-Fu Chang. An exploration of parameter redundancy in deep networks with circulant projections. In ICCV, 2015.<br />
<br />
I. J. Goodfellow, D. Warde-farley, M. Mirza, A. Courville, and Y. Bengio. Maxout networks. In ICML, 2013.<br />
<br />
G. E. Hinton, O. Vinyals, and J. Dean. Distilling the knowledge in a neural network. In arXiv, 2014.<br />
<br />
M. Jaderberg, A. Vedaldi, and A. Zisserman. Deep features for text spotting. In ECCV, 2014a.<br />
<br />
M. Jaderberg, A. Vedaldi, and A. Zisserman. Speeding up convolutional neural networks with low rank expansions. In BMVC, 2014b.<br />
<br />
N. Simon, J. Friedman, T. Hastie, and R. Tibshirani. A sparse-group lasso. Journal of Computational and Graphical Statistics, 2013.<br />
<br />
H. Zhou, J. M. Alvarez, and F. Porikli. Less is more: Towards compact CNNs. In ECCV, 2016.<br />
<br />
Group LASSO - https://pdfs.semanticscholar.org/f677/a011b2a912e3c5c604f6872b9716cc0b8aa0.pdf<br />
<br />
Liu, Baoyuan, et al. "Sparse convolutional neural networks." Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2015.<br />
<br />
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. 2014. Dropout: a simple way to prevent neural networks from overfitting. J. Mach. Learn. Res. 15, 1 (January 2014), 1929-1958.<br />
<br />
<br />
Derivation & Motivation of the Soft Thresholding Operator (Proximal Operator):<br />
# http://www.onmyphd.com/?p=proximal.operator<br />
# https://math.stackexchange.com/questions/471339/derivation-of-soft-thresholding-operator</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Learning_the_Number_of_Neurons_in_Deep_Networks&diff=29733Learning the Number of Neurons in Deep Networks2017-11-08T23:01:20Z<p>H4lyu: /* Critique */</p>
<hr />
<div>='''Introduction'''=<br />
<br />
Due to the availability of large-scale datasets and powerful computation, '''Deep Learning''' has made huge breakthroughs in many areas, like Language Models and Computer Vision. In deep neural networks, we need to determine the number of layers and the number of neurons in each layer, i.e, we need to determine the number of parameters, or complexity of the model. Typically, this is determined by errors manually. Currently, this is mostly achieved by manually tuning these hyper-parameters using validation data or building very deep networks. However, building a very deep model is still challenging, especially for very large datasets, which leads to high cost on memory and reduction in speed.<br />
<br />
In this paper, we used an approach to automatically select the number of neurons in each layer when we learn the network. Our approach introduces a '''group sparsity regularizer''' on the parameters of the network, and each group acts on the parameters of one neuron, rather than trains an initial network as as pre-processing step(training shallow or thin networks to mimic the behaviour of deep ones [Hinton et al., 2014, Romero et al., 2015]). We set those useless parameters to zero, which cancels out the effects of a particular neuron. Therefore, our approach does not need to learn a redundant network successfully and then reduce its parameters, instead, it learns the number of relevant neurons in each layer and the parameters of those neurons simultaneously.<br />
<br />
In the experiments on several image recognition datasets, we showed the effectiveness of our approach, which reduces the number of parameters by up to 80% compared to the complete model, and has no recognition accuracy loss at the same time. Actually, our approach even yields more effective and faster networks, and occupies less memory.<br />
<br />
='''Related Work'''=<br />
<br />
The recent researches tend to build very deep networks. Building very deep networks means we need to learn more parameters, which leads to a significant cost on the memory of the equipment as well as the speed. Even though automatic model selection has developed in the past years by constructive and destructive approaches, there are some drawbacks. For '''constructive method''', it starts a super shallow architecture, and then adds additional parameters [Bello, 1992]. A similar work that adds new layers to the initial shallow networks was successfully employed [Simonyan and Zisserman, 2014] at the process of learning. However, we know shallow networks have fewer parameters, so that it can not handle the non-linearities as effectively as the deep networks [Montufar et al., 2014], so shallow networks may easily get stuck by the bad optima. Therefore, the drawback of this method is that these networks may produce poor initializations for the later processes. The authors make this claim without ever providing any evidence for it. For '''destructive method''', it starts by a deep network to reduce a significant number of redundant parameters [Denil et al., 2013, Cheng et al., 2015] while keeping its behaviour unchanged. Even though this technique has shown removing the redundant parameters [LeCun et al., 1990, Hassibi et al., 1993] or the neurons [Mozer and Smolensky, 1988, Ji et al., 1990, Reed, 1993] has little influence on the output, it requires the analysis of each parameter and neuron by network Hessian, which is very computationally expensive for large architectures. The main motivation of these works was to build a more compact network.<br />
<br />
Particularly, building a compact network is a research focus for '''Convolutional Neural Networks'''(CNNs). Some works has proposed to decompose the filters of a pre-trained network into low-rank filters, which reduces the number of parameters [Jaderberg et al., 2014b, Denton et al., 2014, Gong et al., 2014]. The issue of this proposal is that we need to successfully train an initial deep network, since it acts as as post-processing step. [Weigend et al., 1991] and [Collins and Kohl, 2014] used direct training to develop regularizers that eliminate some of the parameters of the network. The problem is that the number of layers and neurons each layer is determined manually. A very similar work using the group lasso method for CNN was previously done in [Liu et al., 2015]. The big-picture idea appears to be very similar but they differ in details of methodology.<br />
<br />
='''Model Training and Model Selection'''=<br />
<br />
In general, a deep network has L layers containing linear operations on their inputs, intertwined with activation functions. The activation function we generally use is '''Rectified Linear Units(RELU) or sigmoids'''. Suppose each layer l has $N_{l}$ neurons, and each of them has parameters $\Theta=(\theta_{l})_{1\leqslant{l}\leqslant{L}}$, where $\theta_{l}=({\theta^n _{l}})_{1\leqslant{n}\leqslant{N_{l}}}$ and where $\theta^n _{l}=[w_{l}^{n},b_{l}^{n}]$ is consisted of linear component $w_{l}^{n}$ and linear bias $b_{l}^{n}$. Given an input $x$, under the linear, on-linear and pooling operations, we obtain the output $\hat{y}=f(x,\theta)$, where $f(*)$ encodes the succession of linear, non-linear and pooling operations.<br />
<br />
At the step of training, we have N input-output pairs ${(x_{i},y_{i})}_{1\leqslant{i}\leqslant{N}}$, and the loss function is given by $\ell(y_{i},f(x_{i},\Theta))$, which compares the predicted output with the ground-truth output. Generally, we choose logistic loss for classification and the square loss for regression. Therefore, learning the parameters of the network is equivalent to solving the optimization of the following:<br />
$$\displaystyle \min_{\Theta}\frac{1}{N}\sum_{i=1}^{N}\ell(y_{i},f(x_{i},\Theta))+\gamma(\Theta),$$ where $\gamma(\Theta)$ represents a regularizer on the network parameters. Our choice for the regularizer can be $\ell_{2}$-norm(i.e, weight decay) or $\ell_{1}$-norm. $\ell_{2}$-norm usually favours small parameter values, and $\ell_{1}$-norm can only delete those irrelevant parameters, but not the neurons. The goal in this paper is to automatically determine the number of neurons of each layer, but neither of the above techniques achieve this goal. Here, we make use of the '''group sparsity''' [Yuan and Lin., 2007] (starting from an overcomplete network and canceling the influence of some neurons). The regularizer, therefore, can be written as $$\gamma(\Theta)=\sum_{l=1}^{L}\beta_{l}\sqrt{P_{l}}\sum_{n=1}^{N_{l}}||\theta_{l}^{n}||_{2},$$ where $P_{l}$ means the size of the vector that includes the parameters of each neuron in layer $l$, and $\beta_{l}$ balances the influence of the penalty. In practice, we found the most effective way to select $\beta$ is a relatively small one for the first few layers, and a larger weight for the remaining layers. The reason we choose a small weight is that it can prevent deleting too much neurons in the first few layers, so that we have enough information for learning the remaining parameters. The original premise of this paper seemed to suggest a new method that was different from both the constructive and destructive methods described above. However, this approach of starting with an overcomplete network and training with group sparsity appears to be no different from destructive methods. The main contribution here is then the regularization function to act on entire neurons, which is in fairness an interesting approach.<br />
<br />
The group sparsity helps us effectively remove some of the neurons, and also standard regularizers on the individual parameters are effective for the generalization purpose [Bartlett, 19996, Krogh and Hertz, 1992, Theodoridis, 2015, Collins and Kohli, 2014]. By this idea, we introduce '''sparse group Lasso''', which considers a more generalised penalty that merges L1 norm in Lasso with the group lasso (i.e. "two-norm"). This leads to the production of a penalty which specifies solutions that are sparse enough both at an individual and group feature levels [1]. It specifies that the regularizer can be written as $$\gamma(\Theta)=\sum_{l=1}^{L}((1-\alpha)\beta_{l}\sqrt{P_{l}}\sum_{n=1}^{N_{l}}||\theta_{l}^{n}||_{2}+\alpha\beta_{l}||\theta_{l}||_{1},$$ where $\alpha\in[0,1]$. We find that if $\alpha=0$, then we have the group sparsity regularizer. In practice, we use both $\alpha=0$ and $\alpha=0.5$ in the experiments.<br />
<br />
This reminds me of the relationships among Lasso regression, Ridge regression and Elastic Net regression. In lasso regression, the penalized residual sum of squares is composed of the regular residual sum of squared plus a L1 regularizer. In ridge regression, its penalized residual sum of squares is composed of the regular residual sum of squared plus a L2 regularizer. Finally, an elastic net regression is a combination of lasso regularizer and ridge regularizer, where its objective function is to optimize parameters by including both L1 and L2 norms. <br />
<br />
To find the optimization, in this paper we use proximal gradient descent [Parikh and Boyed, 2014]. This approach iteratively takes a gradient step of size t with respect to the loss. The following is the algorithm for it: <br />
<br />
We define proximal operator of f as $$prox_{f}(v)=\displaystyle \min_{x}(\frac{1}{2t}||x-v||_{2}^{2}+f(x))$$ <br />
<br />
<br />
Suppose we want to minimize $f(x)+g(x)$, and the proximal gradient method is given by $$x^{(k+1)}=prox_{t^{k}g}(x^{k}-t^{k}\nabla{f}(x^{k})), k=1,2,3...$$ <br />
<br />
Therefore, we can update our parameter by the above method as $$\tilde{\theta}_{l}^{n}=\displaystyle \min_{\theta_{l}^{n}}\frac{1}{2t}||\theta_{l}^{n}-\hat{\theta}_{l}^{n}||_{2}^{2}+\gamma(\Theta),$$<br />
where $\hat{\theta}_{l}^{n}$ is the solution obtained from the general loss gradient. By the derivative of [Simon et al., 2013], we have a closed-form solution for this problem: <br />
$$\tilde{\theta}_{l}^{n}=(1-\frac{t(1-\alpha)\beta_{l}\sqrt{P_{l}}}{||S(\hat{\theta}_{l}^{n},t\alpha\beta_{l})||_{2})})_{+}S(\hat{\theta}_{l}^{n},t\alpha\beta_{l}),$$<br />
where + refers to taking the maximum between the argument and 0, and $S(*)$ is $$S(a,b)=sign(a)(|a|-b)_{+}$$<br />
In practice, we use stochastic gradient descent and work with mini-batches, and then update the variables of all the groups according to the closed-form of $\tilde{\theta}_{l}^{n}$. When the learning steps terminate, we remove the neurons whose parameters have gone to zero.<br />
<br />
='''Experiment'''=<br />
<br />
==='''Set Up'''===<br />
<br />
They use two large-scale image classification datasets, '''ImageNet''' [Russakovsky et al., 2015] and '''Places2-401''' [Zhou et al., 2015]. They also conducted additional experiments on the '''ICDAR''' character recognition dataset of [Jaderberg et al., 2014a]. <br />
<br />
For ImageNet, they used the subset which contains 1000 categories, with 1.2 million training images and 50000 validation images. For Places2-401, it has more than 10 million images with 401 unique scene categories. 5000 to 30000 images are comprised into per category. Both architectures of these two datasets are based on the VGG-B network(BNet) [Simonyan and Zisserman, 2014] and on DecomposeMe8($Dec_{8}$) [ALvarez and Petersson, 2016]. BNet has 10 convolutional layers followed by 3 fully-connected layers. In the experiment, they remove the first 2 fully-connected layers, which we call $BNet^{C}$. $Dec_{8}$ contains 16 convolutional layers with 1D kernels, which can model 8 2D convolutional layers. Both models were trained for a total of 55 epochs with 12000 batches per epoch and a batch size of 48 and 180 for BNet and $Dec_{8}$, respectively. The learning rate was initialized by 0.01 and then multiplied by 0.1. They set $\beta_{l}$=0.102 for the first three layers and $\beta_{l}$=0.255 for the remaining ones.<br />
<br />
For ICDAR dataset, it consists of 185639 training and 5198 test data split into 36 categories. The architecture here starts 6 1D convolutional layers with max-pooling, rather than 3 convolutional layers with a maxout layer [Goodfellow et al., 2013] after each convolution, followed by one fully-connected layer. They call their architecture as Dec3. The model was trained for a total of 45 epochs with a batch size of 256 and 1000 iterations per epoch. The learning rate was initialized by 0.1 and multiplied by 0.1 in the second, seventh and fifteenth epochs. They set $\beta_{l}$=5.1 for the first layer and $\beta_{l}$=10.2 for the remaining ones.<br />
<br />
==='''Results'''===<br />
<br />
[[File:imageNet.png]]<br />
<br />
The above table show the accuracy comparisons between the original architectures and ours. For $Dec_{8}$ on the ImageNet dataset, we evaluated two additional models: $Dec_{8}-640$ with 640 neurons per layer and $Dec_{8}-768$ with 768 neurons per layer. $Dec_{8}-640_{SGL}$ means the sparse group Lasso regularizer with $\alpha=0.5$ and $Dec_{8}-640_{GS}$ represents the group sparsity regularizer. Note that all our architectures yield an improvement over the original network except $Dec_{8}-768$. For instance, Ours-$Bnet_{GS}^{C}$ increases the performance of 1.6% compared to $BNet^{C}$. <br />
<br />
[[File:44.png]]<br />
<br />
[[File:2.png]]<br />
<br />
The above figures reports the reduced percentage of neurons/parameters with our approach for $BNet^{C}$ and $Dec_{8}$. For example, in the first figure, our approach reduces the number of neurons by over 12% and the number of parameters by around 14%, while improving the generalization ability of 1.6%(as indicated by accuracy gap). The left image in the first figure also shows that reduction in number of neurons is spread all the layers with the largest difference in the L10. For $Dec_{8}$, in the second figure, we find when we increase the number of neurons in each layer, the benefits of our approach become more significant. For instance, $Dec_{8}-640$ with group sparsity regularizer reduces the number of neurons by 10%, and of parameters by 12.48%. The left image in the second figure also shows that reduction in number of neurons is spread all the layers. <br />
<br />
[[File:ICDA.png]]<br />
<br />
Finally, the above figure indicates the experiment results for ICDAR dataset. Here, we used the $Dec_{3}$ architecture, where the last two layers initially contain 512 neurons. The accuracy rate for $MaxPllo_{2Dneurons}$ is 83.8%, and accuracy rate for $Dec_{3}$ is 89.3%, which means 1D filters perform better than a network with 2D kernels. Our model on this dataset reduces 38.64% of neurons and totally up to 80% of the number of parameters with group sparsity regularizer.<br />
<br />
All the above results evidence that our algorithm effectively removes the number of parameters and increases the model accuracy. Our algorithm of automatic model selection effectively performs on the classification task.<br />
<br />
='''Analysis on Testing'''=<br />
<br />
Our algorithm does not remove neurons during the training time, however, we remove those neurons after training, which yields a smaller network at test time. This improvement not only reduces the number of parameters of the network, but also decreases the computational memory cost and increases the speed. <br />
<br />
[[File:table2.png]]<br />
<br />
The above table reports the runtime, memory, as well as the percentage of reduced parameters after removing the zeroed-out neurons. The BNet and $Dec_{8}$ were tested on the dataset of ImageNet, while $Dec_{3-GS}$ was tested on the dataset of ICDAR. From the table, we find that all the models for the ImageNet and ICDAR have speeded up the runtime, for example, $Dec_{8}-768_{GS}$ on ImageNet data speeds up the runtime nearly 16% at the batch size of 8, and $Dec_{3}$ on ICDAR data speeds up nearly 50% at natch size of 16. For the percentage of parameters reduced, we find BNet, $Dec_{8}-640_{GS}$ and $Dec_{8}-768_{GS}$ have reduced 12.06%, 26.51%, and 46.73% respectively. More significantly, for $Dec_{3-GS}$, it reduces 82.35% of the parameters. All of these changes show the benefits at the testing time.<br />
<br />
='''Conclusion'''=<br />
<br />
In this paper, we have introduced an approach that relies on group sparsity regularizer. This approach automatically determines the number of neurons in each layer of a deep network. From the experiments, we found our approach not only reduces the number of parameters in our model, but also saves the computation memory and increases the speed at test time. However, the limitation of our approach is that the number of layers in the network remains fixed.<br />
<br />
='''Critique'''=<br />
The authors of the paper state that "...we assume that the parameters of each neuron in layer $l$ are grouped in a vector of size $P_{l}$ and where $\lambda_{l}$ sets the influence of the penalty. Note that, in the general case, this weight can be different for each layer $l$. In practice, however, we found most effective to have<br />
two different weights: a relatively small one for the first few layers, and a larger weight for the<br />
remaining ones. This effectively prevents killing too many neurons in the first few layers, and thus<br />
retains enough information for the remaining ones." However, the authors fail to present any guidance as to what gets counted as "the first few layers" and what the relative sizes for the two weights should be even after we have chosen the "first few layers". Indeed, such choice seems to be an unaccounted component of tuning the model but this receives scant attention in the current paper.<br />
<br />
The experiments could have included better baseline models to compare against. For example, how do we know the original model was not overly complex to begin with? It might have been a good idea for the authors to compare their group sparse lasso method against the naive method of (blindly) reducing the number of neurons in each layer by 10-20% just for a very preliminary check. Also, the author could have applied the theory of ridge and Lasso regression to analyze the effect of the regularization mathematically.<br />
<br />
A rather reliable method of experimentation to compare the performance and accuracy has been left out. The authors have not stated any comparisons of this method with the Dropout method [Srivastava,2014], which is similar in terms of the physical effects on the network.<br />
<br />
='''References'''=<br />
<br />
P. L. Bartlett. For valid generalization the size of the weights is more important than the size of the network. In NIPS, 1996.<br />
<br />
M. G. Bello. Enhanced training algorithms, and integrated training/architecture selection for multilayer perceptron networks. IEEE Transactions on Neural Networks, 3(6):864–875, Nov 1992.<br />
<br />
Yu Cheng, Felix X. Yu, Rogério Schmidt Feris, Sanjiv Kumar, Alok N. Choudhary, and Shih-Fu Chang. An exploration of parameter redundancy in deep networks with circulant projections. In ICCV, 2015.<br />
<br />
I. J. Goodfellow, D. Warde-farley, M. Mirza, A. Courville, and Y. Bengio. Maxout networks. In ICML, 2013.<br />
<br />
G. E. Hinton, O. Vinyals, and J. Dean. Distilling the knowledge in a neural network. In arXiv, 2014.<br />
<br />
M. Jaderberg, A. Vedaldi, and A. Zisserman. Deep features for text spotting. In ECCV, 2014a.<br />
<br />
M. Jaderberg, A. Vedaldi, and A. Zisserman. Speeding up convolutional neural networks with low rank expansions. In BMVC, 2014b.<br />
<br />
N. Simon, J. Friedman, T. Hastie, and R. Tibshirani. A sparse-group lasso. Journal of Computational and Graphical Statistics, 2013.<br />
<br />
H. Zhou, J. M. Alvarez, and F. Porikli. Less is more: Towards compact CNNs. In ECCV, 2016.<br />
<br />
Group LASSO - https://pdfs.semanticscholar.org/f677/a011b2a912e3c5c604f6872b9716cc0b8aa0.pdf<br />
<br />
Liu, Baoyuan, et al. "Sparse convolutional neural networks." Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2015.<br />
<br />
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. 2014. Dropout: a simple way to prevent neural networks from overfitting. J. Mach. Learn. Res. 15, 1 (January 2014), 1929-1958.<br />
<br />
<br />
Derivation & Motivation of the Soft Thresholding Operator (Proximal Operator):<br />
# http://www.onmyphd.com/?p=proximal.operator<br />
# https://math.stackexchange.com/questions/471339/derivation-of-soft-thresholding-operator</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Learning_the_Number_of_Neurons_in_Deep_Networks&diff=29730Learning the Number of Neurons in Deep Networks2017-11-08T22:42:08Z<p>H4lyu: /* Model Training and Model Selection */</p>
<hr />
<div>='''Introduction'''=<br />
<br />
Due to the availability of large-scale datasets and powerful computation, '''Deep Learning''' has made huge breakthroughs in many areas, like Language Models and Computer Vision. In deep neural networks, we need to determine the number of layers and the number of neurons in each layer, i.e, we need to determine the number of parameters, or complexity of the model. Typically, this is determined by errors manually. Currently, this is mostly achieved by manually tuning these hyper-parameters using validation data or building very deep networks. However, building a very deep model is still challenging, especially for very large datasets, which leads to high cost on memory and reduction in speed.<br />
<br />
In this paper, we used an approach to automatically select the number of neurons in each layer when we learn the network. Our approach introduces a '''group sparsity regularizer''' on the parameters of the network, and each group acts on the parameters of one neuron, rather than trains an initial network as as pre-processing step(training shallow or thin networks to mimic the behaviour of deep ones [Hinton et al., 2014, Romero et al., 2015]). We set those useless parameters to zero, which cancels out the effects of a particular neuron. Therefore, our approach does not need to learn a redundant network successfully and then reduce its parameters, instead, it learns the number of relevant neurons in each layer and the parameters of those neurons simultaneously.<br />
<br />
In the experiments on several image recognition datasets, we showed the effectiveness of our approach, which reduces the number of parameters by up to 80% compared to the complete model, and has no recognition accuracy loss at the same time. Actually, our approach even yields more effective and faster networks, and occupies less memory.<br />
<br />
='''Related Work'''=<br />
<br />
The recent researches tend to build very deep networks. Building very deep networks means we need to learn more parameters, which leads to a significant cost on the memory of the equipment as well as the speed. Even though automatic model selection has developed in the past years by constructive and destructive approaches, there are some drawbacks. For '''constructive method''', it starts a super shallow architecture, and then adds additional parameters [Bello, 1992]. A similar work that adds new layers to the initial shallow networks was successfully employed [Simonyan and Zisserman, 2014] at the process of learning. However, we know shallow networks have fewer parameters, so that it can not handle the non-linearities as effectively as the deep networks [Montufar et al., 2014], so shallow networks may easily get stuck by the bad optima. Therefore, the drawback of this method is that these networks may produce poor initializations for the later processes. The authors make this claim without ever providing any evidence for it. For '''destructive method''', it starts by a deep network to reduce a significant number of redundant parameters [Denil et al., 2013, Cheng et al., 2015] while keeping its behaviour unchanged. Even though this technique has shown removing the redundant parameters [LeCun et al., 1990, Hassibi et al., 1993] or the neurons [Mozer and Smolensky, 1988, Ji et al., 1990, Reed, 1993] has little influence on the output, it requires the analysis of each parameter and neuron by network Hessian, which is very computationally expensive for large architectures. The main motivation of these works was to build a more compact network.<br />
<br />
Particularly, building a compact network is a research focus for '''Convolutional Neural Networks'''(CNNs). Some works has proposed to decompose the filters of a pre-trained network into low-rank filters, which reduces the number of parameters [Jaderberg et al., 2014b, Denton et al., 2014, Gong et al., 2014]. The issue of this proposal is that we need to successfully train an initial deep network, since it acts as as post-processing step. [Weigend et al., 1991] and [Collins and Kohl, 2014] used direct training to develop regularizers that eliminate some of the parameters of the network. The problem is that the number of layers and neurons each layer is determined manually. A very similar work using the group lasso method for CNN was previously done in [Liu et al., 2015]. The big-picture idea appears to be very similar but they differ in details of methodology.<br />
<br />
='''Model Training and Model Selection'''=<br />
<br />
In general, a deep network has L layers containing linear operations on their inputs, intertwined with activation functions. The activation function we generally use is '''Rectified Linear Units(RELU) or sigmoids'''. Suppose each layer l has $N_{l}$ neurons, and each of them has parameters $\Theta=(\theta_{l})_{1\leqslant{l}\leqslant{L}}$, where $\theta_{l}=({\theta^n _{l}})_{1\leqslant{n}\leqslant{N_{l}}}$ and where $\theta^n _{l}=[w_{l}^{n},b_{l}^{n}]$ is consisted of linear component $w_{l}^{n}$ and linear bias $b_{l}^{n}$. Given an input $x$, under the linear, on-linear and pooling operations, we obtain the output $\hat{y}=f(x,\theta)$, where $f(*)$ encodes the succession of linear, non-linear and pooling operations.<br />
<br />
At the step of training, we have N input-output pairs ${(x_{i},y_{i})}_{1\leqslant{i}\leqslant{N}}$, and the loss function is given by $\ell(y_{i},f(x_{i},\Theta))$, which compares the predicted output with the ground-truth output. Generally, we choose logistic loss for classification and the square loss for regression. Therefore, learning the parameters of the network is equivalent to solving the optimization of the following:<br />
$$\displaystyle \min_{\Theta}\frac{1}{N}\sum_{i=1}^{N}\ell(y_{i},f(x_{i},\Theta))+\gamma(\Theta),$$ where $\gamma(\Theta)$ represents a regularizer on the network parameters. Our choice for the regularizer can be $\ell_{2}$-norm(i.e, weight decay) or $\ell_{1}$-norm. $\ell_{2}$-norm usually favours small parameter values, and $\ell_{1}$-norm can only delete those irrelevant parameters, but not the neurons. The goal in this paper is to automatically determine the number of neurons of each layer, but neither of the above techniques achieve this goal. Here, we make use of the '''group sparsity''' [Yuan and Lin., 2007] (starting from an overcomplete network and canceling the influence of some neurons). The regularizer, therefore, can be written as $$\gamma(\Theta)=\sum_{l=1}^{L}\beta_{l}\sqrt{P_{l}}\sum_{n=1}^{N_{l}}||\theta_{l}^{n}||_{2},$$ where $P_{l}$ means the size of the vector that includes the parameters of each neuron in layer $l$, and $\beta_{l}$ balances the influence of the penalty. In practice, we found the most effective way to select $\beta$ is a relatively small one for the first few layers, and a larger weight for the remaining layers. The reason we choose a small weight is that it can prevent deleting too much neurons in the first few layers, so that we have enough information for learning the remaining parameters. The original premise of this paper seemed to suggest a new method that was different from both the constructive and destructive methods described above. However, this approach of starting with an overcomplete network and training with group sparsity appears to be no different from destructive methods. The main contribution here is then the regularization function to act on entire neurons, which is in fairness an interesting approach.<br />
<br />
The group sparsity helps us effectively remove some of the neurons, and also standard regularizers on the individual parameters are effective for the generalization purpose [Bartlett, 19996, Krogh and Hertz, 1992, Theodoridis, 2015, Collins and Kohli, 2014]. By this idea, we introduce '''sparse group Lasso''', which considers a more generalised penalty that merges L1 norm in Lasso with the group lasso (i.e. "two-norm"). This leads to the production of a penalty which specifies solutions that are sparse enough both at an individual and group feature levels [1]. It specifies that the regularizer can be written as $$\gamma(\Theta)=\sum_{l=1}^{L}((1-\alpha)\beta_{l}\sqrt{P_{l}}\sum_{n=1}^{N_{l}}||\theta_{l}^{n}||_{2}+\alpha\beta_{l}||\theta_{l}||_{1},$$ where $\alpha\in[0,1]$. We find that if $\alpha=0$, then we have the group sparsity regularizer. In practice, we use both $\alpha=0$ and $\alpha=0.5$ in the experiments.<br />
<br />
This reminds me of the relationships among Lasso regression, Ridge regression and Elastic Net regression. In lasso regression, the penalized residual sum of squares is composed of the regular residual sum of squared plus a L1 regularizer. In ridge regression, its penalized residual sum of squares is composed of the regular residual sum of squared plus a L2 regularizer. Finally, an elastic net regression is a combination of lasso regularizer and ridge regularizer, where its objective function is to optimize parameters by including both L1 and L2 norms. <br />
<br />
To find the optimization, in this paper we use proximal gradient descent [Parikh and Boyed, 2014]. This approach iteratively takes a gradient step of size t with respect to the loss. The following is the algorithm for it: <br />
<br />
We define proximal operator of f as $$prox_{f}(v)=\displaystyle \min_{x}(\frac{1}{2t}||x-v||_{2}^{2}+f(x))$$ <br />
<br />
<br />
Suppose we want to minimize $f(x)+g(x)$, and the proximal gradient method is given by $$x^{(k+1)}=prox_{t^{k}g}(x^{k}-t^{k}\nabla{f}(x^{k})), k=1,2,3...$$ <br />
<br />
Therefore, we can update our parameter by the above method as $$\tilde{\theta}_{l}^{n}=\displaystyle \min_{\theta_{l}^{n}}\frac{1}{2t}||\theta_{l}^{n}-\hat{\theta}_{l}^{n}||_{2}^{2}+\gamma(\Theta),$$<br />
where $\hat{\theta}_{l}^{n}$ is the solution obtained from the general loss gradient. By the derivative of [Simon et al., 2013], we have a closed-form solution for this problem: <br />
$$\tilde{\theta}_{l}^{n}=(1-\frac{t(1-\alpha)\beta_{l}\sqrt{P_{l}}}{||S(\hat{\theta}_{l}^{n},t\alpha\beta_{l})||_{2})})_{+}S(\hat{\theta}_{l}^{n},t\alpha\beta_{l}),$$<br />
where + refers to taking the maximum between the argument and 0, and $S(*)$ is $$S(a,b)=sign(a)(|a|-b)_{+}$$<br />
In practice, we use stochastic gradient descent and work with mini-batches, and then update the variables of all the groups according to the closed-form of $\tilde{\theta}_{l}^{n}$. When the learning steps terminate, we remove the neurons whose parameters have gone to zero.<br />
<br />
='''Experiment'''=<br />
<br />
==='''Set Up'''===<br />
<br />
They use two large-scale image classification datasets, '''ImageNet''' [Russakovsky et al., 2015] and '''Places2-401''' [Zhou et al., 2015]. They also conducted additional experiments on the '''ICDAR''' character recognition dataset of [Jaderberg et al., 2014a]. <br />
<br />
For ImageNet, they used the subset which contains 1000 categories, with 1.2 million training images and 50000 validation images. For Places2-401, it has more than 10 million images with 401 unique scene categories. 5000 to 30000 images are comprised into per category. Both architectures of these two datasets are based on the VGG-B network(BNet) [Simonyan and Zisserman, 2014] and on DecomposeMe8($Dec_{8}$) [ALvarez and Petersson, 2016]. BNet has 10 convolutional layers followed by 3 fully-connected layers. In the experiment, they remove the first 2 fully-connected layers, which we call $BNet^{C}$. $Dec_{8}$ contains 16 convolutional layers with 1D kernels, which can model 8 2D convolutional layers. Both models were trained for a total of 55 epochs with 12000 batches per epoch and a batch size of 48 and 180 for BNet and $Dec_{8}$, respectively. The learning rate was initialized by 0.01 and then multiplied by 0.1. They set $\beta_{l}$=0.102 for the first three layers and $\beta_{l}$=0.255 for the remaining ones.<br />
<br />
For ICDAR dataset, it consists of 185639 training and 5198 test data split into 36 categories. The architecture here starts 6 1D convolutional layers with max-pooling, rather than 3 convolutional layers with a maxout layer [Goodfellow et al., 2013] after each convolution, followed by one fully-connected layer. They call their architecture as Dec3. The model was trained for a total of 45 epochs with a batch size of 256 and 1000 iterations per epoch. The learning rate was initialized by 0.1 and multiplied by 0.1 in the second, seventh and fifteenth epochs. They set $\beta_{l}$=5.1 for the first layer and $\beta_{l}$=10.2 for the remaining ones.<br />
<br />
==='''Results'''===<br />
<br />
[[File:imageNet.png]]<br />
<br />
The above table show the accuracy comparisons between the original architectures and ours. For $Dec_{8}$ on the ImageNet dataset, we evaluated two additional models: $Dec_{8}-640$ with 640 neurons per layer and $Dec_{8}-768$ with 768 neurons per layer. $Dec_{8}-640_{SGL}$ means the sparse group Lasso regularizer with $\alpha=0.5$ and $Dec_{8}-640_{GS}$ represents the group sparsity regularizer. Note that all our architectures yield an improvement over the original network except $Dec_{8}-768$. For instance, Ours-$Bnet_{GS}^{C}$ increases the performance of 1.6% compared to $BNet^{C}$. <br />
<br />
[[File:44.png]]<br />
<br />
[[File:2.png]]<br />
<br />
The above figures reports the reduced percentage of neurons/parameters with our approach for $BNet^{C}$ and $Dec_{8}$. For example, in the first figure, our approach reduces the number of neurons by over 12% and the number of parameters by around 14%, while improving the generalization ability of 1.6%(as indicated by accuracy gap). The left image in the first figure also shows that reduction in number of neurons is spread all the layers with the largest difference in the L10. For $Dec_{8}$, in the second figure, we find when we increase the number of neurons in each layer, the benefits of our approach become more significant. For instance, $Dec_{8}-640$ with group sparsity regularizer reduces the number of neurons by 10%, and of parameters by 12.48%. The left image in the second figure also shows that reduction in number of neurons is spread all the layers. <br />
<br />
[[File:ICDA.png]]<br />
<br />
Finally, the above figure indicates the experiment results for ICDAR dataset. Here, we used the $Dec_{3}$ architecture, where the last two layers initially contain 512 neurons. The accuracy rate for $MaxPllo_{2Dneurons}$ is 83.8%, and accuracy rate for $Dec_{3}$ is 89.3%, which means 1D filters perform better than a network with 2D kernels. Our model on this dataset reduces 38.64% of neurons and totally up to 80% of the number of parameters with group sparsity regularizer.<br />
<br />
All the above results evidence that our algorithm effectively removes the number of parameters and increases the model accuracy. Our algorithm of automatic model selection effectively performs on the classification task.<br />
<br />
='''Analysis on Testing'''=<br />
<br />
Our algorithm does not remove neurons during the training time, however, we remove those neurons after training, which yields a smaller network at test time. This improvement not only reduces the number of parameters of the network, but also decreases the computational memory cost and increases the speed. <br />
<br />
[[File:table2.png]]<br />
<br />
The above table reports the runtime, memory, as well as the percentage of reduced parameters after removing the zeroed-out neurons. The BNet and $Dec_{8}$ were tested on the dataset of ImageNet, while $Dec_{3-GS}$ was tested on the dataset of ICDAR. From the table, we find that all the models for the ImageNet and ICDAR have speeded up the runtime, for example, $Dec_{8}-768_{GS}$ on ImageNet data speeds up the runtime nearly 16% at the batch size of 8, and $Dec_{3}$ on ICDAR data speeds up nearly 50% at natch size of 16. For the percentage of parameters reduced, we find BNet, $Dec_{8}-640_{GS}$ and $Dec_{8}-768_{GS}$ have reduced 12.06%, 26.51%, and 46.73% respectively. More significantly, for $Dec_{3-GS}$, it reduces 82.35% of the parameters. All of these changes show the benefits at the testing time.<br />
<br />
='''Conclusion'''=<br />
<br />
In this paper, we have introduced an approach that relies on group sparsity regularizer. This approach automatically determines the number of neurons in each layer of a deep network. From the experiments, we found our approach not only reduces the number of parameters in our model, but also saves the computation memory and increases the speed at test time. However, the limitation of our approach is that the number of layers in the network remains fixed.<br />
<br />
='''Critique'''=<br />
The authors of the paper state that ``...we assume that the parameters of each neuron in layer $l$ are grouped in a vector of size $P_{l}$ and where $\lambda_{l}$ sets the influence of the penalty. Note that, in the general case, this weight can be different for each layer $l$. In practice, however, we found most effective to have<br />
two different weights: a relatively small one for the first few layers, and a larger weight for the<br />
remaining ones. This effectively prevents killing too many neurons in the first few layers, and thus<br />
retains enough information for the remaining ones.`` However, the authors fail to present any guidance as to what gets counted as ``the first few layers`` and what the relative sizes for the two weights should be even after we have chosen the ``first few layers``. Indeed, such choice seems to be an unaccounted component of tuning the model but this receives scant attention in the current paper.<br />
<br />
The experiments could have included better baseline models to compare against. For example, how do we know the original model was not overly complex to begin with? It might have been a good idea for the authors to compare their group sparse lasso method against the naive method of (blindly) reducing the number of neurons in each layer by 10-20% just for a very preliminary check.<br />
<br />
A rather reliable method of experimentation to compare the performance and accuracy has been left out. The authors have not stated any comparisons of this method with the Dropout method [Srivastava,2014], which is similar in terms of the physical effects on the network.<br />
<br />
='''References'''=<br />
<br />
P. L. Bartlett. For valid generalization the size of the weights is more important than the size of the network. In NIPS, 1996.<br />
<br />
M. G. Bello. Enhanced training algorithms, and integrated training/architecture selection for multilayer perceptron networks. IEEE Transactions on Neural Networks, 3(6):864–875, Nov 1992.<br />
<br />
Yu Cheng, Felix X. Yu, Rogério Schmidt Feris, Sanjiv Kumar, Alok N. Choudhary, and Shih-Fu Chang. An exploration of parameter redundancy in deep networks with circulant projections. In ICCV, 2015.<br />
<br />
I. J. Goodfellow, D. Warde-farley, M. Mirza, A. Courville, and Y. Bengio. Maxout networks. In ICML, 2013.<br />
<br />
G. E. Hinton, O. Vinyals, and J. Dean. Distilling the knowledge in a neural network. In arXiv, 2014.<br />
<br />
M. Jaderberg, A. Vedaldi, and A. Zisserman. Deep features for text spotting. In ECCV, 2014a.<br />
<br />
M. Jaderberg, A. Vedaldi, and A. Zisserman. Speeding up convolutional neural networks with low rank expansions. In BMVC, 2014b.<br />
<br />
N. Simon, J. Friedman, T. Hastie, and R. Tibshirani. A sparse-group lasso. Journal of Computational and Graphical Statistics, 2013.<br />
<br />
H. Zhou, J. M. Alvarez, and F. Porikli. Less is more: Towards compact CNNs. In ECCV, 2016.<br />
<br />
Group LASSO - https://pdfs.semanticscholar.org/f677/a011b2a912e3c5c604f6872b9716cc0b8aa0.pdf<br />
<br />
Liu, Baoyuan, et al. "Sparse convolutional neural networks." Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2015.<br />
<br />
Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. 2014. Dropout: a simple way to prevent neural networks from overfitting. J. Mach. Learn. Res. 15, 1 (January 2014), 1929-1958.<br />
<br />
<br />
Derivation & Motivation of the Soft Thresholding Operator (Proximal Operator):<br />
# http://www.onmyphd.com/?p=proximal.operator<br />
# https://math.stackexchange.com/questions/471339/derivation-of-soft-thresholding-operator</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Convolutional_Sequence_to_Sequence_Learning&diff=29537Convolutional Sequence to Sequence Learning2017-11-07T05:44:09Z<p>H4lyu: </p>
<hr />
<div>= Introduction=<br />
<br />
'''Sequence to sequence learning''' has been used to solve many tasks such as machine translation, speech recognition and text summarization. Most of the past models employ RNNs for this problem, with bidirectional RNNs with soft attention being the dominant approach.<br />
On the other hand, CNNs have not been used for this task, despite the many advantages they offer:<br />
* Compared to recurrent layers, convolutions create representations for fixed size contexts, however, the effective context size of the network can easily be made larger by stacking several layers on top of each other. This allows to precisely control the maximum length of dependencies to be modeled. <br />
* Convolutional networks do not depend on the computations of the previous time step and therefore allow parallelization over every element in a sequence. This contrasts with RNNs which maintain a hidden state of the entire past that prevents parallel computation within a sequence.<br />
* Multi-layer convolutional neural networks create hierarchical representations over the input sequence in which nearby input elements interact at lower layers while distant elements interact at higher layers. This provides a shorter path to capture long-range dependencies compared to the chain structure modeled by recurrent networks.<br />
<br />
In this paper, the authors introduce an architecture for sequence learning based entirely on convolutional neural networks. Compared to recurrent models, computations over all elements can be fully parallelized during training to better exploit the GPU hardware and optimization is easier since the number of non-linearities is fixed and independent of the input length. The use of gated linear units eases gradient propagation and equipping each decoder layer with a separate attention module adding a negligible amount of overhead.<br />
The combination of these choices enables to tackle large scale problems. They outperform the accuracy of the deep LSTM setup of Wu et al. (2016) and is now the state of the art model for neural machine translation.<br />
<br />
= Related Work =<br />
<br />
Bradbury et al.(2016) introduce a quasi-recurrent neural network (QRNNs), an approach to neural sequence modelling that alternates convolutional layers, which apply in parallel across timesteps, and a minimalist recurrent pooling function that applies in parallel across channels. They use QRNNs for sentiment classification, language modelling and aslo briefly describe about an architecture consisting of QRNNs for sequence to sequence learning.<br />
<br />
Kalchbrenner et al.(2016) introduce an architecture called "bytenet". The ByteNet is a one-dimensional convolutional neural network that is composed of two parts, one to encode the source sequence and the other to decode the target sequence. This network has two core properties: it runs in time that is linear in the length of the sequences and it sidesteps the need for excessive memorization.<br />
<br />
However, none of the above approaches has been demonstrated improvements over state of the art results on large benchmark datasets. Gated convolutions have been previously explored for machine translation by Meng et al. (2015) but their evaluation was restricted to a small dataset. The author himself has explored architectures which used CNN but only in the encoder, the decoder part was still Recurrent.<br />
<br />
= Background =<br />
<br />
* '''Sequence to sequence Learning with RNNs ''' - https://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946f15/Sequence_to_sequence_learning_with_neural_networks<br />
* ''' Perplexity(PPL) ''' - In machine learning, the term perplexity has three closely related meanings. Perplexity is a measure of how easy a probability distribution is to predict. Perplexity is a measure of how variable a prediction model is. And perplexity is a measure of prediction error. The third meaning of perplexity is calculated slightly differently but all three have the same fundamental idea. This is given by <math>2^{{H(p)}}=2^{{-\sum _{x}p(x)\log _{2}p(x)}} </math> Suppose you have a four-sided dice (not sure what that’d be). The dice is fair so all sides are equally likely (0.25, 0.25, 0.25, 0.25) and so it’s value here is 4.00. Now suppose you have a different dice whose sides have probabilities (0.10, 0.40, 0.20, 0.30). This dice has perplexity 3.5961 which is lower than 4.00 because it’s easier to predict (namely, predict the side that has p = 0.40).<br />
<br />
* ''' BLEU (BiLingual Evaluation Understudy)''' - It is an algorithm for evaluating the quality of text which has been machine-translated from one natural language to another. Quality is considered to be the correspondence between a machine's output and that of a human: "the closer a machine translation is to a professional human translation, the better it is" – this is the central idea behind BLEU. Additionally, it was one of the first metrics to claim a high correlation with human judgements of quality [11, 12 and 13] and remains one of the most popular automated and inexpensive metrics. More details at : http://www1.cs.columbia.edu/nlp/sgd/bleu.pdf<br />
<br />
* '''Gating Mechanisms''' Gating mechanisms control information flow within a neural network. For example, LSTMs use gating mechanisms to enable long-term memory by using a separate cell controlled by input and forget gates. These gates prevent information from vanishing through transformations. CNNs are not limited by vanishing gradients and do not require forget gates as LSTMs. Gates are useful in controlling what information is propagated through a hierarchy of layers. <br />
= Convolutional Architecture =<br />
<br />
[[File:cs2s_arch.png|Arch]]<br />
== Position Embeddings == <br />
<br />
The architecture uses both word embeddings as well as positional embeddings as the input for the Convolutional Layer. The position order is used to equip the model to recognize the ordering of the word in the sequence and helps the model to know which element it is dealing with.<br />
<br />
For input words <math>x = (x_1, ...,x_m)</math> we get the word vector representation as <math> w = (w_1,....,w_m)</math> and position vectors as <math>p = (p_1,....,p_m)</math> where <math>p_i</math> denotes the actual position of the word in the input sequence.<br />
<br />
Both the vectors are combined to get the input element representation <math>e = (w_1 + p_1,....,w_m+p_m)</math><br />
<br />
Similarly for output elements that were already generated by the decoder network to yield output element representations that are being fed back into the decoder network <math>g = (g_1,....,g_n) </math><br />
<br />
== Convolutional Block Structure ==<br />
<br />
Both encoder and decoder networks share a simple structure of blocks/layers that computes intermediate states based on a fixed number of input elements. The output of l-th block of decoder is denoted by <math>h^l = (h_1^l,....,h_n^l)</math> and <math>z^l = (z_1^l,....,z_m^l)</math>. Each block contains a one-dimensional convolution followed by a non-linearity.<br />
For a decoder network with a single block and kernel width <i>k</i>, each resulting state <math>h_i^1</math> contains information over <i>k</i> input elements. Stacking several blocks on top of each other increases the number of input elements represented in a state. For instance, stacking 6 blocks with k = 5 results in an input field of 25 elements or we can also say that output depends on 25 input elements.<br />
<br />
A kernal parameters is represented as <math> W ∈ ℝ^{2d x kd}, b_w ∈ ℝ^{2d} </math> and takes as input <math>X ∈ ℝ^{k×d} </math> to produce output element <math> Y ∈ ℝ^{2d} </math>. The non linearity chosen was Gated Linear Unit(GLU) mainly because it was shown to perform better in aspects of langauge modelling. A GLU produes an output <math>v([A B]) = A ⊗ σ(B), v([AB]) ∈ ℝ^{d} </math> and <math>Y = [AB] ∈ ℝ^{2d} </math>.<br />
<br />
A residual connection is added from the input of each block to the output of each block. This is done so that the model can be deep. He et al. (Deep Residual Learning for Image Recognition) showed that adding residual connections improve the model performance by making it deep and prevents degradation of training accuracy. This is given by the equation <math>h_i^l = v(W^l [h_{i-k/2}^{l-1},...,h_{i+k/2}^{l-1}] + b_w^l) + h_i^{l-1}</math><br />
<br />
Padding is performed in the encoder after the convolution step so that the output matches the length of the input. The same cannot be applied to the decoder as we don't know the size of the sequence. To overcome this they pad the input of decoder with k-1 zeroes on both the left and right side and then prune the last k elements from the convolutional output. They add a linear mapping to project between embedding size <math>f</math> and convolutional output of size 2d. They apply such a transform to w when feeding embeddings to the encoder network, to the encoder output <math>z_j^u</math>, to the final layer of the decoder just before the softmax <math>h^l</math>, and to all decoder layers <math>h^l</math> before computing attention scores.<br />
<br />
Finally, a probability distribution is generated over next T possible candidates elements <math>p(y_{i+1} | y_i,...y_1,x) = softmax(W_o h_i^l + b_o) ∈ ℝ^T </math><br />
<br />
== Multi-step Attention ==<br />
<br />
A separate attention mechanism is used for each decoder block. To compute the attention decoder state of current layer is combined with the embedding of the last element generated <math>g_i</math> we can now write state summary as <math>d_i^l = W_d^l + b_i^l + g_i</math>. For a decoder layer l the attention <math>a_{ij}^l</math> with state i and source element j is computed as <math>a_{ij}^l = \frac{exp(d_i^l . z_j^u)}{\sum_{t=1}^m exp(d_i^l . z_t^u)}</math>. The conditional input to the decoder layer is weighted sum of encoder and element embeddings. This can be written as <math>c_i^l = \sum_{j=1}^m a_{i,j}^l (z_j^u + e_j)</math>. This conditional input is then added to the decoder state <math>h_i^l</math>,<br />
<br />
The attention in the first layer provides the source context which is then fed to the next layer which takes this information to compute other information in that layer. The decoder aslo has the history of previous attention as <math>h_i^l = h_i^l + c_i^l</math><br />
<br />
== Normalization Strategy and initialization ==<br />
They use Weight Normalization instead of batch normalization with stabilization of model learning is done through careful weight initialization(For more information, Refer to the appendix of the paper) and by scaling parts of the network to ensure that the variance throughout the network does not change dramatically. They scale the output of residual blocks as well as the attention to preserve the variance of activations. They multiply the sum of the input and output of a residual block by <math>\sqrt{0.5}</math> to halve the variance of the sum. The conditional input <math>c_i^l</math> generated by the attention is a weighted sum of m vectors and we counteract a change in variance through scaling by <math>m\sqrt{1/m}</math>, they multiply by m to scale up the inputs to their original size, assuming the attention scores are uniformly distributed though this is not generally found to be working every time in practice. For convolutional decoders with multiple attention, they scale the gradients for the encoder layers by the number of attention mechanisms they use and exclude source word embeddings.<br />
<br />
All embeddings are initialized from a normal distribution with mean 0 and standard deviation 0.1. For layers whose output is not directly fed to a gated linear unit, initialization of weights is from <math>N(0,\sqrt{1/n_l})</math> where <math>n_l</math> is the number of input connections to each neuron. This ensures that the variance of a normally distributed input is retained. For layers which are followed by a GLU activation, they initialize the weights so that<br />
the input to the GLU activations have 4 times the variance of the layer input. This is achieved by drawing their initial values from <math>N(0,\sqrt{4/n_l})</math><br />
<br />
= Experimental Setup =<br />
==Datasets==<br />
* '''WMT 16 English-Romanian''' - remove sentences having words > 175, 2.8M senetnce pairs for training. Use newstest 2016 for evaluation purposes when using this dataset. <br />
* '''WMT 14 English-German''' - 4.5M sentence pairs. Testing for this dataset is done using newstest 2014.<br />
* '''WMT 14 English- French''' - 36M sentence pairs, remove sentences with length > 175 words and source/target ratio exceeding 1.5. Again for evaluation newstest 2014 is used. <br />
* '''Abstractive SUmmarization''' - Trained on Gigaword Corpus, 3.8M examples for training. Newstest 2014 is used for evaluation purposes.<br />
<br />
== Model Parameters and Optimization ==<br />
* Used 512 hidden units for both encoder and decoder with output embeddings also of the same size.<br />
* Optimizer- Nesterov's accelerated gradient method (Sutskever et al., 2013) using 0.99 momentum. Use gradient clipping if norm > 0.1<br />
* Learning rate - 0.25, once validation perplexity stops improving reduce the Learning rate by a magnitude after each epoch until it reaches <math>10^{-4}</math><br />
* Mini batch with 64 sentences. The maximum number of words in a mini-batch is restricted to make sure that batches with long sentences fit in GPU memory.<br />
* When the threshold is exceeded, the batch is split until the threshold is met and the parts are processed separately.<br />
* Gradients are normalized by the number of non-padding tokens per mini-batch.<br />
* Weight normalization is also used for all layers except for lookup tables<br />
* Use dropout on embeddings, decoder output and input of convolution blocks<br />
* All models were implemented using Torch. <br />
* For training a single Nvidia GPU was used except when training on WMT'14 English French. In this case training was done in a parallel fashion on eight GPUs. The authors maintained eight copies of the model on each card and split the batch so that each node managed 1/80th of the gradient computations. <br />
<br />
== Evaluation ==<br />
<br />
Translations are generated by beam search of width 5 and normalization is log likelihood scores by the length. For word-based models, unknown words are replaced based on attention scores after generation with help of pre-computed attention score dictionary. If the dictionary doesn't contain translation the source word is simply copied. Dictionaries were obtained from a word-aligned training data fast_align where each word is mapped to target word it is most frequently aligned to. The final attention scores are the average of attention scores from all layers. They finally use case-sensitive tokenized BLEU scores for all except WMT 16 where they use detokenized BLEU.<br />
<br />
= Results =<br />
[[File:res1.png|thumb|center|Result 1]]<br />
<br />
* ConvS2S outperforms the WMT’16 winning entry for English-Romanian by 1.9 BLEU with a BPE encoding and by 1.3 BLEU with a word factored vocabulary.<br />
* The results (Result 1) show that the convolutional model outpeforms GNMT by 0.5 BLEU on WMT 14' English to German. <br />
* Finally the model is compared to WMT '14 English to French. The model improves over GNMT in the same setting by 1.6 BLEU on average. It also outperforms their reinforcement (RL) models by 0.5 BLEU.<br />
<br />
[[File:res2.png|thumb|center|Result 2: Accuracy of ensembles with other ensemble models]]<br />
<br />
The authors ensemble eight likelihood-trained models for both WMT’14 English-German and WMT’14 English-French and compare to previous work which also reported ensemble results and find out that they outperform all the models.<br />
<br />
[[File:res3.png|thumb|center|Result 3 :CPU and GPU generation speed in seconds on the development set of WMT’14 English-French]]<br />
<br />
[[File:res4.png|thumb|center|Result 4 :Effect of removing position embeddings from our model in terms of validation perplexity]]<br />
<br />
[[File:res5.png|thumb|center|Result 5: Multi-step attention in all five decoder layers or fewer layers in terms of validation perplexity (PPL) and test BLEU.]]<br />
<br />
[[File:res8.png|thumb|center|Result 6: Encoder with different kernel width in terms of BLEU]]<br />
<br />
[[File:res9.png|thumb|center|Result 7: Decoder with different kernel width in terms of BLEU]]<br />
<br />
[[File:res7.png|thumb|center|Result 8: Accuracy on two summarization tasks in terms of Rouge-1 (RG-1), Rouge-2 (RG-2), and Rouge-L (RG-L)]]<br />
<br />
Attention for different layers<br />
[[File:attention.png|center|700px]]<br />
<br />
From the results of the experiments, we conclude our convolutional approach can easily discover the compositional structure in the sequences, because the representations are built hierarchically and our model relies on gating and it performs multiple attention steps. When trained with standard likelihood method, our trained CNN model outperforms the likelihood trained model (RNN MLE) which either optimize the evaluation metric, and is not far behind the best models on this task which benefit from task-specific optimization and model structure<br />
<br />
<br />
= References = <br />
# Cho, Kyunghyun, Van Merrienboer, Bart, Gulcehre, ¨ Caglar, Bahdanau, Dzmitry, Bougares, Fethi, Schwenk, Holger, and Bengio, Yoshua. Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation. In Proc. of EMNLP, 2014.<br />
# Bradbury, James, Merity, Stephen, Xiong, Caiming, and Socher, Richard. Quasi-Recurrent Neural Networks. arXiv preprint arXiv:1611.01576, 2016.<br />
# He, Kaiming, Zhang, Xiangyu, Ren, Shaoqing, and Sun, Jian. Delving deep into rectifiers: Surpassing humanlevel performance on imagenet classification. In Proceedings of the IEEE International Conference on Computer Vision, pp. 1026–1034, 2015b.<br />
# Meng, Fandong, Lu, Zhengdong, Wang, Mingxuan, Li, Hang, Jiang, Wenbin, and Liu, Qun. Encoding Source Language with Convolutional Neural Network for Machine Translation. In Proc. of ACL, 2015.<br />
# Bahdanau, Dzmitry, Cho, Kyunghyun, and Bengio, Yoshua. Neural machine translation by jointly learning to align and translate. arXiv preprint arXiv:1409.0473, 2014.<br />
# Gehring, Jonas, Auli, Michael, Grangier, David, and Dauphin, Yann N. A Convolutional Encoder Model for Neural Machine Translation. arXiv preprint arXiv:1611.02344, 2016.<br />
# Gehring et.al A Convolutional Encoder Model for Neural Machine Translation, ACL 2017<br />
# Dyer, Chris, Chahuneau, Victor, and Smith, Noah A. A Simple, Fast, and Effective Reparameterization of IBM Model 2. In Proc. of ACL, 2013.<br />
# A Short tutorial on Attention models: https://machinelearningmastery.com/attention-long-short-term-memory-recurrent-neural-networks/<br />
# Standford's Lecture on Neural Machine Translation and Attention Models: https://www.youtube.com/watch?v=IxQtK2SjWWM<br />
# https://en.wikipedia.org/wiki/BLEU<br />
# Papineni, K., Roukos, S., Ward, T., Henderson, J and Reeder, F. (2002). “Corpus-based Comprehensive and Diagnostic MT Evaluation: Initial Arabic, Chinese, French, and Spanish Results” in Proceedings of Human Language Technology 2002, San Diego, pp. 132–137<br />
# Callison-Burch, C., Osborne, M. and Koehn, P. (2006) "Re-evaluating the Role of BLEU in Machine Translation Research" in 11th Conference of the European Chapter of the Association for Computational Linguistics: EACL 2006 pp. 249–256<br />
#Sutskever, Ilya, Martens, James, Dahl, George E., and Hinton, Geoffrey E. On the importance of initialization and momentum in deep learning. In ICML, 2013.<br />
# Dauphin, Y., Fan, A., Auli, M., and Grangier, D. 2017. 'Language Modelling with Gated Convolutional Networks". ICML.<br />
# Shen, Shiqi, Zhao, Yu, Liu, Zhiyuan, Sun, Maosong, etal. Neural headline generation with sentence-wise optimization. arXiv preprint arXiv:1604.01904, 2016. <br />
<br />
Implementation Example: https://github.com/facebookresearch/fairseq</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Understanding_the_Effective_Receptive_Field_in_Deep_Convolutional_Neural_Networks&diff=29535Understanding the Effective Receptive Field in Deep Convolutional Neural Networks2017-11-07T05:28:07Z<p>H4lyu: /* Remarks */</p>
<hr />
<div>= Introduction =<br />
== What is the Receptive Field (RF) of a unit? ==<br />
[[File:understanding_ERF_fig0.png|thumbnail|450px]]<br />
The receptive field of a unit is the region of input where the unit 'sees' and responds to. When dealing with high-dimensional inputs such as images, it is impractical to connect neurons to all neurons in the previous volume. Instead, we connect each neuron to only a local region of the input volume. The spatial extent of this connectivity is a hyper-parameter called the receptive field of the neuron (equivalently this is the filter size) [4].<br />
<br />
An effective introduction to Receptive field arithmetic, including ways to calculate the receptive field of CNNs can be found [https://syncedreview.com/2017/05/11/a-guide-to-receptive-field-arithmetic-for-convolutional-neural-networks/ here]<br />
<br />
== Why is RF important? ==<br />
The concept of receptive field is important for understanding and diagnosing how deep Convolutional neural networks (CNNs) work. Unlike in fully connected networks, where the value of each unit depends on the<br />
entire input to the network, in CNNs, anywhere in an input image outside the receptive field of a unit does not affect the value of that unit and hence it is necessary to carefully control the receptive field, to ensure that it covers the entire relevant image region. The property of receptive field allows for the response to be most sensitive to a local region in the image and to specific stimuli; similar stimuli trigger activations of similar magnitudes [2]. The initialization of each receptive field depends on the neuron's degrees of freedom [2]. One example outlined in this paper is that "the weights can be either of the same sign or centered with zero mean. This latter case favors a response to the contrast between the central and peripheral region of the receptive field." [2]. In many tasks, especially dense prediction tasks like semantic image segmentation, stereo and optical flow estimation, where we make a prediction for every single pixel in the input image, it is critical for each output pixel to have a big receptive field, such that no important information is left out when making the prediction.<br />
<br />
== How to increase RF size? ==<br />
''' Make the network deeper''' by stacking more layers, which increases the receptive field size linearly by theory, as<br />
each extra layer increases the receptive field size by the kernel size (more accurate to say kernel size-1).<br />
<br />
'''Add sub-sampling layers''' to increase the receptive field size multiplicatively. Actually, sub-sampling is simply AveragePooling with learnable weights per feature map. It acts like low pass filtering and then downsampling。<br />
<br />
Modern deep CNN architectures like the VGG networks and Residual Networks use a combination of these techniques.<br />
<br />
== Intuition behind Effective Receptive Fields ==<br />
The pixels at the center of a RF have a much larger impact on an output:<br />
* In the forward pass, central pixels can propagate information to the output through many different paths, while the pixels in the outer area of the receptive field have very few paths to propagate its impact. <br />
* In the backward pass, gradients from an output unit are propagated across all the paths, and therefore the central pixels have a much larger magnitude for the gradient from that output [More paths always mean larger gradient?].<br />
* Not all pixels in a receptive field contribute equally to an output unit's response.<br />
<br />
The authors prove that in many cases the distribution of impact in a receptive field distributes as a Gaussian. Since Gaussian distributions generally decay quickly from the center, the effective receptive field, only occupies a fraction of the theoretical receptive field.<br />
<br />
The authors have correlated the theory of effective receptive field with some empirical observations. One such observation is that the random initializations lead some deep CNNs to start with a small effective receptive field, which then grows on training, which indicates a bad initialization bias.<br />
<br />
= Theoretical Results =<br />
<br />
The authors wanted to mathematically characterize how much each input pixel in a receptive field can impact<br />
the output of a unit $n$ layers up the network, i.e. when $n \rightarrow \infty$. More specifically, assume that pixels on each layer are indexed by $(i,j)$ with their centre at $(0,0)$. If we denote the pixel on the $p$th layer as $x_{i,j}^p$ , with $x_{i,j}^0$ as the input to the network, and $y_{i,j}=x_{i,j}^n$ as the output on the $n$th layer, we want to know how much each $x_{i,j}^0$ contributes to $y_{0,0}$. The effective receptive field (ERF) of this central output unit is then can be defined as the region containing input pixels with a non-negligible impact on it. <br />
<br />
They used the partial derivative $\frac{\partial y_{0,0}}{\partial x_{i,j}^0}$ as the measure of such impact, which can be computed using backpropagation. Assuming $l$ as an arbitrary loss by the chain rule we can write $\frac{\partial l}{\partial x_{i,j}^0} = \sum_{i',j'}\frac{\partial l}{\partial y_{i',j'}}\frac{\partial y_{i',j'}}{\partial x_{i,j}^0}$. Now if $\frac{\partial l}{\partial y_{0,0}} =1$ and $\frac{\partial l}{\partial y_{i,j}}=0$ for all $i \neq 0$ and $j \neq 0$, then $\frac{\partial l}{\partial x_{i,j}^0} =\frac{\partial y_{0,0}}{\partial x_{i,j}^0}$.<br />
<br />
For networks without nonlinearity (i.e., linear networks) this measure is independent of the input and depends only on the weights of the network and (i, j) , which clearly shows how the impact of the pixels in the receptive field distributes.<br />
<br />
===Simplest case: Stack of convolutional layers of weights equal to 1===<br />
<br />
The authors first considered the case of $n$ convolutional layers using $k \times k$ kernels of stride 1 and a single channel on each layer and no nonlinearity, and bias. <br />
<br />
<br />
For this special sub-case, the kernel was a $k \times k$ matrix of 1's. Since this kernel is separable to $k \times 1$ and $1 \times k$ matrices, the $2D$ convolution could be replaced by two $1D$ convolutions. This allowed the authors to focus their analysis on the $1D$ convolutions.<br />
<br />
For this case, if we denote the gradient signal $\frac{\partial l}{\partial y_{i,j}}$ by $u(t)$ and the kernel by $v(t)$, we have<br />
<br />
\begin{equation*}<br />
u(t)=\delta(t),\\ \quad v(t) = \sum_{m=0}^{k-1} \delta(t-m), \quad \text{where} \begin{cases} \delta(t)= 1\ \text{if}\ t=0, \\ \delta(t)= 0\ \text{if}\ t\neq 0, \end{cases}<br />
\end{equation*}<br />
and $t =0,1,-1,2,-2,...$ indexes the pixels.<br />
<br />
The gradient signal $o(t)$ on the input pixels can now be computed by convolving $u(t)$ with $n$ $v(t)$'s so that $o(t) = u *v* ...*v$. <br />
<br />
Since convolution in time domain is equivalent to multiplication in Fourier domain, we can write<br />
<br />
\begin{equation*}<br />
U(w) = \sum_{t=-\infty}^{\infty} u(t) e^{-jwt}=1,\\<br />
V(w) = \sum_{t=-\infty}^{\infty} v(t) e^{-jwt}=\sum_{m=0}^{k-1} e^{-jwm},\\<br />
O(w) = F(o(t))=F(u(t)*v(t)*...*v(t)) = U(w).V(w)^n = \Big ( \sum_{m=0}^{k-1} e^{-jwm} \Big )^n,<br />
\end{equation*}<br />
<br />
where $O(w)$, $U(w)$, and $V(w)$ are discrete Fourier transformations of $o(t)$, $u(t)$, and $v(t)$.<br />
<br />
Now let us consider two non-trivial cases.<br />
<br />
'''Case K=2:''' In this case $( \sum_{m=0}^{k-1} e^{-jwm} )^n = (1 + e^{-jw})^n$. Because $O(w)= \sum_{t=-\infty}^{\infty} o(t) e^{-jwt}= (1 + e^{-jw})^n$, we can think of $o(t)$ as coefficients of $e^{-jwt})$. Therefore, $o(t)= <br />
\begin{pmatrix} n\\t\end{pmatrix}$ is the standard binomial coefficients. As $n$ becomes large binomial coefficients distribute with respect to $t$ like a Gaussian distribution. More specifically, when $n \to \infty$ we can write<br />
<br />
<br />
\begin{equation*}<br />
\begin{pmatrix} n\\t \end{pmatrix} \sim \frac{2^n}{\sqrt{\frac{n\pi}{2}}}e^{-d^{2}/2n}, <br />
\end{equation*}<br />
<br />
where $d = n-2t$ (see [https://en.wikipedia.org/wiki/Binomial_coefficient Binomial coefficient]).<br />
<br />
'''Case K>2:''' In this case the coefficients are known as "extended binomial coefficients" or "polynomial<br />
coefficients", and they too distribute like Gaussian [5].<br />
<br />
=== Random Weights===<br />
Denote $g(i, j, p) = \frac{\partial l}{\partial x_{i,j}^p}$ as the gradient on the $p$th layer, and $g(i, j, p) = \frac{\partial l}{\partial y_{i,j}}$ . Then $g(, , 0)$ is the desired gradient image of the input. The backpropagation convolves $g(, , p)$ with the $k x k$ kernel to get $g(, , p-1)$ for each p. So we can write<br />
<br />
\begin{equation*}<br />
g(i,j,p-1) = \sum_{a=0}^{k-1} \sum_{b=0}^{k-1} w_{a,b}^p g(i+a,i+b,p),<br />
\end{equation*}<br />
<br />
where $w_{a,b}^p$ is the convolution weight at $(a, b)$ in the convolution kernel on layer p. In this case, the initial weights are independently drawn from a fixed distribution with zero mean and variance $C$. By assuming that the gradients g are independent from the<br />
weights (linear networks only) and given that $\mathbb{E}_w[w_{a,b}^p] =0$<br />
<br />
\begin{equation*}<br />
\mathbb{E}_{w,input}[g(i,j,p-1)] = \sum_{a=0}^{k-1} \sum_{b=0}^{k-1} \mathbb{E}_w[w_{a,b}^p] \mathbb{E}_{input}[g(i+a,i+b,p)]=0,\\<br />
Var[g(i,j,p-1)] = \sum_{a=0}^{k-1} \sum_{b=0}^{k-1} Var[w_{a,b}^p] Var[g(i+a,i+b,p)]= C\sum_{a=0}^{k-1} \sum_{b=0}^{k-1} Var[g(i+a,i+b,p)].<br />
\end{equation*}<br />
<br />
Therefore, to get $Var[g(, , p-1)]$ we can convolve the gradient variance image $Var[g(, , p)]$ with a $k \times k$ kernel of 1’s, and then multiply it by $C$. Comparing this to the simplest case of all weights equal to one, we can see that the $g(, , 0)$ has a Gaussian shape, with only a slight<br />
change of having an extra $C^n$ constant factor multiplier on the variance gradient images, which does not affect the relative distribution within a receptive field.<br />
<br />
=== Non-uniform Kernels ===<br />
In the case of non-uniform weighting, when w(m)'s are normalized, we can simply use characteristic function to prove the Central Limit Theorem in this case. For $S_n = \sum_{i=1}^n$ $X_i$ and $X_i$’s are i.i.d. multinomial variables distributed according to $w(m)$’s, i.e. $p(X_i = m) = w(m)$, we have:<br />
<br />
\begin{equation*}<br />
E[S_n] = n\sum_{m=0}^{k-1} mw(m),\\<br />
Var[S_n] = n \left (\sum_{m=0}^{k-1} m^2w(m) - \left (\sum_{m=0}^{k-1} mw(m) \right )^2 \right ),<br />
\end{equation*}<br />
<br />
If we take one standard deviation as the effective receptive field (ERF) size which is roughly the radius of the ERF, then this size is<br />
$\sqrt{Var[S_n]} = \sqrt{nVar[X_i]} = O(\sqrt{n})$.<br />
<br />
On the other hand, stacking more convolutional layers implies that the theoretical receptive field grows linearly, therefore relative to the theoretical receptive field, the ERF actually shrinks at a rate of $O(1/\sqrt{n})$.<br />
<br />
=== Non-linear Activation Functions===<br />
<br />
The math in this section is a bit "hand-wavy", as one of their reviewers wrote, and their conclusion (Gaussian-shape ERF) is not really well backed up by their experiments. The most important point to take way form this part is that by introduction of a nonlinear activation function, the gradients depends on the network's input as well.<br />
<br />
=== Dropout, Subsampling, Dilated Convolution and Skip-Connections ===<br />
The authors show that dropout does not change the Gaussian ERF shape. Subsampling (sample the feature maps of convolution layer with some stride >= 2) and dilated convolutions (where unlike vanilla convolution, convolve the kernel with input pixel with some dilation factor d >= 2) turn out to be effective ways to increase receptive field size quickly. Skip-connections on the other hand make ERFs smaller.<br />
<br />
=== Remarks ===<br />
The authors notice us about three critical assumptions in the analyses above: (1) all layers in the CNN use the same set of convolution weights. This is in general not true, however, when we apply the analysis of variance, the weight variance on all layers are usually the same up to a constant factor. (2) The convergence derived is convergence “in distribution”, as implied by the central limit theorem. So this is neither converge almost surely nor in probability, or rather, we are not able to guarantee convergence on any single model. (3) Although CLT gives the limit distribution of $\frac{1}{\sqrt{n}} S_n$, the distribution of $S_n$ does not have a limit, and its "deviation" from a corresponding normal distribution can be large on some finite set, but it still be Gaussian in terms of overall shape.<br />
<br />
= Verifying Theoretical Results =<br />
In all of the following experiments, a gradient signal of 1 was placed at the center of the output plane and 0 everywhere else, and then this gradient was backpropagated through the network to get input gradients. Also random inputs as well as proper random initialization of the kernels were employed.<br />
<br />
<br />
'''ERFs are Gaussian distributed:''' By looking at the figure, [[File:understanding_ERF_fig1.png|thumbnail||600px]] we can observe Gaussian shapes for uniformly and randomly weighted convolution kernels without nonlinear activations, and near Gaussian shapes for randomly weighted kernels with nonlinearity. Adding the ReLU nonlinearity makes the distribution a bit less Gaussian, as the ERF distribution depends on the input as well. Another reason is that ReLU units output exactly zero for half of its inputs and it is very easy to get a zero output for the center pixel on the output plane, which means no path from the receptive field can reach the output, hence the gradient is all zero. Here the ERFs are averaged over 20 runs with different random seed. <br />
<br />
<br />
<br />
<br />
<br />
Figures below show the ERF for networks with 20 layers of random weights, with different nonlinearities. Here the results are averaged both across 100 runs with different random weights as well as different random inputs. In this setting the receptive fields are a lot more Gaussian-like. <br />
<br />
[[File:understanding_ERF_fig2.png|thumbnail|centre|400px]]<br />
<br />
<br />
''' <math>\sqrt{n}</math> absolute growth and <math>1/\sqrt{n}</math> relative shrinkage:''' The figure [[File:understanding_ERF_fig4.png|thumbnail||600px]] shows the change of ERF size and the relative ratio of ERF over theoretical RF wrt number of convolution layers. The fitted line for ERF size has the slope of 0.56 in log domain, while the line for ERF ratio has the slope of -0.43. This indicates ERF size is growing linearly wrt <math>\sqrt{n}</math> and ERF ratio is shrinking linearly wrt <math>1/\sqrt{n}</math>.<br />
They used 2 standard deviations as the measurement for ERF size, i.e. any pixel with value greater than 1 - 95.45% of center point is considered in ERF. The ERF size is represented by the square root of number of pixels within ERF, while the theoretical RF size is the side length of the square in which all pixel has a non-zero impact on the output pixel, no matter how small. All experiments here are averaged over 20 runs.<br />
<br />
<br />
'''Subsampling & dilated convolution increases receptive field:''' The figure shows that the effect of subsampling and dilated convolution. The reference baseline is a CNN with 15 dense convolution layers. Its ERF is shown in the left-most figure. Replacing 3 of the 15 convolutional layers with stride-2 convolution results in the ERF for the ‘Subsample’ figure. Finally, replacing those 3 convolutional layers with dilated convolution with factor 2,4 and 8 gives the ‘Dilation’ figure. Both of them are able to increase the effect receptive field significantly. Note the ‘Dilation’ figure shows a rectangular ERF shape typical for dilated convolutions (why?).<br />
<br />
[[File:understanding_ERF_fig3.png|thumbnail|centre|400px]]<br />
<br />
== How the ERF evolves during training ==<br />
<br />
The authors looked at how the ERF of units in the top-most convolutional layers of a classification CNN and a semantic segmentation CNN evolve during training. For both tasks, they adopted the ResNet architecture which makes extensive use of skip-connections. As expected their analysis showed the ERF of these networks are significantly smaller than the theoretical receptive field. Also, as the networks learns, the ERF got bigger so that at the end of training was significantly larger than the initial ERF. <br />
<br />
The classification network was a ResNet with 17 residual blocks trained on the CIFAR-10 dataset. Figure shows the ERF on the 32x32 image space at the beginning of training (with randomly initialized weights) and at the end of training when it reaches best validation accuracy. Note that the theoretical receptive field of the network is actually 74x74, bigger than the image size, but the ERF is not filling the image completely. Comparing the results before and after training <br />
demonstrates that ERF has grown significantly.<br />
<br />
[[File:understanding_ERF_fig5.png|thumbnail|centre|500px]]<br />
<br />
The semantic segmentation network was trained on the CamVid dataset for urban scene segmentation. The 'front-end' of the model was a purely convolutional network that predicted the output at a slightly lower resolution. And then, a ResNet with 16 residual blocks interleaved with 4 subsampling operations each with a factor of 2 was implemented. Due to subsampling operations the output was 1/16 of the input size. For this model, the theoretical RF of the top convolutional layer units was 505x505. However, as Figure shows the ERF only got a fraction of that with a diameter of 100 at the beginning of training, and at the end of training reached almost a diameter around 150.<br />
<br />
= Discussion =<br />
The Effective Receptive Field (ERF) usually decays quickly from the centre (like 2D Gaussian) and only takes a small portion of the theoretical Receptive Field (RF). This "Gaussian damage" is undesirable for tasks that require a large RF and to reduce it, the authors suggested two solutions:<br />
#'''New Initialization scheme''' to make the weights at the center of the convolution kernel to be smaller and the weights on the outside larger, which diffuses the concentration on the center out to the periphery. One way to implement this is to initialize the network with any initialization method, and then scale the weights according to a distribution that has a lower scale at the center and higher scale on the outside. They tested this solution for the CIFAR-10 classification task, with several random seeds. In a few cases they get a 30% speed-up of training compared to the more standard initializations. But overall the benefit of this method is not always significant.<br />
#'''Architectural changes of CNNs''' is the 'better' approach that may change the ERF in more fundamental ways. For example, instead of connecting each unit in a CNN to a local rectangular convolution window, we can sparsely connect each unit to a larger area in the lower layer using the same number of connections. Dilated convolution belongs to this category, but we may push even further and use sparse connections that are not grid-like.<br />
<br />
= Summary & Conclusion =<br />
The authors showed, theoretically and experimentally, that the distribution of impact within the receptive field (the effective receptive field) is asymptotically Gaussian, and the ERF only takes up a fraction of the full theoretical receptive field. They also studied the effects of some standard CNN approaches on the effective receptive field. They found that dropout does not change the Gaussian ERF shape. Subsampling and dilated convolutions are effective ways to increase receptive field size quickly but skip-connections make ERFs smaller.<br />
<br />
They argued that since larger ERFs are required for higher performance, new methods to achieve larger ERF will not only help the network to train faster but may also improve performance.<br />
<br />
= Critique = <br />
<br />
The authors' finding on $\sqrt{n}$ absolute growth of Effective Receptive Field (ERF) suffers from discrepancy in ERF definition between their theoretical analysis and their experiments. Namely, in the theoretical analysis for non-uniform-kernel case they considered one standard deviation as the ERF size. However, they used two standard deviations as the measure for ERF size in the experiments.<br />
<br />
It would be more practical if the paper also investigated the ERF for natural images (as opposed to random) as network input at least in the two cases where they examined trained networks. <br />
<br />
The authors claim that the ERF results in the experimental section have Gaussian shapes but they never prove this claim. For example, they could fit different 2D-functions, including 2D-Gaussian, to the kernels and show that 2D-Gaussian gives the best fit. Furthermore, the pictures are given as proof of the claim that the ERF has a Gaussian distribution only show the ERF of the center pixel of the output <math> y_{0,0} </math>. Intuitively, the ERF of a node near the boundary of the output layer may have a significantly different shape. This was not addressed in the paper.<br />
<br />
Another weakness is in the discussion section, where they make a connection to the biological networks. They jumped to disprove a well-observed phenomenon in the brain. The fact that the neurons in the higher areas of the visual hierarchy gradually lose their retinotopic property has been shown in a countless number of neuroscience studies. For example, [https://en.wikipedia.org/wiki/Grandmother_cell grandmother cells] do not care about the position of grandmother's face in the visual field. In general, the similarity between deep CNNs and biological visual systems is not as strong, hence we should take any generalization from CNNs to biological networks with a grain of salt.<br />
<br />
Spectrograms are visual representations of audio where the axes represent time, frequency and amplitude of the frequency. The ERF of a CNN when applied to a spectrogram doesn't necessarily have to be from a Gaussian towards the center. In fact many receptive fields are trained to look for the peaks of troughs and cliffs, which essentially imply that the ERF will have more weightage towards the outside rather than the center.<br />
<br />
The paper talks about what ERF represents and how it can be increased, but doesn't say how ERF can be used for improving the model accuracies by changing the configuration of network, say depth of the network, or kernel size etc.<br />
<br />
= References =<br />
[1] Wenjie Luo, Yujia Li, Raquel Urtasun, and Richard Zemel. "Understanding the effective receptive field in deep convolutional neural networks." In Advances in Neural Information Processing Systems, pp. 4898-4906. 2016.<br />
<br />
[2] Buessler, J.-L., Smagghe, P., & Urban, J.-P. (2014). Image receptive fields for artificial neural networks. Neurocomputing, 144(Supplement C), 258–270. https://doi.org/10.1016/j.neucom.2014.04.045<br />
<br />
[3] Dilated Convolutions in Neural Network - [http://www.erogol.com/dilated-convolution/]<br />
<br />
[4] http://cs231n.github.io/convolutional-networks/<br />
<br />
[5] Thorsten Neuschel. "A note on extended binomial coefficients." Journal of Integer Sequences, 17(2):3, 2014.</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Understanding_the_Effective_Receptive_Field_in_Deep_Convolutional_Neural_Networks&diff=29534Understanding the Effective Receptive Field in Deep Convolutional Neural Networks2017-11-07T05:27:50Z<p>H4lyu: /* Theoretical Results */</p>
<hr />
<div>= Introduction =<br />
== What is the Receptive Field (RF) of a unit? ==<br />
[[File:understanding_ERF_fig0.png|thumbnail|450px]]<br />
The receptive field of a unit is the region of input where the unit 'sees' and responds to. When dealing with high-dimensional inputs such as images, it is impractical to connect neurons to all neurons in the previous volume. Instead, we connect each neuron to only a local region of the input volume. The spatial extent of this connectivity is a hyper-parameter called the receptive field of the neuron (equivalently this is the filter size) [4].<br />
<br />
An effective introduction to Receptive field arithmetic, including ways to calculate the receptive field of CNNs can be found [https://syncedreview.com/2017/05/11/a-guide-to-receptive-field-arithmetic-for-convolutional-neural-networks/ here]<br />
<br />
== Why is RF important? ==<br />
The concept of receptive field is important for understanding and diagnosing how deep Convolutional neural networks (CNNs) work. Unlike in fully connected networks, where the value of each unit depends on the<br />
entire input to the network, in CNNs, anywhere in an input image outside the receptive field of a unit does not affect the value of that unit and hence it is necessary to carefully control the receptive field, to ensure that it covers the entire relevant image region. The property of receptive field allows for the response to be most sensitive to a local region in the image and to specific stimuli; similar stimuli trigger activations of similar magnitudes [2]. The initialization of each receptive field depends on the neuron's degrees of freedom [2]. One example outlined in this paper is that "the weights can be either of the same sign or centered with zero mean. This latter case favors a response to the contrast between the central and peripheral region of the receptive field." [2]. In many tasks, especially dense prediction tasks like semantic image segmentation, stereo and optical flow estimation, where we make a prediction for every single pixel in the input image, it is critical for each output pixel to have a big receptive field, such that no important information is left out when making the prediction.<br />
<br />
== How to increase RF size? ==<br />
''' Make the network deeper''' by stacking more layers, which increases the receptive field size linearly by theory, as<br />
each extra layer increases the receptive field size by the kernel size (more accurate to say kernel size-1).<br />
<br />
'''Add sub-sampling layers''' to increase the receptive field size multiplicatively. Actually, sub-sampling is simply AveragePooling with learnable weights per feature map. It acts like low pass filtering and then downsampling。<br />
<br />
Modern deep CNN architectures like the VGG networks and Residual Networks use a combination of these techniques.<br />
<br />
== Intuition behind Effective Receptive Fields ==<br />
The pixels at the center of a RF have a much larger impact on an output:<br />
* In the forward pass, central pixels can propagate information to the output through many different paths, while the pixels in the outer area of the receptive field have very few paths to propagate its impact. <br />
* In the backward pass, gradients from an output unit are propagated across all the paths, and therefore the central pixels have a much larger magnitude for the gradient from that output [More paths always mean larger gradient?].<br />
* Not all pixels in a receptive field contribute equally to an output unit's response.<br />
<br />
The authors prove that in many cases the distribution of impact in a receptive field distributes as a Gaussian. Since Gaussian distributions generally decay quickly from the center, the effective receptive field, only occupies a fraction of the theoretical receptive field.<br />
<br />
The authors have correlated the theory of effective receptive field with some empirical observations. One such observation is that the random initializations lead some deep CNNs to start with a small effective receptive field, which then grows on training, which indicates a bad initialization bias.<br />
<br />
= Theoretical Results =<br />
<br />
The authors wanted to mathematically characterize how much each input pixel in a receptive field can impact<br />
the output of a unit $n$ layers up the network, i.e. when $n \rightarrow \infty$. More specifically, assume that pixels on each layer are indexed by $(i,j)$ with their centre at $(0,0)$. If we denote the pixel on the $p$th layer as $x_{i,j}^p$ , with $x_{i,j}^0$ as the input to the network, and $y_{i,j}=x_{i,j}^n$ as the output on the $n$th layer, we want to know how much each $x_{i,j}^0$ contributes to $y_{0,0}$. The effective receptive field (ERF) of this central output unit is then can be defined as the region containing input pixels with a non-negligible impact on it. <br />
<br />
They used the partial derivative $\frac{\partial y_{0,0}}{\partial x_{i,j}^0}$ as the measure of such impact, which can be computed using backpropagation. Assuming $l$ as an arbitrary loss by the chain rule we can write $\frac{\partial l}{\partial x_{i,j}^0} = \sum_{i',j'}\frac{\partial l}{\partial y_{i',j'}}\frac{\partial y_{i',j'}}{\partial x_{i,j}^0}$. Now if $\frac{\partial l}{\partial y_{0,0}} =1$ and $\frac{\partial l}{\partial y_{i,j}}=0$ for all $i \neq 0$ and $j \neq 0$, then $\frac{\partial l}{\partial x_{i,j}^0} =\frac{\partial y_{0,0}}{\partial x_{i,j}^0}$.<br />
<br />
For networks without nonlinearity (i.e., linear networks) this measure is independent of the input and depends only on the weights of the network and (i, j) , which clearly shows how the impact of the pixels in the receptive field distributes.<br />
<br />
===Simplest case: Stack of convolutional layers of weights equal to 1===<br />
<br />
The authors first considered the case of $n$ convolutional layers using $k \times k$ kernels of stride 1 and a single channel on each layer and no nonlinearity, and bias. <br />
<br />
<br />
For this special sub-case, the kernel was a $k \times k$ matrix of 1's. Since this kernel is separable to $k \times 1$ and $1 \times k$ matrices, the $2D$ convolution could be replaced by two $1D$ convolutions. This allowed the authors to focus their analysis on the $1D$ convolutions.<br />
<br />
For this case, if we denote the gradient signal $\frac{\partial l}{\partial y_{i,j}}$ by $u(t)$ and the kernel by $v(t)$, we have<br />
<br />
\begin{equation*}<br />
u(t)=\delta(t),\\ \quad v(t) = \sum_{m=0}^{k-1} \delta(t-m), \quad \text{where} \begin{cases} \delta(t)= 1\ \text{if}\ t=0, \\ \delta(t)= 0\ \text{if}\ t\neq 0, \end{cases}<br />
\end{equation*}<br />
and $t =0,1,-1,2,-2,...$ indexes the pixels.<br />
<br />
The gradient signal $o(t)$ on the input pixels can now be computed by convolving $u(t)$ with $n$ $v(t)$'s so that $o(t) = u *v* ...*v$. <br />
<br />
Since convolution in time domain is equivalent to multiplication in Fourier domain, we can write<br />
<br />
\begin{equation*}<br />
U(w) = \sum_{t=-\infty}^{\infty} u(t) e^{-jwt}=1,\\<br />
V(w) = \sum_{t=-\infty}^{\infty} v(t) e^{-jwt}=\sum_{m=0}^{k-1} e^{-jwm},\\<br />
O(w) = F(o(t))=F(u(t)*v(t)*...*v(t)) = U(w).V(w)^n = \Big ( \sum_{m=0}^{k-1} e^{-jwm} \Big )^n,<br />
\end{equation*}<br />
<br />
where $O(w)$, $U(w)$, and $V(w)$ are discrete Fourier transformations of $o(t)$, $u(t)$, and $v(t)$.<br />
<br />
Now let us consider two non-trivial cases.<br />
<br />
'''Case K=2:''' In this case $( \sum_{m=0}^{k-1} e^{-jwm} )^n = (1 + e^{-jw})^n$. Because $O(w)= \sum_{t=-\infty}^{\infty} o(t) e^{-jwt}= (1 + e^{-jw})^n$, we can think of $o(t)$ as coefficients of $e^{-jwt})$. Therefore, $o(t)= <br />
\begin{pmatrix} n\\t\end{pmatrix}$ is the standard binomial coefficients. As $n$ becomes large binomial coefficients distribute with respect to $t$ like a Gaussian distribution. More specifically, when $n \to \infty$ we can write<br />
<br />
<br />
\begin{equation*}<br />
\begin{pmatrix} n\\t \end{pmatrix} \sim \frac{2^n}{\sqrt{\frac{n\pi}{2}}}e^{-d^{2}/2n}, <br />
\end{equation*}<br />
<br />
where $d = n-2t$ (see [https://en.wikipedia.org/wiki/Binomial_coefficient Binomial coefficient]).<br />
<br />
'''Case K>2:''' In this case the coefficients are known as "extended binomial coefficients" or "polynomial<br />
coefficients", and they too distribute like Gaussian [5].<br />
<br />
=== Random Weights===<br />
Denote $g(i, j, p) = \frac{\partial l}{\partial x_{i,j}^p}$ as the gradient on the $p$th layer, and $g(i, j, p) = \frac{\partial l}{\partial y_{i,j}}$ . Then $g(, , 0)$ is the desired gradient image of the input. The backpropagation convolves $g(, , p)$ with the $k x k$ kernel to get $g(, , p-1)$ for each p. So we can write<br />
<br />
\begin{equation*}<br />
g(i,j,p-1) = \sum_{a=0}^{k-1} \sum_{b=0}^{k-1} w_{a,b}^p g(i+a,i+b,p),<br />
\end{equation*}<br />
<br />
where $w_{a,b}^p$ is the convolution weight at $(a, b)$ in the convolution kernel on layer p. In this case, the initial weights are independently drawn from a fixed distribution with zero mean and variance $C$. By assuming that the gradients g are independent from the<br />
weights (linear networks only) and given that $\mathbb{E}_w[w_{a,b}^p] =0$<br />
<br />
\begin{equation*}<br />
\mathbb{E}_{w,input}[g(i,j,p-1)] = \sum_{a=0}^{k-1} \sum_{b=0}^{k-1} \mathbb{E}_w[w_{a,b}^p] \mathbb{E}_{input}[g(i+a,i+b,p)]=0,\\<br />
Var[g(i,j,p-1)] = \sum_{a=0}^{k-1} \sum_{b=0}^{k-1} Var[w_{a,b}^p] Var[g(i+a,i+b,p)]= C\sum_{a=0}^{k-1} \sum_{b=0}^{k-1} Var[g(i+a,i+b,p)].<br />
\end{equation*}<br />
<br />
Therefore, to get $Var[g(, , p-1)]$ we can convolve the gradient variance image $Var[g(, , p)]$ with a $k \times k$ kernel of 1’s, and then multiply it by $C$. Comparing this to the simplest case of all weights equal to one, we can see that the $g(, , 0)$ has a Gaussian shape, with only a slight<br />
change of having an extra $C^n$ constant factor multiplier on the variance gradient images, which does not affect the relative distribution within a receptive field.<br />
<br />
=== Non-uniform Kernels ===<br />
In the case of non-uniform weighting, when w(m)'s are normalized, we can simply use characteristic function to prove the Central Limit Theorem in this case. For $S_n = \sum_{i=1}^n$ $X_i$ and $X_i$’s are i.i.d. multinomial variables distributed according to $w(m)$’s, i.e. $p(X_i = m) = w(m)$, we have:<br />
<br />
\begin{equation*}<br />
E[S_n] = n\sum_{m=0}^{k-1} mw(m),\\<br />
Var[S_n] = n \left (\sum_{m=0}^{k-1} m^2w(m) - \left (\sum_{m=0}^{k-1} mw(m) \right )^2 \right ),<br />
\end{equation*}<br />
<br />
If we take one standard deviation as the effective receptive field (ERF) size which is roughly the radius of the ERF, then this size is<br />
$\sqrt{Var[S_n]} = \sqrt{nVar[X_i]} = O(\sqrt{n})$.<br />
<br />
On the other hand, stacking more convolutional layers implies that the theoretical receptive field grows linearly, therefore relative to the theoretical receptive field, the ERF actually shrinks at a rate of $O(1/\sqrt{n})$.<br />
<br />
=== Non-linear Activation Functions===<br />
<br />
The math in this section is a bit "hand-wavy", as one of their reviewers wrote, and their conclusion (Gaussian-shape ERF) is not really well backed up by their experiments. The most important point to take way form this part is that by introduction of a nonlinear activation function, the gradients depends on the network's input as well.<br />
<br />
=== Dropout, Subsampling, Dilated Convolution and Skip-Connections ===<br />
The authors show that dropout does not change the Gaussian ERF shape. Subsampling (sample the feature maps of convolution layer with some stride >= 2) and dilated convolutions (where unlike vanilla convolution, convolve the kernel with input pixel with some dilation factor d >= 2) turn out to be effective ways to increase receptive field size quickly. Skip-connections on the other hand make ERFs smaller.<br />
<br />
=== Remarks ===<br />
The authors notice us about two critical assumptions in the analyses above: (1) all layers in the CNN use the same set of convolution weights. This is in general not true, however, when we apply the analysis of variance, the weight variance on all layers are usually the same up to a constant factor. (2) The convergence derived is convergence “in distribution”, as implied by the central limit theorem. So this is neither converge almost surely nor in probability, or rather, we are not able to guarantee convergence on any single model. (3) Although CLT gives the limit distribution of $\frac{1}{\sqrt{n}} S_n$, the distribution of $S_n$ does not have a limit, and its "deviation" from a corresponding normal distribution can be large on some finite set, but it still be Gaussian in terms of overall shape.<br />
<br />
= Verifying Theoretical Results =<br />
In all of the following experiments, a gradient signal of 1 was placed at the center of the output plane and 0 everywhere else, and then this gradient was backpropagated through the network to get input gradients. Also random inputs as well as proper random initialization of the kernels were employed.<br />
<br />
<br />
'''ERFs are Gaussian distributed:''' By looking at the figure, [[File:understanding_ERF_fig1.png|thumbnail||600px]] we can observe Gaussian shapes for uniformly and randomly weighted convolution kernels without nonlinear activations, and near Gaussian shapes for randomly weighted kernels with nonlinearity. Adding the ReLU nonlinearity makes the distribution a bit less Gaussian, as the ERF distribution depends on the input as well. Another reason is that ReLU units output exactly zero for half of its inputs and it is very easy to get a zero output for the center pixel on the output plane, which means no path from the receptive field can reach the output, hence the gradient is all zero. Here the ERFs are averaged over 20 runs with different random seed. <br />
<br />
<br />
<br />
<br />
<br />
Figures below show the ERF for networks with 20 layers of random weights, with different nonlinearities. Here the results are averaged both across 100 runs with different random weights as well as different random inputs. In this setting the receptive fields are a lot more Gaussian-like. <br />
<br />
[[File:understanding_ERF_fig2.png|thumbnail|centre|400px]]<br />
<br />
<br />
''' <math>\sqrt{n}</math> absolute growth and <math>1/\sqrt{n}</math> relative shrinkage:''' The figure [[File:understanding_ERF_fig4.png|thumbnail||600px]] shows the change of ERF size and the relative ratio of ERF over theoretical RF wrt number of convolution layers. The fitted line for ERF size has the slope of 0.56 in log domain, while the line for ERF ratio has the slope of -0.43. This indicates ERF size is growing linearly wrt <math>\sqrt{n}</math> and ERF ratio is shrinking linearly wrt <math>1/\sqrt{n}</math>.<br />
They used 2 standard deviations as the measurement for ERF size, i.e. any pixel with value greater than 1 - 95.45% of center point is considered in ERF. The ERF size is represented by the square root of number of pixels within ERF, while the theoretical RF size is the side length of the square in which all pixel has a non-zero impact on the output pixel, no matter how small. All experiments here are averaged over 20 runs.<br />
<br />
<br />
'''Subsampling & dilated convolution increases receptive field:''' The figure shows that the effect of subsampling and dilated convolution. The reference baseline is a CNN with 15 dense convolution layers. Its ERF is shown in the left-most figure. Replacing 3 of the 15 convolutional layers with stride-2 convolution results in the ERF for the ‘Subsample’ figure. Finally, replacing those 3 convolutional layers with dilated convolution with factor 2,4 and 8 gives the ‘Dilation’ figure. Both of them are able to increase the effect receptive field significantly. Note the ‘Dilation’ figure shows a rectangular ERF shape typical for dilated convolutions (why?).<br />
<br />
[[File:understanding_ERF_fig3.png|thumbnail|centre|400px]]<br />
<br />
== How the ERF evolves during training ==<br />
<br />
The authors looked at how the ERF of units in the top-most convolutional layers of a classification CNN and a semantic segmentation CNN evolve during training. For both tasks, they adopted the ResNet architecture which makes extensive use of skip-connections. As expected their analysis showed the ERF of these networks are significantly smaller than the theoretical receptive field. Also, as the networks learns, the ERF got bigger so that at the end of training was significantly larger than the initial ERF. <br />
<br />
The classification network was a ResNet with 17 residual blocks trained on the CIFAR-10 dataset. Figure shows the ERF on the 32x32 image space at the beginning of training (with randomly initialized weights) and at the end of training when it reaches best validation accuracy. Note that the theoretical receptive field of the network is actually 74x74, bigger than the image size, but the ERF is not filling the image completely. Comparing the results before and after training <br />
demonstrates that ERF has grown significantly.<br />
<br />
[[File:understanding_ERF_fig5.png|thumbnail|centre|500px]]<br />
<br />
The semantic segmentation network was trained on the CamVid dataset for urban scene segmentation. The 'front-end' of the model was a purely convolutional network that predicted the output at a slightly lower resolution. And then, a ResNet with 16 residual blocks interleaved with 4 subsampling operations each with a factor of 2 was implemented. Due to subsampling operations the output was 1/16 of the input size. For this model, the theoretical RF of the top convolutional layer units was 505x505. However, as Figure shows the ERF only got a fraction of that with a diameter of 100 at the beginning of training, and at the end of training reached almost a diameter around 150.<br />
<br />
= Discussion =<br />
The Effective Receptive Field (ERF) usually decays quickly from the centre (like 2D Gaussian) and only takes a small portion of the theoretical Receptive Field (RF). This "Gaussian damage" is undesirable for tasks that require a large RF and to reduce it, the authors suggested two solutions:<br />
#'''New Initialization scheme''' to make the weights at the center of the convolution kernel to be smaller and the weights on the outside larger, which diffuses the concentration on the center out to the periphery. One way to implement this is to initialize the network with any initialization method, and then scale the weights according to a distribution that has a lower scale at the center and higher scale on the outside. They tested this solution for the CIFAR-10 classification task, with several random seeds. In a few cases they get a 30% speed-up of training compared to the more standard initializations. But overall the benefit of this method is not always significant.<br />
#'''Architectural changes of CNNs''' is the 'better' approach that may change the ERF in more fundamental ways. For example, instead of connecting each unit in a CNN to a local rectangular convolution window, we can sparsely connect each unit to a larger area in the lower layer using the same number of connections. Dilated convolution belongs to this category, but we may push even further and use sparse connections that are not grid-like.<br />
<br />
= Summary & Conclusion =<br />
The authors showed, theoretically and experimentally, that the distribution of impact within the receptive field (the effective receptive field) is asymptotically Gaussian, and the ERF only takes up a fraction of the full theoretical receptive field. They also studied the effects of some standard CNN approaches on the effective receptive field. They found that dropout does not change the Gaussian ERF shape. Subsampling and dilated convolutions are effective ways to increase receptive field size quickly but skip-connections make ERFs smaller.<br />
<br />
They argued that since larger ERFs are required for higher performance, new methods to achieve larger ERF will not only help the network to train faster but may also improve performance.<br />
<br />
= Critique = <br />
<br />
The authors' finding on $\sqrt{n}$ absolute growth of Effective Receptive Field (ERF) suffers from discrepancy in ERF definition between their theoretical analysis and their experiments. Namely, in the theoretical analysis for non-uniform-kernel case they considered one standard deviation as the ERF size. However, they used two standard deviations as the measure for ERF size in the experiments.<br />
<br />
It would be more practical if the paper also investigated the ERF for natural images (as opposed to random) as network input at least in the two cases where they examined trained networks. <br />
<br />
The authors claim that the ERF results in the experimental section have Gaussian shapes but they never prove this claim. For example, they could fit different 2D-functions, including 2D-Gaussian, to the kernels and show that 2D-Gaussian gives the best fit. Furthermore, the pictures are given as proof of the claim that the ERF has a Gaussian distribution only show the ERF of the center pixel of the output <math> y_{0,0} </math>. Intuitively, the ERF of a node near the boundary of the output layer may have a significantly different shape. This was not addressed in the paper.<br />
<br />
Another weakness is in the discussion section, where they make a connection to the biological networks. They jumped to disprove a well-observed phenomenon in the brain. The fact that the neurons in the higher areas of the visual hierarchy gradually lose their retinotopic property has been shown in a countless number of neuroscience studies. For example, [https://en.wikipedia.org/wiki/Grandmother_cell grandmother cells] do not care about the position of grandmother's face in the visual field. In general, the similarity between deep CNNs and biological visual systems is not as strong, hence we should take any generalization from CNNs to biological networks with a grain of salt.<br />
<br />
Spectrograms are visual representations of audio where the axes represent time, frequency and amplitude of the frequency. The ERF of a CNN when applied to a spectrogram doesn't necessarily have to be from a Gaussian towards the center. In fact many receptive fields are trained to look for the peaks of troughs and cliffs, which essentially imply that the ERF will have more weightage towards the outside rather than the center.<br />
<br />
The paper talks about what ERF represents and how it can be increased, but doesn't say how ERF can be used for improving the model accuracies by changing the configuration of network, say depth of the network, or kernel size etc.<br />
<br />
= References =<br />
[1] Wenjie Luo, Yujia Li, Raquel Urtasun, and Richard Zemel. "Understanding the effective receptive field in deep convolutional neural networks." In Advances in Neural Information Processing Systems, pp. 4898-4906. 2016.<br />
<br />
[2] Buessler, J.-L., Smagghe, P., & Urban, J.-P. (2014). Image receptive fields for artificial neural networks. Neurocomputing, 144(Supplement C), 258–270. https://doi.org/10.1016/j.neucom.2014.04.045<br />
<br />
[3] Dilated Convolutions in Neural Network - [http://www.erogol.com/dilated-convolution/]<br />
<br />
[4] http://cs231n.github.io/convolutional-networks/<br />
<br />
[5] Thorsten Neuschel. "A note on extended binomial coefficients." Journal of Integer Sequences, 17(2):3, 2014.</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Understanding_the_Effective_Receptive_Field_in_Deep_Convolutional_Neural_Networks&diff=29531Understanding the Effective Receptive Field in Deep Convolutional Neural Networks2017-11-07T05:17:02Z<p>H4lyu: /* Non-uniform Kernels */</p>
<hr />
<div>= Introduction =<br />
== What is the Receptive Field (RF) of a unit? ==<br />
[[File:understanding_ERF_fig0.png|thumbnail|450px]]<br />
The receptive field of a unit is the region of input where the unit 'sees' and responds to. When dealing with high-dimensional inputs such as images, it is impractical to connect neurons to all neurons in the previous volume. Instead, we connect each neuron to only a local region of the input volume. The spatial extent of this connectivity is a hyper-parameter called the receptive field of the neuron (equivalently this is the filter size) [4].<br />
<br />
An effective introduction to Receptive field arithmetic, including ways to calculate the receptive field of CNNs can be found [https://syncedreview.com/2017/05/11/a-guide-to-receptive-field-arithmetic-for-convolutional-neural-networks/ here]<br />
<br />
== Why is RF important? ==<br />
The concept of receptive field is important for understanding and diagnosing how deep Convolutional neural networks (CNNs) work. Unlike in fully connected networks, where the value of each unit depends on the<br />
entire input to the network, in CNNs, anywhere in an input image outside the receptive field of a unit does not affect the value of that unit and hence it is necessary to carefully control the receptive field, to ensure that it covers the entire relevant image region. The property of receptive field allows for the response to be most sensitive to a local region in the image and to specific stimuli; similar stimuli trigger activations of similar magnitudes [2]. The initialization of each receptive field depends on the neuron's degrees of freedom [2]. One example outlined in this paper is that "the weights can be either of the same sign or centered with zero mean. This latter case favors a response to the contrast between the central and peripheral region of the receptive field." [2]. In many tasks, especially dense prediction tasks like semantic image segmentation, stereo and optical flow estimation, where we make a prediction for every single pixel in the input image, it is critical for each output pixel to have a big receptive field, such that no important information is left out when making the prediction.<br />
<br />
== How to increase RF size? ==<br />
''' Make the network deeper''' by stacking more layers, which increases the receptive field size linearly by theory, as<br />
each extra layer increases the receptive field size by the kernel size (more accurate to say kernel size-1).<br />
<br />
'''Add sub-sampling layers''' to increase the receptive field size multiplicatively. Actually, sub-sampling is simply AveragePooling with learnable weights per feature map. It acts like low pass filtering and then downsampling。<br />
<br />
Modern deep CNN architectures like the VGG networks and Residual Networks use a combination of these techniques.<br />
<br />
== Intuition behind Effective Receptive Fields ==<br />
The pixels at the center of a RF have a much larger impact on an output:<br />
* In the forward pass, central pixels can propagate information to the output through many different paths, while the pixels in the outer area of the receptive field have very few paths to propagate its impact. <br />
* In the backward pass, gradients from an output unit are propagated across all the paths, and therefore the central pixels have a much larger magnitude for the gradient from that output [More paths always mean larger gradient?].<br />
* Not all pixels in a receptive field contribute equally to an output unit's response.<br />
<br />
The authors prove that in many cases the distribution of impact in a receptive field distributes as a Gaussian. Since Gaussian distributions generally decay quickly from the center, the effective receptive field, only occupies a fraction of the theoretical receptive field.<br />
<br />
The authors have correlated the theory of effective receptive field with some empirical observations. One such observation is that the random initializations lead some deep CNNs to start with a small effective receptive field, which then grows on training, which indicates a bad initialization bias.<br />
<br />
= Theoretical Results =<br />
<br />
The authors wanted to mathematically characterize how much each input pixel in a receptive field can impact<br />
the output of a unit $n$ layers up the network. More specifically, assume that pixels on each layer are indexed by $(i,j)$ with their centre at $(0,0)$. If we denote the pixel on the $p$th layer as $x_{i,j}^p$ , with $x_{i,j}^0$ as the input to the network, and $y_{i,j}=x_{i,j}^n$ as the output on the $n$th layer, we want to know how much each $x_{i,j}^0$ contributes to $y_{0,0}$. The effective receptive field (ERF) of this central output unit is then can be defined as the region containing input pixels with a non-negligible impact on it. <br />
<br />
They used the partial derivative $\frac{\partial y_{0,0}}{\partial x_{i,j}^0}$ as the measure of such impact, which can be computed using backpropagation. Assuming $l$ as an arbitrary loss by the chain rule we can write $\frac{\partial l}{\partial x_{i,j}^0} = \sum_{i',j'}\frac{\partial l}{\partial y_{i',j'}}\frac{\partial y_{i',j'}}{\partial x_{i,j}^0}$. Now if $\frac{\partial l}{\partial y_{0,0}} =1$ and $\frac{\partial l}{\partial y_{i,j}}=0$ for all $i \neq 0$ and $j \neq 0$, then $\frac{\partial l}{\partial x_{i,j}^0} =\frac{\partial y_{0,0}}{\partial x_{i,j}^0}$.<br />
<br />
For networks without nonlinearity (i.e., linear networks) this measure is independent of the input and depends only on the weights of the network and (i, j) , which clearly shows how the impact of the pixels in the receptive field distributes.<br />
<br />
===Simplest case: Stack of convolutional layers of weights equal to 1===<br />
<br />
The authors first considered the case of $n$ convolutional layers using $k \times k$ kernels of stride 1 and a single channel on each layer and no nonlinearity, and bias. <br />
<br />
<br />
For this special sub-case, the kernel was a $k \times k$ matrix of 1's. Since this kernel is separable to $k \times 1$ and $1 \times k$ matrices, the $2D$ convolution could be replaced by two $1D$ convolutions. This allowed the authors to focus their analysis on the $1D$ convolutions.<br />
<br />
For this case, if we denote the gradient signal $\frac{\partial l}{\partial y_{i,j}}$ by $u(t)$ and the kernel by $v(t)$, we have<br />
<br />
\begin{equation*}<br />
u(t)=\delta(t),\\ \quad v(t) = \sum_{m=0}^{k-1} \delta(t-m), \quad \text{where} \begin{cases} \delta(t)= 1\ \text{if}\ t=0, \\ \delta(t)= 0\ \text{if}\ t\neq 0, \end{cases}<br />
\end{equation*}<br />
and $t =0,1,-1,2,-2,...$ indexes the pixels.<br />
<br />
The gradient signal $o(t)$ on the input pixels can now be computed by convolving $u(t)$ with $n$ $v(t)$'s so that $o(t) = u *v* ...*v$. <br />
<br />
Since convolution in time domain is equivalent to multiplication in Fourier domain, we can write<br />
<br />
\begin{equation*}<br />
U(w) = \sum_{t=-\infty}^{\infty} u(t) e^{-jwt}=1,\\<br />
V(w) = \sum_{t=-\infty}^{\infty} v(t) e^{-jwt}=\sum_{m=0}^{k-1} e^{-jwm},\\<br />
O(w) = F(o(t))=F(u(t)*v(t)*...*v(t)) = U(w).V(w)^n = \Big ( \sum_{m=0}^{k-1} e^{-jwm} \Big )^n,<br />
\end{equation*}<br />
<br />
where $O(w)$, $U(w)$, and $V(w)$ are discrete Fourier transformations of $o(t)$, $u(t)$, and $v(t)$.<br />
<br />
Now let us consider two non-trivial cases.<br />
<br />
'''Case K=2:''' In this case $( \sum_{m=0}^{k-1} e^{-jwm} )^n = (1 + e^{-jw})^n$. Because $O(w)= \sum_{t=-\infty}^{\infty} o(t) e^{-jwt}= (1 + e^{-jw})^n$, we can think of $o(t)$ as coefficients of $e^{-jwt})$. Therefore, $o(t)= <br />
\begin{pmatrix} n\\t\end{pmatrix}$ is the standard binomial coefficients. As $n$ becomes large binomial coefficients distribute with respect to $t$ like a Gaussian distribution. More specifically, when $n \to \infty$ we can write<br />
<br />
<br />
\begin{equation*}<br />
\begin{pmatrix} n\\t \end{pmatrix} \sim \frac{2^n}{\sqrt{\frac{n\pi}{2}}}e^{-d^{2}/2n}, <br />
\end{equation*}<br />
<br />
where $d = n-2t$ (see [https://en.wikipedia.org/wiki/Binomial_coefficient Binomial coefficient]).<br />
<br />
'''Case K>2:''' In this case the coefficients are known as "extended binomial coefficients" or "polynomial<br />
coefficients", and they too distribute like Gaussian [5].<br />
<br />
=== Random Weights===<br />
Denote $g(i, j, p) = \frac{\partial l}{\partial x_{i,j}^p}$ as the gradient on the $p$th layer, and $g(i, j, p) = \frac{\partial l}{\partial y_{i,j}}$ . Then $g(, , 0)$ is the desired gradient image of the input. The backpropagation convolves $g(, , p)$ with the $k x k$ kernel to get $g(, , p-1)$ for each p. So we can write<br />
<br />
\begin{equation*}<br />
g(i,j,p-1) = \sum_{a=0}^{k-1} \sum_{b=0}^{k-1} w_{a,b}^p g(i+a,i+b,p),<br />
\end{equation*}<br />
<br />
where $w_{a,b}^p$ is the convolution weight at $(a, b)$ in the convolution kernel on layer p. In this case, the initial weights are independently drawn from a fixed distribution with zero mean and variance $C$. By assuming that the gradients g are independent from the<br />
weights (linear networks only) and given that $\mathbb{E}_w[w_{a,b}^p] =0$<br />
<br />
\begin{equation*}<br />
\mathbb{E}_{w,input}[g(i,j,p-1)] = \sum_{a=0}^{k-1} \sum_{b=0}^{k-1} \mathbb{E}_w[w_{a,b}^p] \mathbb{E}_{input}[g(i+a,i+b,p)]=0,\\<br />
Var[g(i,j,p-1)] = \sum_{a=0}^{k-1} \sum_{b=0}^{k-1} Var[w_{a,b}^p] Var[g(i+a,i+b,p)]= C\sum_{a=0}^{k-1} \sum_{b=0}^{k-1} Var[g(i+a,i+b,p)].<br />
\end{equation*}<br />
<br />
Therefore, to get $Var[g(, , p-1)]$ we can convolve the gradient variance image $Var[g(, , p)]$ with a $k \times k$ kernel of 1’s, and then multiply it by $C$. Comparing this to the simplest case of all weights equal to one, we can see that the $g(, , 0)$ has a Gaussian shape, with only a slight<br />
change of having an extra $C^n$ constant factor multiplier on the variance gradient images, which does not affect the relative distribution within a receptive field.<br />
<br />
=== Non-uniform Kernels ===<br />
In the case of non-uniform weighting, when w(m)'s are normalized, we can simply use characteristic function to prove the Central Limit Theorem in this case. For $S_n = \sum_{i=1}^n$ $X_i$ and $X_i$’s are i.i.d. multinomial variables distributed according to $w(m)$’s, i.e. $p(X_i = m) = w(m)$, we have:<br />
<br />
\begin{equation*}<br />
E[S_n] = n\sum_{m=0}^{k-1} mw(m),\\<br />
Var[S_n] = n \left (\sum_{m=0}^{k-1} m^2w(m) - \left (\sum_{m=0}^{k-1} mw(m) \right )^2 \right ),<br />
\end{equation*}<br />
<br />
If we take one standard deviation as the effective receptive field (ERF) size which is roughly the radius of the ERF, then this size is<br />
$\sqrt{Var[S_n]} = \sqrt{nVar[X_i]} = O(\sqrt{n})$.<br />
<br />
On the other hand, stacking more convolutional layers implies that the theoretical receptive field grows linearly, therefore relative to the theoretical receptive field, the ERF actually shrinks at a rate of $O(1/\sqrt{n})$.<br />
<br />
=== Non-linear Activation Functions===<br />
<br />
The math in this section is a bit "hand-wavy", as one of their reviewers wrote, and their conclusion (Gaussian-shape ERF) is not really well backed up by their experiments. The most important point to take way form this part is that by introduction of a nonlinear activation function, the gradients depends on the network's input as well.<br />
<br />
=== Dropout, Subsampling, Dilated Convolution and Skip-Connections ===<br />
The authors show that dropout does not change the Gaussian ERF shape. Subsampling and dilated convolutions turn out to be effective ways to increase receptive field size quickly. Skip-connections on the other hand make ERFs smaller.<br />
<br />
= Verifying Theoretical Results =<br />
In all of the following experiments, a gradient signal of 1 was placed at the center of the output plane and 0 everywhere else, and then this gradient was backpropagated through the network to get input gradients. Also random inputs as well as proper random initialization of the kernels were employed.<br />
<br />
<br />
'''ERFs are Gaussian distributed:''' By looking at the figure, [[File:understanding_ERF_fig1.png|thumbnail||600px]] we can observe Gaussian shapes for uniformly and randomly weighted convolution kernels without nonlinear activations, and near Gaussian shapes for randomly weighted kernels with nonlinearity. Adding the ReLU nonlinearity makes the distribution a bit less Gaussian, as the ERF distribution depends on the input as well. Another reason is that ReLU units output exactly zero for half of its inputs and it is very easy to get a zero output for the center pixel on the output plane, which means no path from the receptive field can reach the output, hence the gradient is all zero. Here the ERFs are averaged over 20 runs with different random seed. <br />
<br />
<br />
<br />
<br />
<br />
Figures below show the ERF for networks with 20 layers of random weights, with different nonlinearities. Here the results are averaged both across 100 runs with different random weights as well as different random inputs. In this setting the receptive fields are a lot more Gaussian-like. <br />
<br />
[[File:understanding_ERF_fig2.png|thumbnail|centre|400px]]<br />
<br />
<br />
''' <math>\sqrt{n}</math> absolute growth and <math>1/\sqrt{n}</math> relative shrinkage:''' The figure [[File:understanding_ERF_fig4.png|thumbnail||600px]] shows the change of ERF size and the relative ratio of ERF over theoretical RF wrt number of convolution layers. The fitted line for ERF size has the slope of 0.56 in log domain, while the line for ERF ratio has the slope of -0.43. This indicates ERF size is growing linearly wrt <math>\sqrt{n}</math> and ERF ratio is shrinking linearly wrt <math>1/\sqrt{n}</math>.<br />
They used 2 standard deviations as the measurement for ERF size, i.e. any pixel with value greater than 1 - 95.45% of center point is considered in ERF. The ERF size is represented by the square root of number of pixels within ERF, while the theoretical RF size is the side length of the square in which all pixel has a non-zero impact on the output pixel, no matter how small. All experiments here are averaged over 20 runs.<br />
<br />
<br />
'''Subsampling & dilated convolution increases receptive field:''' The figure shows that the effect of subsampling and dilated convolution. The reference baseline is a CNN with 15 dense convolution layers. Its ERF is shown in the left-most figure. Replacing 3 of the 15 convolutional layers with stride-2 convolution results in the ERF for the ‘Subsample’ figure. Finally, replacing those 3 convolutional layers with dilated convolution with factor 2,4 and 8 gives the ‘Dilation’ figure. Both of them are able to increase the effect receptive field significantly. Note the ‘Dilation’ figure shows a rectangular ERF shape typical for dilated convolutions (why?).<br />
<br />
[[File:understanding_ERF_fig3.png|thumbnail|centre|400px]]<br />
<br />
== How the ERF evolves during training ==<br />
<br />
The authors looked at how the ERF of units in the top-most convolutional layers of a classification CNN and a semantic segmentation CNN evolve during training. For both tasks, they adopted the ResNet architecture which makes extensive use of skip-connections. As expected their analysis showed the ERF of these networks are significantly smaller than the theoretical receptive field. Also, as the networks learns, the ERF got bigger so that at the end of training was significantly larger than the initial ERF. <br />
<br />
The classification network was a ResNet with 17 residual blocks trained on the CIFAR-10 dataset. Figure shows the ERF on the 32x32 image space at the beginning of training (with randomly initialized weights) and at the end of training when it reaches best validation accuracy. Note that the theoretical receptive field of the network is actually 74x74, bigger than the image size, but the ERF is not filling the image completely. Comparing the results before and after training <br />
demonstrates that ERF has grown significantly.<br />
<br />
[[File:understanding_ERF_fig5.png|thumbnail|centre|500px]]<br />
<br />
The semantic segmentation network was trained on the CamVid dataset for urban scene segmentation. The 'front-end' of the model was a purely convolutional network that predicted the output at a slightly lower resolution. And then, a ResNet with 16 residual blocks interleaved with 4 subsampling operations each with a factor of 2 was implemented. Due to subsampling operations the output was 1/16 of the input size. For this model, the theoretical RF of the top convolutional layer units was 505x505. However, as Figure shows the ERF only got a fraction of that with a diameter of 100 at the beginning of training, and at the end of training reached almost a diameter around 150.<br />
<br />
= Discussion =<br />
The Effective Receptive Field (ERF) usually decays quickly from the centre (like 2D Gaussian) and only takes a small portion of the theoretical Receptive Field (RF). This "Gaussian damage" is undesirable for tasks that require a large RF and to reduce it, the authors suggested two solutions:<br />
#'''New Initialization scheme''' to make the weights at the center of the convolution kernel to be smaller and the weights on the outside larger, which diffuses the concentration on the center out to the periphery. One way to implement this is to initialize the network with any initialization method, and then scale the weights according to a distribution that has a lower scale at the center and higher scale on the outside. They tested this solution for the CIFAR-10 classification task, with several random seeds. In a few cases they get a 30% speed-up of training compared to the more standard initializations. But overall the benefit of this method is not always significant.<br />
#'''Architectural changes of CNNs''' is the 'better' approach that may change the ERF in more fundamental ways. For example, instead of connecting each unit in a CNN to a local rectangular convolution window, we can sparsely connect each unit to a larger area in the lower layer using the same number of connections. Dilated convolution belongs to this category, but we may push even further and use sparse connections that are not grid-like.<br />
<br />
= Summary & Conclusion =<br />
The authors showed, theoretically and experimentally, that the distribution of impact within the receptive field (the effective receptive field) is asymptotically Gaussian, and the ERF only takes up a fraction of the full theoretical receptive field. They also studied the effects of some standard CNN approaches on the effective receptive field. They found that dropout does not change the Gaussian ERF shape. Subsampling and dilated convolutions are effective ways to increase receptive field size quickly but skip-connections make ERFs smaller.<br />
<br />
They argued that since larger ERFs are required for higher performance, new methods to achieve larger ERF will not only help the network to train faster but may also improve performance.<br />
<br />
= Critique = <br />
<br />
The authors' finding on $\sqrt{n}$ absolute growth of Effective Receptive Field (ERF) suffers from discrepancy in ERF definition between their theoretical analysis and their experiments. Namely, in the theoretical analysis for non-uniform-kernel case they considered one standard deviation as the ERF size. However, they used two standard deviations as the measure for ERF size in the experiments.<br />
<br />
It would be more practical if the paper also investigated the ERF for natural images (as opposed to random) as network input at least in the two cases where they examined trained networks. <br />
<br />
The authors claim that the ERF results in the experimental section have Gaussian shapes but they never prove this claim. For example, they could fit different 2D-functions, including 2D-Gaussian, to the kernels and show that 2D-Gaussian gives the best fit. Furthermore, the pictures are given as proof of the claim that the ERF has a Gaussian distribution only show the ERF of the center pixel of the output <math> y_{0,0} </math>. Intuitively, the ERF of a node near the boundary of the output layer may have a significantly different shape. This was not addressed in the paper.<br />
<br />
Another weakness is in the discussion section, where they make a connection to the biological networks. They jumped to disprove a well-observed phenomenon in the brain. The fact that the neurons in the higher areas of the visual hierarchy gradually lose their retinotopic property has been shown in a countless number of neuroscience studies. For example, [https://en.wikipedia.org/wiki/Grandmother_cell grandmother cells] do not care about the position of grandmother's face in the visual field. In general, the similarity between deep CNNs and biological visual systems is not as strong, hence we should take any generalization from CNNs to biological networks with a grain of salt.<br />
<br />
Spectrograms are visual representations of audio where the axes represent time, frequency and amplitude of the frequency. The ERF of a CNN when applied to a spectrogram doesn't necessarily have to be from a Gaussian towards the center. In fact many receptive fields are trained to look for the peaks of troughs and cliffs, which essentially imply that the ERF will have more weightage towards the outside rather than the center.<br />
<br />
= References =<br />
[1] Wenjie Luo, Yujia Li, Raquel Urtasun, and Richard Zemel. "Understanding the effective receptive field in deep convolutional neural networks." In Advances in Neural Information Processing Systems, pp. 4898-4906. 2016.<br />
<br />
[2] Buessler, J.-L., Smagghe, P., & Urban, J.-P. (2014). Image receptive fields for artificial neural networks. Neurocomputing, 144(Supplement C), 258–270. https://doi.org/10.1016/j.neucom.2014.04.045<br />
<br />
[3] Dilated Convolutions in Neural Network - [http://www.erogol.com/dilated-convolution/]<br />
<br />
[4] http://cs231n.github.io/convolutional-networks/<br />
<br />
[5] Thorsten Neuschel. "A note on extended binomial coefficients." Journal of Integer Sequences, 17(2):3, 2014.</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=meProp:_Sparsified_Back_Propagation_for_Accelerated_Deep_Learning_with_Reduced_Overfitting&diff=29529meProp: Sparsified Back Propagation for Accelerated Deep Learning with Reduced Overfitting2017-11-07T05:03:36Z<p>H4lyu: /* Results */</p>
<hr />
<div>=Introduction=<br />
<br />
A simple and effective technique for neural networks learning is introduced in the current paper. The main technique entails a modification to the vanilla backpropagation algorithm. The idea is that after a forward pass has been carried out in usual fashion, we retain only a subset of the full gradient for computation of model parameters. More precisely, a simple quantization technique is employed to sparsify the gradient vectors, viz., the entries of the first gradient in a backpropagation step are set to zero unless they reach a specified size threshold. The rest of the gradients (the ones with respect to the weights and biases of the neural network) are computed using the chain rule in the typical way using the sparsified gradient obtained from the top layer. Since only a small subset of the weight matrix is modified, we obtain a linear reduction in the computational cost. The experimental results presented in the paper suggest that accuracy is improved rather than being degraded. The name given to the proposed technique is minimal effort back propagation method (meProp).<br />
[[File:20.png|right|650px]]<br />
The backpropagation step in neural network training entails high computational cost since each iteration requires calculation of full gradient vectors and matrices and subsequent update of all model parameters.<br />
The main idea of the paper is to find only a small but critical subset of the gradient information and in each learning step, update only this minimal subset of the parameters. This leads to sparsified gradients because only highly relevant parameters are updated and rest remain untouched. Fig. 1 shows an abstract view of the proposed approach.<br />
<br />
'''Two important questions:'''<br />
<br />
1) In the process of stochastic learning how do we find a highly relevant subset of parameters from the current sample?<br />
<br />
One solution to this question is given by Top-$k$ search method to identify the most important parameters. Experimental results suggest that if we use this technique, then we can only update 1–4% of the weights at each back propagation pass and this does not result in a larger number of training iterations. <br />
<br />
Another likely solution I can think of is the method in the paper: http://papers.nips.cc/paper/6372-learning-the-number-of-neurons-in-deep-networks.pdf. We can use the group sparsity regularizer to identify the neurons that have many nonzero parameters, which are considered highly relevant parameters.<br />
<br />
2) Does this process of selecting a small subset of model parameters hurt accuracy?<br />
<br />
The results demonstrate that rather than reduce, this sparsification actually improves the accuracy in most settings.<br />
This result, while somewhat surprising, is attributed to a dropout-like effect which works to prevent overfitting. Because the minimal effort update does not modify any parameters which are weakly relevant, it seems sensible that this would help avoid overfitting the data.<br />
<br />
The authors demonstrate the proposed approach using deep learning approaches (like LSTM, MLP), optimization approaches (like Adam and Adagrad) and tasks like NLP and Image Recognition.<br />
<br />
=Related Work=<br />
<br />
Some of the notable related work to this paper are as follows:<br />
<br />
In 1990, Tollenaere et al.[1] proposed SuperSAB: an adaptive acceleration strategy for error back propagation learning. They proved that it may converge orders of magnitude faster than the original back propagation algorithm, and is only slightly unstable. In addition, the algorithm is very insensitive to the choice of parameter values, and has excellent scaling properties.<br />
<br />
In 1993, Riedmilller et al.[2] and Braun proposed an algorithm called RPROP, to overcome the inherent disadvantages of pure gradient-descent, it performed a local adaptation of the weight-updates according to the behavior of the error function. To be more specific, the authors defined an individual update-value for each weight. When the update of a weight is too large (mathematically, partial derivative of a weight changes its sign), such update-value of this weight will decrease. Otherwise, it'll increase. After the process of adapting update-values is finished, the weight-update process is carried out as follows: when the partial derivative of error function with respect to a specific weight is positive, the original weight decreases by its corresponding update-value, increase otherwise. <br />
<br />
In 2014, Srivastava et al.[3] proposed dropout. Large networks are also slow to use, making it difficult to deal with overfitting by combining the predictions of many different large neural nets at test time. The key idea is to randomly drop units (along with their connections) from the neural network during training. This prevents units from co-adapting too much.<br />
<br />
The work proposed by the authors of meProp method is quite different from the three related works discussed above.<br />
<br />
In 2017, Shazeer et al.[4] presented a Sparsely-Gated Mixture-of-Experts layer (MoE), consisting of up to thousands of feed-forward sub-networks. A trainable gating network determines a sparse combination of these experts to use for each example. They used this approach for the machine translation task and concluded that it gave significantly better results. Their method is limited to a specific set of mixture of experts however, the meProp method does not have these sort of limitations.<br />
<br />
=Proposed Approach=<br />
[[File:7.png|center|600px]]<br />
The original back propagation computes the "full gradient" for the input vector and the weight matrix. However, in me-Prop, back propagation computes an "approximate gradient" by keeping top-k values of the backward flowed gradient and masking the remaining values to 0. That is, only the top-k elements with the largest absolute values are kept and rest are made 0.<br />
<br />
Authors describe conversion of the traditional back-propagation to the back-propagation used in meProp with a computation unit with one linear transformation and one non-linear transformation as an example as given by eq. (1) and (2):<br />
\begin{align*}<br />
y &= W x \quad \quad \quad (1)<br />
\end{align*}<br />
\begin{align*}<br />
z &= \sigma (y) \quad \quad \quad (2)<br />
\end{align*}<br />
where $W \in R_{n \times m}$, $x \in R_m$, $y \in R_n$, $z \in R_n$, $m$ is the dimension of the input vector, $n$ is the dimension of the output vector, and $\sigma$ is a non-linear function (e.g., ReLU, tanh,<br />
and sigmoid). During back propagation, we need to compute the gradient of the parameter matrix $W$ and the input vector $x$:<br />
<br />
\[<br />
\frac{\partial z}{\partial W_{ij}} = \sigma^{'}_{i}x^{T}_{j},\quad i \quad \epsilon \quad [1,n], \quad j \quad \epsilon \quad [1,m] \quad \quad \quad (3)<br />
\]<br />
<br />
\[<br />
\frac{\partial z}{\partial x_{i}} = \sum\limits_j W^{T}_{ij} \sigma^{'}_{j},\quad i \quad \epsilon \quad [1,n], \quad j \quad \epsilon \quad [1,m] \quad \quad \quad (4)<br />
\]<br />
<br />
Since the proposed meProp keeps only top-k elements based on the magnitude values so eq. (3) and (4) get transformed to (5) and (6), respectively:<br />
<br />
\[<br />
\frac{\partial z}{\partial W_{ij}} \leftarrow \sigma^{'}_{i}x^{T}_{j}, \quad if \quad i \quad \epsilon \quad \{ t_{1}, t_{2},....., t_{k} \} \quad else \quad 0 \quad \quad (5)<br />
\]<br />
<br />
\[<br />
\frac{\partial z}{\partial x_{i}} \leftarrow \sum\limits_j W^{T}_{ij} \sigma^{'}_{j}, \quad if \quad j \quad \epsilon \quad \{ t_{1}, t_{2},....., t_{k} \} \quad else \quad 0 \quad \quad (6)<br />
\]<br />
<br />
<br />
<br />
The original back-propagation computes the gradient of the matrix W and the gradient of the input vector x as shown in eq. (7) and (8), respectively:<br />
<br />
\[<br />
\frac{\partial L}{\partial W} = \frac{\partial L}{\partial y} . \frac{\partial y}{\partial W} \quad \quad (7)<br />
\]<br />
<br />
\[<br />
\frac{\partial L}{\partial x} = \frac{\partial y}{\partial x} . \frac{\partial L}{\partial y} \quad \quad (8)<br />
\]<br />
<br />
Since, the proposed meProp selects top-k elements of the traditional gradient to approximate it, hence the gradient of W and gradient of vector x transform to the one shown in eq. (9) and (10):<br />
<br />
<br />
\[<br />
\frac{\partial L}{\partial W} \leftarrow top_{k}(\frac{\partial L}{\partial y}) . \frac{\partial y}{\partial W} \quad \quad (9)<br />
\]<br />
<br />
\[<br />
\frac{\partial L}{\partial x} \leftarrow \frac{\partial y}{\partial x} . top_{k}(\frac{\partial L}{\partial y}) \quad \quad (10)<br />
\]<br />
<br />
The intuition behind the discussed conversions is depicted in Fig. 2. <br />
<br />
'''Where to apply meProp:'''<br />
In the learning task Matrix to Matrix and Matrix to Vector multiplications consume more than 90% of the backpropagation time. The authors apply meProp:<br />
* only to the back propagation from the output of the multiplication to its inputs;<br />
* they apply meProp (meaning top-k sparsification) to every hidden layer. Because the sparsified gradient will again be dense from one layer to another;<br />
* k for the output layer could be different from the k of the hidden layers because of the difference in the dimensionality of the output layers. Example: For example, there are 10 tags in the MNIST task, so the dimension of the output layer is 10, and we use an MLP with the hidden dimension of 500. Thus, the best k for the output layer could be different from that of the hidden layers.<br />
<br />
'''Exception:''' For other element-wise operations (e.g., activation functions), the original backpropagation procedure remains as it is, because those operations are already fast enough compared<br />
with matrix-matrix or matrix-vector multiplication operations.<br />
<br />
'''Choice of top-k algorithms:''' A variant (focusing on memory reuse) of min heap-based top-k selection method is used. The time complexity is: O(n log k) and space complexity is O(k). This is done to save time on sorting the entire vector. A min-heap is a binary tree such that the data contained in each node is less than (or equal to) the data in that node’s children.<br />
<br />
=Experiments and Configurations=<br />
<br />
To establish that the approach is general purpose, authors performed experiments on different deep learning algorithms(i.e. LSTM, MLP) with different optimizers (i.e. Adam, Adagrad) and different problem sets (i.e. Part of Speech Tagging, Transition based dependency parsing, MNIST Image Recognition). <br />
<br />
'''POS-Tag:''' <br />
Part-of-speech tagging is the process of identifying and assigning the parts of speech such as noun,verb, adjectice etc. in a corpus <br />
Baseline model: LSTM. Benchmark dataset: Penn Treebank Corpus. For training and testing: Wall Street Journal.<br />
<br />
'''Parsing:''' Baseline model: MLP. Benchmark dataset: Penn Treebank Corpus. For training, development, and testing: Wall Street Journal. The most common method for evaluating parsers are labeled and unlabeled attachment scores. In this work the authors use the unlabeled attachment score. Labeled attachment refers to the correct matching of a word to its head along with the correct dependency relation. Unlabeled attachment ignores the dependency relation and focuses on the correctness of the assigned head. <br />
<br />
'''MNIST:'''<br />
The MNIST dataset consists of hand-written digits and the solution involves classifying the images among 10 digit classes.<br />
Baseline model: MLP. For training, development, and testing: MNIST dataset.<br />
<br />
In the configuration for Parsing and MNIST authors use the same k for the output and hidden layers. For POS-Tag authors use different k for the output and hidden layers. Due to low dimensionality of output layer in POS-Tag meProp isn't applied to it.<br />
<br />
The code for the paper can be found on Github : https://github.com/jklj077/meProp<br />
<br />
=Results=<br />
<br />
[[File:13.png|right|750px]]<br />
<br />
meProp is applied to the linear transformations which actually entail the major computational cost. Authors call linear transformation related backprop time as Backprop Time. It does not include the time required for non linear activations which usually entail less than 2% of the computational cost. The total time of back propagation including non linear activations is reported as Overall Backprop Time.<br />
<br />
Through results it was observed that meProp substantially speeds up the backpropagation and provides a linear reduction in computational cost. Authors state the main reason for this reduction to be that meProp does not modify weakly relevant parameters, which makes overfitting less likely similar to the dropout effect. Also, the results depict that the proposed approach is independent of specific optimization methods.<br />
<br />
The graphs shown in Fig. 4 depict that meProp addresses the problem of overfitting and it provides better accuracy if the top-k weights are selected instead of random weights. It suggests that top-k elements contain<br />
the most important information of the gradients. This makes us think, instead of using dropout which randomly turns off few neurons, can it be done more deterministically based on the contribution of neuron to the final prediction or output. Also, it was inferred that meProp can achieve further improvements over dropout for reducing overfitting and a model should take advantage of both meProp and dropout to reduce overfitting. Adding hidden layers does not hurt the performance of the model. Although this may be the case for the current set of test cases, a better understanding of the variation of hidden layer size and choice-of-k can be obtained by varying k with different hidden unit sizes <math>h</math> by keeping <math>k*h</math> or a similarly related term constant. This is better studied in [5] where the authors kept <math>p*n</math> constant to obtain greater reductions in training error for smaller p values ( p being the dropout coefficient. Low p, more units dropped). The relevant numerical results have been shown in table 1-5. <br />
<br />
'''Further speed up:'''<br />
For further speeding up the backpropagation on GPUs authors presented a simple unified top-k approach (implementation in PyTorch). The main idea is to treat the entire mini-batch as a "big training example" where top-k operation is based on the averaged values of all examples in the mini-batch, so that the large consistent sparse matrix of the mini-batch can be converted into a dense small matrix by simply removing the zero values. The authors refer to this method as the simplified unified top-$k$ method. The results are presented in Table 6. This GPU acceleration works much more outstandingly for heavy models, with the relevant numerical results shown in table 7 and 8.<br />
<br />
=List of Tables=<br />
<br />
[[File:11.png|thumb|center|750px]]<br />
[[File:12.png|thumb|center|450px]]<br />
[[File:14.png|thumb|center|350px]]<br />
[[File:15.png|thumb|center|400px]]<br />
[[File:16.png|thumb|center|500px]]<br />
[[File:meProp.PNG|thumb|center|750px]]<br />
[[File:56.png|thumb|center|500px]]<br />
[[File:57.png|thumb|center|400px]]<br />
<br />
=Critiques=<br />
The main idea behind meProp is to wipe out the backprop mechanism of (n-k) nodes where "n" is the number of nodes in the current layer and "k" is the number of nodes contributing to maximum of the loss in that layer. Intuitively, meProp in backpropagation process is actually a threshold w.r.t. k, or an activation function in the gradient backpropagation: only if the gradients are big enough in magnitude that will be passed to previous layer. Referring to equation 10,<br />
\[<br />
\frac{\partial L}{\partial x} \leftarrow \frac{\partial y}{\partial x} . top_{k}(\frac{\partial L}{\partial y}) \quad \quad<br />
\]<br />
#The authors have not proposed any method on how k should be selected, hence it is left to the reader's discretion to possibly take it as a hyperparameter. If so, in a deeply layered architecture, where the weights between each layer are randomly initialized during each execution, "k" might change for each layer since the features learned at each layer may not be the same from the previous layers. However, under the assumption that we only perform top-$k$ selection for the gradient vector associated to the top layer, we do not choose $k$ for each subsequent layer through which we backpropagate. The concern as to whether we may lose valuable feature selection due to hidden layers is a valuable one. Moreover, further study should be carried out to see whether this is in fact the case and if not, whether we can directly sparsify weight matrices of hidden layers.<br />
#If the sum of losses caused by the (n-k) nodes in the current layer exceed any of the losses incurred due to "k" nodes, then it would not be correct to drop the (n-k) nodes as we can assume the aggregate (n-k) nodes as a single opaque node with a composite weight which will incur an aggregated loss greater that any of the "k" nodes.<br />
<br />
In essence, the idea of selecting "k" nodes to drop-out prove to be effective as shown by the authors, but the lack of information on the conditions on selecting "k" for each layer given the current state of the layer might lead to lack of consistency in the results.<br />
<br />
In addition to this, the authors did not include convolutional neural networks in their experiments. It would have been interesting to see whether similar results were observed on that architecture. Theoretically, the method presented in this paper should only update kernels in parts of an image that contribute the most to the loss.<br />
<br />
As the experiment settings, all networks are using Adam and AdaGrad, it is an interesting guess that whether the choice of the optimizer will influence the accuracy. The authors did not include the results with SGD(momentum). Since Adam and AdaGrad are using adaptive learning rate for each weight.<br />
<br />
There has been no mention by the authors on the loss ( significant loss or the insignificance) of using meProp on tasks where preservation of temporal information and contextual data is important. For example, in tasks like using RNNs for Question-Answering tasks, memory of details of earlier regions of the paragraph could be garbled due to non-updation of the weights not belonging to the top-k set in backpropagation. Indeed, the lack of principled methods for sparsification is a major issue in this case since tasks such as machine translation often entail data where certain parts of an input are much more predictive than other parts in a systematic way. There could be a trade-off between knowledge preservation and choice of the hyperparameter k which can be verified by further analysis like correlation/covariance studies.<br />
<br />
The approach can be thought of as a deterministic Dropout giving priority to higher gradient contributing connections during backpropagation. However, unlike dropout (which is random in nature), selecting k-top may permanently exclude some parts of NN from training at all, which has not been mentioned in the paper at all. Authors have also failed to test their approach on bigger datasets such as Imagenet, therefore it might be possible that dataset (MNIST) used by the authors is too simple for the given NN architecture, therefore, meProp approach helped generalizing the model better. It is generally a bad habit to use MNIST results in 2017's research works, as they shed no light on the real world AI problems. Idea is really simple, basically applying only k strongest gradients during backprop which should work for different architectures as well (LSTM, RNNs). Paper has shown advantage of their method empirically, but only only simple dataset and it is lacking its results in real world and more complex dataset.<br />
<br />
=References=<br />
# Tollenaere, Tom. "SuperSAB: fast adaptive back propagation with good scaling properties." Neural networks 3.5 (1990): 561-573.<br />
# Riedmiller, Martin, and Heinrich Braun. "A direct adaptive method for faster backpropagation learning: The RPROP algorithm." Neural Networks, 1993., IEEE International Conference on. IEEE, 1993.<br />
# Srivastava, Nitish, et al. "Dropout: a simple way to prevent neural networks from overfitting." Journal of machine learning research 15.1 (2014): 1929-1958.<br />
# Shazeer, Noam, et al. "Outrageously large neural networks: The sparsely-gated mixture-of-experts layer." arXiv preprint arXiv:1701.06538 (2017).<br />
# Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, Ruslan Salakhutdinov. "Dropout: A Simple Way to Prevent Neural Networks from Overfitting", Journal of Machine Learning Research 15 (2014) 1929-1958<br />
#Speech and Language Processing. Daniel Jurafsky & James H. Martin. 2017. Draft of August 28, 2017.</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Incremental_Boosting_Convolutional_Neural_Network_for_Facial_Action_Unit_Recognition&diff=28864Incremental Boosting Convolutional Neural Network for Facial Action Unit Recognition2017-10-31T04:17:55Z<p>H4lyu: /* References */</p>
<hr />
<div>== Introduction ==<br />
<br />
Facial expression is one of the most natural ways that human beings express the emotions. The Facial Action Coding System (FACS) attempts to systemically categorize each facial expression by specifying a basic set of muscle contractions or relaxations, formally called Action Units (AUs). FACS as we know today consists of five major categories (i) main, (ii) head movement, (iii) eye movement, (iv) visibility and (v) gross behavior . For example, "AU 1" stands for the inner portion of the brows being raised, and "AU 6" stands for the cheeks being raised. Such a framework helps in describing any facial expression/emotion as the combination of different AUs. The complete list of AUs in the FACS can be seen [https://en.wikipedia.org/wiki/Facial_Action_Coding_System#List_of_Action_Units_and_Action_Descriptors_.28with_underlying_facial_muscles.29 here]. However, during the course of an average day, most human beings do not experience drastically varying emotions and therefore their facial expressions might change only subtly. Additionally, there might also be a lot of subjectivity involved if this task were to be done manually. To address these issues, it is imperative to automate this task. In addition to applications in human behavior analysis, automating AU recognition also has potential applications in human-computer interaction (HCI), online education, interactive gaming, among other domains.<br />
<br />
Because of the recent advancements in object detection and categorization tasks, CNNs are an appealing go-to for the facial AU recognition task described above. However, compared to the former, the training sets available for the AU recognition task are not very large, and therefore the learned CNNs suffer from overfitting. To overcome this problem, this work builds on the idea of integrating boosting within CNN. Boosting is a technique wherein multiple weak learners are combined together to yield a strong learner. Moreover, this work also modifies the mechanics of learning a CNN by breaking large datasets in mini-batches. Typically, in a batch strategy, each iteration uses a batch only to update the parameters and the features learned in that iteration are discarded for subsequent iterations. Herein the authors incorporate incremental learning by building a classifier that incrementally learns from all the iterations/batches. Hence the genesis of the name Incremental Boosting - Convolutional Neural Network (IB-CNN) is justified. The IB-CNN introduced here outperforms the state-of-the-art CNNs on four standard AU-coded databases.<br />
<br />
== Related Work ==<br />
<br />
There have been many papers that use manually designed features for the AU-recognition task, unlike the CNN-based methods which integrate feature learning. Some of them are Gabor Wavelets, Local Binary Patterns (LBP), Histogram of Oriented Gradients (HOG), Scale Invariant Feature Transform (SIFT) features, Discriminant Non-negative Matrix Factorization(DNMF), Histograms of Local Phase Quantization (LPQ), and their spatiotemporal extensions. Newer approaches include the one proposed by Tong, Liao and Ji [9], who use a dynamic Bayesian network (DBN) to model the relationships among different AUs. Automated methodologies include the Automatic Facial Analysis system which recognizes both permanent and transient AUs with high accuracy rates [6]. Hand-crafted feature descriptors like LBP as very powerful facial recognition algorithm, and these types of descriptors are very robust and do not require training - hence, there is no "overfitting" of data. LBP, in particular, provides a very dense and concise descriptor (i.e. a histogram feature) of facial features without the use of GPU; in essence, the feature vector for LBPs are small, yet powerful, and the accuracy of which is comparable to that of a trained CNN. One of the major breakthroughs to reduce overfitting in CNNs was the dropout, introduced by Hinton et al [5]. As dropout can be regarded a special bagging ensembling method over "different" networks that share same weights. This work can be seen as a refinement of the previous random dropout approach in the sense that only neurons with no contribution towards classification are dropped, while retaining the more discriminative neurons. Moreover, adopting a batch strategy with incremental learning also circumvents the issue of not-so-large datasets.<br />
<br />
== Methodology ==<br />
<br />
=== CNNs ===<br />
Convolutional Neural Networks are neural networks that contain at least one convolution layer. Typically, the convolution layer is accompanied with a pooling layer. A convolution layer, as the name suggests, convolves the input matrix (or tensor) with a filter (or kernel). This operation produces a feature map (or activation map) that, roughly speaking, shows the presence or absence of the filter in the image. Note that the parameters in the kernel are not preset and are learned. A pooling layer reduces the dimensionality of the data by summarizing the activation map by some means (such as max or average). Following this, the matrix (or tensor) data is then vectorized using a fully-connected (FC) layer. The neurons in the FC layer have connections to all the activations in the previous layer. Finally, there is a decision layer that has as many neurons as the number of classes. The decision layer decides the class based on computing a score function using the activations of the FC layer. Generally, an inner-product score function is used, which is replaced by the boosting score function in this work. <br />
<br />
=== Boosting Method ===<br />
Ever since boosting method was first officially introduced in the 1990s, it has become one of the most powerful learning method to improve model performance. The fundamental boosting algorithm that was introduced by Freund and Schapire in 1995: AdaBoost has evolved into many much more sophisticated boosting methods (Freund & Schapire, 1997). Alternative methods of CNN boosting for images ( in counting tasks) also involve Selective Sampling which has been proven to increase accuracy and reduce processing time [7]. Curse of dimensionality is a big problem in machine learning, where evaluating parameters can reduce the speed of classification, as well as the predictive power. The AdaBoost can only choose those parameters who improve the predictive power of the model, and reduce the dimensionality. The essential idea of boosting is to use a weighted combination of a sequence of weak classifiers to make a strong one. The key point to this process is after each iteration of data modification (through any type of model), the boosting algorithm will place more weight on observations that are difficult to classify correctly. Therefore, those “weak” classifiers can then focus on those observations that are hard to predict (Hastie, Tibshirani, & Friedman, 2017). These explanations can be summarized using the following formula:<br />
<center><br />
<math> G(x) = sign(\sum\limits_{l=1}^L \alpha_l G_l(x)) </math> <br />
Where <math>\alpha_l</math> are computed by boosting algorithm and <math>G_l(x) </math> denotes a weak classifier. <br />
</center><br />
<br />
=== Boosting CNN ===<br />
<br />
Let $X = [x_1, \dots, x_M]$ denote the activation features of a training data batch of size $M$, where each $x_i$ has $K$ activation features. Also, let $Y = [y_1, \dots , y_M]$ denote the corresponding labels, i.e. each $y_i \in \{-1, +1\}$. In other words, the vector $x_i$ contains all the activations of the FC layer, which is typically multiplied with the weights corresponding to the connections between the FC layer and the decision layer. However, as mentioned earlier, this work achieves that same task of scoring via boosting as opposed to computing the inner product. Denoting the weak classifiers by $h(\cdot)$, we obtain the strong classifier as:<br />
\begin{equation}<br />
\label{boosting}<br />
H(x_i) = \sum\limits_{j = 1}^K \alpha_j h(x_{ij}; \lambda_j)<br />
\end{equation}<br />
where $x_{ij}$ is the $j^{th}$ activation feature calculated on the $i^{th}$ image. Moreover, $\alpha_j \geq 0$ denotes the weight of the $j^{th}$ weak classifier and $\sum\limits_{i=1}^K\alpha_i = 1$. Here we define $h(\cdot)$ to be a differentiable surrogate for the sign function as:<br />
\begin{equation}<br />
h(x_{ij}; \lambda_j) = \dfrac {f(x_{ij}; \lambda_j)}{\sqrt{f(x_{ij}; \lambda_j)^2 + \eta^2}}<br />
\end{equation}<br />
where $f(x_{ij}; \lambda_j)$ denotes a decision-stump (one-level decision tree) with $\lambda_j$ being the threshold. The parameter $\eta$ controls the slope of the function $\dfrac {f(\cdot)}{\sqrt{f(\cdot)^2 + \eta^2}}$. The authors set this parameter according to the distribution of $f(\cdot)$ as $\eta = \frac{\sigma}{c}$, where $\sigma$ is the standard deviation of $f(\cdot)$ and $c$ is a constant. This parameter is experimented with extensively in Section 4.4 of the paper. It is shown that $\eta$ can drastically influence the $F1$ score of the model.<br />
Note that for a certain strong classifier $H$ if a certain $\alpha_i = 0$, then the activation feature $x_i$ has no contribution to the output of the classifier $H$. In other words, the corresponding neuron can be considered to be inactive. (Refer to the Figure for the schematic diagram of B-CNN.)<br />
<br />
[[File:ibcnn.png]]<br />
<br />
=== Incremental Boosting ===<br />
In vanilla B-CNN, the information learned in a certain batch, i.e. the weights and the thresholds of the active neurons is discarded for subsequent batches. To address this issue, in this work the idea of incremental learning is incorporated. Formally, the incremental strong classifier at the $t^{th}$ iteration, $H_I^t$, is given as:<br />
\begin{equation}<br />
\label{incr}<br />
H_I^t(x_i^t) = \frac{(t-1)H_I^{t-1}(x_i^t) + H^t(x_i^t)}{t}<br />
\end{equation}<br />
where $H_I^{t-1}$ is the incremental strong classifier obtained at the $(t-1)^{th}$ iteration and $H^t$ is the boosted strong classifier at the $t^{th}$ iteration. Manipulating the equations, we obtain:<br />
\begin{equation}<br />
H_I^t(x_i^t) = \sum\limits_{j=1}^K \alpha_j^t h^t(x_{ij}; \lambda_j^t);\quad \alpha_j^t = \frac{(t-1)\alpha_j^{t-1}+\hat{\alpha}_j^t}{t}<br />
\end{equation}<br />
where $\hat{\alpha}_j^t$ is the weak classifier weight calculated in the $t^{th}$ iteration by boosting and $\alpha_j^t$ is the cumulative weight considering previous iterations. (Refer to the Figure for a schematic diagram of IB-CNN.) <br />
Typically, boosting algorithms minimize an objective function that captures the loss of the strong classifier. However, the loss of the strong classifier may be dominated by some weak classifiers with large weights, which might lead to overfitting. Therefore, to exercise better control, the loss function at a certain iteration $t$, $\epsilon^{IB}$ is expressed as a convex combination of the loss of the incremental strong classifier at iteration $t$ and the loss of the weak classifiers determined at iteration $t$. That is,<br />
\begin{equation}<br />
\epsilon^{IB} = \beta\epsilon_{strong}^{IB} + (1-\beta)\epsilon_{weak}<br />
\end{equation}<br />
where<br />
\begin{equation}<br />
\epsilon_{strong}^{IB} = \frac{1}{M}\sum\limits_{i=1}^M[H_I^t(x_i^t)-y_i^t]^2;\quad \epsilon_{weak} = \frac{1}{MN}\sum\limits_{i=1}^M\sum\limits_{\substack{1 \leq j \leq K \\ \alpha_j > 0}}[h(x_{ij}, \lambda_j)-y_i]^2.<br />
\end{equation}<br />
<br />
To learn the parameters of the IB-CNN, stochastic gradient descent is used, and the descent directions are obtained by differentiating the loss above with respect to the parameters $x_{ij}^t$ and $\lambda_j^t$ as follows:<br />
\begin{equation}<br />
\dfrac{\partial \epsilon^{IB}}{\partial x_{ij}^t} = \beta\dfrac{\partial \epsilon_{strong}^{IB}}{\partial H_I^t(x_i^t)}\dfrac{\partial H_I^t(x_i^t)}{\partial x_{ij}^t} + (1-\beta)\dfrac{\partial \epsilon_{weak}^{IB}}{\partial h^t(x_{ij}^t;\lambda_j^t)}\dfrac{\partial h^t(x_{ij}^t;\lambda_j^t)}{\partial x_{ij}^t} \\<br />
\dfrac{\partial \epsilon^{IB}}{\partial \lambda_j^t} = \beta\sum\limits_{i = 1}^M\dfrac{\partial \epsilon_{strong}^{IB}}{\partial H_I^t(x_i^t)}\dfrac{\partial H_I^t(x_i^t)}{\partial x_{ij}^t} + (1-\beta)\sum\limits_{i = 1}^M\dfrac{\partial \epsilon_{weak}^{IB}}{\partial h^t(x_{ij}^t;\lambda_j^t)}\dfrac{\partial h^t(x_{ij}^t;\lambda_j^t)}{\partial x_{ij}^t}<br />
\end{equation}<br />
where $\frac{\partial \epsilon^{IB}}{\partial x_{ij}^t}$ and $\frac{\partial \epsilon^{IB}}{\partial \lambda_j^t}$ only need to be calculated for the active neurons. Overall, the pseudocode for the incremental boosting algorithm for the IB-CNN is as follows.<br />
<br />
<br />
'''Input:''' The number of iterations (mini-batches) $T$ and activation features $X$ with the size of $M \times K$, where $M$ is the number of images in a mini-batch and $K$ is the dimension of the activation feature vector of one image.<br />
<br />
$\quad$ 1: '''for''' each input activation $j$ from $1$ to $K$ '''do'''<br />
<br />
$\quad$ 2: $\quad \alpha_j^1 = 0$<br />
<br />
$\quad$ 3: '''end for'''<br />
<br />
$\quad$ 4: '''for''' each mini-batch $t$ from $1$ to $T$ '''do'''<br />
<br />
$\quad$ 5: $\quad$ Feed-forward to the fully connected layer;<br />
<br />
$\quad$ 6: $\quad$ Select active features by boosting and calculate weights $\hat{\alpha}^t$ based on the standard AdaBoost;<br />
<br />
$\quad$ 7: $\quad$ Update the incremental strong classifier;<br />
<br />
$\quad$ 8: $\quad$ Calculate the overall loss of IB-CNN;<br />
<br />
$\quad$ 9: $\quad$ Backpropagate the loss based using the SGD equations;<br />
<br />
$\quad$ 10: $\quad$ Continue backpropagation to lower layers;<br />
<br />
$\quad$ 11: '''end for'''<br />
<br />
== Experiments ==<br />
<br />
<br />
Experiments have been conducted on four AU-coded databases whose details are as follows.<br />
<br />
1. CK database: contains 486 image sequences from 97 subjects, and 14 AUs.<br />
<br />
2. FERA2015 SEMAINE database: contains 31 subjects with 93,000 images and 6 AUs.<br />
<br />
3. FERA2015 BP4D database: contains 41 subjects with 146,847 images and 11 AUs.<br />
<br />
4. DISFA database: contains 27 subjects with 130,814 images and 12 AUs.<br />
<br />
=== Data Pre-Processing === <br />
The images are subjected to some preprocessing initially in order to scale and align the face regions, and to also remove out-of-plane rotations. Face regions are aligned based on three fiducial points: the centers of the two eyes and the mouth, and scaled to a size of 128 by 96. Face images are warped to a frontal view based on landmarks. A total of 23 facial landmarks that are less affected by facial movements are selected as control points to warp images. <br />
<br />
===IB-CNN Architecture===<br />
The IB-CNN architecture is as follows. The authors built the network by modifying the architecture of a CNN model that was originally used for CIFAR-10 dataset.<br />
The first two layers are convolutional layers having 32 filters with a size of 5 X 5 with a stride of 1. The activation maps then are sent to a rectified layer which is then followed by an average pooling layer with a stride of 3. This is followed again by a convolutional layer with 64 filters of size 5 X 5, the activation maps from which are sent to an FC layer with 128 nodes. The FC layer, in turn, feeds into the decision layer via the boosting mechanism.<br />
<br />
The following table compares the performance of the IB-CNN with other state-of-the-art methods that include CNN-based methods.<br />
<br />
[[File:table1_1.PNG|thumbnail]]<br />
<br />
Lastly, brief arguments are provided supporting the robustness of the IB-CNN to variations in: the slope parameter $\eta$, the number of input neurons, and the learning rate $\gamma$. A smaller $\eta$ simulates the sign function more accurately and thus strengthen the classifier $h(x_{ij}, \lambda_j)$, but will slow down the convergence of learning process, so there should be a balance between small and large $\eta$. In the paper we set $\eta = \sigma / 2$. As to learning rate $\gamma$, the author show by F1 score that IB-CNN is less sensitive than traditional CNN's in $\gamma$.<br />
<br />
== Conclusion ==<br />
To deal with the issue of relatively small-sized datasets in the domain of facial AU-recognition, the authors incorporate boosting and incremental learning to CNNs. Boosting helps the model to generalize well by preventing overfitting, and incremental learning exploits more information from the mini-batches but retaining more learned information. Moreover, the loss function is also changed so as to have more control over fine-tuning the model. The proposed IB-CNN shows improvement over the existent methods over four standard databases, showing a pronounced improvement in recognizing infrequent AUs.<br />
<br />
There are two immediate extensions to this work. Firstly, the IB-CNN may be applied to other problems wherein the data is limited. Secondly, the model may also be modified to go beyond binary classification and achieve multiclass boosted classification. Modern-day AdaBoost allows the weights $\alpha_i$, where i is the classifier, to be negative as well which gives the intuition that, "do exactly opposite of what this classifier says". In IB-CNN, the authors have restricted the weights of the input (FC units) to the boosting to be $\geq 0$. Considering that the IB-CNN is advantageous to models where datasets of a certain class are less in number, it would help the CNN to identify activation units in the FC layer which consistently contribute to the wrong prediction so that their consistency can be utilized in making correct predictions. From the AdaBoost classifier weight update, we conclude that the classifiers which should be considered to be inactive are those which cannot be relied upon i.e. those which contribute consistently nearly equal amount of correct and incorrect predictions.<br />
<br />
== References ==<br />
[1] Han, S., Meng, H., Khan, A. S., Tong, Y. (2016) "Incremental Boosting Convolutional Neural Network for Facial Action Unit Recognition". NIPS.<br />
<br />
[2] Tian, Y., Kanade, T., Cohn, J. F. (2001) "Recognizing Action Units for Facial Expression Analysis". IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 23., No. 2.<br />
<br />
[3] Facial Action Coding System - https://en.wikipedia.org/wiki/Facial_Action_Coding_System<br />
<br />
[4] YouTube Explanation of the Paper: https://www.youtube.com/watch?v=LZer5dW8h5c<br />
<br />
[5] Srivastava, N., Hinton, G. E., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R. (2014). Dropout: a simple way to prevent neural networks from overfitting. Journal of machine learning research, 15(1), 1929-1958.<br />
<br />
[6] Tian Y, Kanade T, Cohn JF. Recognizing Action Units for Facial Expression Analysis. IEEE transactions on pattern analysis and machine intelligence. 2001;23(2):97-115. doi:10.1109/34.908962.<br />
<br />
[7] Learning to Count with CNN Boosting. Elad Walach and Lior Wolf. The Blavatnik School of Computer Science. Tel Aviv University<br />
<br />
[8] A Brief Introduction to Boosting, Robert E. Schapire, Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence, 1999.<br />
<br />
[9] Tong, Y., Liao, W., & Ji, Q. (2007). Facial action unit recognition by exploiting their dynamic and semantic relationships. IEEE transactions on pattern analysis and machine intelligence, 29(10).</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Incremental_Boosting_Convolutional_Neural_Network_for_Facial_Action_Unit_Recognition&diff=28863Incremental Boosting Convolutional Neural Network for Facial Action Unit Recognition2017-10-31T04:13:15Z<p>H4lyu: /* IB-CNN Architecture */</p>
<hr />
<div>== Introduction ==<br />
<br />
Facial expression is one of the most natural ways that human beings express the emotions. The Facial Action Coding System (FACS) attempts to systemically categorize each facial expression by specifying a basic set of muscle contractions or relaxations, formally called Action Units (AUs). FACS as we know today consists of five major categories (i) main, (ii) head movement, (iii) eye movement, (iv) visibility and (v) gross behavior . For example, "AU 1" stands for the inner portion of the brows being raised, and "AU 6" stands for the cheeks being raised. Such a framework helps in describing any facial expression/emotion as the combination of different AUs. The complete list of AUs in the FACS can be seen [https://en.wikipedia.org/wiki/Facial_Action_Coding_System#List_of_Action_Units_and_Action_Descriptors_.28with_underlying_facial_muscles.29 here]. However, during the course of an average day, most human beings do not experience drastically varying emotions and therefore their facial expressions might change only subtly. Additionally, there might also be a lot of subjectivity involved if this task were to be done manually. To address these issues, it is imperative to automate this task. In addition to applications in human behavior analysis, automating AU recognition also has potential applications in human-computer interaction (HCI), online education, interactive gaming, among other domains.<br />
<br />
Because of the recent advancements in object detection and categorization tasks, CNNs are an appealing go-to for the facial AU recognition task described above. However, compared to the former, the training sets available for the AU recognition task are not very large, and therefore the learned CNNs suffer from overfitting. To overcome this problem, this work builds on the idea of integrating boosting within CNN. Boosting is a technique wherein multiple weak learners are combined together to yield a strong learner. Moreover, this work also modifies the mechanics of learning a CNN by breaking large datasets in mini-batches. Typically, in a batch strategy, each iteration uses a batch only to update the parameters and the features learned in that iteration are discarded for subsequent iterations. Herein the authors incorporate incremental learning by building a classifier that incrementally learns from all the iterations/batches. Hence the genesis of the name Incremental Boosting - Convolutional Neural Network (IB-CNN) is justified. The IB-CNN introduced here outperforms the state-of-the-art CNNs on four standard AU-coded databases.<br />
<br />
== Related Work ==<br />
<br />
There have been many papers that use manually designed features for the AU-recognition task, unlike the CNN-based methods which integrate feature learning. Some of them are Gabor Wavelets, Local Binary Patterns (LBP), Histogram of Oriented Gradients (HOG), Scale Invariant Feature Transform (SIFT) features, Discriminant Non-negative Matrix Factorization(DNMF), Histograms of Local Phase Quantization (LPQ), and their spatiotemporal extensions. Newer approaches include the one proposed by Tong, Liao and Ji [9], who use a dynamic Bayesian network (DBN) to model the relationships among different AUs. Automated methodologies include the Automatic Facial Analysis system which recognizes both permanent and transient AUs with high accuracy rates [6]. Hand-crafted feature descriptors like LBP as very powerful facial recognition algorithm, and these types of descriptors are very robust and do not require training - hence, there is no "overfitting" of data. LBP, in particular, provides a very dense and concise descriptor (i.e. a histogram feature) of facial features without the use of GPU; in essence, the feature vector for LBPs are small, yet powerful, and the accuracy of which is comparable to that of a trained CNN. One of the major breakthroughs to reduce overfitting in CNNs was the dropout, introduced by Hinton et al [5]. As dropout can be regarded a special bagging ensembling method over "different" networks that share same weights. This work can be seen as a refinement of the previous random dropout approach in the sense that only neurons with no contribution towards classification are dropped, while retaining the more discriminative neurons. Moreover, adopting a batch strategy with incremental learning also circumvents the issue of not-so-large datasets.<br />
<br />
== Methodology ==<br />
<br />
=== CNNs ===<br />
Convolutional Neural Networks are neural networks that contain at least one convolution layer. Typically, the convolution layer is accompanied with a pooling layer. A convolution layer, as the name suggests, convolves the input matrix (or tensor) with a filter (or kernel). This operation produces a feature map (or activation map) that, roughly speaking, shows the presence or absence of the filter in the image. Note that the parameters in the kernel are not preset and are learned. A pooling layer reduces the dimensionality of the data by summarizing the activation map by some means (such as max or average). Following this, the matrix (or tensor) data is then vectorized using a fully-connected (FC) layer. The neurons in the FC layer have connections to all the activations in the previous layer. Finally, there is a decision layer that has as many neurons as the number of classes. The decision layer decides the class based on computing a score function using the activations of the FC layer. Generally, an inner-product score function is used, which is replaced by the boosting score function in this work. <br />
<br />
=== Boosting Method ===<br />
Ever since boosting method was first officially introduced in the 1990s, it has become one of the most powerful learning method to improve model performance. The fundamental boosting algorithm that was introduced by Freund and Schapire in 1995: AdaBoost has evolved into many much more sophisticated boosting methods (Freund & Schapire, 1997). Alternative methods of CNN boosting for images ( in counting tasks) also involve Selective Sampling which has been proven to increase accuracy and reduce processing time [7]. Curse of dimensionality is a big problem in machine learning, where evaluating parameters can reduce the speed of classification, as well as the predictive power. The AdaBoost can only choose those parameters who improve the predictive power of the model, and reduce the dimensionality. The essential idea of boosting is to use a weighted combination of a sequence of weak classifiers to make a strong one. The key point to this process is after each iteration of data modification (through any type of model), the boosting algorithm will place more weight on observations that are difficult to classify correctly. Therefore, those “weak” classifiers can then focus on those observations that are hard to predict (Hastie, Tibshirani, & Friedman, 2017). These explanations can be summarized using the following formula:<br />
<center><br />
<math> G(x) = sign(\sum\limits_{l=1}^L \alpha_l G_l(x)) </math> <br />
Where <math>\alpha_l</math> are computed by boosting algorithm and <math>G_l(x) </math> denotes a weak classifier. <br />
</center><br />
<br />
=== Boosting CNN ===<br />
<br />
Let $X = [x_1, \dots, x_M]$ denote the activation features of a training data batch of size $M$, where each $x_i$ has $K$ activation features. Also, let $Y = [y_1, \dots , y_M]$ denote the corresponding labels, i.e. each $y_i \in \{-1, +1\}$. In other words, the vector $x_i$ contains all the activations of the FC layer, which is typically multiplied with the weights corresponding to the connections between the FC layer and the decision layer. However, as mentioned earlier, this work achieves that same task of scoring via boosting as opposed to computing the inner product. Denoting the weak classifiers by $h(\cdot)$, we obtain the strong classifier as:<br />
\begin{equation}<br />
\label{boosting}<br />
H(x_i) = \sum\limits_{j = 1}^K \alpha_j h(x_{ij}; \lambda_j)<br />
\end{equation}<br />
where $x_{ij}$ is the $j^{th}$ activation feature calculated on the $i^{th}$ image. Moreover, $\alpha_j \geq 0$ denotes the weight of the $j^{th}$ weak classifier and $\sum\limits_{i=1}^K\alpha_i = 1$. Here we define $h(\cdot)$ to be a differentiable surrogate for the sign function as:<br />
\begin{equation}<br />
h(x_{ij}; \lambda_j) = \dfrac {f(x_{ij}; \lambda_j)}{\sqrt{f(x_{ij}; \lambda_j)^2 + \eta^2}}<br />
\end{equation}<br />
where $f(x_{ij}; \lambda_j)$ denotes a decision-stump (one-level decision tree) with $\lambda_j$ being the threshold. The parameter $\eta$ controls the slope of the function $\dfrac {f(\cdot)}{\sqrt{f(\cdot)^2 + \eta^2}}$. The authors set this parameter according to the distribution of $f(\cdot)$ as $\eta = \frac{\sigma}{c}$, where $\sigma$ is the standard deviation of $f(\cdot)$ and $c$ is a constant. This parameter is experimented with extensively in Section 4.4 of the paper. It is shown that $\eta$ can drastically influence the $F1$ score of the model.<br />
Note that for a certain strong classifier $H$ if a certain $\alpha_i = 0$, then the activation feature $x_i$ has no contribution to the output of the classifier $H$. In other words, the corresponding neuron can be considered to be inactive. (Refer to the Figure for the schematic diagram of B-CNN.)<br />
<br />
[[File:ibcnn.png]]<br />
<br />
=== Incremental Boosting ===<br />
In vanilla B-CNN, the information learned in a certain batch, i.e. the weights and the thresholds of the active neurons is discarded for subsequent batches. To address this issue, in this work the idea of incremental learning is incorporated. Formally, the incremental strong classifier at the $t^{th}$ iteration, $H_I^t$, is given as:<br />
\begin{equation}<br />
\label{incr}<br />
H_I^t(x_i^t) = \frac{(t-1)H_I^{t-1}(x_i^t) + H^t(x_i^t)}{t}<br />
\end{equation}<br />
where $H_I^{t-1}$ is the incremental strong classifier obtained at the $(t-1)^{th}$ iteration and $H^t$ is the boosted strong classifier at the $t^{th}$ iteration. Manipulating the equations, we obtain:<br />
\begin{equation}<br />
H_I^t(x_i^t) = \sum\limits_{j=1}^K \alpha_j^t h^t(x_{ij}; \lambda_j^t);\quad \alpha_j^t = \frac{(t-1)\alpha_j^{t-1}+\hat{\alpha}_j^t}{t}<br />
\end{equation}<br />
where $\hat{\alpha}_j^t$ is the weak classifier weight calculated in the $t^{th}$ iteration by boosting and $\alpha_j^t$ is the cumulative weight considering previous iterations. (Refer to the Figure for a schematic diagram of IB-CNN.) <br />
Typically, boosting algorithms minimize an objective function that captures the loss of the strong classifier. However, the loss of the strong classifier may be dominated by some weak classifiers with large weights, which might lead to overfitting. Therefore, to exercise better control, the loss function at a certain iteration $t$, $\epsilon^{IB}$ is expressed as a convex combination of the loss of the incremental strong classifier at iteration $t$ and the loss of the weak classifiers determined at iteration $t$. That is,<br />
\begin{equation}<br />
\epsilon^{IB} = \beta\epsilon_{strong}^{IB} + (1-\beta)\epsilon_{weak}<br />
\end{equation}<br />
where<br />
\begin{equation}<br />
\epsilon_{strong}^{IB} = \frac{1}{M}\sum\limits_{i=1}^M[H_I^t(x_i^t)-y_i^t]^2;\quad \epsilon_{weak} = \frac{1}{MN}\sum\limits_{i=1}^M\sum\limits_{\substack{1 \leq j \leq K \\ \alpha_j > 0}}[h(x_{ij}, \lambda_j)-y_i]^2.<br />
\end{equation}<br />
<br />
To learn the parameters of the IB-CNN, stochastic gradient descent is used, and the descent directions are obtained by differentiating the loss above with respect to the parameters $x_{ij}^t$ and $\lambda_j^t$ as follows:<br />
\begin{equation}<br />
\dfrac{\partial \epsilon^{IB}}{\partial x_{ij}^t} = \beta\dfrac{\partial \epsilon_{strong}^{IB}}{\partial H_I^t(x_i^t)}\dfrac{\partial H_I^t(x_i^t)}{\partial x_{ij}^t} + (1-\beta)\dfrac{\partial \epsilon_{weak}^{IB}}{\partial h^t(x_{ij}^t;\lambda_j^t)}\dfrac{\partial h^t(x_{ij}^t;\lambda_j^t)}{\partial x_{ij}^t} \\<br />
\dfrac{\partial \epsilon^{IB}}{\partial \lambda_j^t} = \beta\sum\limits_{i = 1}^M\dfrac{\partial \epsilon_{strong}^{IB}}{\partial H_I^t(x_i^t)}\dfrac{\partial H_I^t(x_i^t)}{\partial x_{ij}^t} + (1-\beta)\sum\limits_{i = 1}^M\dfrac{\partial \epsilon_{weak}^{IB}}{\partial h^t(x_{ij}^t;\lambda_j^t)}\dfrac{\partial h^t(x_{ij}^t;\lambda_j^t)}{\partial x_{ij}^t}<br />
\end{equation}<br />
where $\frac{\partial \epsilon^{IB}}{\partial x_{ij}^t}$ and $\frac{\partial \epsilon^{IB}}{\partial \lambda_j^t}$ only need to be calculated for the active neurons. Overall, the pseudocode for the incremental boosting algorithm for the IB-CNN is as follows.<br />
<br />
<br />
'''Input:''' The number of iterations (mini-batches) $T$ and activation features $X$ with the size of $M \times K$, where $M$ is the number of images in a mini-batch and $K$ is the dimension of the activation feature vector of one image.<br />
<br />
$\quad$ 1: '''for''' each input activation $j$ from $1$ to $K$ '''do'''<br />
<br />
$\quad$ 2: $\quad \alpha_j^1 = 0$<br />
<br />
$\quad$ 3: '''end for'''<br />
<br />
$\quad$ 4: '''for''' each mini-batch $t$ from $1$ to $T$ '''do'''<br />
<br />
$\quad$ 5: $\quad$ Feed-forward to the fully connected layer;<br />
<br />
$\quad$ 6: $\quad$ Select active features by boosting and calculate weights $\hat{\alpha}^t$ based on the standard AdaBoost;<br />
<br />
$\quad$ 7: $\quad$ Update the incremental strong classifier;<br />
<br />
$\quad$ 8: $\quad$ Calculate the overall loss of IB-CNN;<br />
<br />
$\quad$ 9: $\quad$ Backpropagate the loss based using the SGD equations;<br />
<br />
$\quad$ 10: $\quad$ Continue backpropagation to lower layers;<br />
<br />
$\quad$ 11: '''end for'''<br />
<br />
== Experiments ==<br />
<br />
<br />
Experiments have been conducted on four AU-coded databases whose details are as follows.<br />
<br />
1. CK database: contains 486 image sequences from 97 subjects, and 14 AUs.<br />
<br />
2. FERA2015 SEMAINE database: contains 31 subjects with 93,000 images and 6 AUs.<br />
<br />
3. FERA2015 BP4D database: contains 41 subjects with 146,847 images and 11 AUs.<br />
<br />
4. DISFA database: contains 27 subjects with 130,814 images and 12 AUs.<br />
<br />
=== Data Pre-Processing === <br />
The images are subjected to some preprocessing initially in order to scale and align the face regions, and to also remove out-of-plane rotations. Face regions are aligned based on three fiducial points: the centers of the two eyes and the mouth, and scaled to a size of 128 by 96. Face images are warped to a frontal view based on landmarks. A total of 23 facial landmarks that are less affected by facial movements are selected as control points to warp images. <br />
<br />
===IB-CNN Architecture===<br />
The IB-CNN architecture is as follows. The authors built the network by modifying the architecture of a CNN model that was originally used for CIFAR-10 dataset.<br />
The first two layers are convolutional layers having 32 filters with a size of 5 X 5 with a stride of 1. The activation maps then are sent to a rectified layer which is then followed by an average pooling layer with a stride of 3. This is followed again by a convolutional layer with 64 filters of size 5 X 5, the activation maps from which are sent to an FC layer with 128 nodes. The FC layer, in turn, feeds into the decision layer via the boosting mechanism.<br />
<br />
The following table compares the performance of the IB-CNN with other state-of-the-art methods that include CNN-based methods.<br />
<br />
[[File:table1_1.PNG|thumbnail]]<br />
<br />
Lastly, brief arguments are provided supporting the robustness of the IB-CNN to variations in: the slope parameter $\eta$, the number of input neurons, and the learning rate $\gamma$. A smaller $\eta$ simulates the sign function more accurately and thus strengthen the classifier $h(x_{ij}, \lambda_j)$, but will slow down the convergence of learning process, so there should be a balance between small and large $\eta$. In the paper we set $\eta = \sigma / 2$. As to learning rate $\gamma$, the author show by F1 score that IB-CNN is less sensitive than traditional CNN's in $\gamma$.<br />
<br />
== Conclusion ==<br />
To deal with the issue of relatively small-sized datasets in the domain of facial AU-recognition, the authors incorporate boosting and incremental learning to CNNs. Boosting helps the model to generalize well by preventing overfitting, and incremental learning exploits more information from the mini-batches but retaining more learned information. Moreover, the loss function is also changed so as to have more control over fine-tuning the model. The proposed IB-CNN shows improvement over the existent methods over four standard databases, showing a pronounced improvement in recognizing infrequent AUs.<br />
<br />
There are two immediate extensions to this work. Firstly, the IB-CNN may be applied to other problems wherein the data is limited. Secondly, the model may also be modified to go beyond binary classification and achieve multiclass boosted classification. Modern-day AdaBoost allows the weights $\alpha_i$, where i is the classifier, to be negative as well which gives the intuition that, "do exactly opposite of what this classifier says". In IB-CNN, the authors have restricted the weights of the input (FC units) to the boosting to be $\geq 0$. Considering that the IB-CNN is advantageous to models where datasets of a certain class are less in number, it would help the CNN to identify activation units in the FC layer which consistently contribute to the wrong prediction so that their consistency can be utilized in making correct predictions. From the AdaBoost classifier weight update, we conclude that the classifiers which should be considered to be inactive are those which cannot be relied upon i.e. those which contribute consistently nearly equal amount of correct and incorrect predictions.<br />
<br />
== References ==<br />
1. Han, S., Meng, H., Khan, A. S., Tong, Y. (2016) "Incremental Boosting Convolutional Neural Network for Facial Action Unit Recognition". NIPS.<br />
<br />
2. Tian, Y., Kanade, T., Cohn, J. F. (2001) "Recognizing Action Units for Facial Expression Analysis". IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 23., No. 2.<br />
<br />
3. Facial Action Coding System - https://en.wikipedia.org/wiki/Facial_Action_Coding_System<br />
<br />
4. YouTube Explanation of the Paper: https://www.youtube.com/watch?v=LZer5dW8h5c<br />
<br />
5. Srivastava, N., Hinton, G. E., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R. (2014). Dropout: a simple way to prevent neural networks from overfitting. Journal of machine learning research, 15(1), 1929-1958.<br />
<br />
6. Tian Y, Kanade T, Cohn JF. Recognizing Action Units for Facial Expression Analysis. IEEE transactions on pattern analysis and machine intelligence. 2001;23(2):97-115. doi:10.1109/34.908962.<br />
<br />
7. Learning to Count with CNN Boosting. Elad Walach and Lior Wolf. The Blavatnik School of Computer Science. Tel Aviv University<br />
<br />
8. A Brief Introduction to Boosting, Robert E. Schapire, Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence, 1999.<br />
<br />
9. Tong, Y., Liao, W., & Ji, Q. (2007). Facial action unit recognition by exploiting their dynamic and semantic relationships. IEEE transactions on pattern analysis and machine intelligence, 29(10).</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Incremental_Boosting_Convolutional_Neural_Network_for_Facial_Action_Unit_Recognition&diff=28862Incremental Boosting Convolutional Neural Network for Facial Action Unit Recognition2017-10-31T04:11:40Z<p>H4lyu: /* IB-CNN Architecture */</p>
<hr />
<div>== Introduction ==<br />
<br />
Facial expression is one of the most natural ways that human beings express the emotions. The Facial Action Coding System (FACS) attempts to systemically categorize each facial expression by specifying a basic set of muscle contractions or relaxations, formally called Action Units (AUs). FACS as we know today consists of five major categories (i) main, (ii) head movement, (iii) eye movement, (iv) visibility and (v) gross behavior . For example, "AU 1" stands for the inner portion of the brows being raised, and "AU 6" stands for the cheeks being raised. Such a framework helps in describing any facial expression/emotion as the combination of different AUs. The complete list of AUs in the FACS can be seen [https://en.wikipedia.org/wiki/Facial_Action_Coding_System#List_of_Action_Units_and_Action_Descriptors_.28with_underlying_facial_muscles.29 here]. However, during the course of an average day, most human beings do not experience drastically varying emotions and therefore their facial expressions might change only subtly. Additionally, there might also be a lot of subjectivity involved if this task were to be done manually. To address these issues, it is imperative to automate this task. In addition to applications in human behavior analysis, automating AU recognition also has potential applications in human-computer interaction (HCI), online education, interactive gaming, among other domains.<br />
<br />
Because of the recent advancements in object detection and categorization tasks, CNNs are an appealing go-to for the facial AU recognition task described above. However, compared to the former, the training sets available for the AU recognition task are not very large, and therefore the learned CNNs suffer from overfitting. To overcome this problem, this work builds on the idea of integrating boosting within CNN. Boosting is a technique wherein multiple weak learners are combined together to yield a strong learner. Moreover, this work also modifies the mechanics of learning a CNN by breaking large datasets in mini-batches. Typically, in a batch strategy, each iteration uses a batch only to update the parameters and the features learned in that iteration are discarded for subsequent iterations. Herein the authors incorporate incremental learning by building a classifier that incrementally learns from all the iterations/batches. Hence the genesis of the name Incremental Boosting - Convolutional Neural Network (IB-CNN) is justified. The IB-CNN introduced here outperforms the state-of-the-art CNNs on four standard AU-coded databases.<br />
<br />
== Related Work ==<br />
<br />
There have been many papers that use manually designed features for the AU-recognition task, unlike the CNN-based methods which integrate feature learning. Some of them are Gabor Wavelets, Local Binary Patterns (LBP), Histogram of Oriented Gradients (HOG), Scale Invariant Feature Transform (SIFT) features, Discriminant Non-negative Matrix Factorization(DNMF), Histograms of Local Phase Quantization (LPQ), and their spatiotemporal extensions. Newer approaches include the one proposed by Tong, Liao and Ji [9], who use a dynamic Bayesian network (DBN) to model the relationships among different AUs. Automated methodologies include the Automatic Facial Analysis system which recognizes both permanent and transient AUs with high accuracy rates [6]. Hand-crafted feature descriptors like LBP as very powerful facial recognition algorithm, and these types of descriptors are very robust and do not require training - hence, there is no "overfitting" of data. LBP, in particular, provides a very dense and concise descriptor (i.e. a histogram feature) of facial features without the use of GPU; in essence, the feature vector for LBPs are small, yet powerful, and the accuracy of which is comparable to that of a trained CNN. One of the major breakthroughs to reduce overfitting in CNNs was the dropout, introduced by Hinton et al [5]. As dropout can be regarded a special bagging ensembling method over "different" networks that share same weights. This work can be seen as a refinement of the previous random dropout approach in the sense that only neurons with no contribution towards classification are dropped, while retaining the more discriminative neurons. Moreover, adopting a batch strategy with incremental learning also circumvents the issue of not-so-large datasets.<br />
<br />
== Methodology ==<br />
<br />
=== CNNs ===<br />
Convolutional Neural Networks are neural networks that contain at least one convolution layer. Typically, the convolution layer is accompanied with a pooling layer. A convolution layer, as the name suggests, convolves the input matrix (or tensor) with a filter (or kernel). This operation produces a feature map (or activation map) that, roughly speaking, shows the presence or absence of the filter in the image. Note that the parameters in the kernel are not preset and are learned. A pooling layer reduces the dimensionality of the data by summarizing the activation map by some means (such as max or average). Following this, the matrix (or tensor) data is then vectorized using a fully-connected (FC) layer. The neurons in the FC layer have connections to all the activations in the previous layer. Finally, there is a decision layer that has as many neurons as the number of classes. The decision layer decides the class based on computing a score function using the activations of the FC layer. Generally, an inner-product score function is used, which is replaced by the boosting score function in this work. <br />
<br />
=== Boosting Method ===<br />
Ever since boosting method was first officially introduced in the 1990s, it has become one of the most powerful learning method to improve model performance. The fundamental boosting algorithm that was introduced by Freund and Schapire in 1995: AdaBoost has evolved into many much more sophisticated boosting methods (Freund & Schapire, 1997). Alternative methods of CNN boosting for images ( in counting tasks) also involve Selective Sampling which has been proven to increase accuracy and reduce processing time [7]. Curse of dimensionality is a big problem in machine learning, where evaluating parameters can reduce the speed of classification, as well as the predictive power. The AdaBoost can only choose those parameters who improve the predictive power of the model, and reduce the dimensionality. The essential idea of boosting is to use a weighted combination of a sequence of weak classifiers to make a strong one. The key point to this process is after each iteration of data modification (through any type of model), the boosting algorithm will place more weight on observations that are difficult to classify correctly. Therefore, those “weak” classifiers can then focus on those observations that are hard to predict (Hastie, Tibshirani, & Friedman, 2017). These explanations can be summarized using the following formula:<br />
<center><br />
<math> G(x) = sign(\sum\limits_{l=1}^L \alpha_l G_l(x)) </math> <br />
Where <math>\alpha_l</math> are computed by boosting algorithm and <math>G_l(x) </math> denotes a weak classifier. <br />
</center><br />
<br />
=== Boosting CNN ===<br />
<br />
Let $X = [x_1, \dots, x_M]$ denote the activation features of a training data batch of size $M$, where each $x_i$ has $K$ activation features. Also, let $Y = [y_1, \dots , y_M]$ denote the corresponding labels, i.e. each $y_i \in \{-1, +1\}$. In other words, the vector $x_i$ contains all the activations of the FC layer, which is typically multiplied with the weights corresponding to the connections between the FC layer and the decision layer. However, as mentioned earlier, this work achieves that same task of scoring via boosting as opposed to computing the inner product. Denoting the weak classifiers by $h(\cdot)$, we obtain the strong classifier as:<br />
\begin{equation}<br />
\label{boosting}<br />
H(x_i) = \sum\limits_{j = 1}^K \alpha_j h(x_{ij}; \lambda_j)<br />
\end{equation}<br />
where $x_{ij}$ is the $j^{th}$ activation feature calculated on the $i^{th}$ image. Moreover, $\alpha_j \geq 0$ denotes the weight of the $j^{th}$ weak classifier and $\sum\limits_{i=1}^K\alpha_i = 1$. Here we define $h(\cdot)$ to be a differentiable surrogate for the sign function as:<br />
\begin{equation}<br />
h(x_{ij}; \lambda_j) = \dfrac {f(x_{ij}; \lambda_j)}{\sqrt{f(x_{ij}; \lambda_j)^2 + \eta^2}}<br />
\end{equation}<br />
where $f(x_{ij}; \lambda_j)$ denotes a decision-stump (one-level decision tree) with $\lambda_j$ being the threshold. The parameter $\eta$ controls the slope of the function $\dfrac {f(\cdot)}{\sqrt{f(\cdot)^2 + \eta^2}}$. The authors set this parameter according to the distribution of $f(\cdot)$ as $\eta = \frac{\sigma}{c}$, where $\sigma$ is the standard deviation of $f(\cdot)$ and $c$ is a constant. This parameter is experimented with extensively in Section 4.4 of the paper. It is shown that $\eta$ can drastically influence the $F1$ score of the model.<br />
Note that for a certain strong classifier $H$ if a certain $\alpha_i = 0$, then the activation feature $x_i$ has no contribution to the output of the classifier $H$. In other words, the corresponding neuron can be considered to be inactive. (Refer to the Figure for the schematic diagram of B-CNN.)<br />
<br />
[[File:ibcnn.png]]<br />
<br />
=== Incremental Boosting ===<br />
In vanilla B-CNN, the information learned in a certain batch, i.e. the weights and the thresholds of the active neurons is discarded for subsequent batches. To address this issue, in this work the idea of incremental learning is incorporated. Formally, the incremental strong classifier at the $t^{th}$ iteration, $H_I^t$, is given as:<br />
\begin{equation}<br />
\label{incr}<br />
H_I^t(x_i^t) = \frac{(t-1)H_I^{t-1}(x_i^t) + H^t(x_i^t)}{t}<br />
\end{equation}<br />
where $H_I^{t-1}$ is the incremental strong classifier obtained at the $(t-1)^{th}$ iteration and $H^t$ is the boosted strong classifier at the $t^{th}$ iteration. Manipulating the equations, we obtain:<br />
\begin{equation}<br />
H_I^t(x_i^t) = \sum\limits_{j=1}^K \alpha_j^t h^t(x_{ij}; \lambda_j^t);\quad \alpha_j^t = \frac{(t-1)\alpha_j^{t-1}+\hat{\alpha}_j^t}{t}<br />
\end{equation}<br />
where $\hat{\alpha}_j^t$ is the weak classifier weight calculated in the $t^{th}$ iteration by boosting and $\alpha_j^t$ is the cumulative weight considering previous iterations. (Refer to the Figure for a schematic diagram of IB-CNN.) <br />
Typically, boosting algorithms minimize an objective function that captures the loss of the strong classifier. However, the loss of the strong classifier may be dominated by some weak classifiers with large weights, which might lead to overfitting. Therefore, to exercise better control, the loss function at a certain iteration $t$, $\epsilon^{IB}$ is expressed as a convex combination of the loss of the incremental strong classifier at iteration $t$ and the loss of the weak classifiers determined at iteration $t$. That is,<br />
\begin{equation}<br />
\epsilon^{IB} = \beta\epsilon_{strong}^{IB} + (1-\beta)\epsilon_{weak}<br />
\end{equation}<br />
where<br />
\begin{equation}<br />
\epsilon_{strong}^{IB} = \frac{1}{M}\sum\limits_{i=1}^M[H_I^t(x_i^t)-y_i^t]^2;\quad \epsilon_{weak} = \frac{1}{MN}\sum\limits_{i=1}^M\sum\limits_{\substack{1 \leq j \leq K \\ \alpha_j > 0}}[h(x_{ij}, \lambda_j)-y_i]^2.<br />
\end{equation}<br />
<br />
To learn the parameters of the IB-CNN, stochastic gradient descent is used, and the descent directions are obtained by differentiating the loss above with respect to the parameters $x_{ij}^t$ and $\lambda_j^t$ as follows:<br />
\begin{equation}<br />
\dfrac{\partial \epsilon^{IB}}{\partial x_{ij}^t} = \beta\dfrac{\partial \epsilon_{strong}^{IB}}{\partial H_I^t(x_i^t)}\dfrac{\partial H_I^t(x_i^t)}{\partial x_{ij}^t} + (1-\beta)\dfrac{\partial \epsilon_{weak}^{IB}}{\partial h^t(x_{ij}^t;\lambda_j^t)}\dfrac{\partial h^t(x_{ij}^t;\lambda_j^t)}{\partial x_{ij}^t} \\<br />
\dfrac{\partial \epsilon^{IB}}{\partial \lambda_j^t} = \beta\sum\limits_{i = 1}^M\dfrac{\partial \epsilon_{strong}^{IB}}{\partial H_I^t(x_i^t)}\dfrac{\partial H_I^t(x_i^t)}{\partial x_{ij}^t} + (1-\beta)\sum\limits_{i = 1}^M\dfrac{\partial \epsilon_{weak}^{IB}}{\partial h^t(x_{ij}^t;\lambda_j^t)}\dfrac{\partial h^t(x_{ij}^t;\lambda_j^t)}{\partial x_{ij}^t}<br />
\end{equation}<br />
where $\frac{\partial \epsilon^{IB}}{\partial x_{ij}^t}$ and $\frac{\partial \epsilon^{IB}}{\partial \lambda_j^t}$ only need to be calculated for the active neurons. Overall, the pseudocode for the incremental boosting algorithm for the IB-CNN is as follows.<br />
<br />
<br />
'''Input:''' The number of iterations (mini-batches) $T$ and activation features $X$ with the size of $M \times K$, where $M$ is the number of images in a mini-batch and $K$ is the dimension of the activation feature vector of one image.<br />
<br />
$\quad$ 1: '''for''' each input activation $j$ from $1$ to $K$ '''do'''<br />
<br />
$\quad$ 2: $\quad \alpha_j^1 = 0$<br />
<br />
$\quad$ 3: '''end for'''<br />
<br />
$\quad$ 4: '''for''' each mini-batch $t$ from $1$ to $T$ '''do'''<br />
<br />
$\quad$ 5: $\quad$ Feed-forward to the fully connected layer;<br />
<br />
$\quad$ 6: $\quad$ Select active features by boosting and calculate weights $\hat{\alpha}^t$ based on the standard AdaBoost;<br />
<br />
$\quad$ 7: $\quad$ Update the incremental strong classifier;<br />
<br />
$\quad$ 8: $\quad$ Calculate the overall loss of IB-CNN;<br />
<br />
$\quad$ 9: $\quad$ Backpropagate the loss based using the SGD equations;<br />
<br />
$\quad$ 10: $\quad$ Continue backpropagation to lower layers;<br />
<br />
$\quad$ 11: '''end for'''<br />
<br />
== Experiments ==<br />
<br />
<br />
Experiments have been conducted on four AU-coded databases whose details are as follows.<br />
<br />
1. CK database: contains 486 image sequences from 97 subjects, and 14 AUs.<br />
<br />
2. FERA2015 SEMAINE database: contains 31 subjects with 93,000 images and 6 AUs.<br />
<br />
3. FERA2015 BP4D database: contains 41 subjects with 146,847 images and 11 AUs.<br />
<br />
4. DISFA database: contains 27 subjects with 130,814 images and 12 AUs.<br />
<br />
=== Data Pre-Processing === <br />
The images are subjected to some preprocessing initially in order to scale and align the face regions, and to also remove out-of-plane rotations. Face regions are aligned based on three fiducial points: the centers of the two eyes and the mouth, and scaled to a size of 128 by 96. Face images are warped to a frontal view based on landmarks. A total of 23 facial landmarks that are less affected by facial movements are selected as control points to warp images. <br />
<br />
===IB-CNN Architecture===<br />
The IB-CNN architecture is as follows. The authors built the network by modifying the architecture of a CNN model that was originally used for CIFAR-10 dataset.<br />
The first two layers are convolutional layers having 32 filters with a size of 5 X 5 with a stride of 1. The activation maps then are sent to a rectified layer which is then followed by an average pooling layer with a stride of 3. This is followed again by a convolutional layer with 64 filters of size 5 X 5, the activation maps from which are sent to an FC layer with 128 nodes. The FC layer, in turn, feeds into the decision layer via the boosting mechanism.<br />
<br />
The following table compares the performance of the IB-CNN with other state-of-the-art methods that include CNN-based methods.<br />
<br />
[[File:table1_1.PNG|thumbnail]]<br />
<br />
Lastly, brief arguments are provided supporting the robustness of the IB-CNN to variations in: the slope parameter $\eta$, the number of input neurons, and the learning rate $\gamma$. A smaller $\eta$ simulates the sign function more accurately and thus strengthen the classifier $h(x_{ij}, \lambda_j)$, but will slow down the learning process. As to learning rate $\gamma$, the author show by F1 score that IB-CNN is less sensitive than traditional CNN's in $\gamma$.<br />
<br />
== Conclusion ==<br />
To deal with the issue of relatively small-sized datasets in the domain of facial AU-recognition, the authors incorporate boosting and incremental learning to CNNs. Boosting helps the model to generalize well by preventing overfitting, and incremental learning exploits more information from the mini-batches but retaining more learned information. Moreover, the loss function is also changed so as to have more control over fine-tuning the model. The proposed IB-CNN shows improvement over the existent methods over four standard databases, showing a pronounced improvement in recognizing infrequent AUs.<br />
<br />
There are two immediate extensions to this work. Firstly, the IB-CNN may be applied to other problems wherein the data is limited. Secondly, the model may also be modified to go beyond binary classification and achieve multiclass boosted classification. Modern-day AdaBoost allows the weights $\alpha_i$, where i is the classifier, to be negative as well which gives the intuition that, "do exactly opposite of what this classifier says". In IB-CNN, the authors have restricted the weights of the input (FC units) to the boosting to be $\geq 0$. Considering that the IB-CNN is advantageous to models where datasets of a certain class are less in number, it would help the CNN to identify activation units in the FC layer which consistently contribute to the wrong prediction so that their consistency can be utilized in making correct predictions. From the AdaBoost classifier weight update, we conclude that the classifiers which should be considered to be inactive are those which cannot be relied upon i.e. those which contribute consistently nearly equal amount of correct and incorrect predictions.<br />
<br />
== References ==<br />
1. Han, S., Meng, H., Khan, A. S., Tong, Y. (2016) "Incremental Boosting Convolutional Neural Network for Facial Action Unit Recognition". NIPS.<br />
<br />
2. Tian, Y., Kanade, T., Cohn, J. F. (2001) "Recognizing Action Units for Facial Expression Analysis". IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 23., No. 2.<br />
<br />
3. Facial Action Coding System - https://en.wikipedia.org/wiki/Facial_Action_Coding_System<br />
<br />
4. YouTube Explanation of the Paper: https://www.youtube.com/watch?v=LZer5dW8h5c<br />
<br />
5. Srivastava, N., Hinton, G. E., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R. (2014). Dropout: a simple way to prevent neural networks from overfitting. Journal of machine learning research, 15(1), 1929-1958.<br />
<br />
6. Tian Y, Kanade T, Cohn JF. Recognizing Action Units for Facial Expression Analysis. IEEE transactions on pattern analysis and machine intelligence. 2001;23(2):97-115. doi:10.1109/34.908962.<br />
<br />
7. Learning to Count with CNN Boosting. Elad Walach and Lior Wolf. The Blavatnik School of Computer Science. Tel Aviv University<br />
<br />
8. A Brief Introduction to Boosting, Robert E. Schapire, Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence, 1999.<br />
<br />
9. Tong, Y., Liao, W., & Ji, Q. (2007). Facial action unit recognition by exploiting their dynamic and semantic relationships. IEEE transactions on pattern analysis and machine intelligence, 29(10).</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Incremental_Boosting_Convolutional_Neural_Network_for_Facial_Action_Unit_Recognition&diff=28861Incremental Boosting Convolutional Neural Network for Facial Action Unit Recognition2017-10-31T04:01:52Z<p>H4lyu: /* References */</p>
<hr />
<div>== Introduction ==<br />
<br />
Facial expression is one of the most natural ways that human beings express the emotions. The Facial Action Coding System (FACS) attempts to systemically categorize each facial expression by specifying a basic set of muscle contractions or relaxations, formally called Action Units (AUs). FACS as we know today consists of five major categories (i) main, (ii) head movement, (iii) eye movement, (iv) visibility and (v) gross behavior . For example, "AU 1" stands for the inner portion of the brows being raised, and "AU 6" stands for the cheeks being raised. Such a framework helps in describing any facial expression/emotion as the combination of different AUs. The complete list of AUs in the FACS can be seen [https://en.wikipedia.org/wiki/Facial_Action_Coding_System#List_of_Action_Units_and_Action_Descriptors_.28with_underlying_facial_muscles.29 here]. However, during the course of an average day, most human beings do not experience drastically varying emotions and therefore their facial expressions might change only subtly. Additionally, there might also be a lot of subjectivity involved if this task were to be done manually. To address these issues, it is imperative to automate this task. In addition to applications in human behavior analysis, automating AU recognition also has potential applications in human-computer interaction (HCI), online education, interactive gaming, among other domains.<br />
<br />
Because of the recent advancements in object detection and categorization tasks, CNNs are an appealing go-to for the facial AU recognition task described above. However, compared to the former, the training sets available for the AU recognition task are not very large, and therefore the learned CNNs suffer from overfitting. To overcome this problem, this work builds on the idea of integrating boosting within CNN. Boosting is a technique wherein multiple weak learners are combined together to yield a strong learner. Moreover, this work also modifies the mechanics of learning a CNN by breaking large datasets in mini-batches. Typically, in a batch strategy, each iteration uses a batch only to update the parameters and the features learned in that iteration are discarded for subsequent iterations. Herein the authors incorporate incremental learning by building a classifier that incrementally learns from all the iterations/batches. Hence the genesis of the name Incremental Boosting - Convolutional Neural Network (IB-CNN) is justified. The IB-CNN introduced here outperforms the state-of-the-art CNNs on four standard AU-coded databases.<br />
<br />
== Related Work ==<br />
<br />
There have been many papers that use manually designed features for the AU-recognition task, unlike the CNN-based methods which integrate feature learning. Some of them are Gabor Wavelets, Local Binary Patterns (LBP), Histogram of Oriented Gradients (HOG), Scale Invariant Feature Transform (SIFT) features, Discriminant Non-negative Matrix Factorization(DNMF), Histograms of Local Phase Quantization (LPQ), and their spatiotemporal extensions. Newer approaches include the one proposed by Tong, Liao and Ji [9], who use a dynamic Bayesian network (DBN) to model the relationships among different AUs. Automated methodologies include the Automatic Facial Analysis system which recognizes both permanent and transient AUs with high accuracy rates [6]. Hand-crafted feature descriptors like LBP as very powerful facial recognition algorithm, and these types of descriptors are very robust and do not require training - hence, there is no "overfitting" of data. LBP, in particular, provides a very dense and concise descriptor (i.e. a histogram feature) of facial features without the use of GPU; in essence, the feature vector for LBPs are small, yet powerful, and the accuracy of which is comparable to that of a trained CNN. One of the major breakthroughs to reduce overfitting in CNNs was the dropout, introduced by Hinton et al [5]. As dropout can be regarded a special bagging ensembling method over "different" networks that share same weights. This work can be seen as a refinement of the previous random dropout approach in the sense that only neurons with no contribution towards classification are dropped, while retaining the more discriminative neurons. Moreover, adopting a batch strategy with incremental learning also circumvents the issue of not-so-large datasets.<br />
<br />
== Methodology ==<br />
<br />
=== CNNs ===<br />
Convolutional Neural Networks are neural networks that contain at least one convolution layer. Typically, the convolution layer is accompanied with a pooling layer. A convolution layer, as the name suggests, convolves the input matrix (or tensor) with a filter (or kernel). This operation produces a feature map (or activation map) that, roughly speaking, shows the presence or absence of the filter in the image. Note that the parameters in the kernel are not preset and are learned. A pooling layer reduces the dimensionality of the data by summarizing the activation map by some means (such as max or average). Following this, the matrix (or tensor) data is then vectorized using a fully-connected (FC) layer. The neurons in the FC layer have connections to all the activations in the previous layer. Finally, there is a decision layer that has as many neurons as the number of classes. The decision layer decides the class based on computing a score function using the activations of the FC layer. Generally, an inner-product score function is used, which is replaced by the boosting score function in this work. <br />
<br />
=== Boosting Method ===<br />
Ever since boosting method was first officially introduced in the 1990s, it has become one of the most powerful learning method to improve model performance. The fundamental boosting algorithm that was introduced by Freund and Schapire in 1995: AdaBoost has evolved into many much more sophisticated boosting methods (Freund & Schapire, 1997). Alternative methods of CNN boosting for images ( in counting tasks) also involve Selective Sampling which has been proven to increase accuracy and reduce processing time [7]. Curse of dimensionality is a big problem in machine learning, where evaluating parameters can reduce the speed of classification, as well as the predictive power. The AdaBoost can only choose those parameters who improve the predictive power of the model, and reduce the dimensionality. The essential idea of boosting is to use a weighted combination of a sequence of weak classifiers to make a strong one. The key point to this process is after each iteration of data modification (through any type of model), the boosting algorithm will place more weight on observations that are difficult to classify correctly. Therefore, those “weak” classifiers can then focus on those observations that are hard to predict (Hastie, Tibshirani, & Friedman, 2017). These explanations can be summarized using the following formula:<br />
<center><br />
<math> G(x) = sign(\sum\limits_{l=1}^L \alpha_l G_l(x)) </math> <br />
Where <math>\alpha_l</math> are computed by boosting algorithm and <math>G_l(x) </math> denotes a weak classifier. <br />
</center><br />
<br />
=== Boosting CNN ===<br />
<br />
Let $X = [x_1, \dots, x_M]$ denote the activation features of a training data batch of size $M$, where each $x_i$ has $K$ activation features. Also, let $Y = [y_1, \dots , y_M]$ denote the corresponding labels, i.e. each $y_i \in \{-1, +1\}$. In other words, the vector $x_i$ contains all the activations of the FC layer, which is typically multiplied with the weights corresponding to the connections between the FC layer and the decision layer. However, as mentioned earlier, this work achieves that same task of scoring via boosting as opposed to computing the inner product. Denoting the weak classifiers by $h(\cdot)$, we obtain the strong classifier as:<br />
\begin{equation}<br />
\label{boosting}<br />
H(x_i) = \sum\limits_{j = 1}^K \alpha_j h(x_{ij}; \lambda_j)<br />
\end{equation}<br />
where $x_{ij}$ is the $j^{th}$ activation feature calculated on the $i^{th}$ image. Moreover, $\alpha_j \geq 0$ denotes the weight of the $j^{th}$ weak classifier and $\sum\limits_{i=1}^K\alpha_i = 1$. Here we define $h(\cdot)$ to be a differentiable surrogate for the sign function as:<br />
\begin{equation}<br />
h(x_{ij}; \lambda_j) = \dfrac {f(x_{ij}; \lambda_j)}{\sqrt{f(x_{ij}; \lambda_j)^2 + \eta^2}}<br />
\end{equation}<br />
where $f(x_{ij}; \lambda_j)$ denotes a decision-stump (one-level decision tree) with $\lambda_j$ being the threshold. The parameter $\eta$ controls the slope of the function $\dfrac {f(\cdot)}{\sqrt{f(\cdot)^2 + \eta^2}}$. The authors set this parameter according to the distribution of $f(\cdot)$ as $\eta = \frac{\sigma}{c}$, where $\sigma$ is the standard deviation of $f(\cdot)$ and $c$ is a constant. This parameter is experimented with extensively in Section 4.4 of the paper. It is shown that $\eta$ can drastically influence the $F1$ score of the model.<br />
Note that for a certain strong classifier $H$ if a certain $\alpha_i = 0$, then the activation feature $x_i$ has no contribution to the output of the classifier $H$. In other words, the corresponding neuron can be considered to be inactive. (Refer to the Figure for the schematic diagram of B-CNN.)<br />
<br />
[[File:ibcnn.png]]<br />
<br />
=== Incremental Boosting ===<br />
In vanilla B-CNN, the information learned in a certain batch, i.e. the weights and the thresholds of the active neurons is discarded for subsequent batches. To address this issue, in this work the idea of incremental learning is incorporated. Formally, the incremental strong classifier at the $t^{th}$ iteration, $H_I^t$, is given as:<br />
\begin{equation}<br />
\label{incr}<br />
H_I^t(x_i^t) = \frac{(t-1)H_I^{t-1}(x_i^t) + H^t(x_i^t)}{t}<br />
\end{equation}<br />
where $H_I^{t-1}$ is the incremental strong classifier obtained at the $(t-1)^{th}$ iteration and $H^t$ is the boosted strong classifier at the $t^{th}$ iteration. Manipulating the equations, we obtain:<br />
\begin{equation}<br />
H_I^t(x_i^t) = \sum\limits_{j=1}^K \alpha_j^t h^t(x_{ij}; \lambda_j^t);\quad \alpha_j^t = \frac{(t-1)\alpha_j^{t-1}+\hat{\alpha}_j^t}{t}<br />
\end{equation}<br />
where $\hat{\alpha}_j^t$ is the weak classifier weight calculated in the $t^{th}$ iteration by boosting and $\alpha_j^t$ is the cumulative weight considering previous iterations. (Refer to the Figure for a schematic diagram of IB-CNN.) <br />
Typically, boosting algorithms minimize an objective function that captures the loss of the strong classifier. However, the loss of the strong classifier may be dominated by some weak classifiers with large weights, which might lead to overfitting. Therefore, to exercise better control, the loss function at a certain iteration $t$, $\epsilon^{IB}$ is expressed as a convex combination of the loss of the incremental strong classifier at iteration $t$ and the loss of the weak classifiers determined at iteration $t$. That is,<br />
\begin{equation}<br />
\epsilon^{IB} = \beta\epsilon_{strong}^{IB} + (1-\beta)\epsilon_{weak}<br />
\end{equation}<br />
where<br />
\begin{equation}<br />
\epsilon_{strong}^{IB} = \frac{1}{M}\sum\limits_{i=1}^M[H_I^t(x_i^t)-y_i^t]^2;\quad \epsilon_{weak} = \frac{1}{MN}\sum\limits_{i=1}^M\sum\limits_{\substack{1 \leq j \leq K \\ \alpha_j > 0}}[h(x_{ij}, \lambda_j)-y_i]^2.<br />
\end{equation}<br />
<br />
To learn the parameters of the IB-CNN, stochastic gradient descent is used, and the descent directions are obtained by differentiating the loss above with respect to the parameters $x_{ij}^t$ and $\lambda_j^t$ as follows:<br />
\begin{equation}<br />
\dfrac{\partial \epsilon^{IB}}{\partial x_{ij}^t} = \beta\dfrac{\partial \epsilon_{strong}^{IB}}{\partial H_I^t(x_i^t)}\dfrac{\partial H_I^t(x_i^t)}{\partial x_{ij}^t} + (1-\beta)\dfrac{\partial \epsilon_{weak}^{IB}}{\partial h^t(x_{ij}^t;\lambda_j^t)}\dfrac{\partial h^t(x_{ij}^t;\lambda_j^t)}{\partial x_{ij}^t} \\<br />
\dfrac{\partial \epsilon^{IB}}{\partial \lambda_j^t} = \beta\sum\limits_{i = 1}^M\dfrac{\partial \epsilon_{strong}^{IB}}{\partial H_I^t(x_i^t)}\dfrac{\partial H_I^t(x_i^t)}{\partial x_{ij}^t} + (1-\beta)\sum\limits_{i = 1}^M\dfrac{\partial \epsilon_{weak}^{IB}}{\partial h^t(x_{ij}^t;\lambda_j^t)}\dfrac{\partial h^t(x_{ij}^t;\lambda_j^t)}{\partial x_{ij}^t}<br />
\end{equation}<br />
where $\frac{\partial \epsilon^{IB}}{\partial x_{ij}^t}$ and $\frac{\partial \epsilon^{IB}}{\partial \lambda_j^t}$ only need to be calculated for the active neurons. Overall, the pseudocode for the incremental boosting algorithm for the IB-CNN is as follows.<br />
<br />
<br />
'''Input:''' The number of iterations (mini-batches) $T$ and activation features $X$ with the size of $M \times K$, where $M$ is the number of images in a mini-batch and $K$ is the dimension of the activation feature vector of one image.<br />
<br />
$\quad$ 1: '''for''' each input activation $j$ from $1$ to $K$ '''do'''<br />
<br />
$\quad$ 2: $\quad \alpha_j^1 = 0$<br />
<br />
$\quad$ 3: '''end for'''<br />
<br />
$\quad$ 4: '''for''' each mini-batch $t$ from $1$ to $T$ '''do'''<br />
<br />
$\quad$ 5: $\quad$ Feed-forward to the fully connected layer;<br />
<br />
$\quad$ 6: $\quad$ Select active features by boosting and calculate weights $\hat{\alpha}^t$ based on the standard AdaBoost;<br />
<br />
$\quad$ 7: $\quad$ Update the incremental strong classifier;<br />
<br />
$\quad$ 8: $\quad$ Calculate the overall loss of IB-CNN;<br />
<br />
$\quad$ 9: $\quad$ Backpropagate the loss based using the SGD equations;<br />
<br />
$\quad$ 10: $\quad$ Continue backpropagation to lower layers;<br />
<br />
$\quad$ 11: '''end for'''<br />
<br />
== Experiments ==<br />
<br />
<br />
Experiments have been conducted on four AU-coded databases whose details are as follows.<br />
<br />
1. CK database: contains 486 image sequences from 97 subjects, and 14 AUs.<br />
<br />
2. FERA2015 SEMAINE database: contains 31 subjects with 93,000 images and 6 AUs.<br />
<br />
3. FERA2015 BP4D database: contains 41 subjects with 146,847 images and 11 AUs.<br />
<br />
4. DISFA database: contains 27 subjects with 130,814 images and 12 AUs.<br />
<br />
=== Data Pre-Processing === <br />
The images are subjected to some preprocessing initially in order to scale and align the face regions, and to also remove out-of-plane rotations. Face regions are aligned based on three fiducial points: the centers of the two eyes and the mouth, and scaled to a size of 128 by 96. Face images are warped to a frontal view based on landmarks. A total of 23 facial landmarks that are less affected by facial movements are selected as control points to warp images. <br />
<br />
===IB-CNN Architecture===<br />
The IB-CNN architecture is as follows. The authors built the network by modifying the architecture of a CNN model that was originally used for CIFAR-10 dataset.<br />
The first two layers are convolutional layers having 32 filters with a size of 5 X 5 with a stride of 1. The activation maps then are sent to a rectified layer which is then followed by an average pooling layer with a stride of 3. This is followed again by a convolutional layer with 64 filters of size 5 X 5, the activation maps from which are sent to an FC layer with 128 nodes. The FC layer, in turn, feeds into the decision layer via the boosting mechanism.<br />
<br />
The following table compares the performance of the IB-CNN with other state-of-the-art methods that include CNN-based methods.<br />
<br />
[[File:table1_1.PNG|thumbnail]]<br />
<br />
Lastly, brief arguments are provided supporting the robustness of the IB-CNN to variations in: the slope parameter $\eta$, the number of input neurons, and the learning rate $\gamma$.<br />
<br />
== Conclusion ==<br />
To deal with the issue of relatively small-sized datasets in the domain of facial AU-recognition, the authors incorporate boosting and incremental learning to CNNs. Boosting helps the model to generalize well by preventing overfitting, and incremental learning exploits more information from the mini-batches but retaining more learned information. Moreover, the loss function is also changed so as to have more control over fine-tuning the model. The proposed IB-CNN shows improvement over the existent methods over four standard databases, showing a pronounced improvement in recognizing infrequent AUs.<br />
<br />
There are two immediate extensions to this work. Firstly, the IB-CNN may be applied to other problems wherein the data is limited. Secondly, the model may also be modified to go beyond binary classification and achieve multiclass boosted classification. Modern-day AdaBoost allows the weights $\alpha_i$, where i is the classifier, to be negative as well which gives the intuition that, "do exactly opposite of what this classifier says". In IB-CNN, the authors have restricted the weights of the input (FC units) to the boosting to be $\geq 0$. Considering that the IB-CNN is advantageous to models where datasets of a certain class are less in number, it would help the CNN to identify activation units in the FC layer which consistently contribute to the wrong prediction so that their consistency can be utilized in making correct predictions. From the AdaBoost classifier weight update, we conclude that the classifiers which should be considered to be inactive are those which cannot be relied upon i.e. those which contribute consistently nearly equal amount of correct and incorrect predictions.<br />
<br />
== References ==<br />
1. Han, S., Meng, H., Khan, A. S., Tong, Y. (2016) "Incremental Boosting Convolutional Neural Network for Facial Action Unit Recognition". NIPS.<br />
<br />
2. Tian, Y., Kanade, T., Cohn, J. F. (2001) "Recognizing Action Units for Facial Expression Analysis". IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 23., No. 2.<br />
<br />
3. Facial Action Coding System - https://en.wikipedia.org/wiki/Facial_Action_Coding_System<br />
<br />
4. YouTube Explanation of the Paper: https://www.youtube.com/watch?v=LZer5dW8h5c<br />
<br />
5. Srivastava, N., Hinton, G. E., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R. (2014). Dropout: a simple way to prevent neural networks from overfitting. Journal of machine learning research, 15(1), 1929-1958.<br />
<br />
6. Tian Y, Kanade T, Cohn JF. Recognizing Action Units for Facial Expression Analysis. IEEE transactions on pattern analysis and machine intelligence. 2001;23(2):97-115. doi:10.1109/34.908962.<br />
<br />
7. Learning to Count with CNN Boosting. Elad Walach and Lior Wolf. The Blavatnik School of Computer Science. Tel Aviv University<br />
<br />
8. A Brief Introduction to Boosting, Robert E. Schapire, Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence, 1999.<br />
<br />
9. Tong, Y., Liao, W., & Ji, Q. (2007). Facial action unit recognition by exploiting their dynamic and semantic relationships. IEEE transactions on pattern analysis and machine intelligence, 29(10).</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Incremental_Boosting_Convolutional_Neural_Network_for_Facial_Action_Unit_Recognition&diff=28860Incremental Boosting Convolutional Neural Network for Facial Action Unit Recognition2017-10-31T04:00:55Z<p>H4lyu: /* Related Work */</p>
<hr />
<div>== Introduction ==<br />
<br />
Facial expression is one of the most natural ways that human beings express the emotions. The Facial Action Coding System (FACS) attempts to systemically categorize each facial expression by specifying a basic set of muscle contractions or relaxations, formally called Action Units (AUs). FACS as we know today consists of five major categories (i) main, (ii) head movement, (iii) eye movement, (iv) visibility and (v) gross behavior . For example, "AU 1" stands for the inner portion of the brows being raised, and "AU 6" stands for the cheeks being raised. Such a framework helps in describing any facial expression/emotion as the combination of different AUs. The complete list of AUs in the FACS can be seen [https://en.wikipedia.org/wiki/Facial_Action_Coding_System#List_of_Action_Units_and_Action_Descriptors_.28with_underlying_facial_muscles.29 here]. However, during the course of an average day, most human beings do not experience drastically varying emotions and therefore their facial expressions might change only subtly. Additionally, there might also be a lot of subjectivity involved if this task were to be done manually. To address these issues, it is imperative to automate this task. In addition to applications in human behavior analysis, automating AU recognition also has potential applications in human-computer interaction (HCI), online education, interactive gaming, among other domains.<br />
<br />
Because of the recent advancements in object detection and categorization tasks, CNNs are an appealing go-to for the facial AU recognition task described above. However, compared to the former, the training sets available for the AU recognition task are not very large, and therefore the learned CNNs suffer from overfitting. To overcome this problem, this work builds on the idea of integrating boosting within CNN. Boosting is a technique wherein multiple weak learners are combined together to yield a strong learner. Moreover, this work also modifies the mechanics of learning a CNN by breaking large datasets in mini-batches. Typically, in a batch strategy, each iteration uses a batch only to update the parameters and the features learned in that iteration are discarded for subsequent iterations. Herein the authors incorporate incremental learning by building a classifier that incrementally learns from all the iterations/batches. Hence the genesis of the name Incremental Boosting - Convolutional Neural Network (IB-CNN) is justified. The IB-CNN introduced here outperforms the state-of-the-art CNNs on four standard AU-coded databases.<br />
<br />
== Related Work ==<br />
<br />
There have been many papers that use manually designed features for the AU-recognition task, unlike the CNN-based methods which integrate feature learning. Some of them are Gabor Wavelets, Local Binary Patterns (LBP), Histogram of Oriented Gradients (HOG), Scale Invariant Feature Transform (SIFT) features, Discriminant Non-negative Matrix Factorization(DNMF), Histograms of Local Phase Quantization (LPQ), and their spatiotemporal extensions. Newer approaches include the one proposed by Tong, Liao and Ji [9], who use a dynamic Bayesian network (DBN) to model the relationships among different AUs. Automated methodologies include the Automatic Facial Analysis system which recognizes both permanent and transient AUs with high accuracy rates [6]. Hand-crafted feature descriptors like LBP as very powerful facial recognition algorithm, and these types of descriptors are very robust and do not require training - hence, there is no "overfitting" of data. LBP, in particular, provides a very dense and concise descriptor (i.e. a histogram feature) of facial features without the use of GPU; in essence, the feature vector for LBPs are small, yet powerful, and the accuracy of which is comparable to that of a trained CNN. One of the major breakthroughs to reduce overfitting in CNNs was the dropout, introduced by Hinton et al [5]. As dropout can be regarded a special bagging ensembling method over "different" networks that share same weights. This work can be seen as a refinement of the previous random dropout approach in the sense that only neurons with no contribution towards classification are dropped, while retaining the more discriminative neurons. Moreover, adopting a batch strategy with incremental learning also circumvents the issue of not-so-large datasets.<br />
<br />
== Methodology ==<br />
<br />
=== CNNs ===<br />
Convolutional Neural Networks are neural networks that contain at least one convolution layer. Typically, the convolution layer is accompanied with a pooling layer. A convolution layer, as the name suggests, convolves the input matrix (or tensor) with a filter (or kernel). This operation produces a feature map (or activation map) that, roughly speaking, shows the presence or absence of the filter in the image. Note that the parameters in the kernel are not preset and are learned. A pooling layer reduces the dimensionality of the data by summarizing the activation map by some means (such as max or average). Following this, the matrix (or tensor) data is then vectorized using a fully-connected (FC) layer. The neurons in the FC layer have connections to all the activations in the previous layer. Finally, there is a decision layer that has as many neurons as the number of classes. The decision layer decides the class based on computing a score function using the activations of the FC layer. Generally, an inner-product score function is used, which is replaced by the boosting score function in this work. <br />
<br />
=== Boosting Method ===<br />
Ever since boosting method was first officially introduced in the 1990s, it has become one of the most powerful learning method to improve model performance. The fundamental boosting algorithm that was introduced by Freund and Schapire in 1995: AdaBoost has evolved into many much more sophisticated boosting methods (Freund & Schapire, 1997). Alternative methods of CNN boosting for images ( in counting tasks) also involve Selective Sampling which has been proven to increase accuracy and reduce processing time [7]. Curse of dimensionality is a big problem in machine learning, where evaluating parameters can reduce the speed of classification, as well as the predictive power. The AdaBoost can only choose those parameters who improve the predictive power of the model, and reduce the dimensionality. The essential idea of boosting is to use a weighted combination of a sequence of weak classifiers to make a strong one. The key point to this process is after each iteration of data modification (through any type of model), the boosting algorithm will place more weight on observations that are difficult to classify correctly. Therefore, those “weak” classifiers can then focus on those observations that are hard to predict (Hastie, Tibshirani, & Friedman, 2017). These explanations can be summarized using the following formula:<br />
<center><br />
<math> G(x) = sign(\sum\limits_{l=1}^L \alpha_l G_l(x)) </math> <br />
Where <math>\alpha_l</math> are computed by boosting algorithm and <math>G_l(x) </math> denotes a weak classifier. <br />
</center><br />
<br />
=== Boosting CNN ===<br />
<br />
Let $X = [x_1, \dots, x_M]$ denote the activation features of a training data batch of size $M$, where each $x_i$ has $K$ activation features. Also, let $Y = [y_1, \dots , y_M]$ denote the corresponding labels, i.e. each $y_i \in \{-1, +1\}$. In other words, the vector $x_i$ contains all the activations of the FC layer, which is typically multiplied with the weights corresponding to the connections between the FC layer and the decision layer. However, as mentioned earlier, this work achieves that same task of scoring via boosting as opposed to computing the inner product. Denoting the weak classifiers by $h(\cdot)$, we obtain the strong classifier as:<br />
\begin{equation}<br />
\label{boosting}<br />
H(x_i) = \sum\limits_{j = 1}^K \alpha_j h(x_{ij}; \lambda_j)<br />
\end{equation}<br />
where $x_{ij}$ is the $j^{th}$ activation feature calculated on the $i^{th}$ image. Moreover, $\alpha_j \geq 0$ denotes the weight of the $j^{th}$ weak classifier and $\sum\limits_{i=1}^K\alpha_i = 1$. Here we define $h(\cdot)$ to be a differentiable surrogate for the sign function as:<br />
\begin{equation}<br />
h(x_{ij}; \lambda_j) = \dfrac {f(x_{ij}; \lambda_j)}{\sqrt{f(x_{ij}; \lambda_j)^2 + \eta^2}}<br />
\end{equation}<br />
where $f(x_{ij}; \lambda_j)$ denotes a decision-stump (one-level decision tree) with $\lambda_j$ being the threshold. The parameter $\eta$ controls the slope of the function $\dfrac {f(\cdot)}{\sqrt{f(\cdot)^2 + \eta^2}}$. The authors set this parameter according to the distribution of $f(\cdot)$ as $\eta = \frac{\sigma}{c}$, where $\sigma$ is the standard deviation of $f(\cdot)$ and $c$ is a constant. This parameter is experimented with extensively in Section 4.4 of the paper. It is shown that $\eta$ can drastically influence the $F1$ score of the model.<br />
Note that for a certain strong classifier $H$ if a certain $\alpha_i = 0$, then the activation feature $x_i$ has no contribution to the output of the classifier $H$. In other words, the corresponding neuron can be considered to be inactive. (Refer to the Figure for the schematic diagram of B-CNN.)<br />
<br />
[[File:ibcnn.png]]<br />
<br />
=== Incremental Boosting ===<br />
In vanilla B-CNN, the information learned in a certain batch, i.e. the weights and the thresholds of the active neurons is discarded for subsequent batches. To address this issue, in this work the idea of incremental learning is incorporated. Formally, the incremental strong classifier at the $t^{th}$ iteration, $H_I^t$, is given as:<br />
\begin{equation}<br />
\label{incr}<br />
H_I^t(x_i^t) = \frac{(t-1)H_I^{t-1}(x_i^t) + H^t(x_i^t)}{t}<br />
\end{equation}<br />
where $H_I^{t-1}$ is the incremental strong classifier obtained at the $(t-1)^{th}$ iteration and $H^t$ is the boosted strong classifier at the $t^{th}$ iteration. Manipulating the equations, we obtain:<br />
\begin{equation}<br />
H_I^t(x_i^t) = \sum\limits_{j=1}^K \alpha_j^t h^t(x_{ij}; \lambda_j^t);\quad \alpha_j^t = \frac{(t-1)\alpha_j^{t-1}+\hat{\alpha}_j^t}{t}<br />
\end{equation}<br />
where $\hat{\alpha}_j^t$ is the weak classifier weight calculated in the $t^{th}$ iteration by boosting and $\alpha_j^t$ is the cumulative weight considering previous iterations. (Refer to the Figure for a schematic diagram of IB-CNN.) <br />
Typically, boosting algorithms minimize an objective function that captures the loss of the strong classifier. However, the loss of the strong classifier may be dominated by some weak classifiers with large weights, which might lead to overfitting. Therefore, to exercise better control, the loss function at a certain iteration $t$, $\epsilon^{IB}$ is expressed as a convex combination of the loss of the incremental strong classifier at iteration $t$ and the loss of the weak classifiers determined at iteration $t$. That is,<br />
\begin{equation}<br />
\epsilon^{IB} = \beta\epsilon_{strong}^{IB} + (1-\beta)\epsilon_{weak}<br />
\end{equation}<br />
where<br />
\begin{equation}<br />
\epsilon_{strong}^{IB} = \frac{1}{M}\sum\limits_{i=1}^M[H_I^t(x_i^t)-y_i^t]^2;\quad \epsilon_{weak} = \frac{1}{MN}\sum\limits_{i=1}^M\sum\limits_{\substack{1 \leq j \leq K \\ \alpha_j > 0}}[h(x_{ij}, \lambda_j)-y_i]^2.<br />
\end{equation}<br />
<br />
To learn the parameters of the IB-CNN, stochastic gradient descent is used, and the descent directions are obtained by differentiating the loss above with respect to the parameters $x_{ij}^t$ and $\lambda_j^t$ as follows:<br />
\begin{equation}<br />
\dfrac{\partial \epsilon^{IB}}{\partial x_{ij}^t} = \beta\dfrac{\partial \epsilon_{strong}^{IB}}{\partial H_I^t(x_i^t)}\dfrac{\partial H_I^t(x_i^t)}{\partial x_{ij}^t} + (1-\beta)\dfrac{\partial \epsilon_{weak}^{IB}}{\partial h^t(x_{ij}^t;\lambda_j^t)}\dfrac{\partial h^t(x_{ij}^t;\lambda_j^t)}{\partial x_{ij}^t} \\<br />
\dfrac{\partial \epsilon^{IB}}{\partial \lambda_j^t} = \beta\sum\limits_{i = 1}^M\dfrac{\partial \epsilon_{strong}^{IB}}{\partial H_I^t(x_i^t)}\dfrac{\partial H_I^t(x_i^t)}{\partial x_{ij}^t} + (1-\beta)\sum\limits_{i = 1}^M\dfrac{\partial \epsilon_{weak}^{IB}}{\partial h^t(x_{ij}^t;\lambda_j^t)}\dfrac{\partial h^t(x_{ij}^t;\lambda_j^t)}{\partial x_{ij}^t}<br />
\end{equation}<br />
where $\frac{\partial \epsilon^{IB}}{\partial x_{ij}^t}$ and $\frac{\partial \epsilon^{IB}}{\partial \lambda_j^t}$ only need to be calculated for the active neurons. Overall, the pseudocode for the incremental boosting algorithm for the IB-CNN is as follows.<br />
<br />
<br />
'''Input:''' The number of iterations (mini-batches) $T$ and activation features $X$ with the size of $M \times K$, where $M$ is the number of images in a mini-batch and $K$ is the dimension of the activation feature vector of one image.<br />
<br />
$\quad$ 1: '''for''' each input activation $j$ from $1$ to $K$ '''do'''<br />
<br />
$\quad$ 2: $\quad \alpha_j^1 = 0$<br />
<br />
$\quad$ 3: '''end for'''<br />
<br />
$\quad$ 4: '''for''' each mini-batch $t$ from $1$ to $T$ '''do'''<br />
<br />
$\quad$ 5: $\quad$ Feed-forward to the fully connected layer;<br />
<br />
$\quad$ 6: $\quad$ Select active features by boosting and calculate weights $\hat{\alpha}^t$ based on the standard AdaBoost;<br />
<br />
$\quad$ 7: $\quad$ Update the incremental strong classifier;<br />
<br />
$\quad$ 8: $\quad$ Calculate the overall loss of IB-CNN;<br />
<br />
$\quad$ 9: $\quad$ Backpropagate the loss based using the SGD equations;<br />
<br />
$\quad$ 10: $\quad$ Continue backpropagation to lower layers;<br />
<br />
$\quad$ 11: '''end for'''<br />
<br />
== Experiments ==<br />
<br />
<br />
Experiments have been conducted on four AU-coded databases whose details are as follows.<br />
<br />
1. CK database: contains 486 image sequences from 97 subjects, and 14 AUs.<br />
<br />
2. FERA2015 SEMAINE database: contains 31 subjects with 93,000 images and 6 AUs.<br />
<br />
3. FERA2015 BP4D database: contains 41 subjects with 146,847 images and 11 AUs.<br />
<br />
4. DISFA database: contains 27 subjects with 130,814 images and 12 AUs.<br />
<br />
=== Data Pre-Processing === <br />
The images are subjected to some preprocessing initially in order to scale and align the face regions, and to also remove out-of-plane rotations. Face regions are aligned based on three fiducial points: the centers of the two eyes and the mouth, and scaled to a size of 128 by 96. Face images are warped to a frontal view based on landmarks. A total of 23 facial landmarks that are less affected by facial movements are selected as control points to warp images. <br />
<br />
===IB-CNN Architecture===<br />
The IB-CNN architecture is as follows. The authors built the network by modifying the architecture of a CNN model that was originally used for CIFAR-10 dataset.<br />
The first two layers are convolutional layers having 32 filters with a size of 5 X 5 with a stride of 1. The activation maps then are sent to a rectified layer which is then followed by an average pooling layer with a stride of 3. This is followed again by a convolutional layer with 64 filters of size 5 X 5, the activation maps from which are sent to an FC layer with 128 nodes. The FC layer, in turn, feeds into the decision layer via the boosting mechanism.<br />
<br />
The following table compares the performance of the IB-CNN with other state-of-the-art methods that include CNN-based methods.<br />
<br />
[[File:table1_1.PNG|thumbnail]]<br />
<br />
Lastly, brief arguments are provided supporting the robustness of the IB-CNN to variations in: the slope parameter $\eta$, the number of input neurons, and the learning rate $\gamma$.<br />
<br />
== Conclusion ==<br />
To deal with the issue of relatively small-sized datasets in the domain of facial AU-recognition, the authors incorporate boosting and incremental learning to CNNs. Boosting helps the model to generalize well by preventing overfitting, and incremental learning exploits more information from the mini-batches but retaining more learned information. Moreover, the loss function is also changed so as to have more control over fine-tuning the model. The proposed IB-CNN shows improvement over the existent methods over four standard databases, showing a pronounced improvement in recognizing infrequent AUs.<br />
<br />
There are two immediate extensions to this work. Firstly, the IB-CNN may be applied to other problems wherein the data is limited. Secondly, the model may also be modified to go beyond binary classification and achieve multiclass boosted classification. Modern-day AdaBoost allows the weights $\alpha_i$, where i is the classifier, to be negative as well which gives the intuition that, "do exactly opposite of what this classifier says". In IB-CNN, the authors have restricted the weights of the input (FC units) to the boosting to be $\geq 0$. Considering that the IB-CNN is advantageous to models where datasets of a certain class are less in number, it would help the CNN to identify activation units in the FC layer which consistently contribute to the wrong prediction so that their consistency can be utilized in making correct predictions. From the AdaBoost classifier weight update, we conclude that the classifiers which should be considered to be inactive are those which cannot be relied upon i.e. those which contribute consistently nearly equal amount of correct and incorrect predictions.<br />
<br />
== References ==<br />
1. Han, S., Meng, H., Khan, A. S., Tong, Y. (2016) "Incremental Boosting Convolutional Neural Network for Facial Action Unit Recognition". NIPS.<br />
<br />
2. Tian, Y., Kanade, T., Cohn, J. F. (2001) "Recognizing Action Units for Facial Expression Analysis". IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 23., No. 2.<br />
<br />
3. Facial Action Coding System - https://en.wikipedia.org/wiki/Facial_Action_Coding_System<br />
<br />
4. YouTube Explanation of the Paper: https://www.youtube.com/watch?v=LZer5dW8h5c<br />
<br />
5. Srivastava, N., Hinton, G. E., Krizhevsky, A., Sutskever, I., & Salakhutdinov, R. (2014). Dropout: a simple way to prevent neural networks from overfitting. Journal of machine learning research, 15(1), 1929-1958.<br />
<br />
6. Tian Y, Kanade T, Cohn JF. Recognizing Action Units for Facial Expression Analysis. IEEE transactions on pattern analysis and machine intelligence. 2001;23(2):97-115. doi:10.1109/34.908962.<br />
<br />
7.Learning to Count with CNN Boosting. Elad Walach and Lior Wolf. The Blavatnik School of Computer Science. Tel Aviv University<br />
<br />
8. A Brief Introduction to Boosting, Robert E. Schapire, Proceedings of the Sixteenth International Joint Conference on Artificial Intelligence, 1999.</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28597STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-28T03:30:10Z<p>H4lyu: /* References */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small, the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failures of traditional methods ==<br />
Failures of traditional (derivative-based) methods fall into several categories:<br />
<br />
First, derivatives are local and not quite useful for comparative analysis. Since derivatives are local, if the input to compare is far from reference input, it may not be appropriate to assign contribution based on derivatives. An example can be image recognizition, where change in a single pixel makes no sense. In some cases where values in an input are discrete (not continuous), derivatives are not available.<br />
<br />
Second, using derivatives may mislead the analysis when derivatives have completely different behaviour in different parts of the output function. For example, on ReLU and (hard) max layers, taking derivative may lead to the conclusion that changes in all values make no difference as long as they don't touch the boundary.<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y / \Delta x_i} = \Delta y$. <br />
<br />
===Chain Rule===<br />
First, by the appendix of [1] we know the DeepLIFT multiplier $m$ behaves just like derivatives and satisfies the chain rule: If $z = z\left( y(x_1,...,x_N) \right)$ then<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
===Linear Rule===<br />
Second, let's consider a simple neuron $y = f(s)$ where $s = \sum_{i=1}^N w_i x_i$. We separate the positive and negative contribution of each $x_i$ and assign $m_{\Delta s / \Delta x_i}$ by the following rule, named as '''linear rule'''. The positive contribution of $x_i$ to $s$ is defined as<br />
<br />
$ \begin{align}<br />
\Delta s^{+} & = \sum_{i=1}^N 1_{\left\{ w_i \Delta x_i > 0 \right\}} w_i \Delta x_i \\<br />
& = \sum_{i=1}^N \left( 1_{\left\{ w_i > 0 \right\}} w_i \Delta x_{i}^{+} + 1_{\left\{ w_i < 0 \right\}} w_i \Delta x_{i}^{-} \right) \\<br />
\end{align} $<br />
<br />
We then assign the secant as $ m_{\Delta s^{+} / \Delta x_{i}^{+} } = 1_{\left\{ w_i > 0 \right\}} w_i $ and $ m_{\Delta s^{+} / \Delta x_{i}^{-} } = 1_{\left\{ w_i < 0 \right\}} w_i $. Similarly, we have $ m_{\Delta s^{-} / \Delta x_{i}^{+} } = 1_{\left\{ w_i < 0 \right\}} w_i $ and $ m_{\Delta s^{-} / \Delta x_{i}^{-} } = 1_{\left\{ w_i > 0 \right\}} w_i $. As to the occasion when $\Delta x_{i} = 0$, we let $ m_{\Delta s^{\pm} / \Delta x_{i}^{\pm} } = \frac{1}{2} w_i $.<br />
<br />
===Rescale Rule===<br />
For the function $f(s)$, it is possible to use the easiest method called '''rescale rule''':<br />
<br />
$ \Delta y = \frac{\Delta y}{\Delta s} \Delta s = \frac{\Delta y}{\Delta s} \left( \Delta s^{+} + \Delta s^{-} \right) $<br />
<br />
===Reveal-Cancel Rule===<br />
Rescale rule works for simple functions such as ReLU, but it does not always work well especially for some cases like pooling layers. To solve this we introduce '''reveal-cancel rule''':<br />
<br />
Suppose our reference input is $x_0$ and $s_0 = \sum_{i=1}^N w_i x_{0,i}$ is the sum in $y_0 = f(s_0)$. We define:<br />
<br />
$ \Delta y^{+} = \frac{1}{2} \left[ f(s_0 + \Delta s^{+}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{-}) \right] $<br />
<br />
$ \Delta y^{-} = \frac{1}{2} \left[ f(s_0 + \Delta s^{-}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{+}) \right] $<br />
<br />
$ m_{\Delta y / \Delta s^{+} } = \Delta y^{+} / \Delta s^{+} , m_{\Delta y / \Delta s^{-} } = \Delta y^{-} / \Delta s^{-} $<br />
<br />
===Adjustments for softmax layers===<br />
Since softmax layer normalizes its input, we can let the contribution of softmax layer $y=softmax(z)$ be that of its preceding layer $z = z(x)$ minus the average contribution of that preceding layer:<br />
<br />
$ C^{\prime}_{\Delta z_i / \Delta x} = C_{\Delta z_i / \Delta x} - \frac{1}{n} \sum_{j=1}^{N} C_{\Delta z_j / \Delta x} $<br />
<br />
Given the rules above, it is easy to calculate $ m_{\Delta y / \Delta x_i^{\pm} } $ for each $x_i$, and thus to calculate the DeepLIFT multiplier $m$ and contribution $C$ of a certain input and output compared to a given reference input $x_0$. It is suggested by the author that reference input should be case-specific and no general rule for choosing $x_0$ is currently available.<br />
<br />
== Numerical results ==<br />
The authors perform two main examples to test whether the suggestions of DeepLIFT method are based on correct "understanding" of what the model is doing.<br />
<br />
===MNIST handwriting re-morphing===<br />
Suppose we have a well-trained MNIST handwriting recognition model which can identify 0-9 correctly. Now we have a hand-written 8, and we want to know "how can we erase a part of this handwritten 8 to change it, say, to 3?"<br />
<br />
In this test, we take our reference input as all-zeros (black image) as this is the background of the images. We subtract contribution scores of pixels of class 8 to that of class 3, and erase 20% of pixels with highest contribution score. It appears that DeepLIFT can identify the left side of 8 as to be erased, which is far better that other gradient-based methods.<br />
<br />
===DNA motif detection===<br />
Suppose we have a long sequence of DNA $\left\{ x_n \right\}$, and we have a neuron network detecting whether the sequence contains GATA motif (a short DNA sub-sequence) or TAL motif. The training dataset is a mixture of randomly generated DNA sequences and real-world DNA sequences equipped with GATA and TAL, and the network is well-trained. We know the behavior of the neuron network:<br />
<br />
* The network adhere tag (0,0) for these sequences which are believed to be randomly generated.<br />
* The network adhere tag (1,0) for these sequences which are believed to be real-world and contains GATA but not TAL, and (0,1) for these sequences which contains TAL but not GATA.<br />
* The network adhere tag (1,1) for these sequences which are believed to be real-world and contains both GATA and TAL.<br />
<br />
The question is: "What makes sequences with tag (1,0) different from these with tag (0,0)?" The answer should be "Whether it contains GATA sequence". Now we sample a sequence $\left\{ a_n \right\}$ with tag (1,0) and another one randomly generated with tag (0,0), $\left\{ b_n \right\}$, as reference, and assign contribution scores to the sequence by different schemes. We expect a good analyzing scheme to show us the answer above by highlighting the GATA motif in $\left\{ a_n \right\}$. The authors of [1] show that DeepLIFT performs much better than other schemes in highlighting the requested motif.<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685<br />
<br />
[2] Video tutorial to DeepLIFT: http://goo.gl/qKb7pL</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28596STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-28T03:29:46Z<p>H4lyu: /* DNA motif detection */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small, the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failures of traditional methods ==<br />
Failures of traditional (derivative-based) methods fall into several categories:<br />
<br />
First, derivatives are local and not quite useful for comparative analysis. Since derivatives are local, if the input to compare is far from reference input, it may not be appropriate to assign contribution based on derivatives. An example can be image recognizition, where change in a single pixel makes no sense. In some cases where values in an input are discrete (not continuous), derivatives are not available.<br />
<br />
Second, using derivatives may mislead the analysis when derivatives have completely different behaviour in different parts of the output function. For example, on ReLU and (hard) max layers, taking derivative may lead to the conclusion that changes in all values make no difference as long as they don't touch the boundary.<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y / \Delta x_i} = \Delta y$. <br />
<br />
===Chain Rule===<br />
First, by the appendix of [1] we know the DeepLIFT multiplier $m$ behaves just like derivatives and satisfies the chain rule: If $z = z\left( y(x_1,...,x_N) \right)$ then<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
===Linear Rule===<br />
Second, let's consider a simple neuron $y = f(s)$ where $s = \sum_{i=1}^N w_i x_i$. We separate the positive and negative contribution of each $x_i$ and assign $m_{\Delta s / \Delta x_i}$ by the following rule, named as '''linear rule'''. The positive contribution of $x_i$ to $s$ is defined as<br />
<br />
$ \begin{align}<br />
\Delta s^{+} & = \sum_{i=1}^N 1_{\left\{ w_i \Delta x_i > 0 \right\}} w_i \Delta x_i \\<br />
& = \sum_{i=1}^N \left( 1_{\left\{ w_i > 0 \right\}} w_i \Delta x_{i}^{+} + 1_{\left\{ w_i < 0 \right\}} w_i \Delta x_{i}^{-} \right) \\<br />
\end{align} $<br />
<br />
We then assign the secant as $ m_{\Delta s^{+} / \Delta x_{i}^{+} } = 1_{\left\{ w_i > 0 \right\}} w_i $ and $ m_{\Delta s^{+} / \Delta x_{i}^{-} } = 1_{\left\{ w_i < 0 \right\}} w_i $. Similarly, we have $ m_{\Delta s^{-} / \Delta x_{i}^{+} } = 1_{\left\{ w_i < 0 \right\}} w_i $ and $ m_{\Delta s^{-} / \Delta x_{i}^{-} } = 1_{\left\{ w_i > 0 \right\}} w_i $. As to the occasion when $\Delta x_{i} = 0$, we let $ m_{\Delta s^{\pm} / \Delta x_{i}^{\pm} } = \frac{1}{2} w_i $.<br />
<br />
===Rescale Rule===<br />
For the function $f(s)$, it is possible to use the easiest method called '''rescale rule''':<br />
<br />
$ \Delta y = \frac{\Delta y}{\Delta s} \Delta s = \frac{\Delta y}{\Delta s} \left( \Delta s^{+} + \Delta s^{-} \right) $<br />
<br />
===Reveal-Cancel Rule===<br />
Rescale rule works for simple functions such as ReLU, but it does not always work well especially for some cases like pooling layers. To solve this we introduce '''reveal-cancel rule''':<br />
<br />
Suppose our reference input is $x_0$ and $s_0 = \sum_{i=1}^N w_i x_{0,i}$ is the sum in $y_0 = f(s_0)$. We define:<br />
<br />
$ \Delta y^{+} = \frac{1}{2} \left[ f(s_0 + \Delta s^{+}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{-}) \right] $<br />
<br />
$ \Delta y^{-} = \frac{1}{2} \left[ f(s_0 + \Delta s^{-}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{+}) \right] $<br />
<br />
$ m_{\Delta y / \Delta s^{+} } = \Delta y^{+} / \Delta s^{+} , m_{\Delta y / \Delta s^{-} } = \Delta y^{-} / \Delta s^{-} $<br />
<br />
===Adjustments for softmax layers===<br />
Since softmax layer normalizes its input, we can let the contribution of softmax layer $y=softmax(z)$ be that of its preceding layer $z = z(x)$ minus the average contribution of that preceding layer:<br />
<br />
$ C^{\prime}_{\Delta z_i / \Delta x} = C_{\Delta z_i / \Delta x} - \frac{1}{n} \sum_{j=1}^{N} C_{\Delta z_j / \Delta x} $<br />
<br />
Given the rules above, it is easy to calculate $ m_{\Delta y / \Delta x_i^{\pm} } $ for each $x_i$, and thus to calculate the DeepLIFT multiplier $m$ and contribution $C$ of a certain input and output compared to a given reference input $x_0$. It is suggested by the author that reference input should be case-specific and no general rule for choosing $x_0$ is currently available.<br />
<br />
== Numerical results ==<br />
The authors perform two main examples to test whether the suggestions of DeepLIFT method are based on correct "understanding" of what the model is doing.<br />
<br />
===MNIST handwriting re-morphing===<br />
Suppose we have a well-trained MNIST handwriting recognition model which can identify 0-9 correctly. Now we have a hand-written 8, and we want to know "how can we erase a part of this handwritten 8 to change it, say, to 3?"<br />
<br />
In this test, we take our reference input as all-zeros (black image) as this is the background of the images. We subtract contribution scores of pixels of class 8 to that of class 3, and erase 20% of pixels with highest contribution score. It appears that DeepLIFT can identify the left side of 8 as to be erased, which is far better that other gradient-based methods.<br />
<br />
===DNA motif detection===<br />
Suppose we have a long sequence of DNA $\left\{ x_n \right\}$, and we have a neuron network detecting whether the sequence contains GATA motif (a short DNA sub-sequence) or TAL motif. The training dataset is a mixture of randomly generated DNA sequences and real-world DNA sequences equipped with GATA and TAL, and the network is well-trained. We know the behavior of the neuron network:<br />
<br />
* The network adhere tag (0,0) for these sequences which are believed to be randomly generated.<br />
* The network adhere tag (1,0) for these sequences which are believed to be real-world and contains GATA but not TAL, and (0,1) for these sequences which contains TAL but not GATA.<br />
* The network adhere tag (1,1) for these sequences which are believed to be real-world and contains both GATA and TAL.<br />
<br />
The question is: "What makes sequences with tag (1,0) different from these with tag (0,0)?" The answer should be "Whether it contains GATA sequence". Now we sample a sequence $\left\{ a_n \right\}$ with tag (1,0) and another one randomly generated with tag (0,0), $\left\{ b_n \right\}$, as reference, and assign contribution scores to the sequence by different schemes. We expect a good analyzing scheme to show us the answer above by highlighting the GATA motif in $\left\{ a_n \right\}$. The authors of [1] show that DeepLIFT performs much better than other schemes in highlighting the requested motif.<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685<br />
[2] Video tutorial to DeepLIFT: http://goo.gl/qKb7pL</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28595STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-28T03:29:13Z<p>H4lyu: /* References */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small, the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failures of traditional methods ==<br />
Failures of traditional (derivative-based) methods fall into several categories:<br />
<br />
First, derivatives are local and not quite useful for comparative analysis. Since derivatives are local, if the input to compare is far from reference input, it may not be appropriate to assign contribution based on derivatives. An example can be image recognizition, where change in a single pixel makes no sense. In some cases where values in an input are discrete (not continuous), derivatives are not available.<br />
<br />
Second, using derivatives may mislead the analysis when derivatives have completely different behaviour in different parts of the output function. For example, on ReLU and (hard) max layers, taking derivative may lead to the conclusion that changes in all values make no difference as long as they don't touch the boundary.<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y / \Delta x_i} = \Delta y$. <br />
<br />
===Chain Rule===<br />
First, by the appendix of [1] we know the DeepLIFT multiplier $m$ behaves just like derivatives and satisfies the chain rule: If $z = z\left( y(x_1,...,x_N) \right)$ then<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
===Linear Rule===<br />
Second, let's consider a simple neuron $y = f(s)$ where $s = \sum_{i=1}^N w_i x_i$. We separate the positive and negative contribution of each $x_i$ and assign $m_{\Delta s / \Delta x_i}$ by the following rule, named as '''linear rule'''. The positive contribution of $x_i$ to $s$ is defined as<br />
<br />
$ \begin{align}<br />
\Delta s^{+} & = \sum_{i=1}^N 1_{\left\{ w_i \Delta x_i > 0 \right\}} w_i \Delta x_i \\<br />
& = \sum_{i=1}^N \left( 1_{\left\{ w_i > 0 \right\}} w_i \Delta x_{i}^{+} + 1_{\left\{ w_i < 0 \right\}} w_i \Delta x_{i}^{-} \right) \\<br />
\end{align} $<br />
<br />
We then assign the secant as $ m_{\Delta s^{+} / \Delta x_{i}^{+} } = 1_{\left\{ w_i > 0 \right\}} w_i $ and $ m_{\Delta s^{+} / \Delta x_{i}^{-} } = 1_{\left\{ w_i < 0 \right\}} w_i $. Similarly, we have $ m_{\Delta s^{-} / \Delta x_{i}^{+} } = 1_{\left\{ w_i < 0 \right\}} w_i $ and $ m_{\Delta s^{-} / \Delta x_{i}^{-} } = 1_{\left\{ w_i > 0 \right\}} w_i $. As to the occasion when $\Delta x_{i} = 0$, we let $ m_{\Delta s^{\pm} / \Delta x_{i}^{\pm} } = \frac{1}{2} w_i $.<br />
<br />
===Rescale Rule===<br />
For the function $f(s)$, it is possible to use the easiest method called '''rescale rule''':<br />
<br />
$ \Delta y = \frac{\Delta y}{\Delta s} \Delta s = \frac{\Delta y}{\Delta s} \left( \Delta s^{+} + \Delta s^{-} \right) $<br />
<br />
===Reveal-Cancel Rule===<br />
Rescale rule works for simple functions such as ReLU, but it does not always work well especially for some cases like pooling layers. To solve this we introduce '''reveal-cancel rule''':<br />
<br />
Suppose our reference input is $x_0$ and $s_0 = \sum_{i=1}^N w_i x_{0,i}$ is the sum in $y_0 = f(s_0)$. We define:<br />
<br />
$ \Delta y^{+} = \frac{1}{2} \left[ f(s_0 + \Delta s^{+}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{-}) \right] $<br />
<br />
$ \Delta y^{-} = \frac{1}{2} \left[ f(s_0 + \Delta s^{-}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{+}) \right] $<br />
<br />
$ m_{\Delta y / \Delta s^{+} } = \Delta y^{+} / \Delta s^{+} , m_{\Delta y / \Delta s^{-} } = \Delta y^{-} / \Delta s^{-} $<br />
<br />
===Adjustments for softmax layers===<br />
Since softmax layer normalizes its input, we can let the contribution of softmax layer $y=softmax(z)$ be that of its preceding layer $z = z(x)$ minus the average contribution of that preceding layer:<br />
<br />
$ C^{\prime}_{\Delta z_i / \Delta x} = C_{\Delta z_i / \Delta x} - \frac{1}{n} \sum_{j=1}^{N} C_{\Delta z_j / \Delta x} $<br />
<br />
Given the rules above, it is easy to calculate $ m_{\Delta y / \Delta x_i^{\pm} } $ for each $x_i$, and thus to calculate the DeepLIFT multiplier $m$ and contribution $C$ of a certain input and output compared to a given reference input $x_0$. It is suggested by the author that reference input should be case-specific and no general rule for choosing $x_0$ is currently available.<br />
<br />
== Numerical results ==<br />
The authors perform two main examples to test whether the suggestions of DeepLIFT method are based on correct "understanding" of what the model is doing.<br />
<br />
===MNIST handwriting re-morphing===<br />
Suppose we have a well-trained MNIST handwriting recognition model which can identify 0-9 correctly. Now we have a hand-written 8, and we want to know "how can we erase a part of this handwritten 8 to change it, say, to 3?"<br />
<br />
In this test, we take our reference input as all-zeros (black image) as this is the background of the images. We subtract contribution scores of pixels of class 8 to that of class 3, and erase 20% of pixels with highest contribution score. It appears that DeepLIFT can identify the left side of 8 as to be erased, which is far better that other gradient-based methods.<br />
<br />
===DNA motif detection===<br />
Suppose we have a long sequence of DNA $\left{ x_n \right}$, and we have a neuron network detecting whether the sequence contains GATA motif (a short DNA sub-sequence) or TAL motif. The training dataset is a mixture of randomly generated DNA sequences and real-world DNA sequences equipped with GATA and TAL, and the network is well-trained. We know the behavior of the neuron network:<br />
<br />
* The network adhere tag (0,0) for these sequences which are believed to be randomly generated.<br />
* The network adhere tag (1,0) for these sequences which are believed to be real-world and contains GATA but not TAL, and (0,1) for these sequences which contains TAL but not GATA.<br />
* The network adhere tag (1,1) for these sequences which are believed to be real-world and contains both GATA and TAL.<br />
<br />
The question is: "What makes sequences with tag (1,0) different from these with tag (0,0)?" The answer should be "Whether it contains GATA sequence". Now we sample a sequence $\left{ a_n \right}$ with tag (1,0) and another one randomly generated with tag (0,0), $\left{ b_n \right}$, as reference, and assign contribution scores to the sequence by different schemes. We expect a good analyzing scheme to show us the answer above by highlighting the GATA motif in $\left{ a_n \right}$. The authors of [1] show that DeepLIFT performs much better than other schemes in highlighting the requested motif.<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685<br />
[2] Video tutorial to DeepLIFT: http://goo.gl/qKb7pL</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28594STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-28T03:28:37Z<p>H4lyu: /* Numerical results */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small, the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failures of traditional methods ==<br />
Failures of traditional (derivative-based) methods fall into several categories:<br />
<br />
First, derivatives are local and not quite useful for comparative analysis. Since derivatives are local, if the input to compare is far from reference input, it may not be appropriate to assign contribution based on derivatives. An example can be image recognizition, where change in a single pixel makes no sense. In some cases where values in an input are discrete (not continuous), derivatives are not available.<br />
<br />
Second, using derivatives may mislead the analysis when derivatives have completely different behaviour in different parts of the output function. For example, on ReLU and (hard) max layers, taking derivative may lead to the conclusion that changes in all values make no difference as long as they don't touch the boundary.<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y / \Delta x_i} = \Delta y$. <br />
<br />
===Chain Rule===<br />
First, by the appendix of [1] we know the DeepLIFT multiplier $m$ behaves just like derivatives and satisfies the chain rule: If $z = z\left( y(x_1,...,x_N) \right)$ then<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
===Linear Rule===<br />
Second, let's consider a simple neuron $y = f(s)$ where $s = \sum_{i=1}^N w_i x_i$. We separate the positive and negative contribution of each $x_i$ and assign $m_{\Delta s / \Delta x_i}$ by the following rule, named as '''linear rule'''. The positive contribution of $x_i$ to $s$ is defined as<br />
<br />
$ \begin{align}<br />
\Delta s^{+} & = \sum_{i=1}^N 1_{\left\{ w_i \Delta x_i > 0 \right\}} w_i \Delta x_i \\<br />
& = \sum_{i=1}^N \left( 1_{\left\{ w_i > 0 \right\}} w_i \Delta x_{i}^{+} + 1_{\left\{ w_i < 0 \right\}} w_i \Delta x_{i}^{-} \right) \\<br />
\end{align} $<br />
<br />
We then assign the secant as $ m_{\Delta s^{+} / \Delta x_{i}^{+} } = 1_{\left\{ w_i > 0 \right\}} w_i $ and $ m_{\Delta s^{+} / \Delta x_{i}^{-} } = 1_{\left\{ w_i < 0 \right\}} w_i $. Similarly, we have $ m_{\Delta s^{-} / \Delta x_{i}^{+} } = 1_{\left\{ w_i < 0 \right\}} w_i $ and $ m_{\Delta s^{-} / \Delta x_{i}^{-} } = 1_{\left\{ w_i > 0 \right\}} w_i $. As to the occasion when $\Delta x_{i} = 0$, we let $ m_{\Delta s^{\pm} / \Delta x_{i}^{\pm} } = \frac{1}{2} w_i $.<br />
<br />
===Rescale Rule===<br />
For the function $f(s)$, it is possible to use the easiest method called '''rescale rule''':<br />
<br />
$ \Delta y = \frac{\Delta y}{\Delta s} \Delta s = \frac{\Delta y}{\Delta s} \left( \Delta s^{+} + \Delta s^{-} \right) $<br />
<br />
===Reveal-Cancel Rule===<br />
Rescale rule works for simple functions such as ReLU, but it does not always work well especially for some cases like pooling layers. To solve this we introduce '''reveal-cancel rule''':<br />
<br />
Suppose our reference input is $x_0$ and $s_0 = \sum_{i=1}^N w_i x_{0,i}$ is the sum in $y_0 = f(s_0)$. We define:<br />
<br />
$ \Delta y^{+} = \frac{1}{2} \left[ f(s_0 + \Delta s^{+}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{-}) \right] $<br />
<br />
$ \Delta y^{-} = \frac{1}{2} \left[ f(s_0 + \Delta s^{-}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{+}) \right] $<br />
<br />
$ m_{\Delta y / \Delta s^{+} } = \Delta y^{+} / \Delta s^{+} , m_{\Delta y / \Delta s^{-} } = \Delta y^{-} / \Delta s^{-} $<br />
<br />
===Adjustments for softmax layers===<br />
Since softmax layer normalizes its input, we can let the contribution of softmax layer $y=softmax(z)$ be that of its preceding layer $z = z(x)$ minus the average contribution of that preceding layer:<br />
<br />
$ C^{\prime}_{\Delta z_i / \Delta x} = C_{\Delta z_i / \Delta x} - \frac{1}{n} \sum_{j=1}^{N} C_{\Delta z_j / \Delta x} $<br />
<br />
Given the rules above, it is easy to calculate $ m_{\Delta y / \Delta x_i^{\pm} } $ for each $x_i$, and thus to calculate the DeepLIFT multiplier $m$ and contribution $C$ of a certain input and output compared to a given reference input $x_0$. It is suggested by the author that reference input should be case-specific and no general rule for choosing $x_0$ is currently available.<br />
<br />
== Numerical results ==<br />
The authors perform two main examples to test whether the suggestions of DeepLIFT method are based on correct "understanding" of what the model is doing.<br />
<br />
===MNIST handwriting re-morphing===<br />
Suppose we have a well-trained MNIST handwriting recognition model which can identify 0-9 correctly. Now we have a hand-written 8, and we want to know "how can we erase a part of this handwritten 8 to change it, say, to 3?"<br />
<br />
In this test, we take our reference input as all-zeros (black image) as this is the background of the images. We subtract contribution scores of pixels of class 8 to that of class 3, and erase 20% of pixels with highest contribution score. It appears that DeepLIFT can identify the left side of 8 as to be erased, which is far better that other gradient-based methods.<br />
<br />
===DNA motif detection===<br />
Suppose we have a long sequence of DNA $\left{ x_n \right}$, and we have a neuron network detecting whether the sequence contains GATA motif (a short DNA sub-sequence) or TAL motif. The training dataset is a mixture of randomly generated DNA sequences and real-world DNA sequences equipped with GATA and TAL, and the network is well-trained. We know the behavior of the neuron network:<br />
<br />
* The network adhere tag (0,0) for these sequences which are believed to be randomly generated.<br />
* The network adhere tag (1,0) for these sequences which are believed to be real-world and contains GATA but not TAL, and (0,1) for these sequences which contains TAL but not GATA.<br />
* The network adhere tag (1,1) for these sequences which are believed to be real-world and contains both GATA and TAL.<br />
<br />
The question is: "What makes sequences with tag (1,0) different from these with tag (0,0)?" The answer should be "Whether it contains GATA sequence". Now we sample a sequence $\left{ a_n \right}$ with tag (1,0) and another one randomly generated with tag (0,0), $\left{ b_n \right}$, as reference, and assign contribution scores to the sequence by different schemes. We expect a good analyzing scheme to show us the answer above by highlighting the GATA motif in $\left{ a_n \right}$. The authors of [1] show that DeepLIFT performs much better than other schemes in highlighting the requested motif.<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28592STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-27T22:27:35Z<p>H4lyu: /* Numerical results */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small, the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failures of traditional methods ==<br />
Failures of traditional (derivative-based) methods fall into several categories:<br />
<br />
First, derivatives are local and not quite useful for comparative analysis. Since derivatives are local, if the input to compare is far from reference input, it may not be appropriate to assign contribution based on derivatives. An example can be image recognizition, where change in a single pixel makes no sense. In some cases where values in an input are discrete (not continuous), derivatives are not available.<br />
<br />
Second, using derivatives may mislead the analysis when derivatives have completely different behaviour in different parts of the output function. For example, on ReLU and (hard) max layers, taking derivative may lead to the conclusion that changes in all values make no difference as long as they don't touch the boundary.<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y / \Delta x_i} = \Delta y$. <br />
<br />
===Chain Rule===<br />
First, by the appendix of [1] we know the DeepLIFT multiplier $m$ behaves just like derivatives and satisfies the chain rule: If $z = z\left( y(x_1,...,x_N) \right)$ then<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
===Linear Rule===<br />
Second, let's consider a simple neuron $y = f(s)$ where $s = \sum_{i=1}^N w_i x_i$. We separate the positive and negative contribution of each $x_i$ and assign $m_{\Delta s / \Delta x_i}$ by the following rule, named as '''linear rule'''. The positive contribution of $x_i$ to $s$ is defined as<br />
<br />
$ \begin{align}<br />
\Delta s^{+} & = \sum_{i=1}^N 1_{\left\{ w_i \Delta x_i > 0 \right\}} w_i \Delta x_i \\<br />
& = \sum_{i=1}^N \left( 1_{\left\{ w_i > 0 \right\}} w_i \Delta x_{i}^{+} + 1_{\left\{ w_i < 0 \right\}} w_i \Delta x_{i}^{-} \right) \\<br />
\end{align} $<br />
<br />
We then assign the secant as $ m_{\Delta s^{+} / \Delta x_{i}^{+} } = 1_{\left\{ w_i > 0 \right\}} w_i $ and $ m_{\Delta s^{+} / \Delta x_{i}^{-} } = 1_{\left\{ w_i < 0 \right\}} w_i $. Similarly, we have $ m_{\Delta s^{-} / \Delta x_{i}^{+} } = 1_{\left\{ w_i < 0 \right\}} w_i $ and $ m_{\Delta s^{-} / \Delta x_{i}^{-} } = 1_{\left\{ w_i > 0 \right\}} w_i $. As to the occasion when $\Delta x_{i} = 0$, we let $ m_{\Delta s^{\pm} / \Delta x_{i}^{\pm} } = \frac{1}{2} w_i $.<br />
<br />
===Rescale Rule===<br />
For the function $f(s)$, it is possible to use the easiest method called '''rescale rule''':<br />
<br />
$ \Delta y = \frac{\Delta y}{\Delta s} \Delta s = \frac{\Delta y}{\Delta s} \left( \Delta s^{+} + \Delta s^{-} \right) $<br />
<br />
===Reveal-Cancel Rule===<br />
Rescale rule works for simple functions such as ReLU, but it does not always work well especially for some cases like pooling layers. To solve this we introduce '''reveal-cancel rule''':<br />
<br />
Suppose our reference input is $x_0$ and $s_0 = \sum_{i=1}^N w_i x_{0,i}$ is the sum in $y_0 = f(s_0)$. We define:<br />
<br />
$ \Delta y^{+} = \frac{1}{2} \left[ f(s_0 + \Delta s^{+}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{-}) \right] $<br />
<br />
$ \Delta y^{-} = \frac{1}{2} \left[ f(s_0 + \Delta s^{-}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{+}) \right] $<br />
<br />
$ m_{\Delta y / \Delta s^{+} } = \Delta y^{+} / \Delta s^{+} , m_{\Delta y / \Delta s^{-} } = \Delta y^{-} / \Delta s^{-} $<br />
<br />
===Adjustments for softmax layers===<br />
Since softmax layer normalizes its input, we can let the contribution of softmax layer $y=softmax(z)$ be that of its preceding layer $z = z(x)$ minus the average contribution of that preceding layer:<br />
<br />
$ C^{\prime}_{\Delta z_i / \Delta x} = C_{\Delta z_i / \Delta x} - \frac{1}{n} \sum_{j=1}^{N} C_{\Delta z_j / \Delta x} $<br />
<br />
Given the rules above, it is easy to calculate $ m_{\Delta y / \Delta x_i^{\pm} } $ for each $x_i$, and thus to calculate the DeepLIFT multiplier $m$ and contribution $C$ of a certain input and output compared to a given reference input $x_0$. It is suggested by the author that reference input should be case-specific and no general rule for choosing $x_0$ is currently available.<br />
<br />
== Numerical results ==<br />
The authors perform two main examples to test whether the suggestions of DeepLIFT method are correct.<br />
<br />
===MNIST handwriting re-morphing===<br />
Suppose we have a well-trained MNIST handwriting recognition model which can identify 0-9 correctly. Now we have a hand-written 8, and we want to know "how can we erase a part of this handwritten 8 to change it, say, to 3?"<br />
<br />
In this test, we take our reference input as all-zeros (black image) as this is the background of the images. We subtract contribution scores of pixels of class 8 to that of class 3, and erase 20% of pixels with highest contribution score. It appears that DeepLIFT can identify the left side of 8 as to be erased, which is far better that other gradient-based methods.<br />
<br />
===DNA motif detection===<br />
http://goo.gl/ qKb7pL<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28591STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-27T22:11:50Z<p>H4lyu: /* Failures of traditional methods */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small, the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failures of traditional methods ==<br />
Failures of traditional (derivative-based) methods fall into several categories:<br />
<br />
First, derivatives are local and not quite useful for comparative analysis. Since derivatives are local, if the input to compare is far from reference input, it may not be appropriate to assign contribution based on derivatives. An example can be image recognizition, where change in a single pixel makes no sense. In some cases where values in an input are discrete (not continuous), derivatives are not available.<br />
<br />
Second, using derivatives may mislead the analysis when derivatives have completely different behaviour in different parts of the output function. For example, on ReLU and (hard) max layers, taking derivative may lead to the conclusion that changes in all values make no difference as long as they don't touch the boundary.<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y / \Delta x_i} = \Delta y$. <br />
<br />
===Chain Rule===<br />
First, by the appendix of [1] we know the DeepLIFT multiplier $m$ behaves just like derivatives and satisfies the chain rule: If $z = z\left( y(x_1,...,x_N) \right)$ then<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
===Linear Rule===<br />
Second, let's consider a simple neuron $y = f(s)$ where $s = \sum_{i=1}^N w_i x_i$. We separate the positive and negative contribution of each $x_i$ and assign $m_{\Delta s / \Delta x_i}$ by the following rule, named as '''linear rule'''. The positive contribution of $x_i$ to $s$ is defined as<br />
<br />
$ \begin{align}<br />
\Delta s^{+} & = \sum_{i=1}^N 1_{\left\{ w_i \Delta x_i > 0 \right\}} w_i \Delta x_i \\<br />
& = \sum_{i=1}^N \left( 1_{\left\{ w_i > 0 \right\}} w_i \Delta x_{i}^{+} + 1_{\left\{ w_i < 0 \right\}} w_i \Delta x_{i}^{-} \right) \\<br />
\end{align} $<br />
<br />
We then assign the secant as $ m_{\Delta s^{+} / \Delta x_{i}^{+} } = 1_{\left\{ w_i > 0 \right\}} w_i $ and $ m_{\Delta s^{+} / \Delta x_{i}^{-} } = 1_{\left\{ w_i < 0 \right\}} w_i $. Similarly, we have $ m_{\Delta s^{-} / \Delta x_{i}^{+} } = 1_{\left\{ w_i < 0 \right\}} w_i $ and $ m_{\Delta s^{-} / \Delta x_{i}^{-} } = 1_{\left\{ w_i > 0 \right\}} w_i $. As to the occasion when $\Delta x_{i} = 0$, we let $ m_{\Delta s^{\pm} / \Delta x_{i}^{\pm} } = \frac{1}{2} w_i $.<br />
<br />
===Rescale Rule===<br />
For the function $f(s)$, it is possible to use the easiest method called '''rescale rule''':<br />
<br />
$ \Delta y = \frac{\Delta y}{\Delta s} \Delta s = \frac{\Delta y}{\Delta s} \left( \Delta s^{+} + \Delta s^{-} \right) $<br />
<br />
===Reveal-Cancel Rule===<br />
Rescale rule works for simple functions such as ReLU, but it does not always work well especially for some cases like pooling layers. To solve this we introduce '''reveal-cancel rule''':<br />
<br />
Suppose our reference input is $x_0$ and $s_0 = \sum_{i=1}^N w_i x_{0,i}$ is the sum in $y_0 = f(s_0)$. We define:<br />
<br />
$ \Delta y^{+} = \frac{1}{2} \left[ f(s_0 + \Delta s^{+}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{-}) \right] $<br />
<br />
$ \Delta y^{-} = \frac{1}{2} \left[ f(s_0 + \Delta s^{-}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{+}) \right] $<br />
<br />
$ m_{\Delta y / \Delta s^{+} } = \Delta y^{+} / \Delta s^{+} , m_{\Delta y / \Delta s^{-} } = \Delta y^{-} / \Delta s^{-} $<br />
<br />
===Adjustments for softmax layers===<br />
Since softmax layer normalizes its input, we can let the contribution of softmax layer $y=softmax(z)$ be that of its preceding layer $z = z(x)$ minus the average contribution of that preceding layer:<br />
<br />
$ C^{\prime}_{\Delta z_i / \Delta x} = C_{\Delta z_i / \Delta x} - \frac{1}{n} \sum_{j=1}^{N} C_{\Delta z_j / \Delta x} $<br />
<br />
Given the rules above, it is easy to calculate $ m_{\Delta y / \Delta x_i^{\pm} } $ for each $x_i$, and thus to calculate the DeepLIFT multiplier $m$ and contribution $C$ of a certain input and output compared to a given reference input $x_0$. It is suggested by the author that reference input should be case-specific and no general rule for choosing $x_0$ is currently available.<br />
<br />
== Numerical results ==<br />
to be done<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28590STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-27T22:09:43Z<p>H4lyu: /* Failure of traditional methods */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small, the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failures of traditional methods ==<br />
Failures of traditional (derivative-based) methods fall into several categories:<br />
<br />
First, derivatives are local and not quite useful for comparative analysis. Since derivatives are local, if the input to compare is far from reference input, it may not be appropriate to assign contribution based on derivatives. In some cases where values in an input are discrete (not continuous), derivatives are not available.<br />
<br />
Second, using derivatives may mislead the analysis when derivatives have completely different behaviour in different parts of the output function. For example, on ReLU and (hard) max layers, taking derivative may lead to the conclusion that changes in all values make no difference as long as they don't touch the boundary.<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y / \Delta x_i} = \Delta y$. <br />
<br />
===Chain Rule===<br />
First, by the appendix of [1] we know the DeepLIFT multiplier $m$ behaves just like derivatives and satisfies the chain rule: If $z = z\left( y(x_1,...,x_N) \right)$ then<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
===Linear Rule===<br />
Second, let's consider a simple neuron $y = f(s)$ where $s = \sum_{i=1}^N w_i x_i$. We separate the positive and negative contribution of each $x_i$ and assign $m_{\Delta s / \Delta x_i}$ by the following rule, named as '''linear rule'''. The positive contribution of $x_i$ to $s$ is defined as<br />
<br />
$ \begin{align}<br />
\Delta s^{+} & = \sum_{i=1}^N 1_{\left\{ w_i \Delta x_i > 0 \right\}} w_i \Delta x_i \\<br />
& = \sum_{i=1}^N \left( 1_{\left\{ w_i > 0 \right\}} w_i \Delta x_{i}^{+} + 1_{\left\{ w_i < 0 \right\}} w_i \Delta x_{i}^{-} \right) \\<br />
\end{align} $<br />
<br />
We then assign the secant as $ m_{\Delta s^{+} / \Delta x_{i}^{+} } = 1_{\left\{ w_i > 0 \right\}} w_i $ and $ m_{\Delta s^{+} / \Delta x_{i}^{-} } = 1_{\left\{ w_i < 0 \right\}} w_i $. Similarly, we have $ m_{\Delta s^{-} / \Delta x_{i}^{+} } = 1_{\left\{ w_i < 0 \right\}} w_i $ and $ m_{\Delta s^{-} / \Delta x_{i}^{-} } = 1_{\left\{ w_i > 0 \right\}} w_i $. As to the occasion when $\Delta x_{i} = 0$, we let $ m_{\Delta s^{\pm} / \Delta x_{i}^{\pm} } = \frac{1}{2} w_i $.<br />
<br />
===Rescale Rule===<br />
For the function $f(s)$, it is possible to use the easiest method called '''rescale rule''':<br />
<br />
$ \Delta y = \frac{\Delta y}{\Delta s} \Delta s = \frac{\Delta y}{\Delta s} \left( \Delta s^{+} + \Delta s^{-} \right) $<br />
<br />
===Reveal-Cancel Rule===<br />
Rescale rule works for simple functions such as ReLU, but it does not always work well especially for some cases like pooling layers. To solve this we introduce '''reveal-cancel rule''':<br />
<br />
Suppose our reference input is $x_0$ and $s_0 = \sum_{i=1}^N w_i x_{0,i}$ is the sum in $y_0 = f(s_0)$. We define:<br />
<br />
$ \Delta y^{+} = \frac{1}{2} \left[ f(s_0 + \Delta s^{+}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{-}) \right] $<br />
<br />
$ \Delta y^{-} = \frac{1}{2} \left[ f(s_0 + \Delta s^{-}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{+}) \right] $<br />
<br />
$ m_{\Delta y / \Delta s^{+} } = \Delta y^{+} / \Delta s^{+} , m_{\Delta y / \Delta s^{-} } = \Delta y^{-} / \Delta s^{-} $<br />
<br />
===Adjustments for softmax layers===<br />
Since softmax layer normalizes its input, we can let the contribution of softmax layer $y=softmax(z)$ be that of its preceding layer $z = z(x)$ minus the average contribution of that preceding layer:<br />
<br />
$ C^{\prime}_{\Delta z_i / \Delta x} = C_{\Delta z_i / \Delta x} - \frac{1}{n} \sum_{j=1}^{N} C_{\Delta z_j / \Delta x} $<br />
<br />
Given the rules above, it is easy to calculate $ m_{\Delta y / \Delta x_i^{\pm} } $ for each $x_i$, and thus to calculate the DeepLIFT multiplier $m$ and contribution $C$ of a certain input and output compared to a given reference input $x_0$. It is suggested by the author that reference input should be case-specific and no general rule for choosing $x_0$ is currently available.<br />
<br />
== Numerical results ==<br />
to be done<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28589STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-27T21:51:33Z<p>H4lyu: /* DeepLIFT scheme */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small, the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failure of traditional methods ==<br />
to be done<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y / \Delta x_i} = \Delta y$. <br />
<br />
===Chain Rule===<br />
First, by the appendix of [1] we know the DeepLIFT multiplier $m$ behaves just like derivatives and satisfies the chain rule: If $z = z\left( y(x_1,...,x_N) \right)$ then<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
===Linear Rule===<br />
Second, let's consider a simple neuron $y = f(s)$ where $s = \sum_{i=1}^N w_i x_i$. We separate the positive and negative contribution of each $x_i$ and assign $m_{\Delta s / \Delta x_i}$ by the following rule, named as '''linear rule'''. The positive contribution of $x_i$ to $s$ is defined as<br />
<br />
$ \begin{align}<br />
\Delta s^{+} & = \sum_{i=1}^N 1_{\left\{ w_i \Delta x_i > 0 \right\}} w_i \Delta x_i \\<br />
& = \sum_{i=1}^N \left( 1_{\left\{ w_i > 0 \right\}} w_i \Delta x_{i}^{+} + 1_{\left\{ w_i < 0 \right\}} w_i \Delta x_{i}^{-} \right) \\<br />
\end{align} $<br />
<br />
We then assign the secant as $ m_{\Delta s^{+} / \Delta x_{i}^{+} } = 1_{\left\{ w_i > 0 \right\}} w_i $ and $ m_{\Delta s^{+} / \Delta x_{i}^{-} } = 1_{\left\{ w_i < 0 \right\}} w_i $. Similarly, we have $ m_{\Delta s^{-} / \Delta x_{i}^{+} } = 1_{\left\{ w_i < 0 \right\}} w_i $ and $ m_{\Delta s^{-} / \Delta x_{i}^{-} } = 1_{\left\{ w_i > 0 \right\}} w_i $. As to the occasion when $\Delta x_{i} = 0$, we let $ m_{\Delta s^{\pm} / \Delta x_{i}^{\pm} } = \frac{1}{2} w_i $.<br />
<br />
===Rescale Rule===<br />
For the function $f(s)$, it is possible to use the easiest method called '''rescale rule''':<br />
<br />
$ \Delta y = \frac{\Delta y}{\Delta s} \Delta s = \frac{\Delta y}{\Delta s} \left( \Delta s^{+} + \Delta s^{-} \right) $<br />
<br />
===Reveal-Cancel Rule===<br />
Rescale rule works for simple functions such as ReLU, but it does not always work well especially for some cases like pooling layers. To solve this we introduce '''reveal-cancel rule''':<br />
<br />
Suppose our reference input is $x_0$ and $s_0 = \sum_{i=1}^N w_i x_{0,i}$ is the sum in $y_0 = f(s_0)$. We define:<br />
<br />
$ \Delta y^{+} = \frac{1}{2} \left[ f(s_0 + \Delta s^{+}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{-}) \right] $<br />
<br />
$ \Delta y^{-} = \frac{1}{2} \left[ f(s_0 + \Delta s^{-}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{+}) \right] $<br />
<br />
$ m_{\Delta y / \Delta s^{+} } = \Delta y^{+} / \Delta s^{+} , m_{\Delta y / \Delta s^{-} } = \Delta y^{-} / \Delta s^{-} $<br />
<br />
===Adjustments for softmax layers===<br />
Since softmax layer normalizes its input, we can let the contribution of softmax layer $y=softmax(z)$ be that of its preceding layer $z = z(x)$ minus the average contribution of that preceding layer:<br />
<br />
$ C^{\prime}_{\Delta z_i / \Delta x} = C_{\Delta z_i / \Delta x} - \frac{1}{n} \sum_{j=1}^{N} C_{\Delta z_j / \Delta x} $<br />
<br />
Given the rules above, it is easy to calculate $ m_{\Delta y / \Delta x_i^{\pm} } $ for each $x_i$, and thus to calculate the DeepLIFT multiplier $m$ and contribution $C$ of a certain input and output compared to a given reference input $x_0$. It is suggested by the author that reference input should be case-specific and no general rule for choosing $x_0$ is currently available.<br />
<br />
== Numerical results ==<br />
to be done<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28588STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-27T18:49:23Z<p>H4lyu: /* DeepLIFT scheme */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small, the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failure of traditional methods ==<br />
to be done<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y / \Delta x_i} = \Delta y$. <br />
<br />
First, by the appendix of [1] we know the DeepLIFT multiplier $m$ behaves just like derivatives and satisfies the chain rule: If $z = z\left( y(x_1,...,x_N) \right)$ then<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
Second, let's consider a simple neuron $y = f(s)$ where $s = \sum_{i=1}^N w_i x_i$. We separate the positive and negative contribution of each $x_i$ and assign $m_{\Delta s / \Delta x_i}$ by the following rule, named as '''linear rule'''. The positive contribution of $x_i$ to $s$ is defined as<br />
<br />
$ \begin{align}<br />
\Delta s^{+} & = \sum_{i=1}^N 1_{\left\{ w_i \Delta x_i > 0 \right\}} w_i \Delta x_i \\<br />
& = \sum_{i=1}^N \left( 1_{\left\{ w_i > 0 \right\}} w_i \Delta x_{i}^{+} + 1_{\left\{ w_i < 0 \right\}} w_i \Delta x_{i}^{-} \right) \\<br />
\end{align} $<br />
<br />
We then assign the secant as $ m_{\Delta s^{+} / \Delta x_{i}^{+} } = 1_{\left\{ w_i > 0 \right\}} w_i $ and $ m_{\Delta s^{+} / \Delta x_{i}^{-} } = 1_{\left\{ w_i < 0 \right\}} w_i $. Similarly, we have $ m_{\Delta s^{-} / \Delta x_{i}^{+} } = 1_{\left\{ w_i < 0 \right\}} w_i $ and $ m_{\Delta s^{-} / \Delta x_{i}^{-} } = 1_{\left\{ w_i > 0 \right\}} w_i $. As to the occasion when $\Delta x_{i} = 0$, we let $ m_{\Delta s^{\pm} / \Delta x_{i}^{\pm} } = \frac{1}{2} w_i $.<br />
<br />
For the function $f(s)$, it is possible to use the easiest method called '''rescale rule''':<br />
<br />
$ \Delta y = \frac{\Delta y}{\Delta s} \Delta s = \frac{\Delta y}{\Delta s} \left( \Delta s^{+} + \Delta s^{-} \right) $<br />
<br />
This method works for simple functions such as ReLU, but it does not always work well especially for some cases like pooling layers. To solve this we introduce '''reveal-cancel rule''':<br />
<br />
Suppose our reference input is $x_0$ and $s_0 = \sum_{i=1}^N w_i x_{0,i}$ is the sum in $y_0 = f(s_0)$. We define:<br />
<br />
$ \Delta y^{+} = \frac{1}{2} \left[ f(s_0 + \Delta s^{+}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{-}) \right] $<br />
<br />
$ \Delta y^{-} = \frac{1}{2} \left[ f(s_0 + \Delta s^{-}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{+}) \right] $<br />
<br />
$ m_{\Delta y / \Delta s^{+} } = \Delta y^{+} / \Delta s^{+} , m_{\Delta y / \Delta s^{-} } = \Delta y^{-} / \Delta s^{-} $<br />
<br />
Given the rules above, it is easy to calculate $ m_{\Delta y / \Delta x_i^{\pm} } $ for each $x_i$, and thus to calculate the DeepLIFT multiplier $m$ and contribution $C$ of a certain input and output compared to a given reference input $x_0$. It is suggested by the author that reference input should be case-specific and no general rule for choosing $x_0$ is currently available.<br />
<br />
== Numerical results ==<br />
to be done<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28587STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-27T18:48:49Z<p>H4lyu: /* DeepLIFT scheme */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small, the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failure of traditional methods ==<br />
to be done<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y / \Delta x_i} = \Delta y$. <br />
<br />
First, by the appendix of [1] we know the DeepLIFT multiplier $m$ behaves just like derivatives and satisfies the chain rule: If $z = z\left( y(x_1,...,x_N) \right)$ then<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
Second, let's consider a simple neuron $y = f(s)$ where $s = \sum_{i=1}^N w_i x_i$. We separate the positive and negative contribution of each $x_i$ and assign $m_{\Delta s / \Delta x_i}$ by the following rule, named as '''linear rule'''. The positive contribution of $x_i$ to $s$ is defined as<br />
<br />
$ \begin{align}<br />
\Delta s^{+} & = \sum_{i=1}^N 1_{\left\{ w_i \Delta x_i > 0 \right\}} w_i \Delta x_i \\<br />
& = \sum_{i=1}^N \left( 1_{\left\{ w_i > 0 \right\}} w_i \Delta x_{i}^{+} + 1_{\left\{ w_i < 0 \right\}} w_i \Delta x_{i}^{-} \right) \\<br />
\end{align} $<br />
<br />
We then assign the secant as $ m_{\Delta s^{+} / \Delta x_{i}^{+} } = 1_{\left\{ w_i > 0 \right\}} w_i $ and $ m_{\Delta s^{+} / \Delta x_{i}^{-} } = 1_{\left\{ w_i < 0 \right\}} w_i $. Similarly, we have $ m_{\Delta s^{-} / \Delta x_{i}^{+} } = 1_{\left\{ w_i < 0 \right\}} w_i $ and $ m_{\Delta s^{-} / \Delta x_{i}^{-} } = 1_{\left\{ w_i > 0 \right\}} w_i $. As to the occasion when $\Delta x_{i} = 0$, we let $ m_{\Delta s^{\pm} / \Delta x_{i}^{\pm} } = \frac{1}{2} w_i $.<br />
<br />
For the function $f(s)$, it is possible to use the easiest method called '''rescale rule''':<br />
<br />
$ \Delta y = \frac{\Delta y}{\Delta s} \Delta s = \frac{\Delta y}{\Delta s} \left( \Delta s^{+} + \Delta s^{-} \right) $<br />
<br />
This method works for simple functions such as ReLU, but it does not always work well especially for some cases like pooling layers. To solve this we introduce '''reveal-cancel rule''':<br />
<br />
Suppose our reference input is $x_0$ and $s_0 = \sum_{i=1}^N w_i x_{0,i}$ is the sum in $y_0 = f(s_0)$. We define:<br />
<br />
$ \Delta y^{+} = \frac{1}{2} \left[ f(s_0 + \Delta s^{+}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{-}) \right] $<br />
$ \Delta y^{-} = \frac{1}{2} \left[ f(s_0 + \Delta s^{-}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{+}) \right] $<br />
$ m_{\Delta y / \Delta s^{+} } = \Delta y^{+} / \Delta s^{+} , m_{\Delta y / \Delta s^{-} } = \Delta y^{-} / \Delta s^{-} $<br />
<br />
Given the rules above, it is easy to calculate $ m_{\Delta y / \Delta x_i^{\pm} } $ for each $x_i$, and thus to calculate the DeepLIFT multiplier $m$ and contribution $C$ of a certain input and output compared to a given reference input $x_0$. It is suggested by the author that reference input should be case-specific and no general rule for choosing $x_0$ is currently available.<br />
<br />
== Numerical results ==<br />
to be done<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28586STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-27T18:48:05Z<p>H4lyu: /* DeepLIFT scheme */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small, the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failure of traditional methods ==<br />
to be done<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y / \Delta x_i} = \Delta y$. <br />
<br />
First, by the appendix of [1] we know the DeepLIFT multiplier $m$ behaves just like derivatives and satisfies the chain rule: If $z = z\left( y(x_1,...,x_N) \right)$ then<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
So we can utilize back-propagation to calculate $m$ over the whole network.<br />
<br />
Second, let's consider a simple neuron $y = f(s)$ where $s = \sum_{i=1}^N w_i x_i$. We separate the positive and negative contribution of each $x_i$ and assign $m_{\Delta s / \Delta x_i}$ by the following rule, named as '''linear rule'''. The positive contribution of $x_i$ to $s$ is defined as<br />
<br />
$ \begin{align}<br />
\Delta s^{+} & = \sum_{i=1}^N 1_{\left\{ w_i \Delta x_i > 0 \right\}} w_i \Delta x_i \\<br />
& = \sum_{i=1}^N \left( 1_{\left\{ w_i > 0 \right\}} w_i \Delta x_{i}^{+} + 1_{\left\{ w_i < 0 \right\}} w_i \Delta x_{i}^{-} \right) \\<br />
\end{align} $<br />
<br />
We then assign the secant as $ m_{\Delta s^{+} / \Delta x_{i}^{+} } = 1_{\left\{ w_i > 0 \right\}} w_i $ and $ m_{\Delta s^{+} / \Delta x_{i}^{-} } = 1_{\left\{ w_i < 0 \right\}} w_i $. Similarly, we have $ m_{\Delta s^{-} / \Delta x_{i}^{+} } = 1_{\left\{ w_i < 0 \right\}} w_i $ and $ m_{\Delta s^{-} / \Delta x_{i}^{-} } = 1_{\left\{ w_i > 0 \right\}} w_i $. As to the occasion when $\Delta x_{i} = 0$, we let $ m_{\Delta s^{\pm} / \Delta x_{i}^{\pm} } = \frac{1}{2} w_i $.<br />
<br />
For the function $f(s)$, it is possible to use the easiest method called '''rescale rule''':<br />
<br />
$ \Delta y = \frac{\Delta y}{\Delta s} \Delta s = \frac{\Delta y}{\Delta s} \left( \Delta s^{+} + \Delta s^{-} \right) $<br />
<br />
This method works for simple functions such as ReLU, but it does not always work well especially for some cases like pooling layers. To solve this we introduce '''reveal-cancel rule''':<br />
<br />
Suppose our reference input is $x_0$ and $s_0 = \sum_{i=1}^N w_i x_{0,i}$ is the sum in $y_0 = f(s_0)$. We define:<br />
<br />
$ \Delta y^{+} = \frac{1}{2} \left[ f(s_0 + \Delta s^{+}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{-}) \right] $<br />
$ \Delta y^{-} = \frac{1}{2} \left[ f(s_0 + \Delta s^{-}) - f(s_0) \right] + \frac{1}{2} \left[ f(s_0 + \Delta s^{+} + \Delta s^{-}) - f(s_0 + \Delta s^{+}) \right] $<br />
$ m_{\Delta y / \Delta s^{+} } = \Delta y^{+} / \Delta s^{+} , m_{\Delta y / \Delta s^{-} } = \Delta y^{-} / \Delta s^{-} $<br />
<br />
Given the rules above, it is easy to calculate $ m_{\Delta y / \Delta x_i^{\pm} } $ for each $x_i$, and thus to calculate the DeepLIFT multiplier $m$ and contribution $C$ of a certain input and output compared to a given reference input $x_0$. It is suggested by the author that reference input should be case-specific and no general rule for choosing $x_0$ is currently available.<br />
<br />
== Numerical results ==<br />
to be done<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28585STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-27T18:18:54Z<p>H4lyu: /* DeepLIFT scheme */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small, the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failure of traditional methods ==<br />
to be done<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y / \Delta x_i} = \Delta y$. <br />
<br />
First, by the appendix of [1] we know the "DeepLIFT secant" $m$ satisfies the chain rule: If $z = z\left( y(x_1,...,x_N) \right)$ then<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
So we can utilize back-propagation to calculate $m$ over the whole network.<br />
<br />
Second, let's consider a simple neuron $y = f(s)$ where $s = \sum_{i=1}^N w_i x_i$. We separate the positive and negative contribution of each $x_i$ and assign $m_{\Delta s / \Delta x_i}$ by the following scheme. The positive contribution of $x_i$ to $s$ is defined as<br />
<br />
$ \begin{align}<br />
\Delta s^{+} & = \sum_{i=1}^N 1_{\left\{ w_i \Delta x_i > 0 \right\}} w_i \Delta x_i \\<br />
& = \sum_{i=1}^N \left( 1_{\left\{ w_i > 0 \right\}} w_i \Delta x_{i}^{+} + 1_{\left\{ w_i < 0 \right\}} w_i \Delta x_{i}^{-} \right) \\<br />
\end{align} $<br />
<br />
== Numerical results ==<br />
to be done<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28571STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-27T04:42:28Z<p>H4lyu: /* Sensitivity Analysis */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small, the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failure of traditional methods ==<br />
to be done<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y \/ \Delta x_i} = \Delta y$. By the appendix of [1] we know the "DeepLIFT secant" $m$ satisfies the chain rule:<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
== Numerical results ==<br />
to be done<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685</div>H4lyuhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=STAT946F17/_Learning_Important_Features_Through_Propagating_Activation_Differences&diff=28570STAT946F17/ Learning Important Features Through Propagating Activation Differences2017-10-27T04:42:02Z<p>H4lyu: /* Sensitivity Analysis */</p>
<hr />
<div>This is a summary of ICML 2017 paper [1].<br />
<br />
== Introduction ==<br />
Deep neuron network is purported for its "black box" nature which is a barrier to adoption in applications where interpretability is essential. Also, the "black box" nature brings difficulty for analyzing and improving the structure of the model. In our topic paper, DeepLIFT method is presented to decompose the output of a neural network on a specific input by backpropagating the contributions of all neurons in the network to every feature of the input. This is a form of sensitivity analysis and helps understand the model better.<br />
<br />
== Sensitivity Analysis ==<br />
Sensitivity Analysis is a concept in risk management and actuarial science. According to [[http://www.investopedia.com/terms/s/sensitivityanalysis.asp Invectopedia]], a sensitivity analysis is a technique used to determine how changes in an independent variable influence a particular dependent variable under given assumptions. This technique is used within specific boundaries that depend on one or more input variables, such as the effect that changes in interest rates have on bond prices. <br />
<br />
In our topic, we have a well-trained deep neuron network with two high-dimensional input vectors $x_0, x_1$ and output $y_0=f(x_0), y_1=f(x_1)$. Now we know $x_1$ is a perturbation of $x_0$ and we want to know which element in $x_1 - x_0$ contributes the most to $y_1 - y_0$.<br />
<br />
As one can imagine, if $\left| x_1 - x_0 \right|$ is small the most "crude" approximation is to calculate<br />
<br />
$\left . \frac{\partial y}{\partial x} \right|_{x = x_0} $<br />
<br />
and get its largest element in terms of absolute value. This is well feasible because back-propagation enables us to calculate the differentials layer by layer. However, this method doesn't always work well.<br />
<br />
== Failure of traditional methods ==<br />
to be done<br />
<br />
== DeepLIFT scheme ==<br />
DeepLIFT assigns contribution scores $C_{\Delta y / \Delta x_i} = m_{\Delta y / \Delta x_i} \Delta x_i$ to each element $x_i$ so that sum of contribution scores satisfies $\sum_{i=1}^N C_{\Delta y \/ \Delta x_i} = \Delta y$. By the appendix of [1] we know the "DeepLIFT secant" $m$ satisfies the chain rule:<br />
<br />
$m_{\Delta z / \Delta x_i} = m_{\Delta z / \Delta y} m_{\Delta y / \Delta x_i} $<br />
<br />
== Numerical results ==<br />
to be done<br />
<br />
== References ==<br />
[1] Shrikumar, A., Greenside, P., and Kundaje, A. Learning Important Features Through Propagating Activation Differences. arXiv:1704.02685</div>H4lyu