Gradient Episodic Memory for Continual Learning

Presented by

• Yu Xuan Lee
• Tsen Yee Heng

Background and Introduction

Supervised learning consist of a training set [math]\displaystyle{ D_{tx}=(x_i,y_i)^n_{i=1} }[/math], where [math]\displaystyle{ x_i \in \mathcal{X} }[/math] and [math]\displaystyle{ y_i \in \mathcal{Y} }[/math]. Empirical Risk Minimization (ERM) is one of the common supervised learning method used to minimize a loss function by having multiple passes over the training set.

[math]\displaystyle{ \frac{1}{|D_{tr}|}\textstyle \sum_{(x_i,y_i) \in D_{tr}} \ell (f(x_i),y_i) }[/math]

where [math]\displaystyle{ \ell :\mathcal {Y} \times \mathcal {Y} \to [0, \infty) }[/math]

Different to machine learning, datas are being observed sequentially, occurred recurrently, and stored limitedly for learning humans. Thus, the iid assumption is not applicable to ERM. One of the characteristics of ERM is "catastrophic forgetting", which is the problem of recalling past knowledge upon acquiring new ones. To overcome this problem, Gradient Episodic Memory (GEM) is introduced to alleviates forgetting on previous acquired knowledge, while solving new problems more efficiently.

Framework for Continual Learning

The feature vector [math]\displaystyle{ x_i \in \mathcal{X}_t }[/math], task descriptor [math]\displaystyle{ t_i \in \mathcal{T} }[/math], and target vector [math]\displaystyle{ y_i \in \mathcal{Y}_t }[/math] are the three main components of a continuum of data. Note that the continuum is locally iid where for every [math]\displaystyle{ (x_i, t_i, y_i) }[/math]

[math]\displaystyle{ (x_i,y_i) \overset{iid}{\sim} P_{t_i}(X,Y) }[/math]

The main mathematical purpose of continual learning is to obtain [math]\displaystyle{ f: \mathcal{X} \times \mathcal{Y} }[/math] where a target vector [math]\displaystyle{ y }[/math] must be inquired using a test pair [math]\displaystyle{ (x,t) }[/math].

Task Descriptor

Task descriptors are structured objects, describing how to solve each [math]\displaystyle{ i }[/math]-th task. They are integers [math]\displaystyle{ t_i=i \in \mathbb{Z} }[/math] which occurs in a collection where [math]\displaystyle{ t_1,...,t_n \in \mathcal{T} }[/math]. Most importantly, they could distinguish every same input [math]\displaystyle{ x_i }[/math] that have different target. To conclude, task descriptors plays the part of carrying crucial information of the example and distinguishing different learning environment for similar examples.

Training Protocol

The target setting for continual learning are as follow:

• Large task quantity
• Small quantity of training examples for each task
• Examples for each tasks being observed only once
• Outcome of transfer and forgetting being concluded

To perform this, each example were only given once to the learner in one at a time in sequence. In this case, learner gets information in [math]\displaystyle{ (x_i,t_i,y_i) }[/math] form with no duplication.

Evaluation Metrics

The capability of transferring knowledge across tasks are very important in addition to results across each tasks. First of all, transferring knowledge are categorized as follow:

• Backward transfer (BWT) This is the difference of judgement towards previously encountered task [math]\displaystyle{ k }[/math] after learning new task [math]\displaystyle{ t }[/math], noted as [math]\displaystyle{ k \prec t }[/math]. Within backward transfer, there are two categories, positive backward transfer and negative backward transfer. Positive backward transfer shows a better judgement towards previously encountered task [math]\displaystyle{ k }[/math] after learning new task [math]\displaystyle{ t }[/math]. Contrarily, negative backward transfer shows the opposite. Also, do note that catastrophic forgetting happens due to extensive negative backward transfer.
• Forward transfer (FWT) Opposite to BWT, FWT shows judgement towards new task [math]\displaystyle{ t }[/math] after learning task [math]\displaystyle{ t }[/math], noted as [math]\displaystyle{ k \succ t }[/math]. Positive forward transfer is one way of forward transfer.

Given a test set of [math]\displaystyle{ T }[/math], we would learn task [math]\displaystyle{ t_i }[/math] and observe its performance towards all [math]\displaystyle{ T }[/math] tasks. A matrix [math]\displaystyle{ R_{i,j} }[/math] as test classification accuracy of the model on task [math]\displaystyle{ t_j }[/math] after observing the last sample from task [math]\displaystyle{ t_j }[/math] is constructed, where [math]\displaystyle{ R \in \mathbb{R} ^{T \times T} }[/math]. Note that [math]\displaystyle{ \bar b\ }[/math] is the vector of test accuracies for each task at random initialization. The function for Average Accuracy (ACC), Backward Transfer (BWT) and Forward Transfer (FWT) are shown below:

[math]\displaystyle{ ACC = \frac{1}{T} \sum_{i=1}^T R_{T,i} }[/math]

[math]\displaystyle{ BWT = \frac{1}{T-1} \sum_{i=1}^{T-1} R_{T,i} - R_{i,i} }[/math]

[math]\displaystyle{ FWT = \frac{1}{T-1} \sum_{i=2}^{T} R_{i-1,i- \bar b\ _i} }[/math]

Note that if ACC happens to be similar for both models, model with higher BWT and FWT values are more desired.

Gradient Episodic Memory (GEM)

Episodic memory [math]\displaystyle{ M_t }[/math] is very important in GEM, it contains information on examples on task [math]\displaystyle{ t }[/math] and it is indicated from the integer task descriptors. So practically, we would minimize catastrophic forgetting by using the episodic memory efficiently. Note that learner is assumed to have limited memory locations [math]\displaystyle{ M }[/math]. Hence, the amount located for each task is calculated as [math]\displaystyle{ m=\frac{M}{T} }[/math] which results to more memory for the final [math]\displaystyle{ m }[/math] examples for each tasks. To calculate the loss of memories from the [math]\displaystyle{ k }[/math]-th task, assuming predictors [math]\displaystyle{ f_ \theta }[/math] parameterized by [math]\displaystyle{ \theta \in \mathbb{R} ^p }[/math], we have the following equation:

[math]\displaystyle{ \ell (f_\theta, \mathcal{M}_k)=\frac{1}{|\mathcal{M}_k|} \sum_{(x_i,k,y_i) \in \mathcal{M}_k} \ell(f_ \theta (x_i,k),y_i) }[/math]

The above equation will be treated as inequality constraint and a decrease in the equation would be in favour instead of increase. So we would use [math]\displaystyle{ (x,t,y) }[/math] to minimize the following equation:

[math]\displaystyle{ mimimize_\theta \space \space \ell(f_\theta(x,t),y) }[/math]

[math]\displaystyle{ subject\space to \space \space \ell (f_\theta,\mathcal{M}_k) \le \ell(f_\theta^{t-1},\mathcal{M}_k) \space\space for \space all \space k\lt t }[/math]

where [math]\displaystyle{ f_\theta^{t-1} }[/math] is the predictor state at the end of learning of task [math]\displaystyle{ t-1 }[/math].

To efficiently solve the above equation, three ideas are proposed:

• Delete old predictors [math]\displaystyle{ f_\theta^{t-1} }[/math]. This is because the old predictors remain unchanged for each update of g.
• Functions are locally linear.
• Loss of previous tasks could be calculated using the angle between loss gradient vector and proposed update.

With the above ideas, the loss function is further improved as follow:

[math]\displaystyle{ \langle g,g_k \rangle := \langle \frac{\partial \ell(f_\theta(x,t),y)}{\partial \theta}, \frac{\partial \ell(f_\theta,\mathcal{M}_k)}{\partial \theta} \rangle \ge 0, \space for \space all \space k\lt t. }[/math]

However, if there is at least one violation in the equality constraint, we would overcome this by projecting the gradient [math]\displaystyle{ g }[/math] to the closest gradient [math]\displaystyle{ \tilde{g} }[/math] satisfying all the constraints. The optimization problem becomes

[math]\displaystyle{ minimize_{ \tilde{g} } \space \space \frac{1}{2}\parallel g - \tilde{g} \parallel _2^2 }[/math]

[math]\displaystyle{ subject \space to \space \space \langle \tilde{g},g_k \rangle \ge 0 \space \space for \space all \space k\lt t }[/math]

Therefore, the primal GEM Quadratic Program (QP) is

[math]\displaystyle{ minimize_z \space \space \frac{1}{2}z^Tz - g^Tz+\frac{1}{2}g^Tg }[/math]

[math]\displaystyle{ subject \space to \space \space Gz \ge 0, }[/math]

Dual of the GEM QP is

[math]\displaystyle{ minimize_v \space \space \frac{1}{2}v^TGG^Tv + g^TG^Tv }[/math]

[math]\displaystyle{ subject \space to \space \space v\ge 0 }[/math]

By solving [math]\displaystyle{ v^* }[/math], we could obtain the projected gradient update [math]\displaystyle{ \tilde{g}=G^Tv^* + g }[/math].