http://wiki.math.uwaterloo.ca/statwiki/api.php?action=feedcontributions&user=Pa2forsy&feedformat=atomstatwiki - User contributions [US]2023-01-31T21:08:43ZUser contributionsMediaWiki 1.28.3http://wiki.math.uwaterloo.ca/statwiki/index.php?title=MarrNet:_3D_Shape_Reconstruction_via_2.5D_Sketches&diff=36164MarrNet: 3D Shape Reconstruction via 2.5D Sketches2018-04-04T20:55:20Z<p>Pa2forsy: </p>
<hr />
<div>= Introduction =<br />
Humans are able to quickly recognize 3D shapes from images, even in spite of drastic differences in object texture, material, lighting, and background.<br />
<br />
[[File:marrnet_intro_image.png|700px|thumb|center|Objects in real images. The appearance of the same shaped object varies based on colour, texture, lighting, background, etc. However, the 2.5D sketches (e.g. depth or normal maps) of the object remain constant, and can be seen as an abstraction of the object which is used to reconstruct the 3D shape.]]<br />
<br />
In this work, the authors propose a novel end-to-end trainable model that sequentially estimates 2.5D sketches and 3D object shape from images and also enforce re-projection consistency between the 3D shape and the estimated sketch. 2.5D is the construction of a 3D environment using 2D retina projection along with depth perception obtained from the image. The two step approach makes the network more robust to differences in object texture, material, lighting and background. Based on the idea from [Marr, 1982] that human 3D perception relies on recovering 2.5D sketches, which include depth maps (contains information related to the distance of surfaces from a viewpoint) and surface normal maps (technique for adding the illusion of depth details to surfaces using an image's RGB information), the authors design an end-to-end trainable pipeline which they call MarrNet. MarrNet first estimates depth, normal maps, and silhouette, followed by a 3D shape. MarrNet uses an encoder-decoder structure for the sub-components of the framework. <br />
<br />
The authors claim several unique advantages to their method. Single image 3D reconstruction is a highly under-constrained problem, requiring strong prior knowledge of object shapes. As well, accurate 3D object annotations using real images are not common, and many previous approaches rely on purely synthetic data. However, most of these methods suffer from domain adaptation due to imperfect rendering.<br />
<br />
Using 2.5D sketches can alleviate the challenges of domain transfer. It is straightforward to generate perfect object surface normals and depths using a graphics engine. Since 2.5D sketches contain only depth, surface normal, and silhouette information, the second step of recovering 3D shape can be trained purely from synthetic data. As well, the introduction of differentiable constraints between 2.5D sketches and 3D shape makes it possible to fine-tune the system, even without any annotations.<br />
<br />
The framework is evaluated on both synthetic objects from ShapeNet, and real images from PASCAL 3D+, showing good qualitative and quantitative performance in 3D shape reconstruction.<br />
<br />
= Related Work =<br />
<br />
== 2.5D Sketch Recovery ==<br />
Researchers have explored recovering 2.5D information from shading, texture, and colour images in the past. More recently, the development of depth sensors has led to the creation of large RGB-D datasets, and papers on estimating depth, surface normals, and other intrinsic images using deep networks. While this method employs 2.5D estimation, the final output is a full 3D shape of an object.<br />
<br />
[[File:2-5d_example.PNG|700px|thumb|center|Results from the paper: Learning Non-Lambertian Object Intrinsics across ShapeNet Categories. The results show that neural networks can be trained to recover 2.5D information from an image. The top row predicts the albedo and the bottom row predicts the shading. It can be observed that the results are still blurry and the fine details are not fully recovered.]]<br />
<br />
== Single Image 3D Reconstruction ==<br />
The development of large-scale shape repositories like ShapeNet has allowed for the development of models encoding shape priors for single image 3D reconstruction. These methods normally regress voxelized 3D shapes, relying on synthetic data or 2D masks for training. A voxel is an abbreviation for volume element, the three-dimensional version of a pixel. The formulation in the paper tackles domain adaptation better, since the network can be fine-tuned on images without any annotations.<br />
<br />
== 2D-3D Consistency ==<br />
Intuitively, the 3D shape can be constrained to be consistent with 2D observations. This idea has been explored for decades, and has been widely used in 3D shape completion with the use of depths and silhouettes. A few recent papers [5,6,7,8] discussed enforcing differentiable 2D-3D constraints between shape and silhouettes to enable joint training of deep networks for the task of 3D reconstruction. In this work, this idea is exploited to develop differentiable constraints for consistency between the 2.5D sketches and 3D shape.<br />
<br />
= Approach =<br />
The 3D structure is recovered from a single RGB view using three steps, shown in the figure below. The first step estimates 2.5D sketches, including depth, surface normal, and silhouette of the object. The second step estimates a 3D voxel representation of the object. The third step uses a reprojection consistency function to enforce the 2.5D sketch and 3D structure alignment.<br />
<br />
[[File:marrnet_model_components.png|700px|thumb|center|MarrNet architecture. 2.5D sketches of normals, depths, and silhouette are first estimated. The sketches are then used to estimate the 3D shape. Finally, re-projection consistency is used to ensure consistency between the sketch and 3D output.]]<br />
<br />
== 2.5D Sketch Estimation ==<br />
The first step takes a 2D RGB image and predicts the 2.5 sketch with surface normal, depth, and silhouette of the object. The goal is to estimate intrinsic object properties from the image, while discarding non-essential information such as texture and lighting. An encoder-decoder architecture is used. The encoder is a A ResNet-18 network, which takes a 256 x 256 RGB image and produces 512 feature maps of size 8 x 8. The decoder is four sets of 5 x 5 fully convolutional and ReLU layers, followed by four sets of 1 x 1 convolutional and ReLU layers. The output is 256 x 256 resolution depth, surface normal, and silhouette images.<br />
<br />
== 3D Shape Estimation ==<br />
The second step estimates a voxelized 3D shape using the 2.5D sketches from the first step. The focus here is for the network to learn the shape prior that can explain the input well, and can be trained on synthetic data without suffering from the domain adaptation problem since it only takes in surface normal and depth images as input. The network architecture is inspired by the TL[10] network, and 3D-VAE-GAN, with an encoder-decoder structure. The normal and depth image, masked by the estimated silhouette, are passed into 5 sets of convolutional, ReLU, and pooling layers, followed by two fully connected layers, with a final output width of 200. The 200-dimensional vector is passed into a decoder of 5 fully convolutional and ReLU layers, outputting a 128 x 128 x 128 voxelized estimate of the input.<br />
<br />
== Re-projection Consistency ==<br />
The third step consists of a depth re-projection loss and surface normal re-projection loss. Here, <math>v_{x, y, z}</math> represents the value at position <math>(x, y, z)</math> in a 3D voxel grid, with <math>v_{x, y, z} \in [0, 1] ∀ x, y, z</math>. <math>d_{x, y}</math> denotes the estimated depth at position <math>(x, y)</math>, <math>n_{x, y} = (n_a, n_b, n_c)</math> denotes the estimated surface normal. Orthographic projection is used.<br />
<br />
[[File:marrnet_reprojection_consistency.png|700px|thumb|center|Reprojection consistency for voxels. Left and middle: criteria for depth and silhouettes. Right: criterion for surface normals]]<br />
<br />
=== Depths ===<br />
The voxel with depth <math>v_{x, y}, d_{x, y}</math> should be 1, while all voxels in front of it should be 0. This ensures the estimated 3D shape matches the estimated depth values. The projected depth loss and its gradient are defined as follows:<br />
<br />
<math><br />
L_{depth}(x, y, z)=<br />
\left\{<br />
\begin{array}{ll}<br />
v^2_{x, y, z}, & z < d_{x, y} \\<br />
(1 - v_{x, y, z})^2, & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
<math><br />
\frac{∂L_{depth}(x, y, z)}{∂v_{x, y, z}} =<br />
\left\{<br />
\begin{array}{ll}<br />
2v{x, y, z}, & z < d_{x, y} \\<br />
2(v_{x, y, z} - 1), & z = d_{x, y} \\<br />
0, & z > d_{x, y} \\<br />
\end{array}<br />
\right.<br />
</math><br />
<br />
When <math>d_{x, y} = \infty</math>, all voxels in front of it should be 0.<br />
<br />
=== Surface Normals ===<br />
Since vectors <math>n_{x} = (0, −n_{c}, n_{b})</math> and <math>n_{y} = (−n_{c}, 0, n_{a})</math> are orthogonal to the normal vector <math>n_{x, y} = (n_{a}, n_{b}, n_{c})</math>, they can be normalized to obtain <math>n’_{x} = (0, −1, n_{b}/n_{c})</math> and <math>n’_{y} = (−1, 0, n_{a}/n_{c})</math> on the estimated surface plane at <math>(x, y, z)</math>. The projected surface normal tried to guarantee voxels at <math>(x, y, z) ± n’_{x}</math> and <math>(x, y, z) ± n’_{y}</math> should be 1 to match the estimated normal. The constraints are only applied when the target voxels are inside the estimated silhouette.<br />
<br />
The projected surface normal loss is defined as follows, with <math>z = d_{x, y}</math>:<br />
<br />
<math><br />
L_{normal}(x, y, z) =<br />
(1 - v_{x, y-1, z+\frac{n_b}{n_c}})^2 + (1 - v_{x, y+1, z-\frac{n_b}{n_c}})^2 + <br />
(1 - v_{x-1, y, z+\frac{n_a}{n_c}})^2 + (1 - v_{x+1, y, z-\frac{n_a}{n_c}})^2<br />
</math><br />
<br />
Gradients along x are:<br />
<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x-1, y, z+\frac{n_a}{n_c}}} = 2(v_{x-1, y, z+\frac{n_a}{n_c}}-1)<br />
</math><br />
and<br />
<math><br />
\frac{dL_{normal}(x, y, z)}{dv_{x+1, y, z-\frac{n_a}{n_c}}} = 2(v_{x+1, y, z-\frac{n_a}{n_c}}-1)<br />
</math><br />
<br />
Gradients along y are similar to x.<br />
<br />
= Training =<br />
The 2.5D and 3D estimation components are first pre-trained separately on synthetic data from ShapeNet, and then fine-tuned on real images.<br />
<br />
For pre-training, the 2.5D sketch estimator is trained on synthetic ShapeNet depth, surface normal, and silhouette ground truth, using an L2 loss. The 3D estimator is trained with ground truth voxels using a cross-entropy loss.<br />
<br />
Reprojection consistency loss is used to fine-tune the 3D estimation using real images, using the predicted depth, normals, and silhouette. A straightforward implementation leads to shapes that explain the 2.5D sketches well, but lead to unrealistic 3D appearance due to overfitting.<br />
<br />
Instead, the decoder of the 3D estimator is fixed, and only the encoder is fine-tuned. The model is fine-tuned separately on each image for 40 iterations, which takes up to 10 seconds on the GPU. Without fine-tuning, testing time takes around 100 milliseconds. SGD is used for optimization with batch size of 4, learning rate of 0.001, and momentum of 0.9.<br />
<br />
= Evaluation =<br />
Qualitative and quantitative results are provided using different variants of the framework. The framework is evaluated on both synthetic and real images on three datasets; ShapeNet, PASCAL 3D+, and IKEA. Intersection-over-Union (IoU) is the main measurement of comparison between the models. However the authors note that models which focus on the IoU metric fail to capture the details of the object they are trying to model, disregarding details to focus on the overall shape. To counter this drawback they poll people on which reconstruction is preferred. IoU is also computationally inefficient since it has to check over all possible scales.<br />
<br />
== ShapeNet ==<br />
Synthesized images of 6,778 chairs from ShapeNet are rendered from 20 random viewpoints. The chairs are placed in front of random background from the SUN dataset, and the RGB, depth, normal, and silhouette images are rendered using the physics-based renderer Mitsuba for more realistic images.<br />
<br />
=== Method ===<br />
MarrNet is trained without the final fine-tuning stage, since 3D shapes are available. A baseline is created that directly predicts the 3D shape using the same 3D shape estimator architecture with no 2.5D sketch estimation.<br />
<br />
=== Results ===<br />
The baseline output is compared to the full framework, and the figure below shows that MarrNet provides model outputs with more details and smoother surfaces than the baseline. The estimated normal and depth images are able to extract intrinsic information about object shape while leaving behind non-essential information such as textures from the original images. Quantitatively, the full model also achieves 0.57 integer over union score (which compares the overlap of the predicted model and ground truth), which is higher than the direct prediction baseline.<br />
<br />
[[File:marrnet_shapenet_results.png|700px|thumb|center|ShapeNet results.]]<br />
<br />
== PASCAL 3D+ ==<br />
Rough 3D models are provided from real-life images.<br />
<br />
=== Method ===<br />
Each module is pre-trained on the ShapeNet dataset, and then fine-tuned on the PASCAL 3D+ dataset. Three variants of the model are tested. The first is trained using ShapeNet data only with no fine-tuning. The second is fine-tuned without fixing the decoder. The third is fine-tuned with a fixed decoder.<br />
<br />
=== Results ===<br />
The figure below shows the results of the ablation study. The model trained only on synthetic data provides reasonable estimates. However, fine-tuning without fixing the decoder leads to impossible shapes from certain views. The third model keeps the shape prior, providing more details in the final shape.<br />
<br />
[[File:marrnet_pascal_3d_ablation.png|600px|thumb|center|Ablation studies using the PASCAL 3D+ dataset.]]<br />
<br />
Additional comparisons are made with the state-of-the-art (DRC) on the provided ground truth shapes. MarrNet achieves 0.39 IoU, while DRC achieves 0.34. Since PASCAL 3D+ only has rough annotations, with only 10 CAD chair models for all images, computing IoU with these shapes is not very informative. Instead, human studies are conducted and MarrNet reconstructions are preferred 74% of the time over DRC, and 42% of the time to ground truth. This shows how MarrNet produces nice shapes and also highlights the fact that ground truth shapes are not very good.<br />
<br />
[[File:human_studies.png|400px|thumb|center|Human preferences on chairs in PASCAL 3D+ (Xiang et al. 2014). The numbers show the percentage of how often humans prefered the 3D shape from DRC (state-of-the-art), MarrNet, or GT.]]<br />
<br />
<br />
[[File:marrnet_pascal_3d_drc_comparison.png|600px|thumb|center|Comparison between DRC and MarrNet results.]]<br />
<br />
Several failure cases are shown in the figure below. Specifically, the framework does not seem to work well on thin structures.<br />
<br />
[[File:marrnet_pascal_3d_failure_cases.png|500px|thumb|center|Failure cases on PASCAL 3D+. The algorithm cannot recover thin structures.]]<br />
<br />
== IKEA ==<br />
This dataset contains images of IKEA furniture, with accurate 3D shape and pose annotations. Objects are often heavily occluded or truncated.<br />
<br />
=== Results ===<br />
Qualitative results are shown in the figure below. The model is shown to deal with mild occlusions in real life scenarios. Human studes show that MarrNet reconstructions are preferred 61% of the time to 3D-VAE-GAN.<br />
<br />
[[File:marrnet_ikea_results.png|700px|thumb|center|Results on chairs in the IKEA dataset, and comparison with 3D-VAE-GAN.]]<br />
<br />
== Other Data ==<br />
MarrNet is also applied on cars and airplanes. Shown below, smaller details such as the horizontal stabilizer and rear-view mirrors are recovered.<br />
<br />
[[File:marrnet_airplanes_and_cars.png|700px|thumb|center|Results on airplanes and cars from the PASCAL 3D+ dataset, and comparison with DRC.]]<br />
<br />
MarrNet is also jointly trained on three object categories, and successfully recovers the shapes of different categories. Results are shown in the figure below.<br />
<br />
[[File:marrnet_multiple_categories.png|700px|thumb|center|Results when trained jointly on all three object categories (cars, airplanes, and chairs).]]<br />
<br />
= Commentary =<br />
Qualitatively, the results look quite impressive. The 2.5D sketch estimation seems to distill the useful information for more realistic looking 3D shape estimation. The disentanglement of 2.5D and 3D estimation steps also allows for easier training and domain adaptation from synthetic data.<br />
<br />
As the authors mention, the IoU metric is not very descriptive, and most of the comparisons in this paper are only qualitative, mainly being human preference studies. A better quantitative evaluation metric would greatly help in making an unbiased comparison between different results.<br />
<br />
As seen in several of the results, the network does not deal well with objects that have thin structures, which is particularly noticeable with many of the chair arm rests. As well, looking more carefully at some results, it seems that fine-tuning only the 3D encoder does not seem to transfer well to unseen objects, since shape priors have already been learned by the decoder. Therefore, future work should address more "difficult" shapes and forms; it should be more difficult to generalize shapes that are more complex than furniture.<br />
<br />
Also there is ambiguity in terms of how the aforementioned self-supervision can work as the authors claim that the model can be fine-tuned using a single image itself. If the parameters are constrained to a single image, then it means it will not generalize well. It is not clearly explained as to what can be fine-tuned.<br />
<br />
= Conclusion =<br />
The proposed MarrNet employs a novel model to estimate 2.5D sketches for 3D shape reconstruction. The sketches are shown to improve the model’s performance, and make it easy to adapt to images across different domains and categories. Differentiable loss functions are created such that the model can be fine-tuned end-to-end on images without ground truth. The experiments show that the model performs well, and human studies show that the results are preferred over other methods.<br />
<br />
= Implementation =<br />
The following repository provides the source code for the paper. The repository provides the source code as written by the authors: https://github.com/jiajunwu/marrnet<br />
<br />
= References =<br />
# Jiajun Wu, Yifan Wang, Tianfan Xue, Xingyuan Sun, William T. Freeman, Joshua B. Tenenbaum. MarrNet: 3D Shape Reconstruction via 2.5D Sketches, 2017<br />
# David Marr. Vision: A computational investigation into the human representation and processing of visual information. W. H. Freeman and Company, 1982.<br />
# Shubham Tulsiani, Tinghui Zhou, Alexei A Efros, and Jitendra Malik. Multi-view supervision for single-view reconstruction via differentiable ray consistency. In CVPR, 2017.<br />
# JiajunWu, Chengkai Zhang, Tianfan Xue,William T Freeman, and Joshua B Tenenbaum. Learning a Proba- bilistic Latent Space of Object Shapes via 3D Generative-Adversarial Modeling. In NIPS, 2016b.<br />
# Wu, J. (n.d.). Jiajunwu/marrnet. Retrieved March 25, 2018, from https://github.com/jiajunwu/marrnet<br />
# Jiajun Wu, Tianfan Xue, Joseph J Lim, Yuandong Tian, Joshua B Tenenbaum, Antonio Torralba, and William T Freeman. Single image 3d interpreter network. In ECCV, 2016a.<br />
# Xinchen Yan, Jimei Yang, Ersin Yumer, Yijie Guo, and Honglak Lee. Perspective transformer nets: Learning single-view 3d object reconstruction without 3d supervision. In NIPS, 2016.<br />
# Danilo Jimenez Rezende, SM Ali Eslami, Shakir Mohamed, Peter Battaglia, Max Jaderberg, and Nicolas Heess. Unsupervised learning of 3d structure from images. In NIPS, 2016.<br />
# Shubham Tulsiani, Tinghui Zhou, Alexei A Efros, and Jitendra Malik. Multi-view supervision for single-view reconstruction via differentiable ray consistency. In CVPR, 2017.<br />
# Rohit Girdhar, David F. Fouhey, Mikel Rodriguez and Abhinav Gupta, Learning a Predictable and Generative Vector Representation for Objects, in ECCV 2016</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Dynamic_Routing_Between_Capsules_STAT946&diff=36163Dynamic Routing Between Capsules STAT9462018-04-04T20:29:54Z<p>Pa2forsy: </p>
<hr />
<div>= Presented by =<br />
<br />
Yang, Tong(Richard)<br />
<br />
= Contributions =<br />
<br />
This paper introduces the concept of "capsules" and an approach to implement this concept in neural networks. Capsules are groups of neurons used to represent various properties of an entity/object present in the image, such as pose, deformation, and even the existence of the entity. Instead of the obvious representation of a logistic unit for the probability of existence, the paper explores using the length of the capsule output vector to represent existence, and the orientation to represent other properties of the entity. The paper makes the following major contributions:<br />
<br />
* Proposes an alternative to max-pooling called routing-by-agreement.<br />
* Demonstrates a mathematical structure for capsule layers and a routing mechanism. Builds a prototype architecture for capsule networks. <br />
* Presented promising results that confirm the value of Capsnet as a new direction for development in deep learning.<br />
<br />
= Hinton's Critiques on CNN =<br />
<br />
In a past talk, Hinton tried to explain why max-pooling is the biggest problem with current convolutional networks. Here are some highlights from his talk. <br />
<br />
== Four arguments against pooling ==<br />
<br />
* It is a bad fit to the psychology of shape perception: It does not explain why we assign intrinsic coordinate frames to objects and why they have such huge effects.<br />
<br />
* It solves the wrong problem: We want equivariance, not invariance. Disentangling rather than discarding.<br />
<br />
* It fails to use the underlying linear structure: It does not make use of the natural linear manifold that perfectly handles the largest source of variance in images.<br />
<br />
* Pooling is a poor way to do dynamic routing: We need to route each part of the input to the neurons that know how to deal with it. Finding the best routing is equivalent to parsing the image.<br />
<br />
===Intuition Behind Capsules ===<br />
We try to achieve viewpoint invariance in the activities of neurons by doing max-pooling. Invariance here means that by changing the input a little, the output still stays the same while the activity is just the output signal of a neuron. In other words, when in the input image we shift the object that we want to detect by a little bit, networks activities (outputs of neurons) will not change because of max pooling and the network will still detect the object. But the spacial relationships are not taken care of in this approach so instead capsules are used, because they encapsulate all important information about the state of the features they are detecting in a form of a vector. Capsules encode probability of detection of a feature as the length of their output vector. And the state of the detected feature is encoded as the direction in which that vector points to. So when detected feature moves around the image or its state somehow changes, the probability still stays the same (length of vector does not change), but its orientation changes.<br />
<br />
== Equivariance ==<br />
<br />
To deal with the invariance problem of CNN, Hinton proposes the concept called equivariance, which is the foundation of capsule concept.<br />
<br />
=== Two types of equivariance ===<br />
<br />
==== Place-coded equivariance ====<br />
If a low-level part moves to a very different position it will be represented by a different capsule.<br />
<br />
==== Rate-coded equivariance ====<br />
If a part only moves a small distance it will be represented by the same capsule but the pose outputs of the capsule will change.<br />
<br />
Higher-level capsules have bigger domains so low-level place-coded equivariance gets converted into high-level rate-coded equivariance.<br />
<br />
= Dynamic Routing =<br />
<br />
In the second section of this paper, authors give a mathematical representations for two key features in routing algorithm in capsule network, which are squashing and agreement. The general setting for this algorithm is between two arbitrary capsules i and j. Capsule j is assumed to be an arbitrary capsule from the first layer of capsules, and capsule i is an arbitrary capsule from the layer below. The purpose of routing algorithm is generate a vector output for routing decision between capsule j and capsule i. Furthermore, this vector output will be used in the decision for choice of dynamic routing. <br />
<br />
== Routing Algorithm ==<br />
<br />
The routing algorithm is as the following:<br />
<br />
[[File:DRBC_Figure_1.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
In the following sections, each part of this algorithm will be explained in details.<br />
<br />
=== Log Prior Probability ===<br />
<br />
<math>b_{ij}</math> represents the log prior probabilities that capsule i should be coupled to capsule j, and updated in each routing iteration. As line 2 suggests, the initial values of <math>b_{ij}</math> for all possible pairs of capsules are set to 0. In the very first routing iteration, <math>b_{ij}</math> equals to zero. For each routing iteration, <math>b_{ij}</math> gets updated by the value of agreement, which will be explained later.<br />
<br />
=== Coupling Coefficient === <br />
<br />
<math>c_{ij}</math> represents the coupling coefficient between capsule j and capsule i. It is calculated by applying the softmax function on the log prior probability <math>b_{ij}</math>. The mathematical transformation is shown below (Equation 3 in paper): <br />
<br />
\begin{align}<br />
c_{ij} = \frac{exp(b_ij)}{\sum_{k}exp(b_ik)}<br />
\end{align}<br />
<br />
<math>c_{ij}</math> are served as weights for computing the weighted sum and probabilities. Therefore, as probabilities, they have the following properties:<br />
<br />
\begin{align}<br />
c_{ij} \geq 0, \forall i, j<br />
\end{align}<br />
<br />
and, <br />
<br />
\begin{align}<br />
\sum_{i,j}c_{ij} = 1, \forall i, j<br />
\end{align}<br />
<br />
=== Predicted Output from Layer Below === <br />
<br />
<math>u_{i}</math> are the output vector from capsule i in the lower layer, and <math>\hat{u}_{j|i}</math> are the input vector for capsule j, which are the "prediction vectors" from the capsules in the layer below. <math>\hat{u}_{j|i}</math> is produced by multiplying <math>u_{i}</math> by a weight matrix <math>W_{ij}</math>, such as the following:<br />
<br />
\begin{align}<br />
\hat{u}_{j|i} = W_{ij}u_i<br />
\end{align}<br />
<br />
where <math>W_{ij}</math> encodes some spatial relationship between capsule j and capsule i.<br />
<br />
=== Capsule ===<br />
<br />
By using the definitions from previous sections, the total input vector for an arbitrary capsule j can be defined as:<br />
<br />
\begin{align}<br />
s_j = \sum_{i}c_{ij}\hat{u}_{j|i}<br />
\end{align}<br />
<br />
which is a weighted sum over all prediction vectors by using coupling coefficients.<br />
<br />
=== Squashing ===<br />
<br />
The length of <math>s_j</math> is arbitrary, which is needed to be addressed with. The next step is to convert its length between 0 and 1, since we want the length of the output vector of a capsule to represent the probability that the entity represented by the capsule is present in the current input. The "squashing" process is shown below:<br />
<br />
\begin{align}<br />
v_j = \frac{||s_j||^2}{1+||s_j||^2}\frac{s_j}{||s_j||}<br />
\end{align}<br />
<br />
Notice that "squashing" is not just normalizing the vector into unit length. In addition, it does extra non-linear transformation to ensure that short vectors get shrunk to almost zero length and long vectors get shrunk to a length slightly below 1. The reason for doing this is to make decision of routing, which is called "routing by agreement" much easier to make between capsule layers.<br />
<br />
=== Agreement ===<br />
<br />
The final step of a routing iteration is to form an routing agreement <math>a_{ij}</math>, which is represents as a scalar product:<br />
<br />
\begin{align}<br />
a_{ij} = v_{j} \cdot \hat{u}_{j|i}<br />
\end{align}<br />
<br />
As we mentioned in "squashing" section, the length of <math>v_{j}</math> is either close to 0 or close to 1, which will effect the magnitude of <math>a_{ij}</math> in this case. Therefore, the magnitude of <math>a_{ij}</math> indicate the how strong the routing algorithm agrees on taking the route between capsule j and capsule i. For each routing iteration, the log prior probability, <math>b_{ij}</math> will be updated by adding the value of its agreement value, which will effect how the coupling coefficients are computed in the next routing iteration. Because of the "squashing" process, we will eventually end up with a capsule j with its <math>v_{j}</math> close to 1 while all other capsules with its <math>v_{j}</math> close to 0, which indicates that this capsule j should be activated.<br />
<br />
= CapsNet Architecture =<br />
<br />
The second part of this paper discuss the experiment results from a 3-layer CapsNet, the architecture can be divided into two parts, encoder and decoder. <br />
<br />
== Encoder == <br />
<br />
[[File:DRBC_Architecture.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
=== How many routing iteration to use? === <br />
In appendix A of this paper, the authors have shown the empirical results from 500 epochs of training at different choice of routing iterations. According to their observation, more routing iterations increases the capacity of CapsNet but tends to bring additional risk of overfitting. Moreover, CapsNet with routing iterations less than three are not effective in general. As result, they suggest 3 iterations of routing for all experiments.<br />
<br />
=== Marginal loss for digit existence ===<br />
<br />
The experiments performed include segmenting overlapping digits on MultiMINST data set, so the loss function has be adjusted for presents of multiple digits. The marginal lose <math>L_k</math> for each capsule k is calculate by:<br />
<br />
\begin{align}<br />
L_k = T_k max(0, m^+ - ||v_k||)^2 + \lambda(1 - T_k) max(0, ||v_k|| - m^-)^2<br />
\end{align}<br />
<br />
where <math>m^+ = 0.9</math>, <math>m^- = 0.1</math>, and <math>\lambda = 0.5</math>.<br />
<br />
<math>T_k</math> is an indicator for presence of digit of class k, it takes value of 1 if and only if class k is presented. If class k is not presented, <math>\lambda</math> down-weight the loss which shrinks the lengths of the activity vectors for all the digit capsules. By doing this, The loss function penalizes the initial learning for all absent digit class, since we would like the top-level capsule for digit class k to have long instantiation vector if and only if that digit class is present in the input.<br />
<br />
=== Layer 1: Conv1 === <br />
<br />
The first layer of CapsNet. Similar to CNN, this is just convolutional layer that converts pixel intensities to activities of local feature detectors. <br />
<br />
* Layer Type: Convolutional Layer.<br />
* Input: <math>28 \times 28</math> pixels.<br />
* Kernel size: <math>9 \times 9</math>.<br />
* Number of Kernels: 256.<br />
* Activation function: ReLU.<br />
* Output: <math>20 \times 20 \times 256</math> tensor.<br />
<br />
=== Layer 2: PrimaryCapsules ===<br />
<br />
The second layer is formed by 32 primary 8D capsules. By 8D, it means that each primary capsule contains 8 convolutional units with a <math>9 \times 9</math> kernel and a stride of 2. Each capsule will take a <math>20 \times 20 \times 256</math> tensor from Conv1 and produce an output of a <math>6 \times 6 \times 8</math> tensor.<br />
<br />
* Layer Type: Convolutional Layer<br />
* Input: <math>20 \times 20 \times 256</math> tensor.<br />
* Number of capsules: 32.<br />
* Number of convolutional units in each capsule: 8.<br />
* Size of each convolutional unit: <math>6 \times 6</math>.<br />
* Output: <math>6 \times 6 \times 8</math> 8-dimensional vectors.<br />
<br />
=== Layer 3: DigitsCaps ===<br />
<br />
The last layer has 10 16D capsules, one for each digit. Not like the PrimaryCapsules layer, this layer is fully connected. Since this is the top capsule layer, dynamic routing mechanism will be applied between DigitsCaps and PrimaryCapsules. The process begins by taking a transformation of predicted output from PrimaryCapsules layer. Each output is a 8-dimensional vector, which needed to be mapped to a 16-dimensional space. Therefore, the weight matrix, <math>W_{ij}</math> is a <math>8 \times 16</math> matrix. The next step is to acquire coupling coefficients from routing algorithm and to perform "squashing" to get the output. <br />
<br />
* Layer Type: Fully connected layer.<br />
* Input: <math>6 \times 6 \times 8</math> 8-dimensional vectors.<br />
* Output: <math>16 \times 10 </math> matrix.<br />
<br />
=== The loss function ===<br />
<br />
The output of the loss function would be a ten-dimensional one-hot encoded vector with 9 zeros and 1 one at the correct position.<br />
<br />
<br />
== Regularization Method: Reconstruction ==<br />
<br />
This is regularization method introduced in the implementation of CapsNet. The method is to introduce a reconstruction loss (scaled down by 0.0005) to margin loss during training. The authors argue this would encourage the digit capsules to encode the instantiation parameters the input digits. All the reconstruction during training is by using the true labels of the image input. The results from experiments also confirms that adding the reconstruction regularizer enforces the pose encoding in CapsNet and thus boots the performance of routing procedure. <br />
<br />
=== Decoder ===<br />
<br />
The decoder consists of 3 fully connected layers, each layer maps pixel intensities to pixel intensities. The number of parameters in each layer and the activation functions used are indicated in the figure below:<br />
<br />
[[File:DRBC_Decoder.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
=== Result ===<br />
<br />
The authors includes some results for CapsNet classification test accuracy to justify the result of reconstruction. We can see that for CapsNet with 1 routing iteration and CapsNet with 3 routing iterations, implement reconstruction shows significant improvements in both MINIST and MultiMINST data set. These improvements show the importance of routing and reconstruction regularizer. <br />
<br />
[[File:DRBC_Reconstruction.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
= Experiment Results for CapsNet = <br />
<br />
In this part, the authors demonstrate experiment results of CapsNet on different data sets, such as MINIST and different variation of MINST, such as expanded MINST, affNIST, MultiMNIST. Moreover, they also briefly discuss the performance on some other popular data set such CIFAR 10. <br />
<br />
== MINST ==<br />
<br />
=== Highlights ===<br />
<br />
* CapsNet archives state-of-the-art performance on MINST with significantly fewer parameters (3-layer baseline CNN model has 35.4M parameters, compared to 8.2M for CapsNet with reconstruction network).<br />
* CapsNet with shallow structure (3 layers) achieves performance that only achieves by deeper network before.<br />
<br />
=== Interpretation of Each Capsule ===<br />
<br />
The authors suggest that they found evidence that dimension of some capsule always captures some variance of the digit, while some others represents the global combinations of different variations, this would open some possibility for interpretation of capsules in the future. After computing the activity vector for the correct digit capsule, the authors fed perturbed versions of those activity vectors to the decoder to examine the effect on reconstruction. Some results from perturbations are shown below, where each row represents the reconstructions when one of the 16 dimensions in the DigitCaps representation is tweaked by intervals of 0.05 from the range [-0.25, 0.25]: <br />
<br />
[[File:DRBC_Dimension.png|650px|center||Source: Sabour, Frosst, Hinton, 2017]]<br />
<br />
== affNIST == <br />
<br />
affNIT data set contains different affine transformation of original MINST data set. By the concept of capsule, CapsNet should gain more robustness from its equivariance nature, and the result confirms this. Compare the baseline CNN, CapsNet achieves 13% improvement on accuracy.<br />
<br />
== MultiMNIST ==<br />
<br />
The MultiMNIST is basically the overlapped version of MINIST. An important point to notice here is that this data set is generated by overlaying a digit on top of another digit from the same set but different class. In other words, the case of stacking digits from the same class is not allowed in MultiMINST. For example, stacking a 5 on a 0 is allowed, but stacking a 5 on another 5 is not. The reason is that CapsNet suffers from the "crowding" effect which will be discussed in the weakness of CapsNet section.<br />
<br />
The architecture used for the training is same as the one used for MNIST dataset. However, decay step of the learning rate is 10x larger to account for the larger dataset. Even with the overlap in MultiMNIST, the network is able to segment both digits separately and it shows that the network is able to position and style of the object in the image.<br />
<br />
[[File:multimnist.PNG | 700px|thumb|center|This figure shows some sample reconstructions on the MultiMNIST dataset using CapsNet. CapsNet reconstructs both of the digits in the image in different colours (green and red). It can be seen that the right most images have incorrect classifications with the 9 being classified as a 0 and the 7 being classified as an 8. ]]<br />
<br />
== Other data sets ==<br />
<br />
CapsNet is used on other data sets such as CIFAR10, smallNORB and SVHN. The results are not comparable with state-of-the-art performance, but it is still promising since this architecture is the very first, while other networks have been development for a long time. The authors pointed out one drawback of CapsNet is that they tend to account for everything in the input images - in the CIFAR10 dataset, the image backgrounds were too varied to model in a reasonably sized network, which partly explains the poorer results.<br />
<br />
= Conclusion = <br />
<br />
This paper discuss the specific part of capsule network, which is the routing-by-agreement mechanism. <br />
<br />
The authors suggest this is a great approach to solve the current problem with max-pooling in convolutional neural network. We see that the design of the capsule builds up upon the design of artificial neuron, but expands it to the vector form to allow for more powerful representational capabilities. It also introduces matrix weights to encode important hierarchical relationships between features of different layers. The result succeeds to achieve the goal of the designer: neuronal activity equivariance with respect to changes in inputs and invariance in probabilities of feature detection. <br />
<br />
Moreover, as author mentioned, the approach mentioned in this paper is only one possible implementation of the capsule concept. Approaches like [https://openreview.net/pdf?id=HJWLfGWRb/ this] have also been proposed to test other routing techniques.<br />
<br />
The preliminary results from experiment using a simple shallow CapsNet also demonstrate unparalleled performance that indicates the capsules are a direction worth exploring.<br />
<br />
= Weakness of Capsule Network =<br />
<br />
* Routing algorithm introduces internal loops for each capsule. As number of capsules and layers increases, these internal loops may exponentially expand the training time. <br />
* Capsule network suffers a perceptual phenomenon called "crowding", which is common for human vision as well. To address this weakness, capsules have to make a very strong representation assumption that at each location of the image, there is at most one instance of the type of entity that capsule represents. This is also the reason for not allowing overlaying digits from same class in generating process of MultiMINST.<br />
* Other criticisms include that the design of capsule networks requires domain knowledge or feature engineering, contrary to the abstraction-oriented goals of deep learning.<br />
<br />
= Implementations = <br />
1) Tensorflow Implementation : https://github.com/naturomics/CapsNet-Tensorflow<br />
<br />
2) Keras Implementation. : https://github.com/XifengGuo/CapsNet-Keras<br />
<br />
= References =<br />
# S. Sabour, N. Frosst, and G. E. Hinton, “Dynamic routing between capsules,” arXiv preprint arXiv:1710.09829v2, 2017<br />
# “XifengGuo/CapsNet-Keras.” GitHub, 14 Dec. 2017, github.com/XifengGuo/CapsNet-Keras. <br />
# “Naturomics/CapsNet-Tensorflow.” GitHub, 6 Mar. 2018, github.com/naturomics/CapsNet-Tensorflow.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Tensorized_LSTMs&diff=36162stat946w18/Tensorized LSTMs2018-04-04T20:12:59Z<p>Pa2forsy: /* A Quick Introduction to RNN and LSTM */</p>
<hr />
<div>= Presented by =<br />
<br />
Chen, Weishi(Edward)<br />
<br />
= Introduction =<br />
<br />
Long Short-Term Memory (LSTM) is a popular approach to boosting the ability of Recurrent Neural Networks to store longer term temporal information. The capacity of an LSTM network can be increased by widening and adding layers (illustrations will be provided later). <br />
<br />
<br />
However, widening an LSTM or adding layers to it usually introduces additional parameters, which in turn increases the time required for model training and evaluation. As an alternative, this paper <Wider and Deeper, Cheaper and Faster: Tensorized LSTMs for Sequence Learning> has proposed a model based on LSTM called the '''Tensorized LSTM''' in which the hidden states are represented by '''tensors''' and updated via a '''cross-layer convolution'''. <br />
<br />
* By increasing the tensor size, the network can be widened efficiently without additional parameters since the parameters are shared across different locations in the tensor<br />
* By delaying the output, the network can be deepened implicitly with little additional run-time since deep computations for each time step are merged into temporal computations of the sequence. <br />
<br />
<br />
Also, the paper presents experiments that were conducted on five challenging sequence learning tasks to show the potential of the proposed model.<br />
<br />
= A Quick Introduction to RNN and LSTM =<br />
<br />
We consider the time-series prediction task of producing a desired output <math>y_t</math> at each time-step t∈ {1, ..., T} given an observed input sequence <math>x_{1:t} = {x_1,x_2, ···, x_t}</math>, where <math>x_t∈R^R</math> and <math>y_t∈R^S</math> are vectors. RNNs learn how to use a hidden state vector <math>h_t ∈ R^M</math> to encapsulate the relevant features of the entire input history x1:t (indicates all inputs from the initial time-step to the final step before predication - illustration given below) up to time-step t.<br />
<br />
\begin{align}<br />
h_{t-1}^{cat} = [x_t, h_{t-1}] \hspace{2cm} (1)<br />
\end{align}<br />
<br />
Where <math>h_{t-1}^{cat} ∈R^{R+M}</math> is the concatenation of the current input <math>x_t</math> and the previous hidden state <math>h_{t−1}</math>, which expands the dimensionality of intermediate information.<br />
<br />
The update of the hidden state h_t is defined as:<br />
<br />
\begin{align}<br />
a_{t} =h_{t-1}^{cat} W^h + b^h \hspace{2cm} (2)<br />
\end{align}<br />
<br />
and<br />
<br />
\begin{align}<br />
h_t = \Phi(a_t) \hspace{2cm} (3)<br />
\end{align}<br />
<br />
<math>W^h∈R^{(R+M)\times M} </math> guarantees each hidden state provided by the previous step is of dimension M. <math> a_t ∈R^M </math> is the hidden activation, and φ(·) is the element-wise hyperbolic tangent. Finally, the output <math> y_t </math> at time-step t is generated by:<br />
<br />
\begin{align}<br />
y_t = \varphi(h_{t}^{cat} W^y + b^y) \hspace{2cm} (4)<br />
\end{align}<br />
<br />
where <math>W^y∈R^{M×S}</math> and <math>b^y∈R^S</math>, and <math>\varphi(·)</math> can be any differentiable function. Note that the <math>\phi</math> is a non-linear, element-wise function which generates hidden output, while <math>\varphi</math> generates the final network output.<br />
<br />
[[File:StdRNN.png|650px|center||Figure 1: Recurrent Neural Network]]<br />
<br />
One shortfall of RNN is the problem of vanishing/exploding gradients. This shortfall is significant, especially when modeling long-range dependencies. One alternative is to instead use LSTM (Long Short-Term Memory), which alleviates these problems by employing several gates to selectively modulate the information flow across each neuron. Since LSTMs have been successfully used in sequence models, it is natural to consider them for accommodating more complex analytical needs.<br />
<br />
[[File:LSTM_Gated.png|650px|center||Figure 2: LSTM]]<br />
<br />
= Structural Measurement of Sequential Model =<br />
<br />
We can consider the capacity of a network consists of two components: the '''width''' (the amount of information handled in parallel) and the depth (the number of computation steps). <br />
<br />
A way to '''widen''' the LSTM is to increase the number of units in a hidden layer; however, the parameter number scales quadratically with the number of units. To deepen the LSTM, the popular Stacked LSTM (sLSTM) stacks multiple LSTM layers. The drawback of sLSTM, however, is that runtime is proportional to the number of layers and information from the input is potentially lost (due to gradient vanishing/explosion) as it propagates vertically through the layers. This paper introduced a way to both widen and deepen the LSTM whilst keeping the parameter number and runtime largely unchanged. In summary, we make the following contributions:<br />
<br />
'''(a)''' Tensorize RNN hidden state vectors into higher-dimensional tensors, to enable more flexible parameter sharing and can be widened more efficiently without additional parameters.<br />
<br />
'''(b)''' Based on (a), merge RNN deep computations into its temporal computations so that the network can be deepened with little additional runtime, resulting in a Tensorized RNN (tRNN).<br />
<br />
'''(c)''' We extend the tRNN to an LSTM, namely the Tensorized LSTM (tLSTM), which integrates a novel memory cell convolution to help to prevent the vanishing/exploding gradients.<br />
<br />
= Method =<br />
<br />
Go through the methodology.<br />
<br />
== Part 1: Tensorize RNN hidden State vectors ==<br />
<br />
'''Definition:''' Tensorization is defined as the transformation or mapping of lower-order data to higher-order data. For example, the low-order data can be a vector, and the tensorized result is a matrix, a third-order tensor or a higher-order tensor. The ‘low-order’ data can also be a matrix or a third-order tensor, for example. In the latter case, tensorization can take place along one or multiple modes.<br />
<br />
[[File:VecTsor.png|320px|center||Figure 3: Vector Third-order tensorization of a vector]]<br />
<br />
'''Optimization Methodology Part 1:''' It can be seen that in an RNN, the parameter number scales quadratically with the size of the hidden state. A popular way to limit the parameter number when widening the network is to organize parameters as higher-dimensional tensors which can be factorized into lower-rank sub-tensors that contain significantly fewer elements, which is is known as tensor factorization. <br />
<br />
'''Optimization Methodology Part 2:''' Another common way to reduce the parameter number is to share a small set of parameters across different locations in the hidden state, similar to Convolutional Neural Networks (CNNs).<br />
<br />
'''Effects:''' This '''widens''' the network since the hidden state vectors are in fact broadcast to interact with the tensorized parameters. <br />
<br />
<br />
<br />
We adopt parameter sharing to cutdown the parameter number for RNNs, since compared with factorization, it has the following advantages: <br />
<br />
(i) '''Scalability,''' the number of shared parameters can be set independent of the hidden state size<br />
<br />
(ii) '''Separability,''' the information flow can be carefully managed by controlling the receptive field, allowing one to shift RNN deep computations to the temporal domain<br />
<br />
<br />
<br />
We also explicitly tensorize the RNN hidden state vectors, since compared with vectors, tensors have a better: <br />
<br />
(i) '''Flexibility,''' one can specify which dimensions to share parameters and then can just increase the size of those dimensions without introducing additional parameters<br />
<br />
(ii) '''Efficiency,''' with higher-dimensional tensors, the network can be widened faster w.r.t. its depth when fixing the parameter number (explained later). <br />
<br />
<br />
'''Illustration:''' For ease of exposition, we first consider 2D tensors (matrices): we tensorize the hidden state <math>h_t∈R^{M}</math> to become <math>Ht∈R^{P×M}</math>, '''where P is the tensor size,''' and '''M the channel size'''. We locally-connect the first dimension of <math>H_t</math> (which is P - the tensor size) in order to share parameters, and fully-connect the second dimension of <math>H_t</math> (which is M - the channel size) to allow global interactions. This is analogous to the CNN which fully-connects one dimension (e.g., the RGB channel for input images) to globally fuse different feature planes. Also, if one compares <math>H_t</math> to the hidden state of a Stacked RNN (sRNN) (see Figure Blow). <br />
<br />
[[File:Screen_Shot_2018-03-26_at_11.28.37_AM.png|160px|center||Figure 4: Stacked RNN]]<br />
<br />
[[File:ind.png|60px|center||Figure 4: Stacked RNN]]<br />
<br />
Then P is akin to the number of stacked hidden layers (vertical length in the graph), and M the size of each hidden layer (each white node in the graph). We start to describe our model based on 2D tensors, and finally show how to strengthen the model with higher-dimensional tensors.<br />
<br />
== Part 2: Merging Deep Computations ==<br />
<br />
Since an RNN is already deep in its temporal direction, we can deepen an input-to-output computation by associating the input <math>x_t</math> with a (delayed) future output. In doing this, we need to ensure that the output <math>y_t</math> is separable, i.e., not influenced by any future input <math>x_{t^{'}}</math> <math>(t^{'}>t)</math>. Thus, we concatenate the projection of <math>x_t</math> to the top of the previous hidden state <math>H_{t−1}</math>, then gradually shift the input information down when the temporal computation proceeds, and finally generate <math>y_t</math> from the bottom of <math>H_{t+L−1}</math>, where L−1 is the number of delayed time-steps for computations of depth L. <br />
<br />
An example with L= 3 is shown in Figure.<br />
<br />
[[File:tRNN.png|160px|center||Figure 5: skewed sRNN]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: skewed sRNN]]<br />
<br />
<br />
This is in fact a skewed sRNN (or tRNN without feedback). However, the method does not need to change the network structure and also allows different kinds of interactions as long as the output is separable; for example, one can increase the local connections and '''use feedback''' (shown in figure below), which can be beneficial for sRNNs (or tRNN). <br />
<br />
[[File:tRNN_wF.png|160px|center||Figure 5: skewed sRNN with F]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: skewed sRNN with F]]<br />
<br />
'''In order to share parameters, we update <math>H_t</math> using a convolution with a learnable kernel.''' In this manner we increase the complexity of the input-to-output mapping (by delaying outputs) and limit parameter growth (by sharing transition parameters using convolutions).<br />
<br />
To examine the resulting model mathematically, let <math>H^{cat}_{t−1}∈R^{(P+1)×M}</math> be the concatenated hidden state, and <math>p∈Z_+</math> the location at a tensor. The channel vector <math>h^{cat}_{t−1, p }∈R^M</math> at location p of <math>H^{cat}_{t−1}</math> (the p-th channel of H) is defined as:<br />
<br />
\begin{align}<br />
h^{cat}_{t-1, p} = x_t W^x + b^x \hspace{1cm} if p = 1 \hspace{1cm} (5)<br />
\end{align}<br />
<br />
\begin{align}<br />
h^{cat}_{t-1, p} = h_{t-1, p-1} \hspace{1cm} if p > 1 \hspace{1cm} (6)<br />
\end{align}<br />
<br />
where <math>W^x ∈ R^{R×M}</math> and <math>b^x ∈ R^M</math> (recall the dimension of input x is R). Then, the update of tensor <math>H_t</math> is implemented via a convolution:<br />
<br />
\begin{align}<br />
A_t = H^{cat}_{t-1} \circledast \{W^h, b^h \} \hspace{2cm} (7)<br />
\end{align}<br />
<br />
\begin{align}<br />
H_t = \Phi{A_t} \hspace{2cm} (8)<br />
\end{align}<br />
<br />
where <math>W^h∈R^{K×M^i×M^o}</math> is the kernel weight of size K, with <math>M^i =M</math> input channels and <math>M^o =M</math> output channels, <math>b^h ∈ R^{M^o}</math> is the kernel bias, <math>A_t ∈ R^{P×M^o}</math> is the hidden activation, and <math>\circledast</math> is the convolution operator. Since the kernel convolves across different hidden layers, we call it the cross-layer convolution. The kernel enables interaction, both bottom-up and top-down across layers. Finally, we generate <math>y_t</math> from the channel vector <math>h_{t+L−1,P}∈R^M</math> which is located at the bottom of <math>H_{t+L−1}</math>:<br />
<br />
\begin{align}<br />
y_t = \varphi(h_{t+L−1}, _PW^y + b^y) \hspace{2cm} (9)<br />
\end{align}<br />
<br />
Where <math>W^y ∈R^{M×S}</math> and <math>b^y ∈R^S</math>. To guarantee that the receptive field of <math>y_t</math> only covers the current and previous inputs x1:t. (Check the Skewed sRNN again below):<br />
<br />
[[File:tRNN_wF.png|160px|center||Figure 5: skewed sRNN with F]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: skewed sRNN with F]]<br />
<br />
=== Quick Summary of Set of Parameters ===<br />
<br />
'''1. <math> W^x</math> and <math>b_x</math>''' connect input to the first hidden node<br />
<br />
'''2. <math> W^h</math> and <math>b_h</math>''' convolute between layers<br />
<br />
'''3. <math> W^y</math> and <math>b_y</math>''' produce output of each stages<br />
<br />
<br />
== Part 3: Extending to LSTMs==<br />
<br />
Similar to standard RNN, to allow the tRNN (skewed sRNN) to capture long-range temporal dependencies, one can straightforwardly extend it<br />
to a tLSTM by replacing the tRNN tensors:<br />
<br />
\begin{align}<br />
[A^g_t, A^i_t, A^f_t, A^o_t] = H^{cat}_{t-1} \circledast \{W^h, b^h \} \hspace{2cm} (10)<br />
\end{align}<br />
<br />
\begin{align}<br />
[G_t, I_t, F_t, O_t]= [\Phi{(A^g_t)}, σ(A^i_t), σ(A^f_t), σ(A^o_t)] \hspace{2cm} (11)<br />
\end{align}<br />
<br />
Which are pretty similar to tRNN case, the main differences can be observes for memory cells of tLSTM (Ct):<br />
<br />
\begin{align}<br />
C_t= G_t \odot I_t + C_{t-1} \odot F_t \hspace{2cm} (12)<br />
\end{align}<br />
<br />
\begin{align}<br />
H_t= \Phi{(C_t )} \odot O_t \hspace{2cm} (13)<br />
\end{align}<br />
<br />
Note that since the previous memory cell <math>C_{t-1}</math> is only gated along the temporal direction, increasing the tensor size ''P'' might result in the loss of long-range dependencies from the input to the output.<br />
<br />
Summary of the terms: <br />
<br />
1. '''<math>\{W^h, b^h \}</math>:''' Kernel of size K <br />
<br />
2. '''<math>A^g_t, A^i_t, A^f_t, A^o_t \in \mathbb{R}^{P\times M}</math>:''' Activations for the new content <math>G_t</math><br />
<br />
3. '''<math>I_t</math>:''' Input gate<br />
<br />
4. '''<math>F_t</math>:''' Forget gate<br />
<br />
5. '''<math>O_t</math>:''' Output gate<br />
<br />
6. '''<math>C_t \in \mathbb{R}^{P\times M}</math>:''' Memory cell<br />
<br />
Then, see graph below for illustration:<br />
<br />
[[File:tLSTM_wo_MC.png |160px|center||Figure 5: tLSTM wo MC]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: tLSTM wo MC]]<br />
<br />
To further evolve tLSTM, we invoke the '''Memory Cell Convolution''' to capture long-range dependencies from multiple directions, we additionally introduce a novel memory cell convolution, by which the memory cells can have a larger receptive field (figure provided below). <br />
<br />
[[File:tLSTM_w_MC.png |160px|center||Figure 5: tLSTM w MC]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: tLSTM w MC]]<br />
<br />
One can also dynamically generate this convolution kernel so that it is both time - and location-dependent, allowing for flexible control over long-range dependencies from different directions. Mathematically, it can be represented in with the following formulas:<br />
<br />
\begin{align}<br />
[A^g_t, A^i_t, A^f_t, A^o_t, A^q_t] = H^{cat}_{t-1} \circledast \{W^h, b^h \} \hspace{2cm} (14)<br />
\end{align}<br />
<br />
\begin{align}<br />
[G_t, I_t, F_t, O_t, Q_t]= [\Phi{(A^g_t)}, σ(A^i_t), σ(A^f_t), σ(A^o_t), ς(A^q_t)] \hspace{2cm} (15)<br />
\end{align}<br />
<br />
\begin{align}<br />
W_t^c(p) = reshape(q_{t,p}, [K, 1, 1]) \hspace{2cm} (16)<br />
\end{align}<br />
<br />
\begin{align}<br />
C_{t-1}^{conv}= C_{t-1} \circledast W_t^c(p) \hspace{2cm} (17)<br />
\end{align}<br />
<br />
\begin{align}<br />
C_t= G_t \odot I_t + C_{t-1}^{conv} \odot F_t \hspace{2cm} (18)<br />
\end{align}<br />
<br />
\begin{align}<br />
H_t= \Phi{(C_t )} \odot O_t \hspace{2cm} (19)<br />
\end{align}<br />
<br />
where the kernel <math>{W^h, b^h}</math> has additional <K> output channels to generate the activation <math>A^q_t ∈ R^{P×<K>}</math> for the dynamic kernel bank <math>Q_t∈R^{P × <K>}</math>, <math>q_{t,p}∈R^{<K>}</math> is the vectorized adaptive kernel at the location p of <math>Q_t</math>, and <math>W^c_t(p) ∈ R^{K×1×1}</math> is the dynamic kernel of size K with a single input/output channel, which is reshaped from <math>q_{t,p}</math>. Each channel of the previous memory cell <math>C_{t-1}</math> is convolved with <math>W_t^c(p)</math> whose values vary with <math>p</math>, to form a memory cell convolution, which produces a convolved memory cell <math>C_{t-1}^{conv} \in \mathbb{R}^{P\times M}</math>. This convolution is defined by:<br />
<br />
\begin{align}<br />
C_{t-1,p,m}^{conv} = \sum\limits_{k=1}^K C_{t-1,p-\frac{K-1}{2}+k,m} · W_{t,k,1,1}^c(p) \hspace{2cm} (30)<br />
\end{align}<br />
<br />
where <math>C_{t-1}</math> is padded with the boundary values to retain the stored information.<br />
<br />
Note the paper also employed a softmax function ς(·) to normalize the channel dimension of <math>Q_t</math>. which can also stabilize the value of memory cells and help to prevent the vanishing/exploding gradients. An illustration is provided below to better illustrate the process:<br />
<br />
[[File:MCC.png |240px|center||Figure 5: MCC]]<br />
<br />
To improve training, the authors introduced a new normalization technique for ''t''LSTM termed channel normalization (adapted from layer normalization), in which the channel vector are normalized at different locations with their own statistics. Note that layer normalization does not work well with ''t''LSTM, because lower level information is near the input and higher level information is near the output. Channel normalization (CN) is defined as: <br />
<br />
\begin{align}<br />
\mathrm{CN}(\mathbf{Z}; \mathbf{\Gamma}, \mathbf{B}) = \mathbf{\hat{Z}} \odot \mathbf{\Gamma} + \mathbf{B} \hspace{2cm} (20)<br />
\end{align}<br />
<br />
where <math>\mathbf{Z}</math>, <math>\mathbf{\hat{Z}}</math>, <math>\mathbf{\Gamma}</math>, <math>\mathbf{B} \in \mathbb{R}^{P \times M^z}</math> are the original tensor, normalized tensor, gain parameter and bias parameter. The <math>m^z</math>-th channel of <math>\mathbf{Z}</math> is normalized element-wisely: <br />
<br />
\begin{align}<br />
\hat{z_{m^z}} = (z_{m^z} - z^\mu)/z^{\sigma} \hspace{2cm} (21)<br />
\end{align}<br />
<br />
where <math>z^{\mu}</math>, <math>z^{\sigma} \in \mathbb{R}^P</math> are the mean and standard deviation along the channel dimension of <math>\mathbf{Z}</math>, and <math>\hat{z_{m^z}} \in \mathbb{R}^P</math> is the <math>m^z</math>-th channel <math>\mathbf{\hat{Z}}</math>. Channel normalization introduces very few additional parameters compared to the number of other parameters in the model.<br />
<br />
= Results and Evaluation =<br />
<br />
Summary of list of models tLSTM family (may be useful later):<br />
<br />
(a) sLSTM (baseline): the implementation of sLSTM with parameters shared across all layers.<br />
<br />
(b) 2D tLSTM: the standard 2D tLSTM.<br />
<br />
(c) 2D tLSTM–M: removing memory (M) cell convolutions from (b).<br />
<br />
(d) 2D tLSTM–F: removing (–) feedback (F) connections from (b).<br />
<br />
(e) 3D tLSTM: tensorizing (b) into 3D tLSTM.<br />
<br />
(f) 3D tLSTM+LN: applying (+) Layer Normalization.<br />
<br />
(g) 3D tLSTM+CN: applying (+) Channel Normalization.<br />
<br />
=== Efficiency Analysis ===<br />
<br />
'''Fundaments:''' For each configuration, fix the parameter number and increase the tensor size to see if the performance of tLSTM can be boosted without increasing the parameter number. Can also investigate how the runtime is affected by the depth, where the runtime is measured by the average GPU milliseconds spent by a forward and backward pass over one timestep of a single example. <br />
<br />
'''Dataset:''' The Hutter Prize Wikipedia dataset consists of 100 million characters taken from 205 different characters including alphabets, XML markups and special symbols. We model the dataset at the character-level, and try to predict the next character of the input sequence.<br />
<br />
All configurations are evaluated with depths L = 1, 2, 3, 4. Bits-per-character(BPC) is used to measure the model performance and the results are shown in the figure below.<br />
[[File:wiki.png |280px|center||Figure 5: WifiPerf]]<br />
[[File:Wiki_Performance.png |480px|center||Figure 5: WifiPerf]]<br />
<br />
=== Accuracy Analysis ===<br />
<br />
The MNIST dataset [35] consists of 50000/10000/10000 handwritten digit images of size 28×28 for training/validation/test. Two tasks are used for evaluation on this dataset:<br />
<br />
(a) '''Sequential MNIST:''' The goal is to classify the digit after sequentially reading the pixels in a scan-line order. It is therefore a 784 time-step sequence learning task where a single output is produced at the last time-step; the task requires very long range dependencies in the sequence.<br />
<br />
(b) '''Sequential Permuted MNIST:''' We permute the original image pixels in a fixed random order, resulting in a permuted MNIST (pMNIST) problem that has even longer range dependencies across pixels and is harder.<br />
<br />
In both tasks, all configurations are evaluated with M = 100 and L= 1, 3, 5. The model performance is measured by the classification accuracy and results are shown in the figure below.<br />
<br />
[[File:MNISTperf.png |480px|center]]<br />
<br />
<br />
<br />
[[File:Acc_res.png |480px|center||Figure 5: MNIST]]<br />
<br />
[[File:33_mnist.PNG|center|thumb|800px| This figure displays a visualization of the means of the diagonal channels of the tLSTM memory cells per task. The columns indicate the time steps and the rows indicate the diagonal locations. The values are normalized between 0 and 1.]]<br />
<br />
It can be seen in the above figure that tLSTM behaves differently with different tasks:<br />
<br />
- Wikipedia: the input can be carried to the output location with less modification if it is sufficient to determine the next character, and vice versa<br />
<br />
- addition: the first integer is gradually encoded into memories and then interacts (performs addition) with the second integer, producing the sum <br />
<br />
- memorization: the network behaves like a shift register that continues to move the input symbol to the output location at the correct timestep<br />
<br />
- sequential MNIST: the network is more sensitive to the pixel value change (representing the contour, or topology of the digit) and can gradually accumulate evidence for the final prediction <br />
<br />
- sequential pMNIST: the network is sensitive to high value pixels (representing the foreground digit), and we conjecture that this is because the permutation destroys the topology of the digit, making each high value pixel potentially important.<br />
<br />
From the figure above it can can also be observe some common phenomena in all tasks: <br />
# it is clear that wider (larger) tensors can encode more information by observing that at each timestep, the values at different tensor locations are markedly different<br />
# from the input to the output, the values become increasingly distinct and are shifted by time, revealing that deep computations are indeed performed together with temporal computations, with long-range dependencies carried by memory cells.<br />
<br />
<br />
= Conclusions =<br />
<br />
The paper introduced the Tensorized LSTM, which employs tensors to share parameters and utilizes the temporal computation to perform the deep computation for sequential tasks. Then validated the model<br />
on a variety of tasks, showing its potential over other popular approaches. The paper shows a method to widen and deepen the LSTM network at the same time and the following 3 points list their main contributions:<br />
* The RNNs are now tensorized into higher dimensional tensors which are more flexible.<br />
* RNNs' deep computation is merged into the temporal computation, referred to as the tensorizedRNN.<br />
* tRNN is extended to a LSTM architecture and a new architecture is studied: tensorizedLSTM<br />
<br />
= Critique(to be edited) =<br />
<br />
= References =<br />
#Zhen He, Shaobing Gao, Liang Xiao, Daxue Liu, Hangen He, and David Barber. <Wider and Deeper, Cheaper and Faster: Tensorized LSTMs for Sequence Learning> (2017)<br />
#Ali Ghodsi, <Deep Learning: STAT 946 - Winter 2018></div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Tensorized_LSTMs&diff=36161stat946w18/Tensorized LSTMs2018-04-04T20:09:59Z<p>Pa2forsy: </p>
<hr />
<div>= Presented by =<br />
<br />
Chen, Weishi(Edward)<br />
<br />
= Introduction =<br />
<br />
Long Short-Term Memory (LSTM) is a popular approach to boosting the ability of Recurrent Neural Networks to store longer term temporal information. The capacity of an LSTM network can be increased by widening and adding layers (illustrations will be provided later). <br />
<br />
<br />
However, widening an LSTM or adding layers to it usually introduces additional parameters, which in turn increases the time required for model training and evaluation. As an alternative, this paper <Wider and Deeper, Cheaper and Faster: Tensorized LSTMs for Sequence Learning> has proposed a model based on LSTM called the '''Tensorized LSTM''' in which the hidden states are represented by '''tensors''' and updated via a '''cross-layer convolution'''. <br />
<br />
* By increasing the tensor size, the network can be widened efficiently without additional parameters since the parameters are shared across different locations in the tensor<br />
* By delaying the output, the network can be deepened implicitly with little additional run-time since deep computations for each time step are merged into temporal computations of the sequence. <br />
<br />
<br />
Also, the paper presents experiments that were conducted on five challenging sequence learning tasks to show the potential of the proposed model.<br />
<br />
= A Quick Introduction to RNN and LSTM =<br />
<br />
We consider the time-series prediction task of producing a desired output <math>y_t</math> at each time-step t∈ {1, ..., T} given an observed input sequence <math>x_{1:t} = {x_1,x_2, ···, x_t}</math>, where <math>x_t∈R^R</math> and <math>y_t∈R^S</math> are vectors. RNNs learn how to use a hidden state vector <math>h_t ∈ R^M</math> to encapsulate the relevant features of the entire input history x1:t (indicates all inputs from the initial time-step to the final step before predication - illustration given below) up to time-step t.<br />
<br />
\begin{align}<br />
h_{t-1}^{cat} = [x_t, h_{t-1}] \hspace{2cm} (1)<br />
\end{align}<br />
<br />
Where <math>h_{t-1}^{cat} ∈R^{R+M}</math> is the concatenation of the current input <math>x_t</math> and the previous hidden state <math>h_{t−1}</math>, which expands the dimensionality of intermediate information.<br />
<br />
The update of the hidden state h_t is defined as:<br />
<br />
\begin{align}<br />
a_{t} =h_{t-1}^{cat} W^h + b^h \hspace{2cm} (2)<br />
\end{align}<br />
<br />
and<br />
<br />
\begin{align}<br />
h_t = \Phi(a_t) \hspace{2cm} (3)<br />
\end{align}<br />
<br />
<math>W^h∈R^(R+M)xM </math> guarantees each hidden status provided by the previous step is of dimension M. <math> a_t ∈R^M </math> is the hidden activation, and φ(·) is the element-wise hyperbolic tangent. Finally, the output <math> y_t </math> at time-step t is generated by:<br />
<br />
\begin{align}<br />
y_t = \varphi(h_{t}^{cat} W^y + b^y) \hspace{2cm} (4)<br />
\end{align}<br />
<br />
where <math>W^y∈R^{M×S}</math> and <math>b^y∈R^S</math>, and <math>\varphi(·)</math> can be any differentiable function. Note that the <math>\phi</math> is a non-linear, element-wise function which generates hidden output, while <math>\varphi</math> generates the final network output.<br />
<br />
[[File:StdRNN.png|650px|center||Figure 1: Recurrent Neural Network]]<br />
<br />
One shortfall of RNN is the problem of vanishing/exploding gradients. This shortfall is significant, especially when modeling long-range dependencies. One alternative is to instead use LSTM (Long Short-Term Memory), which alleviates these problems by employing several gates to selectively modulate the information flow across each neuron. Since LSTMs have been successfully used in sequence models, it is natural to consider them for accommodating more complex analytical needs.<br />
<br />
[[File:LSTM_Gated.png|650px|center||Figure 2: LSTM]]<br />
<br />
= Structural Measurement of Sequential Model =<br />
<br />
We can consider the capacity of a network consists of two components: the '''width''' (the amount of information handled in parallel) and the depth (the number of computation steps). <br />
<br />
A way to '''widen''' the LSTM is to increase the number of units in a hidden layer; however, the parameter number scales quadratically with the number of units. To deepen the LSTM, the popular Stacked LSTM (sLSTM) stacks multiple LSTM layers. The drawback of sLSTM, however, is that runtime is proportional to the number of layers and information from the input is potentially lost (due to gradient vanishing/explosion) as it propagates vertically through the layers. This paper introduced a way to both widen and deepen the LSTM whilst keeping the parameter number and runtime largely unchanged. In summary, we make the following contributions:<br />
<br />
'''(a)''' Tensorize RNN hidden state vectors into higher-dimensional tensors, to enable more flexible parameter sharing and can be widened more efficiently without additional parameters.<br />
<br />
'''(b)''' Based on (a), merge RNN deep computations into its temporal computations so that the network can be deepened with little additional runtime, resulting in a Tensorized RNN (tRNN).<br />
<br />
'''(c)''' We extend the tRNN to an LSTM, namely the Tensorized LSTM (tLSTM), which integrates a novel memory cell convolution to help to prevent the vanishing/exploding gradients.<br />
<br />
= Method =<br />
<br />
Go through the methodology.<br />
<br />
== Part 1: Tensorize RNN hidden State vectors ==<br />
<br />
'''Definition:''' Tensorization is defined as the transformation or mapping of lower-order data to higher-order data. For example, the low-order data can be a vector, and the tensorized result is a matrix, a third-order tensor or a higher-order tensor. The ‘low-order’ data can also be a matrix or a third-order tensor, for example. In the latter case, tensorization can take place along one or multiple modes.<br />
<br />
[[File:VecTsor.png|320px|center||Figure 3: Vector Third-order tensorization of a vector]]<br />
<br />
'''Optimization Methodology Part 1:''' It can be seen that in an RNN, the parameter number scales quadratically with the size of the hidden state. A popular way to limit the parameter number when widening the network is to organize parameters as higher-dimensional tensors which can be factorized into lower-rank sub-tensors that contain significantly fewer elements, which is is known as tensor factorization. <br />
<br />
'''Optimization Methodology Part 2:''' Another common way to reduce the parameter number is to share a small set of parameters across different locations in the hidden state, similar to Convolutional Neural Networks (CNNs).<br />
<br />
'''Effects:''' This '''widens''' the network since the hidden state vectors are in fact broadcast to interact with the tensorized parameters. <br />
<br />
<br />
<br />
We adopt parameter sharing to cutdown the parameter number for RNNs, since compared with factorization, it has the following advantages: <br />
<br />
(i) '''Scalability,''' the number of shared parameters can be set independent of the hidden state size<br />
<br />
(ii) '''Separability,''' the information flow can be carefully managed by controlling the receptive field, allowing one to shift RNN deep computations to the temporal domain<br />
<br />
<br />
<br />
We also explicitly tensorize the RNN hidden state vectors, since compared with vectors, tensors have a better: <br />
<br />
(i) '''Flexibility,''' one can specify which dimensions to share parameters and then can just increase the size of those dimensions without introducing additional parameters<br />
<br />
(ii) '''Efficiency,''' with higher-dimensional tensors, the network can be widened faster w.r.t. its depth when fixing the parameter number (explained later). <br />
<br />
<br />
'''Illustration:''' For ease of exposition, we first consider 2D tensors (matrices): we tensorize the hidden state <math>h_t∈R^{M}</math> to become <math>Ht∈R^{P×M}</math>, '''where P is the tensor size,''' and '''M the channel size'''. We locally-connect the first dimension of <math>H_t</math> (which is P - the tensor size) in order to share parameters, and fully-connect the second dimension of <math>H_t</math> (which is M - the channel size) to allow global interactions. This is analogous to the CNN which fully-connects one dimension (e.g., the RGB channel for input images) to globally fuse different feature planes. Also, if one compares <math>H_t</math> to the hidden state of a Stacked RNN (sRNN) (see Figure Blow). <br />
<br />
[[File:Screen_Shot_2018-03-26_at_11.28.37_AM.png|160px|center||Figure 4: Stacked RNN]]<br />
<br />
[[File:ind.png|60px|center||Figure 4: Stacked RNN]]<br />
<br />
Then P is akin to the number of stacked hidden layers (vertical length in the graph), and M the size of each hidden layer (each white node in the graph). We start to describe our model based on 2D tensors, and finally show how to strengthen the model with higher-dimensional tensors.<br />
<br />
== Part 2: Merging Deep Computations ==<br />
<br />
Since an RNN is already deep in its temporal direction, we can deepen an input-to-output computation by associating the input <math>x_t</math> with a (delayed) future output. In doing this, we need to ensure that the output <math>y_t</math> is separable, i.e., not influenced by any future input <math>x_{t^{'}}</math> <math>(t^{'}>t)</math>. Thus, we concatenate the projection of <math>x_t</math> to the top of the previous hidden state <math>H_{t−1}</math>, then gradually shift the input information down when the temporal computation proceeds, and finally generate <math>y_t</math> from the bottom of <math>H_{t+L−1}</math>, where L−1 is the number of delayed time-steps for computations of depth L. <br />
<br />
An example with L= 3 is shown in Figure.<br />
<br />
[[File:tRNN.png|160px|center||Figure 5: skewed sRNN]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: skewed sRNN]]<br />
<br />
<br />
This is in fact a skewed sRNN (or tRNN without feedback). However, the method does not need to change the network structure and also allows different kinds of interactions as long as the output is separable; for example, one can increase the local connections and '''use feedback''' (shown in figure below), which can be beneficial for sRNNs (or tRNN). <br />
<br />
[[File:tRNN_wF.png|160px|center||Figure 5: skewed sRNN with F]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: skewed sRNN with F]]<br />
<br />
'''In order to share parameters, we update <math>H_t</math> using a convolution with a learnable kernel.''' In this manner we increase the complexity of the input-to-output mapping (by delaying outputs) and limit parameter growth (by sharing transition parameters using convolutions).<br />
<br />
To examine the resulting model mathematically, let <math>H^{cat}_{t−1}∈R^{(P+1)×M}</math> be the concatenated hidden state, and <math>p∈Z_+</math> the location at a tensor. The channel vector <math>h^{cat}_{t−1, p }∈R^M</math> at location p of <math>H^{cat}_{t−1}</math> (the p-th channel of H) is defined as:<br />
<br />
\begin{align}<br />
h^{cat}_{t-1, p} = x_t W^x + b^x \hspace{1cm} if p = 1 \hspace{1cm} (5)<br />
\end{align}<br />
<br />
\begin{align}<br />
h^{cat}_{t-1, p} = h_{t-1, p-1} \hspace{1cm} if p > 1 \hspace{1cm} (6)<br />
\end{align}<br />
<br />
where <math>W^x ∈ R^{R×M}</math> and <math>b^x ∈ R^M</math> (recall the dimension of input x is R). Then, the update of tensor <math>H_t</math> is implemented via a convolution:<br />
<br />
\begin{align}<br />
A_t = H^{cat}_{t-1} \circledast \{W^h, b^h \} \hspace{2cm} (7)<br />
\end{align}<br />
<br />
\begin{align}<br />
H_t = \Phi{A_t} \hspace{2cm} (8)<br />
\end{align}<br />
<br />
where <math>W^h∈R^{K×M^i×M^o}</math> is the kernel weight of size K, with <math>M^i =M</math> input channels and <math>M^o =M</math> output channels, <math>b^h ∈ R^{M^o}</math> is the kernel bias, <math>A_t ∈ R^{P×M^o}</math> is the hidden activation, and <math>\circledast</math> is the convolution operator. Since the kernel convolves across different hidden layers, we call it the cross-layer convolution. The kernel enables interaction, both bottom-up and top-down across layers. Finally, we generate <math>y_t</math> from the channel vector <math>h_{t+L−1,P}∈R^M</math> which is located at the bottom of <math>H_{t+L−1}</math>:<br />
<br />
\begin{align}<br />
y_t = \varphi(h_{t+L−1}, _PW^y + b^y) \hspace{2cm} (9)<br />
\end{align}<br />
<br />
Where <math>W^y ∈R^{M×S}</math> and <math>b^y ∈R^S</math>. To guarantee that the receptive field of <math>y_t</math> only covers the current and previous inputs x1:t. (Check the Skewed sRNN again below):<br />
<br />
[[File:tRNN_wF.png|160px|center||Figure 5: skewed sRNN with F]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: skewed sRNN with F]]<br />
<br />
=== Quick Summary of Set of Parameters ===<br />
<br />
'''1. <math> W^x</math> and <math>b_x</math>''' connect input to the first hidden node<br />
<br />
'''2. <math> W^h</math> and <math>b_h</math>''' convolute between layers<br />
<br />
'''3. <math> W^y</math> and <math>b_y</math>''' produce output of each stages<br />
<br />
<br />
== Part 3: Extending to LSTMs==<br />
<br />
Similar to standard RNN, to allow the tRNN (skewed sRNN) to capture long-range temporal dependencies, one can straightforwardly extend it<br />
to a tLSTM by replacing the tRNN tensors:<br />
<br />
\begin{align}<br />
[A^g_t, A^i_t, A^f_t, A^o_t] = H^{cat}_{t-1} \circledast \{W^h, b^h \} \hspace{2cm} (10)<br />
\end{align}<br />
<br />
\begin{align}<br />
[G_t, I_t, F_t, O_t]= [\Phi{(A^g_t)}, σ(A^i_t), σ(A^f_t), σ(A^o_t)] \hspace{2cm} (11)<br />
\end{align}<br />
<br />
Which are pretty similar to tRNN case, the main differences can be observes for memory cells of tLSTM (Ct):<br />
<br />
\begin{align}<br />
C_t= G_t \odot I_t + C_{t-1} \odot F_t \hspace{2cm} (12)<br />
\end{align}<br />
<br />
\begin{align}<br />
H_t= \Phi{(C_t )} \odot O_t \hspace{2cm} (13)<br />
\end{align}<br />
<br />
Note that since the previous memory cell <math>C_{t-1}</math> is only gated along the temporal direction, increasing the tensor size ''P'' might result in the loss of long-range dependencies from the input to the output.<br />
<br />
Summary of the terms: <br />
<br />
1. '''<math>\{W^h, b^h \}</math>:''' Kernel of size K <br />
<br />
2. '''<math>A^g_t, A^i_t, A^f_t, A^o_t \in \mathbb{R}^{P\times M}</math>:''' Activations for the new content <math>G_t</math><br />
<br />
3. '''<math>I_t</math>:''' Input gate<br />
<br />
4. '''<math>F_t</math>:''' Forget gate<br />
<br />
5. '''<math>O_t</math>:''' Output gate<br />
<br />
6. '''<math>C_t \in \mathbb{R}^{P\times M}</math>:''' Memory cell<br />
<br />
Then, see graph below for illustration:<br />
<br />
[[File:tLSTM_wo_MC.png |160px|center||Figure 5: tLSTM wo MC]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: tLSTM wo MC]]<br />
<br />
To further evolve tLSTM, we invoke the '''Memory Cell Convolution''' to capture long-range dependencies from multiple directions, we additionally introduce a novel memory cell convolution, by which the memory cells can have a larger receptive field (figure provided below). <br />
<br />
[[File:tLSTM_w_MC.png |160px|center||Figure 5: tLSTM w MC]]<br />
<br />
[[File:ind.png|60px|center||Figure 5: tLSTM w MC]]<br />
<br />
One can also dynamically generate this convolution kernel so that it is both time - and location-dependent, allowing for flexible control over long-range dependencies from different directions. Mathematically, it can be represented in with the following formulas:<br />
<br />
\begin{align}<br />
[A^g_t, A^i_t, A^f_t, A^o_t, A^q_t] = H^{cat}_{t-1} \circledast \{W^h, b^h \} \hspace{2cm} (14)<br />
\end{align}<br />
<br />
\begin{align}<br />
[G_t, I_t, F_t, O_t, Q_t]= [\Phi{(A^g_t)}, σ(A^i_t), σ(A^f_t), σ(A^o_t), ς(A^q_t)] \hspace{2cm} (15)<br />
\end{align}<br />
<br />
\begin{align}<br />
W_t^c(p) = reshape(q_{t,p}, [K, 1, 1]) \hspace{2cm} (16)<br />
\end{align}<br />
<br />
\begin{align}<br />
C_{t-1}^{conv}= C_{t-1} \circledast W_t^c(p) \hspace{2cm} (17)<br />
\end{align}<br />
<br />
\begin{align}<br />
C_t= G_t \odot I_t + C_{t-1}^{conv} \odot F_t \hspace{2cm} (18)<br />
\end{align}<br />
<br />
\begin{align}<br />
H_t= \Phi{(C_t )} \odot O_t \hspace{2cm} (19)<br />
\end{align}<br />
<br />
where the kernel <math>{W^h, b^h}</math> has additional <K> output channels to generate the activation <math>A^q_t ∈ R^{P×<K>}</math> for the dynamic kernel bank <math>Q_t∈R^{P × <K>}</math>, <math>q_{t,p}∈R^{<K>}</math> is the vectorized adaptive kernel at the location p of <math>Q_t</math>, and <math>W^c_t(p) ∈ R^{K×1×1}</math> is the dynamic kernel of size K with a single input/output channel, which is reshaped from <math>q_{t,p}</math>. Each channel of the previous memory cell <math>C_{t-1}</math> is convolved with <math>W_t^c(p)</math> whose values vary with <math>p</math>, to form a memory cell convolution, which produces a convolved memory cell <math>C_{t-1}^{conv} \in \mathbb{R}^{P\times M}</math>. This convolution is defined by:<br />
<br />
\begin{align}<br />
C_{t-1,p,m}^{conv} = \sum\limits_{k=1}^K C_{t-1,p-\frac{K-1}{2}+k,m} · W_{t,k,1,1}^c(p) \hspace{2cm} (30)<br />
\end{align}<br />
<br />
where <math>C_{t-1}</math> is padded with the boundary values to retain the stored information.<br />
<br />
Note the paper also employed a softmax function ς(·) to normalize the channel dimension of <math>Q_t</math>. which can also stabilize the value of memory cells and help to prevent the vanishing/exploding gradients. An illustration is provided below to better illustrate the process:<br />
<br />
[[File:MCC.png |240px|center||Figure 5: MCC]]<br />
<br />
To improve training, the authors introduced a new normalization technique for ''t''LSTM termed channel normalization (adapted from layer normalization), in which the channel vector are normalized at different locations with their own statistics. Note that layer normalization does not work well with ''t''LSTM, because lower level information is near the input and higher level information is near the output. Channel normalization (CN) is defined as: <br />
<br />
\begin{align}<br />
\mathrm{CN}(\mathbf{Z}; \mathbf{\Gamma}, \mathbf{B}) = \mathbf{\hat{Z}} \odot \mathbf{\Gamma} + \mathbf{B} \hspace{2cm} (20)<br />
\end{align}<br />
<br />
where <math>\mathbf{Z}</math>, <math>\mathbf{\hat{Z}}</math>, <math>\mathbf{\Gamma}</math>, <math>\mathbf{B} \in \mathbb{R}^{P \times M^z}</math> are the original tensor, normalized tensor, gain parameter and bias parameter. The <math>m^z</math>-th channel of <math>\mathbf{Z}</math> is normalized element-wisely: <br />
<br />
\begin{align}<br />
\hat{z_{m^z}} = (z_{m^z} - z^\mu)/z^{\sigma} \hspace{2cm} (21)<br />
\end{align}<br />
<br />
where <math>z^{\mu}</math>, <math>z^{\sigma} \in \mathbb{R}^P</math> are the mean and standard deviation along the channel dimension of <math>\mathbf{Z}</math>, and <math>\hat{z_{m^z}} \in \mathbb{R}^P</math> is the <math>m^z</math>-th channel <math>\mathbf{\hat{Z}}</math>. Channel normalization introduces very few additional parameters compared to the number of other parameters in the model.<br />
<br />
= Results and Evaluation =<br />
<br />
Summary of list of models tLSTM family (may be useful later):<br />
<br />
(a) sLSTM (baseline): the implementation of sLSTM with parameters shared across all layers.<br />
<br />
(b) 2D tLSTM: the standard 2D tLSTM.<br />
<br />
(c) 2D tLSTM–M: removing memory (M) cell convolutions from (b).<br />
<br />
(d) 2D tLSTM–F: removing (–) feedback (F) connections from (b).<br />
<br />
(e) 3D tLSTM: tensorizing (b) into 3D tLSTM.<br />
<br />
(f) 3D tLSTM+LN: applying (+) Layer Normalization.<br />
<br />
(g) 3D tLSTM+CN: applying (+) Channel Normalization.<br />
<br />
=== Efficiency Analysis ===<br />
<br />
'''Fundaments:''' For each configuration, fix the parameter number and increase the tensor size to see if the performance of tLSTM can be boosted without increasing the parameter number. Can also investigate how the runtime is affected by the depth, where the runtime is measured by the average GPU milliseconds spent by a forward and backward pass over one timestep of a single example. <br />
<br />
'''Dataset:''' The Hutter Prize Wikipedia dataset consists of 100 million characters taken from 205 different characters including alphabets, XML markups and special symbols. We model the dataset at the character-level, and try to predict the next character of the input sequence.<br />
<br />
All configurations are evaluated with depths L = 1, 2, 3, 4. Bits-per-character(BPC) is used to measure the model performance and the results are shown in the figure below.<br />
[[File:wiki.png |280px|center||Figure 5: WifiPerf]]<br />
[[File:Wiki_Performance.png |480px|center||Figure 5: WifiPerf]]<br />
<br />
=== Accuracy Analysis ===<br />
<br />
The MNIST dataset [35] consists of 50000/10000/10000 handwritten digit images of size 28×28 for training/validation/test. Two tasks are used for evaluation on this dataset:<br />
<br />
(a) '''Sequential MNIST:''' The goal is to classify the digit after sequentially reading the pixels in a scan-line order. It is therefore a 784 time-step sequence learning task where a single output is produced at the last time-step; the task requires very long range dependencies in the sequence.<br />
<br />
(b) '''Sequential Permuted MNIST:''' We permute the original image pixels in a fixed random order, resulting in a permuted MNIST (pMNIST) problem that has even longer range dependencies across pixels and is harder.<br />
<br />
In both tasks, all configurations are evaluated with M = 100 and L= 1, 3, 5. The model performance is measured by the classification accuracy and results are shown in the figure below.<br />
<br />
[[File:MNISTperf.png |480px|center]]<br />
<br />
<br />
<br />
[[File:Acc_res.png |480px|center||Figure 5: MNIST]]<br />
<br />
[[File:33_mnist.PNG|center|thumb|800px| This figure displays a visualization of the means of the diagonal channels of the tLSTM memory cells per task. The columns indicate the time steps and the rows indicate the diagonal locations. The values are normalized between 0 and 1.]]<br />
<br />
It can be seen in the above figure that tLSTM behaves differently with different tasks:<br />
<br />
- Wikipedia: the input can be carried to the output location with less modification if it is sufficient to determine the next character, and vice versa<br />
<br />
- addition: the first integer is gradually encoded into memories and then interacts (performs addition) with the second integer, producing the sum <br />
<br />
- memorization: the network behaves like a shift register that continues to move the input symbol to the output location at the correct timestep<br />
<br />
- sequential MNIST: the network is more sensitive to the pixel value change (representing the contour, or topology of the digit) and can gradually accumulate evidence for the final prediction <br />
<br />
- sequential pMNIST: the network is sensitive to high value pixels (representing the foreground digit), and we conjecture that this is because the permutation destroys the topology of the digit, making each high value pixel potentially important.<br />
<br />
From the figure above it can can also be observe some common phenomena in all tasks: <br />
# it is clear that wider (larger) tensors can encode more information by observing that at each timestep, the values at different tensor locations are markedly different<br />
# from the input to the output, the values become increasingly distinct and are shifted by time, revealing that deep computations are indeed performed together with temporal computations, with long-range dependencies carried by memory cells.<br />
<br />
<br />
= Conclusions =<br />
<br />
The paper introduced the Tensorized LSTM, which employs tensors to share parameters and utilizes the temporal computation to perform the deep computation for sequential tasks. Then validated the model<br />
on a variety of tasks, showing its potential over other popular approaches. The paper shows a method to widen and deepen the LSTM network at the same time and the following 3 points list their main contributions:<br />
* The RNNs are now tensorized into higher dimensional tensors which are more flexible.<br />
* RNNs' deep computation is merged into the temporal computation, referred to as the tensorizedRNN.<br />
* tRNN is extended to a LSTM architecture and a new architecture is studied: tensorizedLSTM<br />
<br />
= Critique(to be edited) =<br />
<br />
= References =<br />
#Zhen He, Shaobing Gao, Liang Xiao, Daxue Liu, Hangen He, and David Barber. <Wider and Deeper, Cheaper and Faster: Tensorized LSTMs for Sequence Learning> (2017)<br />
#Ali Ghodsi, <Deep Learning: STAT 946 - Winter 2018></div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Training_And_Inference_with_Integers_in_Deep_Neural_Networks&diff=36160Training And Inference with Integers in Deep Neural Networks2018-04-04T19:43:17Z<p>Pa2forsy: </p>
<hr />
<div>== Introduction ==<br />
<br />
Deep neural networks have enjoyed much success in all manners of tasks, but it is common for these networks to be complicated and have high memory requirements while performing many floating-point operations (FLOPs). As a result, running many of these models are very expensive in terms of energy use, and using state-of-the-art networks in applications where energy is limited can be very difficult. In order to overcome this and allow use of these networks in situations with low energy availability, the energy costs must be reduced while trying to maintain as high network performance as possible and/or practical.<br />
<br />
Most existing methods focus on reducing the energy requirements during inference rather than training. Since training with SGD requires accumulation, training usually has higher precision demand than inference. Most of the existing methods focus on how to compress a model for inference, rather than during training. This paper proposes a framework to reduce complexity both during training and inference through the use of integers instead of floats. The authors address how to quantize all operations and operands as well as examining the bitwidth requirement for SGD computation & accumulation. Using integers instead of floats results in energy-savings because integer operations are more efficient than floating point (see the table below). Also, there already exists dedicated hardware for deep learning that uses integer operations (such as the 1st generation of Google TPU) so understanding the best way to use integers is well-motivated. A TPU is a Tensor Processing Unit developed by Google for Tensor operations. TPU is comparative to a GPU but produces higher IO per second for low precision computations.<br />
{| class="wikitable"<br />
|+Rough Energy Costs in 45nm 0.9V [1]<br />
!<br />
! colspan="2" |Energy(pJ)<br />
! colspan="2" |Area(<math>\mu m^2</math>)<br />
|-<br />
!Operation<br />
!MUL<br />
!ADD<br />
!MUL<br />
!ADD<br />
|-<br />
|8-bit INT<br />
|0.2<br />
|0.03<br />
|282<br />
|36<br />
|-<br />
|16-bit FP<br />
|1.1<br />
|0.4<br />
|1640<br />
|1360<br />
|-<br />
|32-bit FP<br />
|3.7<br />
|0.9<br />
|7700<br />
|4184<br />
|}<br />
The authors call the framework WAGE because they consider how best to handle the '''W'''eights, '''A'''ctivations, '''G'''radients, and '''E'''rrors separately.<br />
<br />
== Related Work ==<br />
<br />
=== Weight and Activation ===<br />
Existing works to train DNNs on binary weights and activations [2] add noise to weights and activations as a form of regularization. The use of high-precision accumulation is required for SGD optimization since real-valued gradients are obtained from real-valued variables. XNOR-Net [11] uses bitwise operations to approximate convolutions in a highly memory-efficient manner, and applies a filter-wise scaling factor for weights to improve performance. However, these floating-point factors are calculated simultaneously during training, which aggravates the training effort. Ternary weight networks (TWN) [3] and Trained ternary quantization (TTQ) [9] offer more expressive ability than binary weight networks by constraining the weights to be ternary-valued {-1,0,1} using two symmetric thresholds. Tang et al. [14] achieve impressive results by using a binarization scheme according to which floating-point activation vectors are approximated as linear combinations of binary vectors, where the weights in the linear combination are floating-point. Still other approaches rely on relative quantization [13]; however, an efficient implementation is difficult to apply in practice due to the requirements of persisting and applying a codebook.<br />
<br />
=== Gradient Computation and Accumulation ===<br />
The DoReFa-Net quantizes gradients to low-bandwidth floating point numbers with discrete states in the backwards pass. In order to reduce the overhead of gradient synchronization in distributed training the TernGrad method quantizes the gradient updates to ternary values. In both works the weights are still stored and updated with float32, and the quantization of batch normalization and its derivative is ignored.<br />
<br />
== WAGE Quantization ==<br />
The core idea of the proposed method is to constrain the following to low-bitwidth integers on each layer:<br />
* '''W:''' weight in inference<br />
* '''a:''' activation in inference<br />
* '''e:''' error in backpropagation<br />
* '''g:''' gradient in backpropagation<br />
[[File:p32fig1.PNG|center|thumb|800px|Four operators QW (·), QA(·), QG(·), QE(·) added in WAGE computation dataflow to reduce precision, bitwidth of signed integers are below or on the right of arrows, activations are included in MAC for concision.]]<br />
The error and gradient are defined as:<br />
<br />
<math>e^i = \frac{\partial L}{\partial a^i}, g^i = \frac{\partial L}{\partial W^i}</math><br />
<br />
where L is the loss function.<br />
<br />
The precision in bits of the errors, activations, gradients, and weights are <math>k_E</math>, <math>k_A</math>, <math>k_G</math>, and <math>k_W</math> respectively. As shown in the above figure, each quantity also has a quantization operators to reduce bitwidth increases caused by multiply-accumulate (MAC) operations. Also, note that since this is a layer-by-layer approach, each layer may be followed or preceded by a layer with different precision, or even a layer using floating point math.<br />
<br />
=== Shift-Based Linear Mapping and Stochastic Mapping ===<br />
The proposed method makes use of a linear mapping where continuous, unbounded values are discretized for each bitwidth <math>k</math> with a uniform spacing of<br />
<br />
<math>\sigma(k) = 2^{1-k}, k \in Z_+ </math><br />
With this, the full quantization function is<br />
<br />
<math>Q(x,k) = Clip\left \{ \sigma(k) \cdot round\left [ \frac{x}{\sigma(k)} \right ], -1 + \sigma(k), 1 - \sigma(k) \right \}</math>, <br />
<br />
where <math>round</math> approximates continuous values to their nearest discrete state, and <math>Clip</math> is the saturation function that clips unbounded values to <math>[-1 + \sigma, 1 - \sigma]</math>. Note that this function is only using when simulating integer operations on floating-point hardware, on native integer hardware, this is done automatically. In addition to this quantization function, a distribution scaling factor is used in some quantization operators to preserve as much variance as possible when applying the quantization function above. The scaling factor is defined below.<br />
<br />
<math>Shift(x) = 2^{round(log_2(x))}</math><br />
<br />
Finally, stochastic rounding is substituted for small or real-valued updates during gradient accumulation.<br />
<br />
A visual representation of these operations is below.<br />
[[File:p32fig2.PNG|center|thumb|800px|Quantization methods used in WAGE. The notation <math>P, x, \lfloor \cdot \rfloor, \lceil \cdot \rceil</math> denotes probability, vector, floor and ceil, respectively. <math>Shift(\cdot)</math> refers to distribution shifting with a certain argument]]<br />
<br />
=== Weight Initialization ===<br />
In this work, batch normalization is simplified to a constant scaling layer in order to sidestep the problem of normalizing outputs without floating point math, and to remove the extra memory requirement with batch normalization. As such, some care must be taken when initializing weights. The authors use a modified initialization method base on MSRA [4].<br />
<br />
<math>W \thicksim U(-L, +L),L = max \left \{ \sqrt{6/n_{in}}, L_{min} \right \}, L_{min} = \beta \sigma</math><br />
<br />
<math>n_{in}</math> is the layer fan-in number, <math>U</math> denotes uniform distribution. The original initialization method for <math>\eta</math> is modified by adding the condition that the distribution width should be at least <math>\beta \sigma</math>, where <math>\beta</math> is a constant greater than 1 and <math>\sigma</math> is the minimum step size seen already. This prevents weights being initialised to all-zeros in the case where the bitwidth is low, or the fan-in number is high.<br />
<br />
=== Quantization Details ===<br />
<br />
==== Weight <math>Q_W(\cdot)</math> ====<br />
<math>W_q = Q_W(W) = Q(W, k_W)</math><br />
<br />
The quantization operator is simply the quantization function previously introduced. <br />
<br />
==== Activation <math>Q_A(\cdot)</math> ====<br />
The authors say that the variance of the weights passed through this function will be scaled compared to the variance of the weights as initialized. To prevent this effect from blowing up the network outputs, they introduce a scaling factor <math>\alpha</math>. Notice that it is constant for each layer.<br />
<br />
<math>\alpha = max \left \{ Shift(L_{min} / L), 1 \right \}</math><br />
<br />
The quantization operator is then<br />
<br />
<math>a_q = Q_A(a) = Q(a/\alpha, k_A)</math><br />
<br />
The scaling factor approximates batch normalization.<br />
<br />
==== Error <math>Q_E(\cdot)</math> ====<br />
The magnitude of the error can vary greatly, and that a previous approach (DoReFa-Net [5]) solves the issue by using an affine transform to map the error to the range <math>[-1, 1]</math>, apply quantization, and then applying the inverse transform. However, the authors claim that this approach still requires using float32, and that the magnitude of the error is unimportant: rather it is the orientation of the error. Thus, they only scale the error distribution to the range <math>\left [ -\sqrt2, \sqrt2 \right ]</math> and quantise:<br />
<br />
<math>e_q = Q_E(e) = Q(e/Shift(max\{|e|\}), k_E)</math><br />
<br />
Max is the element-wise maximum. Note that this discards any error elements less than the minimum step size.<br />
<br />
==== Gradient <math>Q_G(\cdot)</math> ====<br />
Similar to the activations and errors, the gradients are rescaled:<br />
<br />
<math>g_s = \eta \cdot g/Shift(max\{|g|\})</math><br />
<br />
<math> \eta </math> is a shift-based learning rate. It is an integer power of 2. The shifted gradients are represented in units of minimum step sizes <math> \sigma(k) </math>. When reducing the bitwidth of the gradients (remember that the gradients are coming out of a MAC operation, so the bitwidth may have increased) stochastic rounding is used as a substitute for small gradient accumulation.<br />
<br />
<math>\Delta W = Q_G(g) = \sigma(k_G) \cdot sgn(g_s) \cdot \left \{ \lfloor | g_s | \rfloor + Bernoulli(|g_s|<br />
- \lfloor | g_s | \rfloor) \right \}</math><br />
<br />
This randomly rounds the result of the MAC operation up or down to the nearest quantization for the given gradient bitwidth. The weights are updated with the resulting discrete increments:<br />
<br />
<math>W_{t+1} = Clip \left \{ W_t - \Delta W_t, -1 + \sigma(k_G), 1 - \sigma(k_G) \right \}</math><br />
<br />
=== Miscellaneous ===<br />
To train WAGE networks, the authors used pure SGD exclusively because more complicated techniques such as Momentum or RMSProp increase memory consumption and are complicated by the rescaling that happens within each quantization operator.<br />
<br />
The quantization and stochastic rounding are a form of regularization.<br />
<br />
The authors didn't use a traditional softmax with cross-entropy loss for the experiments because there does not yet exist a softmax layer for low-bit integers. Instead, they use a sum of squared error loss. This works for tasks with a small number of categories, but does not scale well.<br />
<br />
== Experiments ==<br />
For all experiments, the default layer bitwidth configuration is 2-8-8-8 for Weights, Activations, Gradients, and Error bits. The weight bitwidth is set to 2 because that results in ternary weights, and therefore no multiplication during inference. They authors argue that the bitwidth for activation and errors should be the same because the computation graph for each is similar and might use the same hardware. During training, the weight bitwidth is 8. For inference the weights are ternarized.<br />
<br />
=== Implementation Details ===<br />
MNIST: Network is LeNet-5 variant [6] with 32C5-MP2-64C5-MP2-512FC-10SSE.<br />
<br />
SVHN & CIFAR10: VGG variant [7] with 2×(128C3)-MP2-2×(256C3)-MP2-2×(512C3)-MP2-1024FC-10SSE. For CIFAR10 dataset, the data augmentation is followed in Lee et al. (2015) [10] for training.<br />
<br />
ImageNet: AlexNet variant [8] on ILSVRC12 dataset.<br />
{| class="wikitable"<br />
|+Test or validation error rates (%) in previous works and WAGE on multiple datasets. Opt denotes gradient descent optimizer, withM means SGD with momentum, BN represents batch normalization, 32 bit refers to float32, and ImageNet top-k format: top1/top5.<br />
!Method<br />
!<math>k_W</math><br />
!<math>k_A</math><br />
!<math>k_G</math><br />
!<math>k_E</math><br />
!Opt<br />
!BN<br />
!MNIST<br />
!SVHN<br />
!CIFAR10<br />
!ImageNet<br />
|-<br />
|BC<br />
|1<br />
|32<br />
|32<br />
|32<br />
|Adam<br />
|yes<br />
|1.29<br />
|2.30<br />
|9.90<br />
|<br />
|-<br />
|BNN<br />
|1<br />
|1<br />
|32<br />
|32<br />
|Adam<br />
|yes <br />
|0.96<br />
|2.53<br />
|10.15<br />
|<br />
|-<br />
|BWN<br />
|1<br />
|32<br />
|32<br />
|32<br />
|withM<br />
|yes<br />
|<br />
|<br />
|<br />
|43.2/20.6<br />
|-<br />
|XNOR<br />
|1<br />
|1<br />
|32<br />
|32<br />
|Adam<br />
|yes<br />
|<br />
|<br />
|<br />
|55.8/30.8<br />
|-<br />
|TWN<br />
|2<br />
|32<br />
|32<br />
|32<br />
|withM<br />
|yes<br />
|0.65<br />
|<br />
|7.44<br />
|'''34.7/13.8'''<br />
|-<br />
|TTQ<br />
|2<br />
|32<br />
|32<br />
|32<br />
|Adam<br />
|yes<br />
|<br />
|<br />
|6.44<br />
|42.5/20.3<br />
|-<br />
|DoReFa<br />
|8<br />
|8<br />
|32<br />
|8<br />
|Adam<br />
|yes<br />
|<br />
|2.30<br />
|<br />
|47.0/<br />
|-<br />
|TernGrad<br />
|32<br />
|32<br />
|2<br />
|32<br />
|Adam<br />
|yes<br />
|<br />
|<br />
|14.36<br />
|42.4/19.5<br />
|-<br />
|WAGE<br />
|2<br />
|8<br />
|8<br />
|8<br />
|SGD<br />
|no<br />
|'''0.40'''<br />
|'''1.92'''<br />
|'''6.78'''<br />
|51.6/27.8<br />
|}<br />
<br />
=== Training Curves and Regularization ===<br />
The authors compare the 2-8-8-8 WAGE configuration introduced above, a 2-8-f-f (meaning float32) configuration, and a completely floating point version on CIFAR10. The test error is plotted against epoch. For training these networks, the learning rate is divided by 8 at the 200th epoch and again at the 250th epoch.<br />
[[File:p32fig3.PNG|center|thumb|800px|Training curves of WAGE variations and a vanilla CNN on CIFAR10]]<br />
The convergence of the 2-8-8-8 has comparable convergence to the vanilla CNN and outperforms the 2-8-f-f variant. The authors speculate that this is because the extra discretization acts as a regularizer.<br />
<br />
=== Bitwidth of Errors ===<br />
The CIFAR10 test accuracy is plotted against bitwidth below and the error density for a single layer is compared with the Vanilla network.<br />
[[File:p32fig4.PNG|center|thumb|520x522px|The 10 run accuracies of different <math>k_E</math>]]<br />
<br />
[[File:32_error.png|center|thumb|520x522px|Histogram of errors for Vanilla network and Wage network. After being quantized and shifted each layer, the error is reshaped and so most orientation information is retained. ]]<br />
<br />
The table below shows the test error rates on CIFAR10 when left-shift upper boundary with factor γ. From this table we could see that large values play critical roles for backpropagation training even though they don't have many while the majority with small values actually just noises.<br />
<br />
[[File:testerror_rate.png|center]]<br />
<br />
=== Bitwidth of Gradients ===<br />
<br />
The authors next investigated the choice of a proper <math>k_G</math> for gradients using the CIFAR10 dataset. <br />
<br />
{| class="wikitable"<br />
|+Test error rates (%) on CIFAR10 with different <math>k_G</math><br />
!<math>k_G</math><br />
!2<br />
!3<br />
!4<br />
!5<br />
!6<br />
!7<br />
!8<br />
!9<br />
!10<br />
!11<br />
!12<br />
|-<br />
|error<br />
|54.22<br />
|51.57<br />
|28.22<br />
|18.01<br />
|11.48<br />
|7.61<br />
|6.78<br />
|6.63<br />
|6.43<br />
|6.55<br />
|6.57<br />
|}<br />
<br />
The results show similar bitwidth requirements as the last experiment for <math>k_E</math>.<br />
<br />
The authors also examined the effect of bitwidth on the ImageNet implementation.<br />
<br />
Here, C denotes 12 bits (Hexidecimal) and BN refers to batch normalization being added. 7 models are used: 2888 from the first experiment, 288C for more accurate errors (12 bits), 28C8 for larger buffer space, 28f8 for non-quantization of gradients, 28ff for errors and gradients in float32, and 28ff with BN added. The baseline vanilla model refers to the original AlexNet architecture. <br />
<br />
{| class="wikitable"<br />
|+Top-5 error rates (%) on ImageNet with different <math>k_G</math>and <math>k_E</math><br />
!Pattern<br />
!vanilla<br />
!28ff-BN<br />
!28ff<br />
!28f8<br />
!28C8<br />
!288C<br />
!2888<br />
|-<br />
|error<br />
|19.29<br />
|20.67<br />
|24.14<br />
|23.92<br />
|26.88<br />
|28.06<br />
|27.82<br />
|}<br />
<br />
The comparison between 28C8 and 288C shows that the model may perform better if it has more buffer space <math>k_G</math> for gradient accumulation than if it has high-resolution orientation <math>k_E</math>. The authors also noted that batch normalization and <math>k_G</math> are more important for ImageNet because the training set samples are highly variant.<br />
<br />
== Discussion ==<br />
The authors have a few areas they believe this approach could be improved.<br />
<br />
'''MAC Operation:''' The 2-8-8-8 configuration was chosen because the low weight bitwidth means there aren't any multiplication during inference. However, this does not remove the requirement for multiplication during training. 2-2-8-8 configuration satisfies this requirement, but it is difficult to train and detrimental to the accuracy.<br />
<br />
'''Non-linear Quantization:''' The linear mapping used in this approach is simple, but there might be a more effective mapping. For example, a logarithmic mapping could be more effective if the weights and activations have a log-normal distribution.<br />
<br />
'''Normalization:''' Normalization layers (softmax, batch normalization) were not used in this paper. Quantized versions are an area of future work<br />
<br />
== Conclusion ==<br />
<br />
A framework for training and inference without the use of floating-point representation is presented. By quantizing all operations and operands of a network, the authors successfully reduce the energy costs of both training and inference with deep learning architectures. Future work may further improve compression and memory requirements.<br />
<br />
== Implementation ==<br />
The following repository provides the source code for the paper: https://github.com/boluoweifenda/WAGE. The repository provides the source code as written by the authors, in Tensorflow.<br />
<br />
<br />
<br />
== References ==<br />
<br />
# Sze, Vivienne; Chen, Yu-Hsin; Yang, Tien-Ju; Emer, Joel (2017-03-27). [http://arxiv.org/abs/1703.09039 "Efficient Processing of Deep Neural Networks: A Tutorial and Survey"]. arXiv:1703.09039 [cs].<br />
# Courbariaux, Matthieu; Bengio, Yoshua; David, Jean-Pierre (2015-11-01). [http://arxiv.org/abs/1511.00363 "BinaryConnect: Training Deep Neural Networks with binary weights during propagations"]. arXiv:1511.00363 [cs].<br />
# Li, Fengfu; Zhang, Bo; Liu, Bin (2016-05-16). [http://arxiv.org/abs/1605.04711 "Ternary Weight Networks"]. arXiv:1605.04711 [cs].<br />
# He, Kaiming; Zhang, Xiangyu; Ren, Shaoqing; Sun, Jian (2015-02-06). [http://arxiv.org/abs/1502.01852 "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification"]. arXiv:1502.01852 [cs].<br />
# Zhou, Shuchang; Wu, Yuxin; Ni, Zekun; Zhou, Xinyu; Wen, He; Zou, Yuheng (2016-06-20). [http://arxiv.org/abs/1606.06160 "DoReFa-Net: Training Low Bitwidth Convolutional Neural Networks with Low Bitwidth Gradients"]. arXiv:1606.06160 [cs].<br />
# Lecun, Y.; Bottou, L.; Bengio, Y.; Haffner, P. (November 1998). [http://ieeexplore.ieee.org/document/726791/?reload=true "Gradient-based learning applied to document recognition"]. Proceedings of the IEEE. 86 (11): 2278–2324. doi:10.1109/5.726791. ISSN 0018-9219.<br />
# Simonyan, Karen; Zisserman, Andrew (2014-09-04). [http://arxiv.org/abs/1409.1556 "Very Deep Convolutional Networks for Large-Scale Image Recognition"]. arXiv:1409.1556 [cs].<br />
# Krizhevsky, Alex; Sutskever, Ilya; Hinton, Geoffrey E (2012). Pereira, F.; Burges, C. J. C.; Bottou, L.; Weinberger, K. Q., eds. [http://papers.nips.cc/paper/4824-imagenet-classification-with-deep-convolutional-neural-networks.pdf Advances in Neural Information Processing Systems 25 (PDF)]. Curran Associates, Inc. pp. 1097–1105.<br />
# Chenzhuo Zhu, Song Han, Huizi Mao, and William J Dally. Trained ternary quantization. arXiv preprint arXiv:1612.01064, 2016.<br />
# Chen-Yu Lee, Saining Xie, Patrick Gallagher, Zhengyou Zhang, and Zhuowen Tu. Deeplysupervisednets. In Artificial Intelligence and Statistics, pp. 562–570, 2015.<br />
# Mohammad Rastegari, Vicente Ordonez, Joseph Redmon, and Ali Farhadi. Xnor-net: Imagenet classification using binary convolutional neural networks. In European Conference on Computer Vision, pp. 525–542. Springer, 2016.<br />
# “Boluoweifenda/WAGE.” GitHub, github.com/boluoweifenda/WAGE.<br />
# Han, S., Mao, H., & Dally, W. J. (2015). Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149.<br />
# Tang, Wei, Gang Hua, and Liang Wang. "How to train a compact binary neural network with high accuracy?." AAAI. 2017.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Multi-scale_Dense_Networks_for_Resource_Efficient_Image_Classification&diff=36159Multi-scale Dense Networks for Resource Efficient Image Classification2018-04-04T17:22:45Z<p>Pa2forsy: </p>
<hr />
<div>= Introduction = <br />
<br />
Multi-Scale Dense Networks, MSDNets, are designed to address the growing demand for efficient object recognition. The issue with existing recognition networks is that they are either efficient networks, but don't do well on hard examples, or large networks that do well on all examples but require a large amount of resources. For example, the winner of the COCO 2016 competition was an [http://image-net.org/challenges/talks/2016/GRMI-COCO-slidedeck.pdf ensemble of CNNs], which are likely far too resource-heavy to be used in any resource-limited application.<br />
<br />
In order to be efficient on all difficulties MSDNets propose a structure that can accurately output classifications for varying levels of computational requirements. The two cases that are used to evaluate the network are:<br />
* Anytime Prediction: What is the best prediction the network can provide when suddenly prompted?<br />
* Budget Batch Predictions: Given a maximum amount of computational resources, how well does the network do on the batch?<br />
<br />
= Related Networks =<br />
<br />
== Computationally Efficient Networks ==<br />
<br />
Much of the existing work on convolution networks that are computationally efficient at test time focus on reducing model size after training. Many existing methods for refining an accurate network to be more efficient include weight pruning [3,4,5], quantization of weights [6,7] (during or after training), and knowledge distillation [8,9], which trains smaller student networks to reproduce the output of a much larger teacher network. The proposed work differs from these approaches as it trains a single model which trades computation efficiency for accuracy at test time without re-training or finetuning.<br />
<br />
== Resource Efficient Networks == <br />
<br />
Unlike the above, resource efficient concepts consider limited resources as a part of the structure/loss.<br />
Examples of work in this area include: <br />
* Efficient variants to existing state of the art networks<br />
* Gradient boosted decision trees, which incorporate computational limitations into the training<br />
* Fractal nets<br />
* Adaptive computation time method<br />
<br />
== Related architectures ==<br />
<br />
MSDNets pull on concepts from a number of existing networks:<br />
* Neural fabrics and others, are used to quickly establish a low resolution feature map, which is integral for classification.<br />
* Deeply supervised nets, introduced the incorporation of multiple classifiers throughout the network. (For example, a Branchynet (Teerapittayanon et al., 2016) is a deeply supervised network explicitly designed for efficiency. A Branchynet has multiple exit branches at various depths, each leading to a softmax classifier. At test time, if a classifier on an early exit branch makes a confident prediction, the rest of network need not be evaluated. However, unlike in MSDnets, in Branchynets early classifiers to not have access to low-resolution features. )<br />
* The feature concatenation method from DenseNets allows the later classifiers to not be disrupted by the weight updates from earlier classifiers.<br />
<br />
= Problem Setup =<br />
The authors consider two settings that impose computational constraints at prediction time.<br />
<br />
== Anytime Prediction ==<br />
In the anytime prediction setting (Grubb & Bagnell, 2012), there is a finite computational budget <math>B > 0</math> available for each test example <math>x</math>. Once the budget is exhausted, the prediction for the class is output using early exit. The budget is nondeterministic and varies per test instance.<br />
They assume that the budget is drawn from some joint distribution <math>P(x,B)</math>. They denote the loss of a model <math>f(x)</math> that has to produce a prediction for instance x with a budget of <math>B</math> by <math>L(f(x),B)</math>. The goal of the anytime learner is to minimize the expected loss under the budget distribution <math>L(f)=\mathop{\mathbb{E}}[L(f(x),B)]_{P(x,B)}</math>.<br />
<br />
== Budgeted Batch Classification ==<br />
In the budgeted batch classification setting, the model needs to classify a set of examples <math>D_{test} = {x_1, . . . , x_M}</math> within a finite computational budget <math>B > 0</math> that is known in advance. The learner aims to minimize the loss across all examples in the <math>D_{test}</math>, within a cumulative cost bounded by <math>B</math>, which is denoted as <math>L(f(D_{test}),B)</math> for some suitable loss function <math>L</math>.<br />
<br />
= Multi-Scale Dense Networks =<br />
<br />
== Integral Contributions ==<br />
<br />
The way MSDNets aims to provide efficient classification with varying computational costs is to create one network that outputs results at depths. While this may seem trivial, as intermediate classifiers can be inserted into any existing network, two major problems arise.<br />
<br />
=== Coarse Level Features Needed For Classification ===<br />
<br />
[[File:paper29 fig3.png | 700px|thumb|center]]<br />
<br />
The term coarse level feature refers to a set of filters in a CNN with low resolution. There are several ways to create such features. These methods are typically refereed to as down sampling. Some example of layers that perform this function are: max pooling, average pooling and convolution with strides. In this architecture, convolution with strides will be used to create coarse features. <br />
<br />
Coarse level features are needed to gain context of scene. In typical CNN based networks, the features propagate from fine to coarse. Classifiers added to the early, fine featured, layers do not output accurate predictions due to the lack of context.<br />
<br />
Figure 3 depicts relative accuracies of the intermediate classifiers and shows that the accuracy of a classifier is highly correlated with its position in the network. It is easy to see, specifically with the case of ResNet, that the classifiers improve in a staircase pattern. All of the experiments were performed on Cifar-100 dataset and it can be seen that the intermediate classifiers perform worst than the final classifiers, thus highlighting the problem with the lack of coarse level features early on.<br />
<br />
To address this issue, MSDNets proposes an architecture in which uses multi scaled feature maps. The feature maps at a particular layer and scale are computed by concatenating results from up to two convolutions: a standard convolution is first applied to same-scale features from the previous layer to pass on high-resolution information that subsequent layers can use to construct better coarse features, and if possible, a strided convolution is also applied on the finer-scale feature map from the previous layer to produce coarser features amenable to classification. The network is quickly formed to contain a set number of scales ranging from fine to coarse. These scales are propagated throughout, so that for the length of the network there are always coarse level features for classification and fine features for learning more difficult representations.<br />
<br />
=== Training of Early Classifiers Interferes with Later Classifiers ===<br />
<br />
When training a network containing intermediate classifiers, the training of early classifiers will cause the early layers to focus on features for that classifier. These learned features may not be as useful to the later classifiers and degrade their accuracy.<br />
<br />
MSDNets use dense connectivity to avoid this issue. By concatenating all prior layers to learn future layers, the gradient propagation is spread throughout the available features. This allows later layers to not be reliant on any single prior, providing opportunities to learn new features that priors have ignored.<br />
<br />
== Architecture ==<br />
<br />
[[File:MSDNet_arch.png | 700px|thumb|center|Left: the MSDNet architecture. Right: example calculations for each output given 3 scales and 4 layers.]]<br />
<br />
The architecture of MSDNet is a structure of convolutions with a set number of layers and a set number of scales. Layers allow the network to build on the previous information to generate more accurate predictions, while the scales allow the network to maintain coarse level features throughout.<br />
<br />
The first layer is a special, mini-CNN-network, that quickly fills all required scales with features. The following layers are generated through the convolutions of the previous layers and scales.<br />
<br />
Each output at a given s scale is given by the convolution of all prior outputs of the same scale, and the strided-convolution of all prior outputs from the previous scale. <br />
<br />
The classifiers consists of two convolutional layers, an average pooling layer and a linear layer and are run on the concatenation of all of the coarsest outputs from the preceding layers.<br />
<br />
=== Loss Function ===<br />
<br />
The loss is calculated as a weighted sum of each classifier's logistic loss: <br />
<br />
<math>\frac{1}{|\mathcal{D}|} \sum_{x,y \in \mathcal{D}} \sum_{k}w_k L(f_k) </math><br />
<br />
Here <math>w_i</math> represents the weights and <math>L(f_k)</math> represents the logistic loss of each classifier. The weighted loss is taken as an average over a set of training samples. The weights can be determined from a budget of computational power, but results also show that setting all to 1 is also acceptable.<br />
<br />
=== Computational Limit Inclusion ===<br />
<br />
When running in a budgeted batch scenario, the network attempts to provide the best overall accuracy. To do this with a set limit on computational resources, it works to use less of the budget on easy detections in order to allow more time to be spent on hard ones. <br />
In order to facilitate this, the classifiers are designed to exit when the confidence of the classification exceeds a preset threshold. To determine the threshold for each classifier, <math>|D_{test}|\sum_{k}(q_k C_k) \leq B </math> must be true. Where <math>|D_{test}|</math> is the total number of test samples, <math>C_k</math> is the computational requirement to get an output from the <math>k</math>th classifier, and <math>q_k </math> is the probability that a sample exits at the <math>k</math>th classifier. Assuming that all classifiers have the same base probability, <math>q</math>, then <math>q_k</math> can be used to find the threshold.<br />
<br />
=== Network Reduction and Lazy Evaluation ===<br />
There are two ways to reduce the computational needs of MSDNets:<br />
<br />
# Reduce the size of the network by splitting it into <math>S</math> blocks along the depth dimension and keeping the <math>(S-i+1)</math> scales in the <math>i^{\text{th}}</math> block.Whenever a scale is removed, a transition layer merges the concatenated features using 1x1 convolution and feeds the fine grained features to coarser scales.<br />
# Remove unnecessary computations: Group the computation in "diagonal blocks"; this propagates the example along paths that are required for the evaluation of the next classifier.<br />
<br />
The strategy of minimizing unnecessary computations when the computational budget is over is known as the ''lazy evaluation''.<br />
<br />
= Experiments = <br />
<br />
When evaluating on CIFAR-10 and CIFAR-100 ensembles and multi-classifier versions of ResNets and DenseNets, as well as FractalNet are used to compare with MSDNet. <br />
<br />
When evaluating on ImageNet ensembles and individual versions of ResNets and DenseNets are compared with MSDNets.<br />
<br />
== Anytime Prediction ==<br />
<br />
In anytime prediction MSDNets are shown to have highly accurate with very little budget, and continue to remain above the alternate methods as the budget increases. The authors attributed this to the fact that MSDNets are able to produce low-resolution feature maps well-suited for classification after just a few layers, in contrast to the high-resolution feature maps in early layers of ResNets or DenseNets. Ensemble networks need to repeat computations of similar low-level features repeatedly when new models need to be evaluated, so their accuracy results do not increase as fast when computational budget increases. <br />
<br />
[[File:MSDNet_anytime.png | 700px|thumb|center|Accuracy of the anytime classification models.]] [[File:cifar10msdnet.png | 700px|thumb|center|CIFAR-10 results.]]<br />
<br />
== Budget Batch ==<br />
<br />
For budget batch 3 MSDNets are designed with classifiers set-up for varying ranges of budget constraints. On both dataset options the MSDNets exceed all alternate methods with a fraction of the budget required.<br />
<br />
[[File:MSDNet_budgetbatch.png | 700px|thumb|center|Accuracy of the budget batch classification models.]]<br />
<br />
The following figure shows examples of what was deemed "easy" and "hard" examples by the network. The top row contains images of either red wine or volcanos that were easily classified, thus exiting the network early and reducing required computations. The bottom row contains examples of "hard" images that were incorrectly classified by the first classifier but were correctly classified by the last layer.<br />
<br />
[[File:MSDNet_visualizingearlyclassifying.png | 700px|thumb|center|Examples of "hard"/"easy" classification]]<br />
<br />
= Ablation study =<br />
Additional experiments were performed to shed light on multi-scale feature maps, dense connectivity, and intermediate classifiers. This experiment started with an MSDNet with six intermediate classifiers and each of these components were removed, one at a time. To make our comparisons fair, the computational costs of the full networks were kept similar by adapting the network width. After removing all the three components, a VGG-like convolutional network is obtained. The classification accuracy of all classifiers is shown in the image below.<br />
<br />
[[File:Screenshot_from_2018-03-29_14-58-03.png]]<br />
<br />
= Critique = <br />
<br />
The problem formulation and scenario evaluation were very well formulated, and according to independent reviews, the results were reproducible. Where the paper could improve is on explaining how to implement the threshold; it isn't very well explained how the use of the validation set can be used to set the threshold value.<br />
<br />
= Implementation =<br />
The following repository provides the source code for the paper, written by the authors: https://github.com/gaohuang/MSDNet<br />
<br />
= Sources =<br />
# Huang, G., Chen, D., Li, T., Wu, F., Maaten, L., & Weinberger, K. Q. (n.d.). Multi-Scale Dense Networks for Resource Efficient Image Classification. ICLR 2018. doi:1703.09844 <br />
# Huang, G. (n.d.). Gaohuang/MSDNet. Retrieved March 25, 2018, from https://github.com/gaohuang/MSDNet<br />
# LeCun, Yann, John S. Denker, and Sara A. Solla. "Optimal brain damage." Advances in neural information processing systems. 1990.<br />
# Hassibi, Babak, David G. Stork, and Gregory J. Wolff. "Optimal brain surgeon and general network pruning." Neural Networks, 1993., IEEE International Conference on. IEEE, 1993.<br />
# Li, Hao, et al. "Pruning filters for efficient convnets." arXiv preprint arXiv:1608.08710 (2016).<br />
# Hubara, Itay, et al. "Binarized neural networks." Advances in neural information processing systems. 2016.<br />
# Rastegari, Mohammad, et al. "Xnor-net: Imagenet classification using binary convolutional neural networks." European Conference on Computer Vision. Springer, Cham, 2016.<br />
# Cristian Bucilua, Rich Caruana, and Alexandru Niculescu-Mizil. Model compression. In ACM SIGKDD, pp. 535–541. ACM, 2006.<br />
# Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. In NIPS Deep Learning Workshop, 2014.<br />
# Teerapittayanon, Surat, Bradley McDanel, and H. T. Kung. "Branchynet: Fast inference via early exiting from deep neural networks." Pattern Recognition (ICPR), 2016 23rd International Conference on. IEEE, 2016.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Multi-scale_Dense_Networks_for_Resource_Efficient_Image_Classification&diff=36158Multi-scale Dense Networks for Resource Efficient Image Classification2018-04-04T17:20:54Z<p>Pa2forsy: /* Related architectures */</p>
<hr />
<div>= Introduction = <br />
<br />
Multi-Scale Dense Networks, MSDNets, are designed to address the growing demand for efficient object recognition. The issue with existing recognition networks is that they are either efficient networks, but don't do well on hard examples, or large networks that do well on all examples but require a large amount of resources. For example, the winner of the COCO 2016 competition was an [http://image-net.org/challenges/talks/2016/GRMI-COCO-slidedeck.pdf ensemble of CNNs], which are likely far too resource-heavy to be used in any resource-limited application.<br />
<br />
In order to be efficient on all difficulties MSDNets propose a structure that can accurately output classifications for varying levels of computational requirements. The two cases that are used to evaluate the network are:<br />
* Anytime Prediction: What is the best prediction the network can provide when suddenly prompted?<br />
* Budget Batch Predictions: Given a maximum amount of computational resources, how well does the network do on the batch?<br />
<br />
= Related Networks =<br />
<br />
== Computationally Efficient Networks ==<br />
<br />
Much of the existing work on convolution networks that are computationally efficient at test time focus on reducing model size after training. Many existing methods for refining an accurate network to be more efficient include weight pruning [3,4,5], quantization of weights [6,7] (during or after training), and knowledge distillation [8,9], which trains smaller student networks to reproduce the output of a much larger teacher network. The proposed work differs from these approaches as it trains a single model which trades computation efficiency for accuracy at test time without re-training or finetuning.<br />
<br />
== Resource Efficient Networks == <br />
<br />
Unlike the above, resource efficient concepts consider limited resources as a part of the structure/loss.<br />
Examples of work in this area include: <br />
* Efficient variants to existing state of the art networks<br />
* Gradient boosted decision trees, which incorporate computational limitations into the training<br />
* Fractal nets<br />
* Adaptive computation time method<br />
<br />
== Related architectures ==<br />
<br />
MSDNets pull on concepts from a number of existing networks:<br />
* Neural fabrics and others, are used to quickly establish a low resolution feature map, which is integral for classification.<br />
* Deeply supervised nets, introduced the incorporation of multiple classifiers throughout the network. (For example, a Branchynet is a deeply supervised network explicitly designed for efficiency. A Branchynet has multiple exit branches at various depths, each leading to a softmax classifier. At test time, if a classifier on an early exit branch makes a confident prediction, the rest of network need not be evaluated. However, unlike in MSDnets, in Branchynets early classifiers to not have access to low-resolution features. )<br />
* The feature concatenation method from DenseNets allows the later classifiers to not be disrupted by the weight updates from earlier classifiers.<br />
<br />
= Problem Setup =<br />
The authors consider two settings that impose computational constraints at prediction time.<br />
<br />
== Anytime Prediction ==<br />
In the anytime prediction setting (Grubb & Bagnell, 2012), there is a finite computational budget <math>B > 0</math> available for each test example <math>x</math>. Once the budget is exhausted, the prediction for the class is output using early exit. The budget is nondeterministic and varies per test instance.<br />
They assume that the budget is drawn from some joint distribution <math>P(x,B)</math>. They denote the loss of a model <math>f(x)</math> that has to produce a prediction for instance x with a budget of <math>B</math> by <math>L(f(x),B)</math>. The goal of the anytime learner is to minimize the expected loss under the budget distribution <math>L(f)=\mathop{\mathbb{E}}[L(f(x),B)]_{P(x,B)}</math>.<br />
<br />
== Budgeted Batch Classification ==<br />
In the budgeted batch classification setting, the model needs to classify a set of examples <math>D_{test} = {x_1, . . . , x_M}</math> within a finite computational budget <math>B > 0</math> that is known in advance. The learner aims to minimize the loss across all examples in the <math>D_{test}</math>, within a cumulative cost bounded by <math>B</math>, which is denoted as <math>L(f(D_{test}),B)</math> for some suitable loss function <math>L</math>.<br />
<br />
= Multi-Scale Dense Networks =<br />
<br />
== Integral Contributions ==<br />
<br />
The way MSDNets aims to provide efficient classification with varying computational costs is to create one network that outputs results at depths. While this may seem trivial, as intermediate classifiers can be inserted into any existing network, two major problems arise.<br />
<br />
=== Coarse Level Features Needed For Classification ===<br />
<br />
[[File:paper29 fig3.png | 700px|thumb|center]]<br />
<br />
The term coarse level feature refers to a set of filters in a CNN with low resolution. There are several ways to create such features. These methods are typically refereed to as down sampling. Some example of layers that perform this function are: max pooling, average pooling and convolution with strides. In this architecture, convolution with strides will be used to create coarse features. <br />
<br />
Coarse level features are needed to gain context of scene. In typical CNN based networks, the features propagate from fine to coarse. Classifiers added to the early, fine featured, layers do not output accurate predictions due to the lack of context.<br />
<br />
Figure 3 depicts relative accuracies of the intermediate classifiers and shows that the accuracy of a classifier is highly correlated with its position in the network. It is easy to see, specifically with the case of ResNet, that the classifiers improve in a staircase pattern. All of the experiments were performed on Cifar-100 dataset and it can be seen that the intermediate classifiers perform worst than the final classifiers, thus highlighting the problem with the lack of coarse level features early on.<br />
<br />
To address this issue, MSDNets proposes an architecture in which uses multi scaled feature maps. The feature maps at a particular layer and scale are computed by concatenating results from up to two convolutions: a standard convolution is first applied to same-scale features from the previous layer to pass on high-resolution information that subsequent layers can use to construct better coarse features, and if possible, a strided convolution is also applied on the finer-scale feature map from the previous layer to produce coarser features amenable to classification. The network is quickly formed to contain a set number of scales ranging from fine to coarse. These scales are propagated throughout, so that for the length of the network there are always coarse level features for classification and fine features for learning more difficult representations.<br />
<br />
=== Training of Early Classifiers Interferes with Later Classifiers ===<br />
<br />
When training a network containing intermediate classifiers, the training of early classifiers will cause the early layers to focus on features for that classifier. These learned features may not be as useful to the later classifiers and degrade their accuracy.<br />
<br />
MSDNets use dense connectivity to avoid this issue. By concatenating all prior layers to learn future layers, the gradient propagation is spread throughout the available features. This allows later layers to not be reliant on any single prior, providing opportunities to learn new features that priors have ignored.<br />
<br />
== Architecture ==<br />
<br />
[[File:MSDNet_arch.png | 700px|thumb|center|Left: the MSDNet architecture. Right: example calculations for each output given 3 scales and 4 layers.]]<br />
<br />
The architecture of MSDNet is a structure of convolutions with a set number of layers and a set number of scales. Layers allow the network to build on the previous information to generate more accurate predictions, while the scales allow the network to maintain coarse level features throughout.<br />
<br />
The first layer is a special, mini-CNN-network, that quickly fills all required scales with features. The following layers are generated through the convolutions of the previous layers and scales.<br />
<br />
Each output at a given s scale is given by the convolution of all prior outputs of the same scale, and the strided-convolution of all prior outputs from the previous scale. <br />
<br />
The classifiers consists of two convolutional layers, an average pooling layer and a linear layer and are run on the concatenation of all of the coarsest outputs from the preceding layers.<br />
<br />
=== Loss Function ===<br />
<br />
The loss is calculated as a weighted sum of each classifier's logistic loss: <br />
<br />
<math>\frac{1}{|\mathcal{D}|} \sum_{x,y \in \mathcal{D}} \sum_{k}w_k L(f_k) </math><br />
<br />
Here <math>w_i</math> represents the weights and <math>L(f_k)</math> represents the logistic loss of each classifier. The weighted loss is taken as an average over a set of training samples. The weights can be determined from a budget of computational power, but results also show that setting all to 1 is also acceptable.<br />
<br />
=== Computational Limit Inclusion ===<br />
<br />
When running in a budgeted batch scenario, the network attempts to provide the best overall accuracy. To do this with a set limit on computational resources, it works to use less of the budget on easy detections in order to allow more time to be spent on hard ones. <br />
In order to facilitate this, the classifiers are designed to exit when the confidence of the classification exceeds a preset threshold. To determine the threshold for each classifier, <math>|D_{test}|\sum_{k}(q_k C_k) \leq B </math> must be true. Where <math>|D_{test}|</math> is the total number of test samples, <math>C_k</math> is the computational requirement to get an output from the <math>k</math>th classifier, and <math>q_k </math> is the probability that a sample exits at the <math>k</math>th classifier. Assuming that all classifiers have the same base probability, <math>q</math>, then <math>q_k</math> can be used to find the threshold.<br />
<br />
=== Network Reduction and Lazy Evaluation ===<br />
There are two ways to reduce the computational needs of MSDNets:<br />
<br />
# Reduce the size of the network by splitting it into <math>S</math> blocks along the depth dimension and keeping the <math>(S-i+1)</math> scales in the <math>i^{\text{th}}</math> block.Whenever a scale is removed, a transition layer merges the concatenated features using 1x1 convolution and feeds the fine grained features to coarser scales.<br />
# Remove unnecessary computations: Group the computation in "diagonal blocks"; this propagates the example along paths that are required for the evaluation of the next classifier.<br />
<br />
The strategy of minimizing unnecessary computations when the computational budget is over is known as the ''lazy evaluation''.<br />
<br />
= Experiments = <br />
<br />
When evaluating on CIFAR-10 and CIFAR-100 ensembles and multi-classifier versions of ResNets and DenseNets, as well as FractalNet are used to compare with MSDNet. <br />
<br />
When evaluating on ImageNet ensembles and individual versions of ResNets and DenseNets are compared with MSDNets.<br />
<br />
== Anytime Prediction ==<br />
<br />
In anytime prediction MSDNets are shown to have highly accurate with very little budget, and continue to remain above the alternate methods as the budget increases. The authors attributed this to the fact that MSDNets are able to produce low-resolution feature maps well-suited for classification after just a few layers, in contrast to the high-resolution feature maps in early layers of ResNets or DenseNets. Ensemble networks need to repeat computations of similar low-level features repeatedly when new models need to be evaluated, so their accuracy results do not increase as fast when computational budget increases. <br />
<br />
[[File:MSDNet_anytime.png | 700px|thumb|center|Accuracy of the anytime classification models.]] [[File:cifar10msdnet.png | 700px|thumb|center|CIFAR-10 results.]]<br />
<br />
== Budget Batch ==<br />
<br />
For budget batch 3 MSDNets are designed with classifiers set-up for varying ranges of budget constraints. On both dataset options the MSDNets exceed all alternate methods with a fraction of the budget required.<br />
<br />
[[File:MSDNet_budgetbatch.png | 700px|thumb|center|Accuracy of the budget batch classification models.]]<br />
<br />
The following figure shows examples of what was deemed "easy" and "hard" examples by the network. The top row contains images of either red wine or volcanos that were easily classified, thus exiting the network early and reducing required computations. The bottom row contains examples of "hard" images that were incorrectly classified by the first classifier but were correctly classified by the last layer.<br />
<br />
[[File:MSDNet_visualizingearlyclassifying.png | 700px|thumb|center|Examples of "hard"/"easy" classification]]<br />
<br />
= Ablation study =<br />
Additional experiments were performed to shed light on multi-scale feature maps, dense connectivity, and intermediate classifiers. This experiment started with an MSDNet with six intermediate classifiers and each of these components were removed, one at a time. To make our comparisons fair, the computational costs of the full networks were kept similar by adapting the network width. After removing all the three components, a VGG-like convolutional network is obtained. The classification accuracy of all classifiers is shown in the image below.<br />
<br />
[[File:Screenshot_from_2018-03-29_14-58-03.png]]<br />
<br />
= Critique = <br />
<br />
The problem formulation and scenario evaluation were very well formulated, and according to independent reviews, the results were reproducible. Where the paper could improve is on explaining how to implement the threshold; it isn't very well explained how the use of the validation set can be used to set the threshold value.<br />
<br />
= Implementation =<br />
The following repository provides the source code for the paper, written by the authors: https://github.com/gaohuang/MSDNet<br />
<br />
= Sources =<br />
# Huang, G., Chen, D., Li, T., Wu, F., Maaten, L., & Weinberger, K. Q. (n.d.). Multi-Scale Dense Networks for Resource Efficient Image Classification. ICLR 2018. doi:1703.09844 <br />
# Huang, G. (n.d.). Gaohuang/MSDNet. Retrieved March 25, 2018, from https://github.com/gaohuang/MSDNet<br />
# LeCun, Yann, John S. Denker, and Sara A. Solla. "Optimal brain damage." Advances in neural information processing systems. 1990.<br />
# Hassibi, Babak, David G. Stork, and Gregory J. Wolff. "Optimal brain surgeon and general network pruning." Neural Networks, 1993., IEEE International Conference on. IEEE, 1993.<br />
# Li, Hao, et al. "Pruning filters for efficient convnets." arXiv preprint arXiv:1608.08710 (2016).<br />
# Hubara, Itay, et al. "Binarized neural networks." Advances in neural information processing systems. 2016.<br />
# Rastegari, Mohammad, et al. "Xnor-net: Imagenet classification using binary convolutional neural networks." European Conference on Computer Vision. Springer, Cham, 2016.<br />
# Cristian Bucilua, Rich Caruana, and Alexandru Niculescu-Mizil. Model compression. In ACM SIGKDD, pp. 535–541. ACM, 2006.<br />
# Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. In NIPS Deep Learning Workshop, 2014.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Neural_Audio_Synthesis_of_Musical_Notes_with_WaveNet_autoencoders&diff=35422Neural Audio Synthesis of Musical Notes with WaveNet autoencoders2018-03-25T02:46:20Z<p>Pa2forsy: </p>
<hr />
<div>= Introduction =<br />
The authors of this paper have pointed out that the method in which most notes are created are hand-designed instruments modifying pitch, velocity and filter parameters to produce the required tone, timbre and dynamics of a sound. The authors suggest that this may be a problem and thus suggest a data-driven approach to audio synthesis. To train such a data expensive model the authors highlight the need for a large dataset much like imagenet for music. <br />
<br />
= Contributions =<br />
To solve the problem highlighted above the authors propose two main contributions of their paper: <br />
* Wavenet-style autoencoder that learn to encode temural data over a long term audio structures without requiring external conditioning<br />
* NSynth: a large dataset of musical notes inspired by the emerging of large image datasets<br />
<br />
<br />
= Models =<br />
<br />
[[File:paper26-figure1-models.png|center]]<br />
<br />
== WaveNet Autoencoder ==<br />
<br />
While the proposed autoencoder structure is very similar to that of WaveNet the authors argue that the algorithm is novel in two ways:<br />
* It is able to attain consistent long-term structure without any external conditioning <br />
* Creating meaningful embedding which can be interpolated between<br />
The authors accomplish this by passing the raw audio throw the encoder to produce an embedding <math>Z = f(x) </math>, next the input is shifted and feed into the decoder which reproduces the input. The resulting probability distribution: <br />
<br />
\begin{align}<br />
p(x) = \prod_{i=1}^N\{x_i | x_1, … , x_N-1, f(x) \}<br />
\end{align}<br />
<br />
A detailed block diagram of the modified WaveNet structure can be seen in figure 1b. This diagram demonstrates the encoder as a 30 layer network in each each node is a ReLU nonlinearity followed by a non-causal dilated convolution. Dilated convolution (aka convolutions with holes) is a type of convolution in which the filter skips input values with a certain step (step size of 1 is equivalent to the standard convolution), effectively allowing the network to operate at a coarser scale compared to traditional convolutional layers and have very large receptive fields. The resulting convolution is 128 channels all feed into another ReLU nonlinearity which is feed into another 1x1 convolution before getting down sampled with average pooling to produce a 16 dimension <math>Z </math> distribution. Each <math>Z </math> encoding is for a specific temporal resolution which the authors of the paper tuned to 32ms. This means that there are 125, 16 dimension <math>Z </math> encodings for each 4 second note present in the NSynth database (1984 embeddings). <br />
Before the <math>Z </math> embedding enters the decoder it is first upsampled to the original audio rate using nearest neighbor interpolation. The embedding then passes through the decoder to recreate the original audio note. The input audio data is first quantized using 8-bit mu-law encoding into 256 possible values, and the output prediction is the softmax over the possible values.<br />
<br />
== Baseline: Spectral Autoencoder ==<br />
Being unable to find an alternative fully deep model which the authors could use to compare to there proposed WaveNet autoencoder to, the authors just made a strong baseline. The baseline algorithm that the authors developed is a spectral autoencoder. The block diagram of its architecture can be seen in figure 1a. The baseline network is 10 layer deep. Each layer has a 4x4 kernels with 2x2 strides followed by a leaky-ReLU (0.1) and batch normalization. The final hidden vector(Z) was set to 1984 to exactly match the hidden vector of the WaveNet autoencoder. <br />
<br />
The authors attempted to train the baseline on multiple input: raw waveforms, FFT, and log magnitude of spectrum finding the latter to be best correlated with perceptual distortion. The authors also explored several representations of phase, finding that estimating magnitude and using established iterative techniques to reconstruct phase to be most effective. (The technique to reconstruct the phase from the magnitude comes from (Griffin and Lim 1984). It can be summarized as follows. In each iteration, generate a Fourier signal z by taking the Short Time Fourier transform of the current estimate of the complete time-domain signal, and replacing its magnitude component with the known true magnitude. Then find the time-domain signal whose Short Time Fourier transform is closest to z in the least-squares sense. This is the estimate of the complete signal for the next iteration. ) A final heuristic that was used by the authors to increase the accuracy of the baseline was weighting the mean square error (MSE) loss starting at 10 for 0 HZ and decreasing linearly to 1 at 4000 Hz and above. This is valid as the fundamental frequency of most instrument are found at lower frequencies. <br />
<br />
== Training ==<br />
Both the modified WaveNet and the baseline autoencoder used stochastic gradient descent with an Adam optimizer. The authors trained the baseline autoencoder model asynchronously for 1800000 epocs with a batch size of 8 with a learning rate of 1e-4. Where as the WaveNet modules were trained synchronously for 250000 epocs with a batch size of 32 with a decaying learning rate ranging from 2e-4 to 6e-6.<br />
<br />
= The NSynth Dataset =<br />
To evaluate the WaveNet autoencoder model, the authors' wanted an audio dataset that let them explore the learned embeddings. Musical notes are an ideal setting for this study. While several smaller music datasets exist, as deep networks train better on abundant, high-quality data, the authors decided on the development of a new dataset - NSynth Dataset.<br />
<br />
The NSynth dataset has 306 043 unique musical notes all 4 seconds in length sampled at 16,000 Hz. The data set consists of 1006 different instruments playing on average of 65.4 different pitches across on average 4.75 different velocities. Average pitches and velocities are used as not all instruments, can reach all 88 MIDI frequencies, or the 5 velocities desired by the authors. The dataset has the following split: training set with 289,205 notes, validation set with 12,678 notes, and test set with 4,096 notes.<br />
<br />
Along with each note the authors also included the following annotations:<br />
* Source - The way each sound was produced. There were 3 classes ‘acoustic’, ‘electronic’ and ‘synthetic’<br />
* Family - The family class of instruments that produced each note. There is 11 classes which include: {‘bass’, ‘brass’, ‘vocal’ ext.}<br />
* Qualities - Sonic qualities about each note<br />
<br />
The full dataset is publicly available here: https://magenta.tensorflow.org/datasets/nsynth.<br />
<br />
<br />
<br />
= Evaluation =<br />
<br />
To fully analyze all aspects of WaveNet the authors proposed three evaluations:<br />
* Reconstruction - Both Quantitative and Qualitative analysis were considered<br />
* Interpolation in Timbre and Dynamics<br />
* Entanglement of Pitch and Timbre <br />
<br />
Sound is historically very difficult to quantify from a picture representation as it requires training and expertise to analyze. Even with expertise it can be difficult to complete a full analyses as two very different sound can look quite similar in the respective pictorial representation. This is why the authors recommend all readers to listen to the created notes which can be sound here: https://magenta.tensorflow.org/nsynth.<br />
<br />
However, even when taking this under consideration the authors do pictorially demonstrate differences in the two proposed algorithms along with the original note, as it is hard to publish a paper with sound included. To demonstrate the pictorial difference the authors demonstrate each note using constant-q transform (CQT) which is able to capture the dynamics of timbre along with representing the frequencies of the sound.<br />
<br />
== Reconstruction ==<br />
<br />
[[File:paper27-figure2-reconstruction.png|center]]<br />
<br />
=== Qualitative Comparison ===<br />
In the Glockenspiel the WaveNet autoencoder is able to reproduce the magnitude, phase of the fundamental frequency (A and C in figure 2), and the attack (B in figure 2) of the instrument; Whereas the Baseline autoencoder introduces non existing harmonics (D in figure 2). The flugelhorn on the other hand, presents the starkest difference between the WaveNet and baseline autoencoders. The WaveNet while not perfect is able to reproduce the verbarto (I and J in figure 2) across multiple frequencies, which results in a natural sounding note. The baseline not only fails to do this but also adds extra noise (K in figure 2). The authors do add that the WaveNet produces some strikes (L in figure 2) however they argue that they are inaudible.<br />
<br />
[[File:paper27-table1.png|center]]<br />
<br />
=== Quantitative Comparison ===<br />
For a quantitative comparison the authors trained a separate multi-task classifier to classify a note using given pitch or quality of a note. The results of both the Baseline and the WaveNet where then inputted and attempted to be classified. As seen in table 1 WaveNet significantly outperformed the Baseline in both metrics posting a ~70% increase when only considering pitch.<br />
<br />
== Interpolation in Timbre and Dynamics ==<br />
<br />
[[File:paper27-figure3-interpolation.png|center]]<br />
<br />
For this evaluation the authors reconstructed from linear interpolations in Z space among different instruments and compared these to superimposed position of the original two instruments. Not surprisingly the model fuse aspects of both instruments during the recreation. The authors claim however, that WaveNet produces much more realistic sounding results. <br />
To support their claim the authors the authors point to WaveNet ability to create dynamic mixing of overtone in time, even jumping to higher harmonics (A in figure 3), capturing the timbre and dynamics of both the bass and flute. This can be once again seen in (B in figure 3) where Wavenet adds additional harmonics as well as a sub-harmonics to the original flute note. <br />
<br />
<br />
== Entanglement of Pitch and Timbre ==<br />
<br />
[[File:paper27-table2.png|center]]<br />
<br />
[[File:paper27-figure4-entanglement.png|center]]<br />
<br />
To study the entanglement between pitch and Z space the authors constructed a classifier which was expected to drop in accuracy if the representation of pitch and timbre is disentangled as it relies heavily on the pitch information. This is clearly demonstrated by the first two rows of table 2 where WaveNet relies more strongly on pitch then the baseline algorithm. The authors provide a more qualitative demonstrating in figure 4. They demonstrate a situation in which a classifier may be confused; a note with pitch of +12 is almost exactly the same as the original apart from an emergence of sub-harmonics.<br />
<br />
Further insight can be gained on the relationship between pitch and timbre by studying the trend amongst the network embeddings among the pitches for specific instruments. This is depicted in figure 5 for several instruments across their entire 88 note range at 127 velocity. It can be noted from the figure that the instruments have unique separation of two or more registers over which the embeddings of notes with different pitches are similar. This is expected since instrumental dynamics and timbre varies dramatically over the range of the instrument.<br />
<br />
= Future Directions =<br />
<br />
One significant area which the authors claim great improvement is needed is the large memory constraints required by there algorithm. Due to the large memory requirement the current WaveNet must rely on down sampling thus being unable to fully capture the global context. <br />
<br />
= Open Source Code base =<br />
<br />
Google has released all code related to this paper at the following open source repository: https://github.com/tensorflow/magenta/tree/master/magenta/models/nsynth<br />
<br />
= References =<br />
<br />
# Engel, J., Resnick, C., Roberts, A., Dieleman, S., Norouzi, M., Eck, D. & Simonyan, K.. (2017). Neural Audio Synthesis of Musical Notes with WaveNet Autoencoders. Proceedings of the 34th International Conference on Machine Learning, in PMLR 70:1068-1077<br />
# Griffin, Daniel, and Jae Lim. "Signal estimation from modified short-time Fourier transform." IEEE Transactions on Acoustics, Speech, and Signal Processing 32.2 (1984): 236-243.<br />
# NSynth: Neural Audio Synthesis. (2017, April 06). Retrieved March 19, 2018, from https://magenta.tensorflow.org/nsynth <br />
# The NSynth Dataset. (2017, April 05). Retrieved March 19, 2018, from https://magenta.tensorflow.org/datasets/nsynth</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Spherical_CNNs&diff=35045Spherical CNNs2018-03-21T22:10:19Z<p>Pa2forsy: note on Haar measure</p>
<hr />
<div>= Introduction =<br />
Convolutional Neural Networks (CNNs), or network architectures involving CNNs, are the current state of the art for learning 2D image processing tasks such as semantic segmentation and object detection. CNNs work well in large part due to the property of being translationally equivariant. This property allows a network trained to detect a certain type of object to still detect the object even if it is translated to another position in the image. However, this does not correspond well to spherical signals since projecting a spherical signal onto a plane will result in distortions, as demonstrated in Figure 1. There are many different types of spherical projections onto a 2D plane, as most people know from the various types of world maps, none of which provide all the necessary properties for rotation-invariant learning.<br />
<br />
[[File:paper26-fig1.png|center]]<br />
<br />
= Notation =<br />
Below are listed several important terms:<br />
* '''Unit Sphere''' <math>S^2</math> is defined as a sphere where all of its points are distance of 1 from the origin. The unit sphere can be parameterized by the spherical coordinates <math>\alpha ∈ [0, 2π]</math> and <math>β ∈ [0, π]</math>. This is a two-dimensional manifold with respect to <math>\alpha</math> and <math>β</math>.<br />
* '''<math>S^2</math> Sphere''' The three dimensional surface from a 3D sphere<br />
* '''Spherical Signals''' In the paper spherical images and filters are modeled as continuous functions <math>f : s^2 → \mathbb{R}^K</math>. K is the number of channels. Such as how RGB images have 3 channels a spherical signal can have numerous channels describing the data. Examples of channels which were used can be found in the experiments section.<br />
* '''Rotations - SO(3)''' The group of 3D rotations on an <math>S^2</math> sphere. Sometimes called the "special orthogonal group". In this paper the ZYZ-Euler parameterization is used to represent SO(3) rotations with <math>\alpha, \beta</math>, and <math>\gamma</math>. Any rotation can be broken down into first a rotation (<math>\alpha</math>) about the Z-axis, then a rotation (<math>\beta</math>) about the new Y-axis (Y'), followed by a rotation (<math>\gamma</math>) about the new Z axis (Z"). [In the rest of this paper, to integrate functions on SO(3), the authors use a rotationally invariant probability measure on the Borel subsets of SO(3). This measure is an example of a Haar measure. Haar measures generalize the idea of rotationally invariant probability measures to general topological groups. For more on Haar measures, see (Feldman 2002) ]<br />
<br />
= Related Work =<br />
The related work presented in this paper is very brief, in large part due to the novelty of spherical CNNs and the length of the rest of the paper. The authors enumerate numerous papers which attempt to exploit larger groups of symmetries such as the translational symmetries of CNNs but do not go into specific details for any of these attempts. They do state that all the previous works are limited to discrete groups with the exception of SO(2)-steerable networks.<br />
The authors also mention that previous works exist that analyze spherical images but that these do not have an equivariant architecture. They claim that Spherical CNNs are "the first to achieve equivariance to a continuous, non-commutative group (SO(3))". They also claim to be the first to use the generalized Fourier transform for speed effective performance of group correlation.<br />
<br />
= Correlations on the Sphere and Rotation Group =<br />
Spherical correlation is like planar correlation except instead of translation, there is rotation. The definitions for each are provided as follows:<br />
<br />
'''Planar correlation''' The value of the output feature map at translation <math>\small x ∈ Z^2</math> is computed as an inner product between the input feature map and a filter, shifted by <math>\small x</math>.<br />
<br />
'''Spherical correlation''' The value of the output feature map evaluated at rotation <math>\small R ∈ SO(3)</math> is computed as an inner product between the input feature map and a filter, rotated by <math>\small R</math>.<br />
<br />
'''Rotation of Spherical Signals''' The paper introduces the rotation operator <math>L_R</math>. The rotation operator simply rotates a function (which allows us to rotate the the spherical filters) by <math>R^{-1}</math>. With this definition we have the property that <math>L_{RR'} = L_R L_{R'}</math>.<br />
<br />
'''Inner Products''' The inner product of spherical signals is simply the integral summation on the vector space over the entire sphere.<br />
<br />
<math>\langle\psi , f \rangle = \int_{S^2} \sum_{k=1}^K \psi_k (x)f_k (x)dx</math><br />
<br />
<math>dx</math> here is SO(3) rotation invariant and is equivalent to <math>d \alpha sin(\beta) d \beta / 4 \pi </math> in spherical coordinates. This comes from the ZYZ-Euler paramaterization where any rotation can be broken down into first a rotation about the Z-axis, then a rotation about the new Y-axis (Y'), followed by a rotation about the new Z axis (Z"). More details on this are given in Appendix A in the paper.<br />
<br />
By this definition, the invariance of the inner product is then guaranteed for any rotation <math>R ∈ SO(3)</math>. In other words, when subjected to rotations, the volume under a spherical heightmap does not change. The following equations show that <math>L_R</math> has a distinct adjoint (<math>L_{R^{-1}}</math>) and that <math>L_R</math> is unitary and thus preserves orthogonality and distances.<br />
<br />
<math>\langle L_R \psi \,, f \rangle = \int_{S^2} \sum_{k=1}^K \psi_k (R^{-1} x)f_k (x)dx</math><br />
<br />
::::<math>= \int_{S^2} \sum_{k=1}^K \psi_k (x)f_k (Rx)dx</math><br />
<br />
::::<math>= \langle \psi , L_{R^{-1}} f \rangle</math><br />
<br />
'''Spherical Correlation''' With the above knowledge the definition of spherical correlation of two signals <math>f</math> and <math>\psi</math> is:<br />
<br />
<math>[\psi \star f](R) = \langle L_R \psi \,, f \rangle = \int_{S^2} \sum_{k=1}^K \psi_k (R^{-1} x)f_k (x)dx</math><br />
<br />
The output of the above equation is a function on SO(3). This can be thought of as for each rotation combination of <math>\alpha , \beta , \gamma </math> there is a different volume under the correlation. The authors make a point of noting that previous work by Driscoll and Healey only ensures circular symmetries about the Z axis and their new formulation ensures symmetry about any rotation.<br />
<br />
'''Rotation of SO(3) Signals''' The first layer of Spherical CNNs take a function on the sphere (<math>S^2</math>) and output a function on SO(3). Therefore, if a Spherical CNN with more than one layer is going to be built there needs to be a way to find the correlation between two signals on SO(3). The authors then generalize the rotation operator (<math>L_R</math>) to encompass acting on signals from SO(3). This new definition of <math>L_R</math> is as follows: (where <math>R^{-1}Q</math> is a composition of rotations, i.e. multiplication of rotation matrices)<br />
<br />
<math>[L_Rf](Q)=f(R^{-1} Q)</math><br />
<br />
'''Rotation Group Correlation''' The correlation of two signals (<math>f,\psi</math>) on SO(3) with K channels is defined as the following:<br />
<br />
<math>[\psi \star f](R) = \langle L_R \psi , f \rangle = \int_{SO(3)} \sum_{k=1}^K \psi_k (R^{-1} Q)f_k (Q)dQ</math><br />
<br />
where dQ represents the ZYZ-Euler angles <math>d \alpha sin(\beta) d \beta d \gamma / 8 \pi^2 </math>. A complete derivation of this can be found in Appendix A.<br />
<br />
The equivariance for the rotation group correlation is similarly demonstrated.<br />
<br />
= Implementation with GFFT =<br />
The authors leverage the Generalized Fourier Transform (GFT) and Generalized Fast Fourier Transform (GFFT) algorithms to compute the correlations outlined in the previous section. The Fast Fourier Transform (FFT) can compute correlations and convolutions efficiently by means of the Fourier theorem. The Fourier theorem states that a continuous periodic function can be expressed as a sum of a series of sine or cosine terms (called Fourier coefficients). The FFT can be generalized to <math>S^2</math> and SO(3) and is then called the GFT. The GFT is a linear projection of a function onto orthogonal basis functions. The basis functions are a set of irreducible unitary representations for a group (such as for <math>S^2</math> or SO(3)). For <math>S^2</math> the basis functions are the spherical harmonics <math>Y_m^l(x)</math>. For SO(3) these basis functions are called the Wigner D-functions <math>D_{mn}^l(R)</math>. For both sets of functions the indices are restricted to <math>l\geq0</math> and <math>-l \leq m,n \geq l</math>. The Wigner D-functions are also orthogonal so the Fourier coefficients can be computed by the inner product with the Wigner D-functions (See Appendix C for complete proof). The Wigner D-functions are complete which means that any function (which is well behaved) on SO(3) can be expressed as a linear combination of the Wigner D-functions. The GFT of a function on SO(3) is thus:<br />
<br />
<math>\hat{f^l} = \int_X f(x) D^l(x)dx</math><br />
<br />
where <math>\hat{f}</math> represents the Fourier coefficients. For <math>S^2</math> we have the same equation but with the basis functions <math>Y^l</math>.<br />
<br />
The inverse SO(3) Fourier transform is:<br />
<br />
<math>f(R)=[\mathcal{F}^{-1} \hat{f}](R) = \sum_{l=0}^b (2l + 1) \sum_{m=-l}^l \sum_{n=-l}^l \hat{f_{mn}^l} D_{mn}^l(R) </math><br />
<br />
The bandwidth b represents the maximum frequency and is related to the resolution of the spatial grid. Kostelec and Rockmore are referenced for more knowledge on this topic.<br />
<br />
The authors give proofs (Appendix D) that the SO(3) correlation satisfies the Fourier theorem and the <math>S^2</math> correlation of spherical signals can be computed by the outer products of the <math>S^2</math>-FTs (Shown in Figure 2).<br />
<br />
[[File:paper26-fig2.png|center]]<br />
<br />
The GFFT algorithm details are taken from Kostelec and Rockmore. The authors claim they have the first automatically differentiable implementation of the GFT for <math>S^2</math> and SO(3). The authors do not provide any run time comparisons for real time applications or any comparisons on training times with/without GFFT. However, they do provide the source code of their implementation at: https://github.com/jonas-koehler/s2cnn<br />
<br />
= Experiments =<br />
The authors provide several experiments. The first set of experiments are designed to show the numerical stability and accuracy of the outlined methods. The second group of experiments demonstrates how the algorithms can be applied to current problem domains.<br />
<br />
==Equivariance Error==<br />
In this experiment the authors try to show experimentally that their theory of equivariance holds. They express that they had doubts about the equivariance in practice due to potential discretization artifacts since equivariance was proven for the continuous case. The experiment is set up by first testing the equivariance of the SO(3) correlation at different resolutions. 500 random rotations and feature maps (with 10 channels) are sampled. They then calculate <math>\small\Delta = \dfrac{1}{n} \sum_{i=1}^n std(L_{R_i} \Phi(f_i) - \phi(L_{R_i} f_i))/std(\Phi(f_i))</math><br />
Note: The authors do not mention what the std function is however it is likely the standard deviation function as 'std' is the command for standard deviation in MATLAB.<br />
<math>\Phi</math> is a composition of SO(3) correlation layers with filters which have been randomly initialized. The authors mention that they were expecting <math>\Delta</math> to be zero in the case of perfect equivariance. This is due to, as proven earlier, the following two terms equaling each other in the continuous case: <math>\small L_{R_i} \Phi(f_i) - \phi(L_{R_i} f_i)</math>. The results are shown in Figure 3. <br />
<br />
[[File:paper26-fig3.png|center]]<br />
<br />
<math>\Delta</math> only grows with resolution/layers when there is no activation function. With ReLU activation the error stays constant once slightly higher than 0 resolution. The authors indicate that the error must therefore be from the feature map rotation since this type of error is exact only for bandlimited functions.<br />
<br />
==MNIST Data==<br />
The experiment using MNIST data was created by projecting MNIST digits onto a sphere using stereographic projection to create the resulting images as seen in Figure 4.<br />
<br />
[[File:paper26-fig4.png|center]]<br />
<br />
The authors created two datasets, one with the projected digits and the other with the same projected digits which were then subjected to a random rotation. The spherical CNN architecture used was <math>\small S^2</math>conv-ReLU-SO(3)conv-ReLU-FC-softmax and was attempted with bandwidths of 30,10,6 and 20,40,10 channels for each layer respectively. This model was compared to a baseline CNN with layers conv-ReLU-conv-ReLU-FC-softmax with 5x5 filters, 32,64,10 channels and stride of 3. For comparison this leads to approximately 68K parameters for the baseline and 58K parameters for the spherical CNN. Results can be seen in Table 1. It is clear from the results that the spherical CNN architecture made the network rotationally invariant. Performance on the rotated set is almost identical to the non-rotated set. This is true even when trained on the non-rotated set and tested on the rotated set. Compare this to the non-spherical architecture which becomes unusable when rotating the digits.<br />
<br />
[[File:paper26-tab1.png|center]]<br />
<br />
==SHREC17==<br />
The SHREC dataset contains 3D models from the ShapeNet dataset which are classified into categories. It consists of a regularly aligned dataset and a rotated dataset. The models from the SHREC17 dataset were projected onto a sphere by means of raycasting. Different properties of the objects obtained from the raycast of the original model and the convex hull of the model make up the different channels which are input into the spherical CNN.<br />
<br />
<br />
[[File:paper26-fig5.png|center]]<br />
<br />
<br />
The network architecture used is an initial <math>\small S^2</math>conv-BN-ReLU block which is followed by two SO(3)conv-BN-ReLU blocks. The output is then fed into a MaxPool-BN block then a linear layer to the output for final classification. The architecture for this experiment has ~1.4M parameters, far exceeding the scale of the spherical CNNs in the other experiments.<br />
<br />
This architecture achieves state of the art results on the SHREC17 tasks. The model places 2nd or 3rd in all categories but was not submitted as the SHREC17 task is closed. Table 2 shows the comparison of results with the top 3 submissions in each category. The authors claim the results show empirical proof of the usefulness of spherical CNNs. They elaborate that this is largely due to the fact that most architectures on the SHREC17 competition are highly specialized whereas their model is fairly general.<br />
<br />
<br />
[[File:paper26-tab2.png|center]]<br />
<br />
==Molecular Atomization==<br />
In this experiment a spherical CNN is implemented with an architecture resembling that of ResNet. They use the QM7 dataset which has the task of predicting atomization energy of molecules. The positions and charges given in the dataset are projected onto the sphere using potential functions. A summary of their results is shown in Table 3 along with some of the spherical CNN architecture details. It shows the different RMSE obtained from different methods. The results from this final experiment also seem to be promising as the network the authors present achieves the second best score. They also note that the first place method grows exponentially with the number of atoms per molecule so is unlikely to scale well.<br />
<br />
[[File:paper26-tab3.png|center]]<br />
<br />
= Conclusions =<br />
This paper presents a novel architecture called Spherical CNNs. The paper defines <math>\small S^2</math> and SO(3) cross correlations, shows the theory behind their rotational invariance for continuous functions, and demonstrates that the invariance also applies to the discrete case. An effective GFFT algorithm was implemented and evaluated on two very different datasets with close to state of the art results, demonstrating that there are practical applications to Spherical CNNs.<br />
<br />
For future work the authors believe that improvements can be obtained by generalizing the algorithms to the SE(3) group (SE(3) simply adds translations in 3D space to the SO(3) group). The authors also briefly mention their excitement for applying Spherical CNNs to omnidirectional vision such as in drones and autonomous cars. They state that there is very little publicly available omnidirectional image data which could be why they did not conduct any experiments in this area.<br />
<br />
= Commentary =<br />
The reviews on Spherical CNNs are very positive and it is ranked in the top 1% of papers submitted to ICLR 2018. Positive points are the novelty of the architecture, the wide variety of experiments performed, and the writing. One critique of the original submission is that the related works section only lists, instead of describing, previous methods and that a description of the methods would have provided more clarity. The authors have since expanded the section however I found that it is still limited which the authors attribute to length limitations. Another critique is that the evaluation does not provide enough depth. For example, it would have been great to see an example of omnidirectional vision for spherical networks. However, this is to be expected as it is just the introduction of spherical CNNs and more work is sure to come.<br />
<br />
= Source Code =<br />
Source code is available at:<br />
https://github.com/jonas-koehler/s2cnn<br />
<br />
= Sources =<br />
* T. Cohen et al. Spherical CNNs, 2018.<br />
* J. Feldman. Haar Measure. http://www.math.ubc.ca/~feldman/m606/haar.pdf<br />
* P. Kostelec, D. Rockmore. FFTs on the Rotation Group, 2008.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/MaskRNN:_Instance_Level_Video_Object_Segmentation&diff=35022stat946w18/MaskRNN: Instance Level Video Object Segmentation2018-03-21T17:16:26Z<p>Pa2forsy: </p>
<hr />
<div>== Introduction ==<br />
Deep Learning has produced state of the art results in many computer vision tasks like image classification, object localization, object detection, object segmentation, semantic segmentation and instance level video object segmentation. Image classification classify the image based on the prominent objects. Object localization is the task of finding objects’ location in the frame. Object Segmentation task involves providing a pixel map which represents the pixel wise location of the objects in the image. Semantic segmentation task attempts at segmenting the image into meaningful parts. Instance level video object segmentation is the task of consistent object segmentation in video sequences.<br />
<br />
There are 2 different types of video object segmentation: Unsupervised and Semi-supervised. In unsupervised video object segmentation, the task is to find the salient objects and track the main objects in the video. In an unsupervised setting, the ground truth mask of the salient objects is provided for the first frame. The task is thus simplified to only track the objects required. In this paper we look at an unsupervised video object segmentation technique.<br />
<br />
== Background Papers ==<br />
Video object segmentation has been performed using spatio-temporal graphs and deep learning. The Graph based methods construct 3D spatio-temporal graphs in order to model the inter- and the intra-frame relationship of pixels or superpixels in a video.Hence they are computationally slower than deep learning methods and are unable to run at real-time. There are 2 main deep learning techniques for semi-supervised video object segmentation: One Shot Video Object Segmentation (OSVOS) and Learning Video Object Segmentation from Static Images (MaskTrack). Following a brief description of the new techniques introduced by these papers for semi-supervised video object segmentation task.<br />
<br />
=== OSVOS (One-Shot Video Object Segmentation) ===<br />
<br />
[[File:OSVOS.jpg | 1000px]]<br />
<br />
This paper introduces the technique of using a frame-by-frame object segmentation without any temporal information from the previous frames of the video. The paper uses a VGG-16 network with pre-trained weights from image classification task. This network is then converted into a fully-connected network (FCN) by removing the fully connected dense layers at the end and adding convolution layers to generate a segment mask of the input. This network is then trained on the DAVIS 2016 dataset.<br />
<br />
During testing, the trained VGG-16 FCN is fine-tuned using the first frame of the video using the ground truth. Because this is a semi-supervised case, the segmented mask (ground truth) for the first frame is available. The first frame data is augmented by zooming/rotating/flipping the first frame and the associated segment mask.<br />
<br />
=== MaskTrack (Learning Video Object Segmentation from Static Images) ===<br />
<br />
[[File:MaskTrack.jpg | 500px]]<br />
<br />
MaskTrack takes the output of the previous frame to improve its predictions to generate the segmentation mask for the next frame. Thus the input to the network is 4 channel wide (3 RGB channels from the frame at time <math>t</math> plus one binary segmentation mask from frame <math>t-1</math>). The output of the network is the binary segmentation mask for frame at time <math>t</math>. Using the binary segmentation mask (referred to as guided object segmentation in the paper), the network is able to use some temporal information from the previous frame to improve its segmentation mask prediction for the next frame.<br />
<br />
The model of the MaskTrack network is similar to a modular VGG-16 and is referred to as MaskTrack ConvNet in the paper. The network is trained offline on saliency segmentation datasets: ECSSD, MSRA 10K, SOD and PASCAL-S. The input mask for the binary segmentation mask channel is generated via non-rigid deformation and affine transformation of the ground truth segmentation mask. Similar data-augmentation techniques are also used during online training. Just like OSVOS, MaskTrack uses the first frame ground truth (with augmented images) to fine-tune the network to improve prediction score for the particular video sequence.<br />
<br />
A parallel ConvNet network is used to generate predicted segment mask based on the optical flow magnitude. The optical flow between 2 frames is calculated using the EpicFlow algorithm. The output of the two networks is combined using averaging operation to generate the final predicted segmented mask.<br />
<br />
Table 1 gives a summary comparison of the different state of the art algorithms. The noteworthy information included in this table is that the technique presented in this paper is the only one which takes into account long-term temporal information. This is accomplished with a recurrent neural net. Furthermore, the bounding box is also estimated instead of just a segmentation mask. The authors claim that this allows the incorporation of a location prior from the tracked object.<br />
<br />
[[File:Paper19-SegmentationComp.png]]<br />
<br />
== Dataset ==<br />
The three major datasets used in this paper are DAVIS-2016, DAVIS-2017 and Segtrack v2. DAVIS-2016 dataset provides video sequences with only one segment mask for all salient objects. DAVIS-2017 improves the ground truth data by providing segmentation mask for each salient object as a separate color segment mask. Segtrack v2 also provides multiple segmentation mask for all salient objects in the video sequence. These datasets try to recreate real-life scenarios like occlusions, low resolution videos, background clutter, motion blur, fast motion etc.<br />
<br />
== MaskRNN: Introduction ==<br />
Most techniques mentioned above don’t work directly on instance level segmentation of the objects through the video sequence. The above approaches focus on image segmentation on each frame and using additional information (mask propagation and optical flow) from the preceding frame perform predictions for the current frame. To address the instance level segmentation problem, MaskRNN proposes a framework where the salient objects are tracked and segmented by capturing the temporal information in the video sequence using a recurrent neural network.<br />
<br />
== MaskRNN: Overview ==<br />
In a video sequence <math>I = \{I_1, I_2, …, I_T\}</math>, the sequence of <math>T</math> frames are given as input to the network, where the video sequence contains <math>N</math> salient objects. The ground truth for the first frame <math>y_1^*</math> is also provided for <math>N</math> salient objects.<br />
In this paper, the problem is formulated as a time dependency problem and using a recurrent neural network, the prediction of the previous frame influences the prediction of the next frame. The approach also computes the optical flow between frames (optical flow is the apparent motion of objects between two consecutive frames in the form of a 2D vector field representing the displacement in brightness patterns for each pixel, apparent because it depends on the relative motion between the observer and the scene) and uses that as the input to the neural network. The optical flow is also used to align the output of the predicted mask. “The warped prediction, the optical flow itself, and the appearance of the current frame are then used as input for <math>N</math> deep nets, one for each of the <math>N</math> objects.”[1 - MaskRNN] Each deep net is a made of an object localization network and a binary segmentation network. The binary segmentation network is used to generate the segmentation mask for an object. The object localization network is used to alleviate outliers from the predictions. The final prediction of the segmentation mask is generated by merging the predictions of the 2 networks. For <math>N</math> objects, there are N deep nets which predict the mask for each salient object. The predictions are then merged into a single prediction using an <math>\text{argmax}</math> operation at test time.<br />
<br />
== MaskRNN: Multiple Instance Level Segmentation ==<br />
<br />
[[File:2ObjectSeg.jpg | 850px]]<br />
<br />
Image segmentation requires producing a pixel level segmentation mask and this can become a multi-class problem. Instead, using the approach from [2- Mask R-CNN] this approach is converted into a multiple binary segmentation problem. A separate segmentation mask is predicted separately for each salient object and thus we get a binary segmentation problem. The binary segments are combined using an <math>\text{argmax}</math> operation where each pixel is assigned to the object containing the largest predicted probability.<br />
<br />
=== MaskRNN: Binary Segmentation Network ===<br />
<br />
[[File:MaskRNNDeepNet.jpg | 850px]]<br />
<br />
The above picture shows a single deep net employed for predicting the segment mask for one salient object in the video frame. The network consists of 2 networks: binary segmentation network and object localization network. The binary segmentation network is split into two streams: appearance and flow stream. The input of the appearance stream is the RGB frame at time t and the wrapped prediction of the binary segmentation mask from time <math>t-1</math>. The wrapping function uses the optical flow between frame <math>t-1</math> and frame <math>t</math> to generate a new binary segmentation mask for frame <math>t</math>. The input to the flow stream is the concatenation of the optical flow magnitude between frames <math>t-1</math> to <math>t</math> and frames <math>t</math> to <math>t+1</math> and the wrapped prediction of the segmentation mask from frame <math>t-1</math>. The magnitude of the optical flow is replicated into an RBG format before feeding it to the flow stream. The network architecture closely resembles a VGG-16 network without the pooling or fully connected layers at the end. The fully connected layers are replaced with convolutional and bilinear interpolation upsampling layers which are then linearly combined to form a feature representation that is the same size of the input image. This feature representation is then used to generate a binary segment mask. This technique is borrowed from the Fully Convolutional Network mentioned above. The output of the flow stream and the appearance stream is linearly combined and sigmoid function is applied to the result to generate binary mask for ith object. All parts of the network are fully differentiable and thus it can be fully trained in every pass.<br />
<br />
=== MaskRNN: Object Localization Network: ===<br />
Using a similar technique to the Fast-RCNN method of object localization, where the region of interest (RoI) pooling of the features of the region proposals (i.e. the bounding box proposals here) is performed and passed through fully connected layers to perform regression, the Object localization network generates a bounding box of the salient object in the frame. This bounding box is enlarged by a factor of 1.25 and combined with the output of binary segmentation mask. Only the segment mask available in the bounding box is used for prediction and the pixels outside of the bounding box are marked as zero. MaskRNN uses the convolutional feature output of the appearance stream as the input to the RoI-pooling layer to generate the predicted bounding box. A pixel is classified as foreground if it is both predicted to be in the foreground by the binary segmentation net and within the enlarged estimated bounding box from the object localization net.<br />
<br />
=== Training and Finetuning ===<br />
For training the network depicted in Figure 1, backpropagation through time is used in order to preserve the recurrence relationship connecting the frames of the video sequence. Predictive performance is further improved by following the algorithm for semi-supervised setting for video object segmentation with fine-tuning achieved by using the first frame segmentation mask of the ground truth. In this way, the network is further optimized using the ground truth data.<br />
<br />
== MaskRNN: Implementation Details ==<br />
=== Offline Training ===<br />
The deep net is first trained offline on a set of static images. The ground truth is randomly perturbed locally to generate the imperfect mask from frame <math>t-1</math>. Two different networks are trained offline separately for DAVIS-2016 and DAVIS-2017 datasets for a fair evaluation of both datasets. After both the object localization net and binary segmentation networks have trained, the temporal information in the network is used to further improve the segmented prediction results. Because of GPU memory constraints, the RNN is only able to backpropagate the gradients back 7 frames and learn long-term temporal information. <br />
<br />
For optical flow, a pre-trained flowNet2.0 is used to compute the optical flow between frames. (A flowNet (Dosovitskiy 2015) is a deep neural network trained to predict optical flow. The simplest form of flowNet has an architecture consisting of two parts. The first part accepts the two images between which the optical flow is to be computed as input, as applies a sequence of convolution and max-pooling operations, as in a standard convolutional neural network. In the second part, repeated up-convolution operations are applied, increasing the dimensions of the feature-maps. Besides the output of the previous upconvolution, each upconvolution is also fed as input the output of the corresponding down-convolution from the first part of the network. Thus part of the architecture resembles that of a U-net (Ronneberger, 2015). The output of the network is the predicted optical flow. ) <br />
<br />
=== Online Finetuning ===<br />
The deep nets (without the RNN) are then fine-tuned during test time by online training the networks on the ground truth of the first frame and some augmentations of the first frame data. The learning rate is set to 10-5 for online training for 200 iterations.<br />
<br />
== MaskRNN: Experimental Results ==<br />
=== Evaluation Metrics ===<br />
There are 3 different techniques for performance analysis for Video Object Segmentation techniques:<br />
<br />
1. Region Similarity (Jaccard Index): Region similarity or Intersection-over-union is used to capture precision of the area covered by the prediction segmentation mask compared to the ground truth segmentation mask.<br />
<br />
[[File:IoU.jpg | 200px]]<br />
<br />
2. Contour Accuracy (F-score): This metric measures the accuracy in the boundary of the predicted segment mask and the ground truth segment mask using bipartite matching between the bounding pixels of the masks. <br />
<br />
[[File:Fscore.jpg | 200px]]<br />
<br />
3. Temporal Stability : This estimates the degree of deformation needed to transform the segmentation masks from one frame to the next and is measured by the dissimilarity of the set of points on the contours of the segmentation between two adjacent frames.<br />
<br />
Temporal Stability measures how well the pixels of the two masks match, while Contour Accuracy measures the accuracy of the contours.<br />
<br />
=== Ablation Study ===<br />
<br />
The ablation study summarized how the different components contributed to the algorithm evaluated on DAVIS-2016 and DAVIS-2017 datasets.<br />
<br />
[[File:MaskRNNTable2.jpg | 700px]]<br />
<br />
The above table presents the contribution of each component of the network to the final prediction score. We observe that online fine-tuning improves the performance by a large margin. Addition of RNN/Localization Net and FStream all seem to positively affect the performance of the deep net.<br />
<br />
=== Quantitative Evaluation ===<br />
<br />
The authors use DAVIS-2016, DAVIS-2017 and Segtrack v2 to compare the performance of the proposed approach to other methods based on foreground-background video object segmentation and multiple instance-level video object segmentation.<br />
<br />
[[File:MaskRNNTable3.jpg | 700px]]<br />
<br />
The above table shows the results for contour accuracy mean and region similarity. The MaskRNN method seems to outperform all previously proposed methods. The performance gain is significant by employing a Recurrent Neural Network for learning recurrence relationship and using a object localization network to improve prediction results.<br />
<br />
The following table shows the improvements in the state of the art achieved by MaskRNN on the DAVIS-2017 and the SegTrack v2 dataset.<br />
<br />
[[File:MaskRNNTable4.jpg | 700px]]<br />
<br />
=== Qualitative Evaluation ===<br />
The authors showed example qualitative results from the DAVIS and Segtrack datasets. <br />
<br />
Below are some success cases of object segmentation under complex motion, cluttered background, and/or multiple object occlusion.<br />
<br />
[[File:maskrnn_example.png | 700px]]<br />
<br />
Below are a few failure cases. The authors explain two reasons for failure: a) when similar objects of interest are contained in the frame (left two images), and b) when there are large variations in scale and viewpoint (right two images).<br />
<br />
[[File:maskrnn_example_fail.png | 700px]]<br />
<br />
== Conclusion ==<br />
In this paper a novel approach to instance level video object segmentation task is presented which performs better than current state of the art. The long-term recurrence relationship is learnt using an RNN. The object localization network is added to improve accuracy of the system. Using online fine-tuning the network is adjusted to predict better for the current video sequence.<br />
<br />
== Implementation ==<br />
<br />
The implementation of this paper was produced as part of the NIPS Paper Implementation Challenge. This implementation can be found at the following open source project [2].<br />
<br />
== References ==<br />
# Dosovitskiy, Alexey, et al. "Flownet: Learning optical flow with convolutional networks." Proceedings of the IEEE International Conference on Computer Vision. 2015.<br />
# Hu, Y., Huang, J., & Schwing, A. "MaskRNN: Instance level video object segmentation". Conference on Neural Information Processing Systems (NIPS). 2017<br />
# Ferriere, P. (n.d.). Semi-Supervised Video Object Segmentation (VOS) with Tensorflow. Retrieved March 20, 2018, from https://github.com/philferriere/tfvos<br />
# Ronneberger, Olaf, Philipp Fischer, and Thomas Brox. "U-net: Convolutional networks for biomedical image segmentation." International Conference on Medical image computing and computer-assisted intervention. Springer, Cham, 2015.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/MaskRNN:_Instance_Level_Video_Object_Segmentation&diff=35021stat946w18/MaskRNN: Instance Level Video Object Segmentation2018-03-21T17:11:59Z<p>Pa2forsy: /* MaskRNN: Implementation Details */</p>
<hr />
<div>== Introduction ==<br />
Deep Learning has produced state of the art results in many computer vision tasks like image classification, object localization, object detection, object segmentation, semantic segmentation and instance level video object segmentation. Image classification classify the image based on the prominent objects. Object localization is the task of finding objects’ location in the frame. Object Segmentation task involves providing a pixel map which represents the pixel wise location of the objects in the image. Semantic segmentation task attempts at segmenting the image into meaningful parts. Instance level video object segmentation is the task of consistent object segmentation in video sequences.<br />
<br />
There are 2 different types of video object segmentation: Unsupervised and Semi-supervised. In unsupervised video object segmentation, the task is to find the salient objects and track the main objects in the video. In an unsupervised setting, the ground truth mask of the salient objects is provided for the first frame. The task is thus simplified to only track the objects required. In this paper we look at an unsupervised video object segmentation technique.<br />
<br />
== Background Papers ==<br />
Video object segmentation has been performed using spatio-temporal graphs and deep learning. The Graph based methods construct 3D spatio-temporal graphs in order to model the inter- and the intra-frame relationship of pixels or superpixels in a video.Hence they are computationally slower than deep learning methods and are unable to run at real-time. There are 2 main deep learning techniques for semi-supervised video object segmentation: One Shot Video Object Segmentation (OSVOS) and Learning Video Object Segmentation from Static Images (MaskTrack). Following a brief description of the new techniques introduced by these papers for semi-supervised video object segmentation task.<br />
<br />
=== OSVOS (One-Shot Video Object Segmentation) ===<br />
<br />
[[File:OSVOS.jpg | 1000px]]<br />
<br />
This paper introduces the technique of using a frame-by-frame object segmentation without any temporal information from the previous frames of the video. The paper uses a VGG-16 network with pre-trained weights from image classification task. This network is then converted into a fully-connected network (FCN) by removing the fully connected dense layers at the end and adding convolution layers to generate a segment mask of the input. This network is then trained on the DAVIS 2016 dataset.<br />
<br />
During testing, the trained VGG-16 FCN is fine-tuned using the first frame of the video using the ground truth. Because this is a semi-supervised case, the segmented mask (ground truth) for the first frame is available. The first frame data is augmented by zooming/rotating/flipping the first frame and the associated segment mask.<br />
<br />
=== MaskTrack (Learning Video Object Segmentation from Static Images) ===<br />
<br />
[[File:MaskTrack.jpg | 500px]]<br />
<br />
MaskTrack takes the output of the previous frame to improve its predictions to generate the segmentation mask for the next frame. Thus the input to the network is 4 channel wide (3 RGB channels from the frame at time <math>t</math> plus one binary segmentation mask from frame <math>t-1</math>). The output of the network is the binary segmentation mask for frame at time <math>t</math>. Using the binary segmentation mask (referred to as guided object segmentation in the paper), the network is able to use some temporal information from the previous frame to improve its segmentation mask prediction for the next frame.<br />
<br />
The model of the MaskTrack network is similar to a modular VGG-16 and is referred to as MaskTrack ConvNet in the paper. The network is trained offline on saliency segmentation datasets: ECSSD, MSRA 10K, SOD and PASCAL-S. The input mask for the binary segmentation mask channel is generated via non-rigid deformation and affine transformation of the ground truth segmentation mask. Similar data-augmentation techniques are also used during online training. Just like OSVOS, MaskTrack uses the first frame ground truth (with augmented images) to fine-tune the network to improve prediction score for the particular video sequence.<br />
<br />
A parallel ConvNet network is used to generate predicted segment mask based on the optical flow magnitude. The optical flow between 2 frames is calculated using the EpicFlow algorithm. The output of the two networks is combined using averaging operation to generate the final predicted segmented mask.<br />
<br />
Table 1 gives a summary comparison of the different state of the art algorithms. The noteworthy information included in this table is that the technique presented in this paper is the only one which takes into account long-term temporal information. This is accomplished with a recurrent neural net. Furthermore, the bounding box is also estimated instead of just a segmentation mask. The authors claim that this allows the incorporation of a location prior from the tracked object.<br />
<br />
[[File:Paper19-SegmentationComp.png]]<br />
<br />
== Dataset ==<br />
The three major datasets used in this paper are DAVIS-2016, DAVIS-2017 and Segtrack v2. DAVIS-2016 dataset provides video sequences with only one segment mask for all salient objects. DAVIS-2017 improves the ground truth data by providing segmentation mask for each salient object as a separate color segment mask. Segtrack v2 also provides multiple segmentation mask for all salient objects in the video sequence. These datasets try to recreate real-life scenarios like occlusions, low resolution videos, background clutter, motion blur, fast motion etc.<br />
<br />
== MaskRNN: Introduction ==<br />
Most techniques mentioned above don’t work directly on instance level segmentation of the objects through the video sequence. The above approaches focus on image segmentation on each frame and using additional information (mask propagation and optical flow) from the preceding frame perform predictions for the current frame. To address the instance level segmentation problem, MaskRNN proposes a framework where the salient objects are tracked and segmented by capturing the temporal information in the video sequence using a recurrent neural network.<br />
<br />
== MaskRNN: Overview ==<br />
In a video sequence <math>I = \{I_1, I_2, …, I_T\}</math>, the sequence of <math>T</math> frames are given as input to the network, where the video sequence contains <math>N</math> salient objects. The ground truth for the first frame <math>y_1^*</math> is also provided for <math>N</math> salient objects.<br />
In this paper, the problem is formulated as a time dependency problem and using a recurrent neural network, the prediction of the previous frame influences the prediction of the next frame. The approach also computes the optical flow between frames (optical flow is the apparent motion of objects between two consecutive frames in the form of a 2D vector field representing the displacement in brightness patterns for each pixel, apparent because it depends on the relative motion between the observer and the scene) and uses that as the input to the neural network. The optical flow is also used to align the output of the predicted mask. “The warped prediction, the optical flow itself, and the appearance of the current frame are then used as input for <math>N</math> deep nets, one for each of the <math>N</math> objects.”[1 - MaskRNN] Each deep net is a made of an object localization network and a binary segmentation network. The binary segmentation network is used to generate the segmentation mask for an object. The object localization network is used to alleviate outliers from the predictions. The final prediction of the segmentation mask is generated by merging the predictions of the 2 networks. For <math>N</math> objects, there are N deep nets which predict the mask for each salient object. The predictions are then merged into a single prediction using an <math>\text{argmax}</math> operation at test time.<br />
<br />
== MaskRNN: Multiple Instance Level Segmentation ==<br />
<br />
[[File:2ObjectSeg.jpg | 850px]]<br />
<br />
Image segmentation requires producing a pixel level segmentation mask and this can become a multi-class problem. Instead, using the approach from [2- Mask R-CNN] this approach is converted into a multiple binary segmentation problem. A separate segmentation mask is predicted separately for each salient object and thus we get a binary segmentation problem. The binary segments are combined using an <math>\text{argmax}</math> operation where each pixel is assigned to the object containing the largest predicted probability.<br />
<br />
=== MaskRNN: Binary Segmentation Network ===<br />
<br />
[[File:MaskRNNDeepNet.jpg | 850px]]<br />
<br />
The above picture shows a single deep net employed for predicting the segment mask for one salient object in the video frame. The network consists of 2 networks: binary segmentation network and object localization network. The binary segmentation network is split into two streams: appearance and flow stream. The input of the appearance stream is the RGB frame at time t and the wrapped prediction of the binary segmentation mask from time <math>t-1</math>. The wrapping function uses the optical flow between frame <math>t-1</math> and frame <math>t</math> to generate a new binary segmentation mask for frame <math>t</math>. The input to the flow stream is the concatenation of the optical flow magnitude between frames <math>t-1</math> to <math>t</math> and frames <math>t</math> to <math>t+1</math> and the wrapped prediction of the segmentation mask from frame <math>t-1</math>. The magnitude of the optical flow is replicated into an RBG format before feeding it to the flow stream. The network architecture closely resembles a VGG-16 network without the pooling or fully connected layers at the end. The fully connected layers are replaced with convolutional and bilinear interpolation upsampling layers which are then linearly combined to form a feature representation that is the same size of the input image. This feature representation is then used to generate a binary segment mask. This technique is borrowed from the Fully Convolutional Network mentioned above. The output of the flow stream and the appearance stream is linearly combined and sigmoid function is applied to the result to generate binary mask for ith object. All parts of the network are fully differentiable and thus it can be fully trained in every pass.<br />
<br />
=== MaskRNN: Object Localization Network: ===<br />
Using a similar technique to the Fast-RCNN method of object localization, where the region of interest (RoI) pooling of the features of the region proposals (i.e. the bounding box proposals here) is performed and passed through fully connected layers to perform regression, the Object localization network generates a bounding box of the salient object in the frame. This bounding box is enlarged by a factor of 1.25 and combined with the output of binary segmentation mask. Only the segment mask available in the bounding box is used for prediction and the pixels outside of the bounding box are marked as zero. MaskRNN uses the convolutional feature output of the appearance stream as the input to the RoI-pooling layer to generate the predicted bounding box. A pixel is classified as foreground if it is both predicted to be in the foreground by the binary segmentation net and within the enlarged estimated bounding box from the object localization net.<br />
<br />
=== Training and Finetuning ===<br />
For training the network depicted in Figure 1, backpropagation through time is used in order to preserve the recurrence relationship connecting the frames of the video sequence. Predictive performance is further improved by following the algorithm for semi-supervised setting for video object segmentation with fine-tuning achieved by using the first frame segmentation mask of the ground truth. In this way, the network is further optimized using the ground truth data.<br />
<br />
== MaskRNN: Implementation Details ==<br />
=== Offline Training ===<br />
The deep net is first trained offline on a set of static images. The ground truth is randomly perturbed locally to generate the imperfect mask from frame <math>t-1</math>. Two different networks are trained offline separately for DAVIS-2016 and DAVIS-2017 datasets for a fair evaluation of both datasets. After both the object localization net and binary segmentation networks have trained, the temporal information in the network is used to further improve the segmented prediction results. Because of GPU memory constraints, the RNN is only able to backpropagate the gradients back 7 frames and learn long-term temporal information. <br />
<br />
For optical flow, a pre-trained flowNet2.0 is used to compute the optical flow between frames. (A flowNet is a deep neural network trained to predict optical flow. The simplest form of flowNet has an architecture consisting of two parts. The first part accepts the two images between which the optical flow is to be computed as input, as applies a sequence of convolution and max-pooling operations, as in a standard convolutional neural network. In the second part, repeated up-convolution operations are applied, increasing the dimensions of the feature-maps. Besides the output of the previous upconvolution, each upconvolution is also fed as input the output of the corresponding down-convolution from the first part of the network. Thus part of the architecture resembles that of a U-net. The output of the network is the predicted optical flow. ) <br />
<br />
=== Online Finetuning ===<br />
The deep nets (without the RNN) are then fine-tuned during test time by online training the networks on the ground truth of the first frame and some augmentations of the first frame data. The learning rate is set to 10-5 for online training for 200 iterations.<br />
<br />
== MaskRNN: Experimental Results ==<br />
=== Evaluation Metrics ===<br />
There are 3 different techniques for performance analysis for Video Object Segmentation techniques:<br />
<br />
1. Region Similarity (Jaccard Index): Region similarity or Intersection-over-union is used to capture precision of the area covered by the prediction segmentation mask compared to the ground truth segmentation mask.<br />
<br />
[[File:IoU.jpg | 200px]]<br />
<br />
2. Contour Accuracy (F-score): This metric measures the accuracy in the boundary of the predicted segment mask and the ground truth segment mask using bipartite matching between the bounding pixels of the masks. <br />
<br />
[[File:Fscore.jpg | 200px]]<br />
<br />
3. Temporal Stability : This estimates the degree of deformation needed to transform the segmentation masks from one frame to the next and is measured by the dissimilarity of the set of points on the contours of the segmentation between two adjacent frames.<br />
<br />
Temporal Stability measures how well the pixels of the two masks match, while Contour Accuracy measures the accuracy of the contours.<br />
<br />
=== Ablation Study ===<br />
<br />
The ablation study summarized how the different components contributed to the algorithm evaluated on DAVIS-2016 and DAVIS-2017 datasets.<br />
<br />
[[File:MaskRNNTable2.jpg | 700px]]<br />
<br />
The above table presents the contribution of each component of the network to the final prediction score. We observe that online fine-tuning improves the performance by a large margin. Addition of RNN/Localization Net and FStream all seem to positively affect the performance of the deep net.<br />
<br />
=== Quantitative Evaluation ===<br />
<br />
The authors use DAVIS-2016, DAVIS-2017 and Segtrack v2 to compare the performance of the proposed approach to other methods based on foreground-background video object segmentation and multiple instance-level video object segmentation.<br />
<br />
[[File:MaskRNNTable3.jpg | 700px]]<br />
<br />
The above table shows the results for contour accuracy mean and region similarity. The MaskRNN method seems to outperform all previously proposed methods. The performance gain is significant by employing a Recurrent Neural Network for learning recurrence relationship and using a object localization network to improve prediction results.<br />
<br />
The following table shows the improvements in the state of the art achieved by MaskRNN on the DAVIS-2017 and the SegTrack v2 dataset.<br />
<br />
[[File:MaskRNNTable4.jpg | 700px]]<br />
<br />
=== Qualitative Evaluation ===<br />
The authors showed example qualitative results from the DAVIS and Segtrack datasets. <br />
<br />
Below are some success cases of object segmentation under complex motion, cluttered background, and/or multiple object occlusion.<br />
<br />
[[File:maskrnn_example.png | 700px]]<br />
<br />
Below are a few failure cases. The authors explain two reasons for failure: a) when similar objects of interest are contained in the frame (left two images), and b) when there are large variations in scale and viewpoint (right two images).<br />
<br />
[[File:maskrnn_example_fail.png | 700px]]<br />
<br />
== Conclusion ==<br />
In this paper a novel approach to instance level video object segmentation task is presented which performs better than current state of the art. The long-term recurrence relationship is learnt using an RNN. The object localization network is added to improve accuracy of the system. Using online fine-tuning the network is adjusted to predict better for the current video sequence.<br />
<br />
== Implementation ==<br />
<br />
The implementation of this paper was produced as part of the NIPS Paper Implementation Challenge. This implementation can be found at the following open source project [2].<br />
<br />
== References ==<br />
# Hu, Y., Huang, J., & Schwing, A. "MaskRNN: Instance level video object segmentation". Conference on Neural Information Processing Systems (NIPS). 2017<br />
# Ferriere, P. (n.d.). Semi-Supervised Video Object Segmentation (VOS) with Tensorflow. Retrieved March 20, 2018, from https://github.com/philferriere/tfvos</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Continuous_Adaptation_via_Meta-Learning_in_Nonstationary_and_Competitive_Environments&diff=34839Continuous Adaptation via Meta-Learning in Nonstationary and Competitive Environments2018-03-20T21:26:19Z<p>Pa2forsy: /* Model Agnostic Meta-Learning */</p>
<hr />
<div>= Introduction =<br />
<br />
Typically, the basic goal of machine learning is to train a model to perform a task. In Meta-learning, the goal is to train a model to perform the task of training a model to perform a task. Hence in this case the term "Meta-Learning" has the exact meaning you would expect; the word "Meta" has the precise function of introducing a layer of abstraction.<br />
<br />
The meta-learning task can be made more concrete by a simple example. Consider the CIFAR-100 classification task that we used for our data competition. We can alter this task from being a 100-class classification problem to a collection of 100 binary classification problems. The goal of Meta-Learning here is to design and train a single binary classifier for each class that will perform well on a randomly sampled task given a limited amount of training data for that specific task. In other words, we would like to train a model to perform the following procedure:<br />
<br />
# A task is sampled. The task is "Is X a dog?".<br />
# A small set of labeled training data is provided to the model. The labels represent whether or not the image is a picture of a dog.<br />
# The model uses the training data to adjust itself to the specific task of checking whether or not an image is a picture of a dog.<br />
<br />
This example also highlights the intuition that the skill of sight is distinct and separable from the skill of knowing what a dog looks like.<br />
<br />
In this paper, a probabilistic framework for meta-learning is derived, then applied to tasks involving simulated robotic spiders. This framework generalizes the typical machine learning set up using Markov Decision Processes. This paper focuses on a multi-agent non-stationary environment which requires reinforcement learning (RL) agents to do continuous adaptation in such an environment. Non-stationarity breaks the standard assumptions and requires agents to continuously adapt, both at training and execution time, in order to earn more rewards, hence the approach is to break this into a sequence of stationary tasks and present it as a multi-task learning problem.<br />
<br />
[[File:paper19_fig1.png|600px|frame|none|alt=Alt text| '''Figure 1'''. a) Illustrates a probabilistic model for Model Agnostic Meta-Learning (MAML) in a multi-task RL setting, where the tasks <math>T</math>, policies <math>\pi</math>, and trajectories <math>\tau</math> are all random variables with dependencies encoded in the edges of a given graph. b) The proposed extension to MAML by the authors suitable for continuous adaptation to a task changing dynamically due to non-stationarity of the environment. The distribution of tasks is represented by a Markov chain, whereby policies from a previous step are used to construct a new policy for the current step. c) The computation graph for the meta-update from <math>\phi_i</math> to <math>\phi_{i+1}</math>. Boxes represent replicas of the policy graphs with the specified parameters. The model is optimized using truncated backpropagation through time starting from <math>L_{T_{i+1}}</math>.]]<br />
<br />
= Background =<br />
== Markov Decision Process (MDP) ==<br />
A MDP is defined by the tuple <math>(S,A,P,r,\gamma)</math>, where S is a set of states, A is a set of actions, P is the transition probability distribution, r is the reward function, and <math>\gamma</math> is the discount factor. More information ([https://www.cs.cmu.edu/~katef/DeepRLControlCourse/lectures/lecture2_mdps.pdf here]).<br />
<br />
<br />
= Model Agnostic Meta-Learning =<br />
<br />
An initial framework for meta-learning is given in "Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks" (Finn et al, 2017):<br />
<br />
"In our approach, the parameters of<br />
the model are explicitly trained such that a small<br />
number of gradient steps with a small amount<br />
of training data from a new task will produce<br />
good generalization performance on that task" (Finn et al, 2017).<br />
<br />
[[File:MAML.png | 500px]]<br />
<br />
In this training algorithm, the parameter vector <math>\theta</math> belonging to the model <math>f_{\theta}</math> is trained such that the meta-objective function <math>\mathcal{L} (\theta) = \sum_{\tau_i \sim P(\tau)} \mathcal{L}_{\tau_i} (f_{\theta_i' }) </math> is minimized. The sum in the objective function is over a sampled batch of training tasks. <math>\mathcal{L}_{\tau_i} (f_{\theta_i'})</math> is the training loss function corresponding to the <math>i^{th}</math> task in the batch evaluated at the model <math>f_{\theta_i'}</math>. The parameter vector <math>\theta_i'</math> is obtained by updating the general parameter <math>\theta</math> using the loss function <math>\mathcal{L}_{\tau_i}</math> and set of K training examples specific to the <math>i^{th}</math> task. Note that in alternate versions of this algorithm, additional testing sets are sampled from <math>\tau_i</math> and used to update <math>\theta</math> using testing loss functions instead of training loss functions.<br />
<br />
One of the important difference between this algorithm and more typical fine-tuning methods is that <math>\theta</math> is explicitly trained to be easily adjusted to perform well on different tasks rather than perform well on any specific tasks then fine tuned as the environment changes. (Sutton et al., 2007). In essence, the model is trained so that gradient steps are highly productive at adapting the model parameters to a new enviroment.<br />
<br />
= Probabilistic Framework for Meta-Learning =<br />
<br />
This paper puts the meta-learning problem into a Markov Decision Process (MDP) framework common to RL, see Figure 1a. Instead of training examples <math>\{(x, y)\}</math>, we have trajectories <math>\tau = (x_0, a_1, x_1, R_1, a_2,x_2,R_2, ... a_H, x_H, R_H)</math>. A trajectory is sequence of states/observations <math>x_t</math>, actions <math>a_t</math> and rewards <math>R_t</math> that is sampled from a task <math> T </math> according to a policy <math>\pi_{\theta}</math>. Included with said task is a method for assigning loss values to trajectories <math>L_T(\tau)</math> which is typically the negative cumulative reward. A policy is a deterministic function that takes in a state and returns an action. Our goal here is to train a policy <math>\pi_{\theta}</math> with parameter vector <math>\theta</math>. This is analougous to training a function <math>f_{\theta}</math> that assigns labels <math>y</math> to feature vectors <math>x</math>. More precisely we have the following definitions:<br />
<br />
* <math>\tau = (x_0, a_1, x_1, R_1, x_2, ... a_H, x_H, R_H)</math> trajectories.<br />
* <math>T :=(L_T, P_T(x), P_T(x_t | x_{t-1}, a_{t-1}), H )</math> (A Task)<br />
* <math>D(T)</math> : A distribution over tasks.<br />
* <math>L_T</math>: A loss function for the task T that assigns numeric loss values to trajectories.<br />
* <math>P_T(x), P_T(x_t | x_{t-1}, a_{t-1})</math>: Probability measures specifying the markovian dynamics of the observations <math>x_t</math><br />
* <math>H</math>: The horizon of the MDP. This is a fixed natural number specifying the lengths of the tasks trajectories.<br />
<br />
The papers goes further to define a Markov dynamic for sequences of tasks as shown in Figure 1b. Thus the policy that we would like to meta learn <math>\pi_{\theta}</math>, after being exposed to a sample of K trajectories <math>\tau_\theta^{1:K}</math> from the task <math>T_i</math>, should produce a new policy <math>\pi_{\phi}</math> that will perform well on the next task <math>T_{i+1}</math>. Thus we seek to minimize the following expectation:<br />
<br />
<math>\mathrm{E}_{P(T_0), P(T_{i+1} | T_i)}\bigg(\sum_{i=1}^{l} \mathcal{L}_{T_i, T_{i+1}}(\theta)\bigg)</math>, <br />
<br />
where <math>\mathcal{L}_{T_i, T_{i + 1}}(\theta) := \mathrm{E}_{\tau_{i, \theta}^{1:K} } \bigg( \mathrm{E}_{\tau_{i+1, \phi}}\Big( L_{T_{i+1}}(\tau_{i+1, \phi} | \tau_{i, \theta}^{1:K}, \theta) \Big) \bigg) </math> and <math>l</math> is the number of tasks.<br />
<br />
The meta-policy <math>\pi_{\theta}</math> is trained and then adapted at test time using the following procedures. The computational graph is given in Figure 1c.<br />
<br />
[[File:MAML2.png | 800px]]<br />
<br />
The mathematics of calculating loss gradients is omitted.<br />
<br />
= Training Spiders to Run with Dynamic Handicaps (Robotic Locomotion in Non-Stationary Environments) =<br />
<br />
The authors used the MuJoCo physics simulator to create a simulated environment where robotic spiders with 6 legs are faced with the task of running due east as quickly as possible. The robotic spider observes the location and velocity of its body, and the angles and velocities of its legs. It interacts with the environment by exerting torque on the joints of its legs. Each leg has two joints, the joint closer to the body rotates horizontally while the joint farther from the body rotates vertically. The environment is made non-stationary by gradually paralyzing two legs of the spider across training and testing episodes.<br />
Putting this example into the above probabilistic framework yields:<br />
<br />
* <math>T_i</math>: The task of walking east with the torques of two legs scaled by <math> (i-1)/6 </math><br />
* <math>\{T_i\}_{i=1}^{7}</math>: A sequence of tasks with the same two legs handicapped in each task. Note there are 15 different ways to choose such legs resulting in 15 sequences of tasks. 12 are used for training and 3 for testing.<br />
* A Markov Descision process composed of<br />
** Observations <math> x_t </math> containing information about the state of the spider.<br />
** Actions <math> a_t </math> containing information about the torques to apply to the spiders legs.<br />
** Rewards <math> R_t </math> corresponding to the speed at which the spider is moving east.<br />
<br />
Three differently structured policy neural networks are trained in this set up using both meta-learning and three different previously developed adaption methods.<br />
<br />
At testing time, the spiders following meta learned policies initially perform worse than the spiders using non-adaptive policies. However, by the third episode (<math> i=3 </math>), the meta-learners perform on par. And by the sixth episode, when the selected legs are mostly immobile, the meta-learners significantly out perform. These results can be seen in the graphs below.<br />
<br />
[[File:locomotion_results.png | 800px]]<br />
<br />
= Training Spiders to Fight Each Other (Adversarial Meta-Learning) =<br />
<br />
The authors created an adversarial environment called RoboSumo where pairs of agents with 4 (named Ants), 6 (named Bugs),or 8 legs (named spiders) sumo wrestle. The agents observe the location and velocity of their bodies and the bodies of their opponent, the angles and velocities of their legs, and the forces being exerted on them by their opponent (equivalent of tactile sense). The game is organized into episodes and rounds. Episodes are single wrestling matches with 500 time steps and win/lose/draw outcomes. Agents win by pushing their opponent out of the ring or making their opponent's body touch the ground. Rounds are batches of episodes. An episode results in a draw when neither of these things happen after 500 time steps. Rounds have possible outcomes win, lose, and draw that are decided based on majority of episodes won. K rounds will be fought. Both agents may update their policies between rounds. The agent that wins the majority of rounds is deemed the winner of the game.<br />
<br />
== Setup ==<br />
Similar to the Robotic locomotion example, this game can be phrased in terms of the RL MDP framework.<br />
<br />
* <math>T_i</math>: The task of fighting a round.<br />
* <math>\{T_i\}_{i=1}^{K}</math>: A sequence of rounds against the same opponent. Note that the opponent may update their policy between rounds but the anatomy of both wrestlers will be constant across rounds.<br />
* A Markov Descision process composed of<br />
** A horizon <math>H = 500*n</math> where <math>n</math> is the number of episodes per round.<br />
** Observations <math> x_t </math> containing information about the state of the agent and its opponent.<br />
** Actions <math> a_t </math> containing information about the torques to apply to the agents legs.<br />
** Rewards <math> R_t </math> rewards given to the agent based on its wrestling performance. <math>R_{500*n} = </math> +2000 if win episode, -2000 if lose, and -1000 if draw.<br />
<br />
Note that the above reward set up is quite sparse, therefore in order to encourage fast training, rewards are introduced at every time step for the following.<br />
* For staying close to the center of the ring.<br />
* For exerting force on the opponents body.<br />
* For moving towards the opponent.<br />
* For the distance of the opponent to the center of the ring.<br />
<br />
In addition to the sparse win/lose rewards, the following dense rewards are also introduced in the early training stages to encourage faster learning:<br />
* Quickly push the opponent outside - penalty proportional to the distance of there opponent from the center of the ring.<br />
* Moving towards the opponent - reward proportional to the velocity component towards the opponent.<br />
* Hit the opponent - reward proportional to square root of the total forces exerted on the opponent.<br />
* Control penalty - penalty denoted by <math> l_2 </math> on actions which lead to jittery/unnatural movements.<br />
<br />
<br />
This makes sense intuitively as these are reasonable goals for agents to explore when they are learning to wrestle.<br />
<br />
== Training ==<br />
The same combinations of policy networks and adaptation methods that were used in the locomotion example are trained and tested here. A family of non-adaptive policies are first trained via self-play and saved at all stages. Self-play simply means the two agents in the training environment use the same policy. All policy versions are saved so that agents of various skill levels can be sampled when training meta-learners. The weights of the different insects were calibrated such that the test win rate between two insects of differing anatomy, who have been trained for the same number of epochs via self-play, is close to 50%.<br />
<br />
[[File:weight_cal.png | 800px]]<br />
<br />
We can see in the above figure that the weight of the spider had to be increased by almost four times in order for the agents to be evenly matched.<br />
<br />
[[File:robosumo_results.png | 800px]]<br />
<br />
The above figure shows testing results for various adaptation strategies. The agent and opponent both start with the self-trained policies. The opponent uses all of its testing experience to continue training. The agent uses only the last 75 episodes to adapt its policy network. This shows that metal learners need only a limited amount of experience in order to hold their own against a constantly improving opponent.<br />
<br />
= Future Work =<br />
The authors noted that the meta-learning adaptation rule they proposed is similar to backpropagation through time with a unit time lag, so a potential area for future research would be to introduce fully-recurrent meta-updates based on the full interaction history with the environment. Secondly, the algorithm proposed involves computing second-order derivatives at training time (see Figure 1c), which resulted in much slower training processes compared to baseline models during experiments, so they suggested finding a method to utilize information from the loss function without explicit backpropagation to speed up computations. The authors also mention that their approach likely will not work well with sparse rewards. This is because the meta-updates, which use policy gradients, are very dependent on the reward signal. They mention that this is an issue they would like to address in the future. A potential solution they have outlined for this is to introduce auxiliary dense rewards which could enable meta-learning.<br />
<br />
= Sources =<br />
# Chelsea Finn, Pieter Abbeel, Sergey Levine. "Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks." arXiv preprint arXiv:1703.03400v3 (2017).<br />
# Richard S Sutton, Anna Koop, and David Silver. On the role of tracking in stationary environments. In Proceedings of the 24th international conference on Machine learning, pp. 871–878. ACM, 2007.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Wasserstein_Auto-Encoders&diff=34656Wasserstein Auto-Encoders2018-03-19T17:10:05Z<p>Pa2forsy: More detail on Fréchet Inception Distance</p>
<hr />
<div><br />
= Introduction =<br />
Recent years have seen a convergence of two previously distinct approaches: representation learning from high dimensional data, and unsupervised generative modeling. In the field that formed at their intersection, Variational Auto-Encoders (VAEs) and Generative Adversarial Networks (GANs) have emerged to become well-established. VAEs are theoretically elegant but with the drawback that they tend to generate blurry samples when applied to natural images. GANs on the other hand produce better visual quality of sampled images, but come without an encoder, are harder to train and suffer from the mode-collapse problem when the trained model is unable to capture all the variability in the true data distribution. Thus there has been a push to come up with the best way to combine them together, but a principled unifying framework is yet to be discovered.<br />
<br />
This work proposes a new family of regularized auto-encoders called the Wasserstein Auto-Encoder (WAE). The proposed method provides a novel theoretical insight into setting up an objective function for auto-encoders from the point of view of of optimal transport (OT). This theoretical formulation leads the authors to examine adversarial and maximum mean discrepancy based regularizers for matching a prior and the distribution of encoded data points in the latent space. An empirical evaluation is performed on MNIST and CelebA datasets, where WAE is found to generate samples of better quality than VAE while preserving training stability, encoder-decoder structure and nice latent manifold structure.<br />
<br />
The main contribution of the proposed algorithm is to provide theoretical foundations for using optimal transport cost as the auto-encoder objective function, while blending auto-encoders and GANs in a principled way. It also theoretically and experimentally explores the interesting relationships between WAEs, VAEs and adversarial auto-encoders.<br />
<br />
= Proposed Approach =<br />
==Theory of Optimal Transport and Wasserstein Distance==<br />
Wasserstein Distance is a measure of the distance between two probability distributions. It is also called Earth Mover’s distance, short for EM distance, because informally it can be interpreted as moving piles of dirt that follow one probability distribution at a minimum cost to follow the other distribution. The cost is quantified by the amount of dirt moved times the moving distance. <br />
A simple case where the probability domain is discrete is presented below.<br />
<br />
<br />
[[File:em_distance.PNG|thumb|upright=1.4|center|Step-by-step plan of moving dirt between piles in ''P'' and ''Q'' to make them match (''W'' = 5).]]<br />
<br />
<br />
When dealing with the continuous probability domain, the EM distance or the minimum one among the costs of all dirt moving solutions becomes:<br />
\begin{align}<br />
\small W(p_r, p_g) = \underset{\gamma\sim\Pi(p_r, p_g)} {\inf}\pmb{\mathbb{E}}_{(x,y)\sim\gamma}[\parallel x-y\parallel]<br />
\end{align}<br />
<br />
Where <math>\Pi(p_r, p_g)</math> is the set of all joint probability distributions with marginals <math>p_r</math> and <math>p_g</math>. Here the distribution <math>\gamma</math> is called a transport plan because it's marginal structure gives some intuition that it represents the amount of probability mass to be moved from x to y. This intuition can be explained by looking at the following equation.<br />
<br />
\begin{align}<br />
\int\gamma(x, y)dx = p_g(y)<br />
\end{align}<br />
<br />
The Wasserstein distance or the cost of Optimal Transport (OT) provides a much weaker topology, which informally means that it makes it easier for a sequence of distribution to converge as compared to other ''f''-divergences. This is particularly important in applications where data is supported on low dimensional manifolds in the input space. As a result, stronger notions of distances such as KL-divergence, often max out, providing no useful gradients for training. In contrast, OT has a much nicer linear behaviour even upon saturation. It can be shown that the Wasserstein distance has guarantees of continuity and differentiability (Arjovsky et al., 2017). Moreover, Arjovsky et al. show there is a nice relationship between the magnitude of the Wasserstein distance and the distance between distributions; a smaller distance nicely corresponds to a smaller distance between the two distributions, and vice versa.<br />
<br />
==Problem Formulation and Notation==<br />
In this paper, calligraphic letters, i.e. <math>\small {\mathcal{X}}</math>, are used for sets, capital letters, i.e. <math>\small X</math>, are used for random variables and lower case letters, i.e. <math>\small x</math>, for their values. Probability distributions are denoted with capital letters, i.e. <math>\small P(X)</math>, and corresponding densities with lower case letters, i.e. <math>\small p(x)</math>.<br />
<br />
This work aims to minimize OT <math>\small W_c(P_X, P_G)</math> between the true (but unknown) data distribution <math>\small P_X</math> and a latent variable model <math>\small P_G</math> specified by the prior distribution <math>\small P_Z</math> of latent codes <math>\small Z \in \pmb{\mathbb{Z}}</math> and the generative model <math>\small P_G(X|Z)</math> of the data points <math>\small X \in \pmb{\mathbb{X}}</math> given <math>\small Z</math>. <br />
<br />
Kantorovich's formulation of the OT problem is given by:<br />
\begin{align}<br />
\small W_c(P_X, P_G) := \underset{\Gamma\sim {\mathcal{P}}(X \sim P_X, Y \sim P_G)}{\inf} {\pmb{\mathbb{E}}_{(X,Y)\sim\Gamma}[c(X,Y)]}<br />
\end{align}<br />
where <math>\small c(x,y)</math> is any measurable cost function and <math>\small {\mathcal{P}(X \sim P_X,Y \sim P_G)}</math> is a set of all joint distributions of <math>\small (X,Y)</math> with marginals <math>\small P_X</math> and <math>\small P_G</math>. When <math>\small c(x,y)=d(x,y)</math>, the following Kantorovich-Rubinstein duality holds for the <math>\small 1^{st}</math> root of <math>\small W_c</math>:<br />
\begin{align}<br />
\small W_1(P_X, P_G) := \underset{f \in {\mathcal{F_L}}} {\sup} {\pmb{\mathbb{E}}_{X \sim P_X}[f(X)]} -{\pmb{\mathbb{E}}_{Y \sim P_G}[f(Y)]}<br />
\end{align}<br />
where <math>\small {\mathcal{F_L}}</math> is the class of all bounded [https://en.wikipedia.org/wiki/Lipschitz_continuity Lipschitz continuous functions].<br />
<br />
==Wasserstein Auto-Encoders==<br />
The proposed method focuses on latent variables <math>\small P_G </math> defined by a two step procedure, where first a code <math>\small Z</math> is sampled from a fixed prior distribution <math>\small P_Z</math> on a latent space <math>\small {\mathcal{Z}}</math> and then <math>\small Z</math> is mapped to the image <math>\small X \in {\mathcal{X}}</math> with a transformation. This results in a density of the form<br />
\begin{align}<br />
\small p_G(x) := \int_{{\mathcal{Z}}} p_G(x|z)p_z(z)dz, \forall x\in{\mathcal{X}}<br />
\end{align}<br />
assuming all the densities are properly defined. It turns out that if the focus is only on generative models deterministically mapping <math>\small Z </math> to <math>\small X = G(Z) </math>, then the OT cost takes a much simpler form as stated below by Theorem 1.<br />
<br />
'''Theorem 1''' For any function <math>\small G:{\mathcal{Z}} \rightarrow {\mathcal{X}}</math>, where <math>\small Q(Z) </math> is the marginal distribution of <math>\small Z </math> when <math>\small X \in P_X </math> and <math>\small Z \in Q(Z|X) </math>,<br />
\begin{align}<br />
\small \underset{\Gamma\sim {\mathcal{P}}(X \sim P_X, Y \sim P_G)}{\inf} {\pmb{\mathbb{E}}_{(X,Y)\sim\Gamma}[c(X,Y)]} = \underset{Q : Q_z=P_z}{\inf} {{\pmb{\mathbb{E}}_{P_X}}{\pmb{\mathbb{E}}_{Q(Z|X)}}[c(X,G(Z))]}<br />
\end{align}<br />
This essentially means that instead of finding a coupling <math>\small \Gamma </math> between two random variables living in the <math>\small {\mathcal{X}} </math> space, one distributed according to <math>\small P_X </math> and the other one according to <math>\small P_G </math>, it is sufficient to find a conditional distribution <math>\small Q(Z|X) </math> such that its <math>\small Z </math> marginal <math>\small Q_Z(Z) := {\pmb{\mathbb{E}}_{X \sim P_X}[Q(Z|X)]} </math> is identical to the prior distribution <math>\small P_Z </math>. In order to implement a numerical solution to Theorem 1, the constraints on <math>\small Q(Z|X) </math> and <math>\small P_Z </math> are relaxed and a penalty function is added to the objective leading to the WAE objective function given by:<br />
<br />
\begin{align}<br />
\small D_{WAE}(P_X, P_G):= \underset{Q(Z|X) \in Q}{\inf} {{\pmb{\mathbb{E}}_{P_X}}{\pmb{\mathbb{E}}_{Q(Z|X)}}[c(X,G(Z))]} + {\lambda} {{\mathcal{D}}_Z(Q_Z,P_Z)}<br />
\end{align}<br />
where <math>\small Q </math> is any non-parametric set of probabilistic encoders, <math>\small {\mathcal{D}}_Z </math> is an arbitrary divergence between <br />
<math>\small Q_Z </math> and <math>\small P_Z </math>, and <math>\small \lambda > 0 </math> is a hyperparameter. The authors propose two different penalties <math>\small {\mathcal{D}}_Z(Q_Z,P_Z) </math> based on adversarial training (GANs) and maximum mean discrepancy (MMD).<br />
<br />
===WAE-GAN: GAN-based===<br />
The first option is to choose <math>\small {\mathcal{D}}_Z(Q_Z,P_Z) = D_{JS}(Q_Z,P_Z)</math>, where <math>\small D_{JS} </math> is the Jensen-Shannon divergence metric, and use adversarial training to estimate it. Specifically a discriminator is introduced in the latent space <math>\small {\mathcal{Z}} </math> trying to separate true points sampled from <math>\small P_Z </math> from fake ones sampled from <math>\small Q_Z </math>. This results in Algorithm 1. It is interesting that the min-max problem is moved from the input pixel space to the latent space.<br />
<br />
<br />
[[File:wae-gan.PNG|270px|center]]<br />
<br />
===WAE-MMD: MMD-based===<br />
For a positive definite kernel <math>\small k: {\mathcal{Z}} \times {\mathcal{Z}} \rightarrow {\mathcal{R}}</math>, the following expression is called the maximum mean discrepancy:<br />
\begin{align}<br />
\small {MMD}_k(P_Z,Q_Z) = \parallel \int_{{\mathcal{Z}}} k(z,\cdot)dP_z(z) - \int_{{\mathcal{Z}}} k(z,\cdot)dQ_z(z) \parallel_{\mathcal{H}_k},<br />
\end{align}<br />
<br />
where <math>\mathcal{H}_k</math> is the reproducing kernel Hilbert space of real-valued functions mapping <math>\mathcal{Z}</math> to <math>\mathcal{R}</math>. This can be used as a divergence measure and the authors propose to use <math>\small {\mathcal{D}}_Z(Q_Z,P_Z) = MMD_k(P_Z,Q_Z) </math>, which leads to Algorithm 2.<br />
<br />
<br />
[[File:wae-mmd.PNG|270px|center]]<br />
<br />
= Comparison with Related Work =<br />
==Auto-Encoders, VAEs and WAEs==<br />
Classical unregularized encoders only minimized the reconstruction cost, and resulted in training points being chaotically scattered across the latent space with holes in between, where the decoder had never been trained. They were hard to sample from and did not provide a useful representation. VAEs circumvented this problem by maximizing a variational lower-bound term comprising of a reconstruction cost and a KL-divergence measure which captures how distinct each training example is from the prior <math>\small P_Z</math>. This however does not guarantee that the overall encoded distribution <math>\small {{\pmb{\mathbb{E}}_{P_X}}}[Q(Z|X)]</math> matches <math>\small P_Z</math>. This is ensured by WAE however, is a direct consequence of our objective function derived from Theorem 1, and is visually represented in the figure below. It is also interesting to note that this also allows WAE to have deterministic encoder-decoder pairs.<br />
<br />
<br />
[[File:vae-wae.PNG|500px|thumb|center|WAE and VAE regularization]]<br />
<br />
<br />
It is also shown that if <math>\small c(x,y)={\parallel x-y \parallel}_2^2</math>, WAE-GAN is equivalent to adversarial autoencoders (AAE). Thus the theory suggests that AAE minimize the 2-Wasserstein distance between <math>\small P_X</math> and <math>\small P_G</math>.<br />
<br />
==OT, W-GAN and WAE==<br />
The Wasserstein GAN (W-GAN) minimizes the 1-Wasserstein distance <math>\small W_1(P_X,P_G)</math> for generative modeling. The W-GAN formulation is approached from the dual form and thus it cannot be applied to another other cost <math>\small W_c</math> as the neat form of the Kantorovich-Rubinstein duality holds only for <math>\small W_1</math>. WAE approaches the same problem from the primal form, can be applied to any cost function <math>\small c</math> and comes naturally with an encoder. The constraint on OT in Theorem 1, is relaxed in line with theory on unbalanced optimal transport by adding a penalty or additional divergences to the objective.<br />
<br />
==GANs and WAEs==<br />
Many of the GAN variations including f-GAN and W-GAN come without an encoder. Often it may be desirable to reconstruct the latent codes and use the learned manifold in which case they won't be applicable. For works which try to blend adversarial auto-encoder structures, encoders and decoders do not have incentive to be reciprocal. WAE does not necessarily lead to a min-max game and has a clear theoretical foundation for using penalties for regularization.<br />
<br />
=Experimental Results=<br />
The authors empirically evaluate the proposed WAE generative model by specifically testing if data points are accurately reconstructed, if the latent manifold has reasonable geometry, and if random samples of good visual quality are generated. <br />
<br />
'''Experimental setup:'''<br />
Gaussian prior distribution <math> \small P_Z</math> and squared cost function <math> \small c(x,y)</math> are used for data-points. The encoder-decoder pairs were deterministic. The convolutional deep neural network for encoder mapping and decoder mapping are similar to DC-GAN with batch normalization. Real world datasets, MNIST with 70k images and CelebA with 203k images were used for training and testing.<br />
<br />
'''WAE-GAN and WAE-MMD:'''<br />
In WAE-GAN, the discriminator <math> \small D </math> composed of several fully connected layers with ReLu activations. For WAE-MMD, the RBF kernel failed to penalize outliers and thus the authors resorted to using inverse multiquadratics kernel <math> \small k(x,y)=C/(C+\parallel{x-y}_2^2\parallel) </math>. Trained models are presented in the figure below.<br />
As far as random sampled results are concerned, WAE-GAN seems to be highly unstable but do lead to better matching scores among WAE-GAN, WAE-MMD and VAE. WAE-MMD on the other hand has much more stable training and fairly good quality of sampled results.<br />
<br />
'''Qualitative assessment:'''<br />
In order to quantitatively assess the quality of the generated images, they use the Fréchet Inception Distance and report the results on CelebA (The Fréchet Inception Distance measures the similarity between two sets of images, by comparing the Fréchet distance of multivariate Gaussian distributions fitted to their feature representations. In more detail, let <math> (m,C) </math> denote the mean vector and covariance matrix of the features of the inception network (Szegedy et al. 2017) applied to model samples. Let <math>(m_w,C_w) </math> denote the mean vector and covariance matrix of the features of the inception network applied to real data. Then the Fréchet Inception Distance between the model samples and the real data is <math> ||m-m_w||^2 +\mathrm{tr}(C+C_w-2(CC_w)^{\frac{1}{2}} )\,</math> (Heusel et al. 2017). ) These results confirm that the sampled images from WAE are of better quality than from VAE (score: 82), and WAE-GAN gets a slightly better score (score:42) than WAE-MMD (score:55), which correlates with visual inspection of the images.<br />
<br />
[[File:results.png|800px|thumb|center|Results on MNIST and Celeb-A dataset. In "test reconstructions" (middle row of images), odd rows correspond to the real test points.]]<br />
<br />
<br />
<br />
The authors also heuristically evaluate the sharpness of generated samples using the Laplace filter. The numbers, summarized in Table1, show that WAE-MMD has samples of slightly better quality than VAE, while WAE-GAN achieves the best results overall.<br />
[[File: paper17_Table.png|300px|thumb|right|Qualitative Assessment of Images]]<br />
<br />
= Commentary and Conclusion =<br />
This paper presents an interesting theoretical justification for a new family of auto-encoders called Wasserstein Auto-Encoders (WAE). The objective function minimizes the optimal transport cost in the form of the Wasserstein distance, but relaxes theoretical constraints to separate it into a reconstruction cost and a regularization penalty. The regularization penalizes divergences between a prior and the distribution of encoded latent space training data, and is estimated by means of adversarial training (WAE-GAN), or kernel-based techniques (WAE-MMD). They show that they achieve samples of better visual quality than VAEs, while achieving stable training at the same time. They also theoretically show that WAEs are a generalization of adversarial auto-encoders (AAEs).<br />
<br />
Although the paper mentions that encoder-decoder pairs can be deterministic, they do not show the geometry of the latent space that is obtained. It is necessary to study the effect of randomness of encoders on the quality of obtained samples. While this method is evaluated on MNIST and CelebA datasets, it is also important to see their performance on other real world data distributions. The authors do not provide a comprehensive evaluation of WAE-GAN regularization, thus making it hard to comment on whether moving an adversarial problem to the latent space results in less instability. Reasons for better sample quality of WAE-GAN over WAE-MMD also need to be inspected. In the future it would be interesting to investigate different ways to compute the divergences between the encoded distribution and the prior distribution.<br />
<br />
=Open Source Code=<br />
1. https://github.com/tolstikhin/wae <br />
<br />
2. https://github.com/maitek/waae-pytorch<br />
<br />
=Sources=<br />
1. M. Arjovsky, S. Chintala, and L. Bottou. Wasserstein GAN, 2017<br />
<br />
2. Martin Heusel et al. "Gans trained by a two time-scale update rule converge to a local nash equilibrium." Advances in Neural Information Processing Systems. 2017.<br />
<br />
3. Christian Szegedy et al. "Inception-v4, inception-resnet and the impact of residual connections on learning." AAAI. Vol. 4. 2017.<br />
<br />
4. Ilya Tolstikhin, Olivier Bousquet, Sylvain Gelly, Bernhard Scholkopf. Wasserstein Auto-Encoders, 2017<br />
<br />
5. https://lilianweng.github.io/lil-log/2017/08/20/from-GAN-to-WGAN.html</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=On_The_Convergence_Of_ADAM_And_Beyond&diff=33126On The Convergence Of ADAM And Beyond2018-03-11T21:51:33Z<p>Pa2forsy: /* Notation */</p>
<hr />
<div>= Introduction =<br />
Somewhat different to the presentation I gave in class, this paper focuses strictly on the pitfalls in convergance of the ADAM training algorithm for neural networks from a theoretical standpoint and proposes a novel improvement to ADAM called AMSGrad. The result essentially introduces the idea that it is possible for ADAM to get itself "stuck" in it's weighted average history which must be prevented somehow.<br />
<br />
== Notation ==<br />
The paper presents the following framework that generalizes training algorithms to allow us to define a specific variant such as AMSGrad or SGD entirely within it:<br />
<br />
<br />
[[File:training_algo_framework.png]]<br />
<br />
Where we have <math> x_t </math> as our network parameters defined within a vector space <math> \mathcal{F} </math>. <math> \prod_{\mathcal{F}} (y) = </math> the projection of <math> y </math> on to the set <math> \mathcal{F} </math>.<br />
<math> \psi_t </math> and <math> \phi_t </math> correspond to arbitrary functions we will provide later, The former maps from the history of gradients to <math> \mathds{R}^d </math> and the latter maps from the history of the gradients to positive semi definite matrices. And finally <math> f_t </math> is our loss function at some time <math> t </math>, the rest should be pretty self explanatory. Using this framework and defining different <math> \psi_t </math> , <math> \phi_t </math> will allow us to recover all different kinds of training algorithms under this one roof.<br />
<br />
<br />
=== SGD As An Example ===<br />
To recover SGD using this framework we simply select <math> \phi_t (g_1, \dotsc, g_t) = g_t</math> and <math> \psi_t (g_1, \dotsc, g_t) = I </math>. It's easy to see that no transformations are ultimately applied to any of the parameters based on any gradient history other than the most recent from <math> \phi_t </math> and that <math> \psi_t </math> in no way transforms any of the parameters by any specific amount as <math> V_t = I </math> has no impact later on.<br />
<br />
<br />
=== ADAM As Another Example ===<br />
Once you can convince yourself that SGD is correct, you should understand the framework enough to see why the following setup for ADAM will allow us to recover the behavior we want. ADAM has the ability to define a "learning rate" for every parameter based on how much that parameter moves over time (a.k.a it's momentum) supposedly to help with the learning process.<br />
<br />
In order to do this we will choose <math> \phi_t (g_1, \dotsc, g_t) = (1 - \beta_1) \sum_{i=0}^{t} {\beta_1}^{t - i} g_t </math><br />
And we will choose psi to be <math> \psi_t (g_1, \dotsc, g_t) = (1 - \beta_2)</math>diag<math>( \sum_{i=0}^{t} {\beta_2}^{t - i} {g_t}^2) </math>.<br />
<br />
From this we can now see that <math>m_t </math> gets filled up with the exponentially weighted average of the history of our gradients that we've come across so far in the algorithm. And that as we proceed to update we scale each one of our paramaters by dividing out <math> V_t </math> (in the case of diagonal it's just 1/the diagonal entry) which contains the exponentially weighted average of each parameters momentum (<math> {g_t}^2 </math>) across our training so far in the algorithm. Thus giving each parameter it's own unique scaling by it's second moment or momentum. Intuitively from a physical perspective if each parameter is a ball rolling around in the optimization landscape what we are now doing is instead of having the ball changed positions on the landscape at a fixed velocity (i.e. momentum of 0) the ball now has the ability to accelerate and speed up or slow down if it's on a steep hill or flat trough in the landscape (i.e. a momentum that can change with time).<br />
<br />
= <math> \Gamma_t </math> An Interesting Quantity =<br />
Now that we have an idea of what ADAM looks like in this framework, let us now investigate the following:<br />
<br />
<math> \Gamma_{t + 1} = \frac{\sqrt{V_{t+1}}}{\alpha_{t+1}} - \frac{\sqrt{V_t}}{\alpha_t} </math><br />
<br />
<br />
Which essentially measure the change of the "Inverse of the learning rate" across time (since we are using alpha's as step sizes). Looking back to our example of SGD it's not hard to see that this quantity is strictly positive, which leads to "non-increasing" learning rates a desired property. However that is not the case with ADAM, and can pose a problem in a theoretical and applied setting. The problem ADAM can face is that <math> \Gamma_t </math> can be indefinite, which the original proof assumed it could not be. The math for this proof is VERY long so instead we will opt for an example to showcase why this could be an issue.<br />
<br />
Consider the loss function <math> f_t(x) = \begin{cases} <br />
Cx & \text{for } t \text{ mod 3} = 1 \\<br />
-x & \text{otherwise}<br />
\end{cases} </math><br />
<br />
Where we have <math> C > 2 </math> and <math> \mathcal{F} </math> is <math> [-1,1] </math><br />
Additionally we choose <math> \beta_1 = 0 </math> and <math> \beta_2 = 1/(1+C^2) </math>. We then proceed to plug this into our framework from before. <br />
This function is periodic and it's easy to see that it has the gradient of C once and then a gradient of -1 twice every period. It has an optimal solution of <math> x = -1 </math> (from a regret standpoint), but using ADAM we would eventually converge at <math> x = 1 </math> since <math> C </math> would "overpower" the -1's.<br />
<br />
= AMSGrad as an improvement to ADAM =<br />
<br />
There is a very simple intuitive fix to ADAM to handle this problem. We simply scale our historical weighted average by the maximum we have seen so far to avoid the negative sign problem. There is a very simple one liner adaptation of ADAM to get to AMSGRAD:<br />
[[File:AMSGrad_algo.png]]<br />
<br />
Below are some simple plots comparing ADAM and AMSGrad, the first are from the paper and the second are from another individual who attempted to recreate the experiments. The two plots somewhat disagree with one another so take this heuristic improvement with a grain of salt.<br />
[[File:AMSGrad_vs_adam.png]]<br />
[[File:AMSGrad_vs_adam2.png]]<br />
<br />
<br />
= Conclusion =<br />
We have introduced a framework for which we could view several different trainging algorithms. From there we used it to recover SGD as well as ADAM. In our recovery of ADAM we investigated the change of the inverse of the learning rate over time to discover in certain cases there were convergence issues. We proposed a new heuristic AMSGrad to help deal with this problem and presented some empirical results that show it may have helped ADAM slightly. Thanks for your time.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Robust_Imitation_of_Diverse_Behaviors&diff=33106Robust Imitation of Diverse Behaviors2018-03-11T01:23:53Z<p>Pa2forsy: Add citation about brittleness</p>
<hr />
<div>=Introduction=<br />
One of the longest standing challenges in AI is building versatile embodied agents, both in the form of real robots and animated avatars, capable of a wide and diverse set of behaviors. State-of-the-art robots cannot compete with the effortless variety and adaptive flexibility of motor behaviors produced by toddlers. Towards addressing this challenge, the authors combine several deep generative approaches to imitation learning in a way that accentuates their individual strengths and addresses their limitations. The end product is a robust neural network policy that can imitate a large and diverse set of behaviors using few training demonstrations.<br />
<br />
=Motivation=<br />
Some of the models that have recently shown great promise in imitation learning for motor control are the deep generative models. The authors primarily talk about two approaches viz. supervised approaches that condition on demonstrations and Generative Adversarial Imitation Learning (GAIL) and their limitations and try to combine those two approaches in order to address these limitations. Some of these limitations are as follows:<br />
<br />
* Supervised approaches that condition on demonstrations (VAE):<br />
** They require large training datasets in order to work for non-trivial tasks<br />
** They tend to be brittle and fail when the agent diverges too much from the demonstration trajectories (As proof of this brittleness, the authors cite Ross et al. (2010), who provide a theorem showing that the cost incurred by this kind of model when it deviates from a demonstration trajectory with a small probability can be amplified in a manner quadratic in the number of time steps. )<br />
* Generative Adversarial Imitation Learning (GAIL)<br />
** Adversarial training leads to mode-collapse (the tendency of adversarial generative models to cover only a subset of modes of a probability distribution, resulting in a failure to produce adequately diverse samples)<br />
** More difficult and slow to train as they do not immediately provide a latent representation of the data<br />
<br />
Thus, the former approach can model diverse behaviors without dropping modes but does not learn robust policies, while the latter approach gives robust policies but insufficiently diverse behaviors. Thus, the authors combine the favorable aspects of these two approaches. The base of their model is a new type of variational autoencoder on demonstration trajectories that learns semantic policy embeddings. Leveraging these policy representations, they develop a new version of GAIL that <br />
# is much more robust than the purely-supervised controller, especially with few demonstrations, and <br />
# avoids mode collapse, capturing many diverse behaviors when GAIL on its own does not.<br />
<br />
=Model=<br />
The authors first introduce a variational autoencoder (VAE) for supervised imitation, consisting of a bi-directional LSTM encoder mapping demonstration sequences to embedding vectors, and two decoders. The first decoder is a multi-layer perceptron (MLP) policy mapping a trajectory embedding and the current state to a continuous action vector. The second is a dynamics model mapping the embedding and previous state to the present state while modeling correlations among states with a WaveNet.<br />
<br />
[[File: Model_Architecture.png|700px|center|]]<br />
<br />
==Behavioral cloning with VAE suited for control==<br />
<br />
In this section, the authors follow a similar approach to Duan et al., but opt for stochastic VAEs as having a distribution <math display="inline">q_\phi(z|x_{1:T})</math> to better regularize the latent space. In their VAE, an encoder maps a demonstration sequence to an embedding vector <math display="inline">z</math>. Given <math display="inline">z</math>, they decode both the state and action trajectories as shown in the figure above. To train the model, the following loss is minimized:<br />
<br />
\begin{align}<br />
L\left( \alpha, w, \phi; \tau_i \right) = - \pmb{\mathbb{E}}_{q_{\phi}(z|x_{1:T_i}^i)} \left[ \sum_{t=1}^{T_i} log \pi_\alpha \left( a_t^i|x_t^i, z \right) + log p_w \left( x_{t+1}^i|x_t^i, z\right) \right] +D_{KL}\left( q_\phi(z|x_{1:T_i}^i)||p(z) \right)<br />
\end{align}<br />
<br />
The encoder <math display="inline">q</math> uses a bi-directional LSTM. To produce the final embedding, it calculates the average of all the outputs of the second layer of this LSTM before applying a final linear transformation to<br />
generate the mean and standard deviation of a Gaussian. Then, one sample from this Gaussian is taken as the demonstration encoding.<br />
<br />
The action decoder is an MLP that maps the concatenation of the state and the embedding of the parameters of a Gaussian policy. The state decoder is similar to a conditional WaveNet model. In particular, it conditions on the embedding <math display="inline">z</math> and previous state <math display="inline">x_{t-1}</math> to generate the vector <math display="inline">x_t</math> autoregressively. That is, the autoregression is over the components of the vector <math display="inline">x_t</math>. Finally, instead of a Softmax, the model uses a mixture of Gaussians as the output of the WaveNet.<br />
<br />
==Diverse generative adversarial imitiation learning==<br />
To enable GAIL to produce diverse solutions, the authors condition the discriminator on the embeddings generated by the VAE encoder and integrate out the GAIL objective with respect to the variational posterior <math display="inline">q_\phi(z|x_{1:T})</math>. Specifically, the authors train the discriminator by optimizing the following objective:<br />
<br />
\begin{align}<br />
{max}_{\psi} \pmb{\mathbb{E}}_{\tau_i \sim \pi_E} \left( \pmb{\mathbb{E}}_{q(z|x_{1:T_i}^i)} \left[\frac{1}{T_i} \sum_{t=1}^{T_i} logD_{\psi} \left( x_t^i, a_t^i | z \right) + \pmb{\mathbb{E}}_{\pi_\theta} \left[ log(1 - D_\psi(x, a | z)) \right] \right] \right)<br />
\end{align}<br />
<br />
The authors condition on unlabeled trajectories, which have been passed through a powerful encoder, and hence this approach is capable of one-shot imitation learning. Moreover, the VAE encoder enables to obtain a continuous latent embedding space where interpolation is possible.<br />
<br />
To better motivate the objective, the authors propose on temporarily leaving the context of imitation learning and considering an alternative objective for training GANs<br />
<br />
\begin{align}<br />
{min}_{G}{max}_{D} V (G, D) = \int_{y} p(y) \int_{z} q(z|y) \left[ log D(y | z) + \int_{\hat{y}} G(\hat{y} | z) log (1 - D(\hat{y} | z)) d\hat{y} \right] dy dz<br />
\end{align}<br />
<br />
This function is a simplification of the previous objective function. Furthermore, it satisfies the following property.<br />
<br />
===Lemma 1===<br />
Assuming that <math display="inline">q</math> computes the true posterior distribution that is <math display="inline">q(z|y) = \frac{p(y|z)p(z)}{p(y)}</math> then<br />
<br />
\begin{align}<br />
V (G, D) = \int_{z} p(z) \left[ \int_{y} p(y|z) log D(y|z) dy + \int_{\hat{y}} G(\hat{y} | z) log (1 - D(\hat{y} | z)) d\hat{y} \right] dz<br />
\end{align}<br />
<br />
If an optimal discriminator is further assumed, the cost-optimized by the generator then becomes<br />
<br />
\begin{align}<br />
C(G) = 2 \int_ p p(z) JSD[p(\cdot|z) || G(\cdot|z)] dz - log4<br />
\end{align}<br />
<br />
where <math display="inline">JSD</math> stands for the Jensen-Shannon divergence.<br />
<br />
=Experiments=<br />
<br />
The primary focus of the paper's experimental evaluation is to demonstrate that the architecture allows learning of robust controllers capable of producing the full spectrum of demonstration behaviors for a diverse range of challenging control problems. The authors consider three bodies: a 9 DoF robotic arm, a 9 DoF planar walker, and a 62 DoF complex humanoid (56-actuated joint angles, and a freely translating and rotating 3d root joint). While for the reaching task BC is sufficient to obtain a working controller, for the other two problems the full learning procedure is critical.<br />
<br />
The authors analyze the resulting embedding spaces and demonstrate that they exhibit a rich and sensible structure that can be exploited for control. Finally, the authors show that the encoder can be used to capture the gist of novel demonstration trajectories which can then be reproduced by the controller.<br />
<br />
==Robotic arm reaching==<br />
In this experiment, the authors demonstrate the effectiveness of their VAE architecture and investigate the nature of the learned embedding space on a reaching task with a simulated Jaco arm.<br />
<br />
To obtain demonstrations, the authors trained 60 independent policies to reach to random target locations in the workspace starting from the same initial configuration. 30 trajectories from each of<br />
the first 50 policies were generated. These served as training data for the VAE model (1500 training trajectories in total). The remaining 10 policies were used to generate test data.<br />
<br />
The reaching task is relatively simple, so with this amount of data the VAE policy is fairly robust. After training, the VAE encodes and reproduces the demonstrations as shown in the figure below.<br />
<br />
[[File: Robotic_arm_reaching.png|650px|center|]]<br />
<br />
==2D Walker==<br />
<br />
As a more challenging test compared to the reaching task, the authors consider bipedal locomotion. Here, the authors train 60 neural network policies for a 2d walker to serve as demonstrations. These policies are each trained to move at different speeds both forward and backward depending on a label provided as additional input to the policy. Target speeds for training were chosen from a set of four different speeds (m/s): -1, 0, 1, 3.<br />
<br />
[[File: 2D_Walker.png|650px|center|]]<br />
<br />
The authors trained their model with 20 episodes per policy (1200 demonstration trajectories in total, each with a length of 400 steps or 10s of simulated time). In this experiment a full approach is required: training the VAE with BC alone can imitate some of the trajectories, but it performs poorly in general, presumably because the relatively small training set does not cover the space of trajectories sufficiently densely. On this generated dataset, they also train policies with GAIL using the same architecture and hyper-parameters. Due to the lack of conditioning, GAIL does not reproduce coherently trajectories. Instead, it simply meshes different behaviors together. In addition, the policies trained with GAIL also, exhibit dramatically less diversity; see [https://www.youtube.com/watch?v=kIguLQ4OwuM video].<br />
<br />
[[File: 2D_Walker_Optimized.gif|frame|center|In the left panel, the planar walker demonstrates a particular walking style. In the right panel, the model's agent imitates this walking style using a single policy network.]]<br />
<br />
==Complex humanoid==<br />
For this experiment, the authors consider a humanoid body of high dimensionality that poses a hard control problem. They generate training trajectories with the existing controllers, which can produce instances of one of six different movement styles. Examples of such trajectories are shown in Fig. 5.<br />
<br />
[[File: Complex_humanoid.png|650px|center|]]<br />
<br />
The training set consists of 250 random trajectories from 6 different neural network controllers that were trained to match 6 different movement styles from the CMU motion capture database. Each trajectory is 334 steps or 10s long. The authors use a second set of 5 controllers from which they generate trajectories for evaluation (3 of these policies were trained on the same movement styles as the policies used for generating training data).<br />
<br />
Surprisingly, despite the complexity of the body, supervised learning is quite effective at producing sensible controllers: The VAE policy is reasonably good at imitating the demonstration trajectories, although it lacks the robustness to be practically useful. Adversarial training dramatically improves the stability of the controller. The authors analyze the improvement quantitatively by computing the percentage of the humanoid falling down before the end of an episode while imitating either training or test policies. The results are summarized in Figure 5 right. The figure further shows sequences of frames of representative demonstration and associated imitation trajectories. Videos of demonstration and imitation behaviors can be found in the [https://www.youtube.com/watch?v=NaohsyUxpxw video].<br />
<br />
[[File: Complex_humanoid_optimized.gif|frame|center|In the left and middle panels we show two demonstrated behaviors. In the right panel, the model's agent produces an unseen transition between those behaviors.]]<br />
<br />
Also, for practical purposes, it is desirable to allow the controller to transition from one behavior to another. The authors test this possibility in an experiment similar to the one for the Jaco arm: They determine the<br />
embedding vectors of pairs of demonstration trajectories, start the trajectory by conditioning on the first embedding vector, and then transition from one behavior to the other half-way through the episode by linearly interpolating the embeddings of the two demonstration trajectories over a window of 20 control steps. Although not always successful the learned controller often transitions robustly, despite not having been trained to do so. An example of this can be seen in the gif above. More examples can be seen in this [https://www.youtube.com/watch?v=VBrIll0B24o video].<br />
<br />
=Conclusions=<br />
The authors have proposed an approach for imitation learning that combines the favorable properties of techniques for density modeling with latent variables (VAEs) with those of GAIL. The result is a model that learns, from a moderate number of demonstration trajectories <br />
# a semantically well structured embedding of behaviors, <br />
# a corresponding multi-task controller that allows to robustly execute diverse behaviors from this embedding space, as well as <br />
# an encoder that can map new trajectories into the embedding space and hence allows for one-shot imitation. <br />
The experimental results demonstrate that this approach can work on a variety of control problems and that it scales even to very challenging ones such as the control of a simulated humanoid with a large number of degrees of freedoms.<br />
<br />
=Critique=<br />
The paper proposes a deep-learning-based approach to imitation learning which is sample-efficient and is able to imitate many diverse behaviors. The architecture can be seen as conditional generative adversarial imitation learning (GAIL). The conditioning vector is an embedding of a demonstrated trajectory, provided by a variational autoencoder. This results in one-shot imitation learning: at test time, a new demonstration can be embedded and provided as a conditioning vector to the imitation policy. The authors evaluate the method on several simulated motor control tasks.<br />
<br />
Pros:<br />
* Addresses a challenging problem of learning complex dynamics controllers / control policies<br />
* Well-written introduction / motivation<br />
* The proposed approach is able to learn complex and diverse behaviors and outperforms both the VAE alone (quantitatively) and GAIL alone (qualitatively).<br />
* Appealing qualitative results on the three evaluation problems. Interesting experiments with motion transitioning. <br />
<br />
Cons:<br />
* Comparisons to baselines could be more detailed.<br />
* Many key details are omitted (either on purpose, placed in the appendix, or simply absent, like the lack of definitions of terms in the modeling section, details of the planner model, simulation process, or the details of experimental settings)<br />
* Experimental evaluation is largely subjective (videos of robotic arm/biped/3D human motion)<br />
* A discussion of sample efficiency compared to GAIL and VAE would be interesting.<br />
* The presentation is not always clear, in particular, I had a hard time figuring out the notation in Section 3.<br />
* There has been some work on hybrids of VAEs and GANs, which seem worth mentioning when generative models are discussed, like:<br />
# Autoencoding beyond pixels using a learned similarity metric, Larsen et al., ICML 2016<br />
# Generating Images with Perceptual Similarity Metrics based on Deep Networks, Dosovitskiy&Brox. NIPS 2016<br />
These works share the intuition that good coverage of VAEs can be combined with sharp results generated by GANs.<br />
* Some more extensive analysis of the approach would be interesting. How sensitive is it to hyperparameters? How important is it to use VAE, not usual AE or supervised learning? How difficult will it be for others to apply it to new tasks?<br />
<br />
=References=<br />
# Ross, Stéphane, and Drew Bagnell. "Efficient reductions for imitation learning." Proceedings of the thirteenth international conference on artificial intelligence and statistics. 2010.<br />
# Wang, Z., Merel, J. S., Reed, S. E., de Freitas, N., Wayne, G., & Heess, N. (2017). Robust imitation of diverse behaviors. In Advances in Neural Information Processing Systems (pp. 5326-5335).<br />
# https://deepmind.com/blog/producing-flexible-behaviours-simulated-environments/<br />
# https://www.youtube.com/watch?v=NaohsyUxpxw<br />
# https://www.youtube.com/watch?v=VBrIll0B24o</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=A_Neural_Representation_of_Sketch_Drawings&diff=33105A Neural Representation of Sketch Drawings2018-03-11T00:25:00Z<p>Pa2forsy: Source for information on HyperLSTMs</p>
<hr />
<div>= Introduction =<br />
<br />
There have been many recent advances in neural generative models for low resolution pixel-based images. Humans, however, do not see the world in a grid of pixels and more typically communicate drawings of the things we see using a series of pen strokes that represent components of objects. These pen strokes are similar to the way vector-based images store data. This paper proposes a new method for creating conditional and unconditional generative models for creating these kinds of vector sketch drawings based on recurrent neural networks (RNNs). The paper explores many applications of these kinds of models, especially creative applications and makes available their unique dataset of vector images.<br />
<br />
= Related Work =<br />
<br />
Previous work related to sketch drawing generation includes methods that focussed primarily on converting input photographs into equivalent vector line drawings. Image generating models using neural networks also exist but focussed more on generation of pixel-based imagery. Some recent work has focussed on handwritten character generation using RNNs and Mixture Density Networks to generate continuous data points. This work has been extended somewhat recently to conditionally and unconditionally generate handwritten vectorized Chinese Kanji characters by modeling them as a series of pen strokes. Furthermore, this paper builds on work that employed Sequence-to-Sequence models with Variational Autencoders to model English sentences in latent vector space.<br />
<br />
One of the limiting factors for creating models that operate on vector datasets has been the dearth of publicly available data. Previously available datasets include Sketch, a set of 20K vector drawings; Sketchy, a set of 70K vector drawings; and ShadowDraw, a set of 30K raster images with extracted vector drawings.<br />
<br />
= Methodology =<br />
<br />
=== Dataset ===<br />
<br />
The “QuickDraw” dataset used in this research was assembled from 75K user drawings extracted from the game “Quick, Draw!” where users drew objects from one of hundreds of classes in 20 seconds or less. The dataset is split into 70K training samples and 2.5K validation and test samples each and represents each sketch a set of “pen stroke actions”. Each action is provided as a vector in the form <math>(\Delta x, \Delta y, p_{1}, p_{2}, p_{3})</math>. For each vector, <math>\Delta x</math> and <math>\Delta y</math> give the movement of the pen from the previous point, with the initial location being the origin. The last three vector elements are a one-hot representation of pen states; <math>p_{1}</math> indicates that the pen is down and a line should be drawn between the current point and the next point, <math>p_{2}</math> indicates that the pen is up and no line should be drawn between the current point and the next point, and <math>p_{3}</math> indicates that the drawing is finished and subsequent points and the current point should not be drawn.<br />
<br />
=== Sketch-RNN ===<br />
[[File:sketchrnn.PNG]]<br />
<br />
The model is a Sequence-to-Sequence Variational Autoencoder (VAE). The encoder model is a symmetric and parallel set of two RNNs that individually process the sketch drawings in forward and reverse order, respectively. The hidden state produced by each encoder model is then concatenated into a single hidden state <math>h</math>. <br />
<br />
The concatenated hidden state <math>h</math> is then projected into two vectors <math>\mu</math> and <math>\hat{\sigma}</math> each of size <math>N_{z}</math> using a fully connected layer. <math>\hat{\sigma}</math> is then converted into a non-negative standard deviation parameter <math>\sigma</math> using an exponential operator. These two parameters <math>\mu</math> and <math>\sigma</math> are then used along with an IID Gaussian vector distributed as <math>\mathcal{N}(0, I)</math> of size <math>N_{z}</math> to construct a random vector <math>z \in ℝ^{N_{z}}</math>, similar to the method used for VAE:<br />
\begin{align}<br />
\mu = W_{\mu}h + b_{mu}\textrm{, }\hat{\sigma} = W_{\sigma}h + b_{\sigma}\textrm{, }\sigma = exp\bigg{(}\frac{\hat{\sigma}}{2}\bigg{)}\textrm{, }z = \mu + \sigma \odot \mathcal{N}(0,I)<br />
\end{align}<br />
<br />
The decoder model is another RNN that samples output sketches from the latent vector <math>z</math>. The initial hidden states of each recurrent neuron are determined using <math>[h_{0}, c_{0}] = tanh(W_{z}z + b_{z})</math>. Each step of the decoder RNN accepts the previous point <math>S_{i-1}</math> and the latent vector <math>z</math> as concatenated input. The initial point given is the origin point with pen state down. The output at each step are the parameters for a probability distribution of the next point <math>S_{i}</math>. Outputs <math>\Delta x</math> and <math>\Delta y</math> are modelled using a Gaussian Mixture Model (GMM) with M normal distributions and output pen states <math>(q_{1}, q_{2}, q_{3})</math> modelled as a categorical distribution with one-hot encoding.<br />
\begin{align}<br />
P(\Delta x, \Delta y) = \sum_{j=1}^{M}\Pi_{j}\mathcal{N}(\Delta x, \Delta y | \mu_{x, j}, \mu_{y, j}, \sigma_{x, j}, \sigma_{y, j}, \rho_{xy, j})\textrm{, where }\sum_{j=1}^{M}\Pi_{j} = 1<br />
\end{align}<br />
<br />
For each of the M distributions in the GMM, parameters <math>\mu</math> and <math>\sigma</math> are output for both the x and y locations signifying the mean location of the next point and the standard deviation, respectively. Also output from each model is parameter <math>\rho_{xy}</math> signifying correlation of each bivariate normal distribution. An additional vector <math>\Pi</math> is an output giving the mixture weights for the GMM. The output <math>S_{i}</math> is determined from each of the mixture models using softmax sampling from these distributions.<br />
<br />
One of the key difficulties in training this model is the highly imbalanced class distribution of pen states. In particular, the state that signifies a drawing is complete will only appear one time per each sketch and is difficult to incorporate into the model. In order to have the model stop drawing, the authors introduce a hyperparameter that limits the number of points per drawing to being no more than <math>N_{max}</math>, after which all output states form the model are set to (0, 0, 0, 0, 1) to force the drawing to stop.<br />
<br />
To sample from the model, the parameters required by the GMM and categorical distributions are generated at each time step and the model is sampled until a “stop drawing” state appears or the time state reaches time <math>N_{max}</math>. The authors also introduce a “temperature” parameter <math>\tau</math> that controls the randomness of the drawings by modifying the pen states, model standard deviations, and mixture weights as follows:<br />
<br />
\begin{align}<br />
\hat{q}_{k} \rightarrow \frac{\hat{q}_{k}}{\tau}\textrm{, }\hat{\Pi}_{k} \rightarrow \frac{\hat{\Pi}_{k}}{\tau}\textrm{, }\sigma^{2}_{x} \rightarrow \sigma^{2}_{x}\tau\textrm{, }\sigma^{2}_{y} \rightarrow \sigma^{2}_{y}\tau<br />
\end{align}<br />
<br />
This parameter <math>\tau</math> lies in the range (0, 1]. As the parameter approaches 0, the model becomes more deterministic and always produces the point locations with the maximum likelihood for a given timestep.<br />
<br />
=== Unconditional Generation ===<br />
The authors also explored unconditional generation of sketch drawings by only training the decoder RNN module. To do this, the initial hidden states of the RNN were set to 0, and only vectors from the drawing input are used as input without any conditional latent variable <math>z</math>. Different sketches are sampled from the network by only varying the temperature parameter <math>\tau</math> between 0.2 and 0.9<br />
<br />
=== Training ===<br />
The training procedure follows the same approach as training for VAE and uses a loss function that consists of the sum of Reconstruction Loss <math>L_{R}</math> and KL Divergence Loss <math>L_{KL}</math>. The reconstruction loss term is composed of two terms; <math>L_{s}</math>, which tries to maximize the log-likelihood of the generated probability distribution explaining the training data <math>S</math> and <math>L_{p}</math> which is the log loss of the pen state terms.<br />
\begin{align}<br />
L_{s} = -\frac{1}{N_{max}}\sum_{i=1}^{N_{S}}log\bigg{(}\sum_{j=1}^{M}\Pi_{j,i}\mathcal{N}(\Delta x_{i},\Delta y_{i} | \mu_{x,j,i},\mu_{y,j,i},\sigma_{x,j,i},\sigma_{y,j,i},\rho_{xy,j,i})\bigg{)}<br />
\end{align}<br />
\begin{align}<br />
L_{p} = -\frac{1}{N_{max}}\sum_{i=1}^{N_{max}} \sum_{k=1}^{3}p_{k,i}log(q_{k,i})<br />
\end{align}<br />
\begin{align}<br />
L_{R} = L_{s} + L{p}<br />
\end{align}<br />
<br />
The KL divergence loss <math>L_{KL}</math> measures the difference between the latent vector <math>z</math> and an IID Gaussian distribution with 0 mean and unit variance. This term, normalized by the number of dimensions <math>N_{z}</math> is calculated as:<br />
\begin{align}<br />
L_{KL} = -\frac{1}{2N_{z}}\big{(}1 + \hat{\sigma} - \mu^{2} – exp(\hat{\sigma})\big{)}<br />
\end{align}<br />
<br />
The loss for the entire model is thus the weighted sum:<br />
\begin{align}<br />
Loss = L_{R} + w_{KL}L_{KL}<br />
\end{align}<br />
<br />
The value of the weight parameter <math>w_{KL}</math> has the effect that as <math>w_{KL} \rightarrow 0</math>, there is a loss in ability to enforce a prior over the latent space and the model assumes the form of a pure autoencoder.<br />
<br />
= Experiments =<br />
The authors trained multiple conditional and unconditional models using varying values of <math>w_{KL}</math> and recorded the different <math>L_{R}</math> and <math>L_{KL}</math> values at convergence. The network used LSTM as it’s encoder RNN and HyperLSTM as the decoder network. The HyperLSTM model was used for decoding because it has a history of being useful in sequence generation tasks. (A HyperLSTM consists of two coupled LSTMS: an auxiliary LSTM and a main LSTM. At every time step, the auxiliary LSTM reads the previous hidden state and the current input vector, and computes an intermediate vector <math display="inline"> z </math>. The weights of the main LSTM used in the current time step are then a learned function of this intermediate vector <math display="inline"> z </math>. That is, the weights of the main LSTM are allowed to vary between time steps as a function of the output of the auxiliary LSTM. See Ha et al. (2016) for details)<br />
<br />
=== Conditional Reconstruction ===<br />
[[File:conditional_generation.PNG]]<br />
<br />
The authors qualitatively assessed the reconstructed images <math>S’</math> given input sketch <math>S</math> using different values for the temperature hyperparameter <math>\tau</math>. The figure above shows the results for different values of <math>\tau</math> starting with 0.01 at the far left and increasing to 1.0 on the far right. Interestingly, sketches with extra features like a cat with 3 eyes are reproduced as a sketch of a cat with two eyes and sketches of object of a different class such as a toothbrush are reproduced as a sketch of a cat that maintains several of the input toothbrush sketches features.<br />
<br />
=== Latent Space Interpolation ===<br />
[[File:latent_space_interp.PNG]]<br />
<br />
The latent space vectors <math>z</math> have few “gaps” between encoded latent space vectors due to the enforcement of a Guassian prior. This allowed the authors to do simple arithmetic on the latent vectors from different sketches and produce logical resulting images in the same style as latent space arithmetic on Word2Vec vectors.<br />
<br />
=== Sketch Drawing Analogies ===<br />
Given the latent space arithmetic possible, it was found that features of a sketch could be added after some sketch input was encoded. For example, a drawing of a cat with a body could be produced by providing the network with a drawing of a cat’s head, and then adding a latent vector to the embedding layer that represents “body”. As an example, this “body” vector might be produced by taking a drawing of a pig with a body and subtracting a vector representing the pigs head.<br />
<br />
=== Predicting Different Endings of Incomplete Sketches ===<br />
[[File:predicting_endings.PNG]]<br />
<br />
Using the decoder RNN only, it is possible to finish sketches by conditioning future vector line predictions on the previous points. To do this, the decoder RNN is first used to encode some existing points into the hidden state of the decoder network and then generating the remaining points of the sketch.<br />
<br />
= Applications and Future Work =<br />
Sketch-rnn may enable the production of several creative applications. These might include applications that help suggest ways an artist could finish a sketch, enable artists to explore latent space arithmetic to find interesting outputs given different sketch inputs, or allow the production of multiple different sketches of some object as a purely generative application. The authors suggest that providing some conditional sketch of an object to a model designed to produce output of a different class might be useful for producing sketches that morph the two different object classes into one sketch. For example, the image below was trained on drawing cats, but a chair was used as the input. This results in a chair looking cat.<br />
<br />
[[File:cat-chair.png]]<br />
<br />
Sketch-rnn may also be useful as a teaching tool to help people learn how to draw, especially if it were to be trained on higher quality images. Teaching tools might suggest to students how to procede to finish a sketch, or intake low fidelity sketches to produce a higher quality and “more coherent” output sketch.<br />
<br />
Finally the authors suggest that combining this model with another that produces photorealistic pixel-based images using sketch input, such as Pix2Pix may be an interesting direction for future research. In this case, the output from the sketch-rnn model would be used as input for Pix2Pix and could produce photorealistic images given some crude sketch from a user.<br />
<br />
= Conclusion =<br />
The authors presented sketch-rnn, a RNN model for modelling and generating vector-based sketch drawings. The VAE inspired architecture allows sampling the latent space to generate new drawings and also allows for applications that use latent space arithmetic in the style of Word2Vec to produce new drawings given operations on embedded sketch vectors. The authors also made available a large dataset of sketch drawings in the hope of encouraging more research in the area of vector-based image modelling.<br />
<br />
= Criticisms =<br />
The paper produces an interesting model that can effectively model vector-based images instead of traditional pixel-based images. This is an interesting problem because vector based images require producing a new way to encode the data. While the results from this paper are interesting, most of the techniques used are borrowed ideas from Variational Autoencoders and the main architecture is not terribly groundbreaking. <br />
<br />
One novel part about the architecture presented was the way the authors used GMMs in the decoder network. While this was interesting and seemed to allow the authors to produce different outputs given the same latent vector input <math>z</math> by manipulating the <math>\tau</math> hyperparameter, it was not that clear in the article why GMMs were used instead of a more simple architecture. Much time was spent explaining basics about GMM parameters like <math>\mu</math> and <math>\sigma</math>, but there was comparatively little explanation about how points were actually sampled from these mixture models.<br />
<br />
Finally, the authors gloss somewhat over how they were able to encode previous sketch points using only the decoder network into the hidden state of the decoder RNN to finish partially finished sketches. I can only assume that some kind of back-propagation was used to encode the expected sketch points into the hidden parameters of the decoder, but no explanation was given in the paper.<br />
<br />
= Source =<br />
<br />
Ha, D., & Eck, D. A neural representation of sketch drawings. In Proc. International Conference on Learning Representations (2018).<br />
<br />
Ha, David, Andrew Dai, and Quoc V. Le. "Hypernetworks." arXiv preprint arXiv:1609.09106 (2016).</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=A_Neural_Representation_of_Sketch_Drawings&diff=33104A Neural Representation of Sketch Drawings2018-03-11T00:24:36Z<p>Pa2forsy: Details on HyperLSTMs</p>
<hr />
<div>= Introduction =<br />
<br />
There have been many recent advances in neural generative models for low resolution pixel-based images. Humans, however, do not see the world in a grid of pixels and more typically communicate drawings of the things we see using a series of pen strokes that represent components of objects. These pen strokes are similar to the way vector-based images store data. This paper proposes a new method for creating conditional and unconditional generative models for creating these kinds of vector sketch drawings based on recurrent neural networks (RNNs). The paper explores many applications of these kinds of models, especially creative applications and makes available their unique dataset of vector images.<br />
<br />
= Related Work =<br />
<br />
Previous work related to sketch drawing generation includes methods that focussed primarily on converting input photographs into equivalent vector line drawings. Image generating models using neural networks also exist but focussed more on generation of pixel-based imagery. Some recent work has focussed on handwritten character generation using RNNs and Mixture Density Networks to generate continuous data points. This work has been extended somewhat recently to conditionally and unconditionally generate handwritten vectorized Chinese Kanji characters by modeling them as a series of pen strokes. Furthermore, this paper builds on work that employed Sequence-to-Sequence models with Variational Autencoders to model English sentences in latent vector space.<br />
<br />
One of the limiting factors for creating models that operate on vector datasets has been the dearth of publicly available data. Previously available datasets include Sketch, a set of 20K vector drawings; Sketchy, a set of 70K vector drawings; and ShadowDraw, a set of 30K raster images with extracted vector drawings.<br />
<br />
= Methodology =<br />
<br />
=== Dataset ===<br />
<br />
The “QuickDraw” dataset used in this research was assembled from 75K user drawings extracted from the game “Quick, Draw!” where users drew objects from one of hundreds of classes in 20 seconds or less. The dataset is split into 70K training samples and 2.5K validation and test samples each and represents each sketch a set of “pen stroke actions”. Each action is provided as a vector in the form <math>(\Delta x, \Delta y, p_{1}, p_{2}, p_{3})</math>. For each vector, <math>\Delta x</math> and <math>\Delta y</math> give the movement of the pen from the previous point, with the initial location being the origin. The last three vector elements are a one-hot representation of pen states; <math>p_{1}</math> indicates that the pen is down and a line should be drawn between the current point and the next point, <math>p_{2}</math> indicates that the pen is up and no line should be drawn between the current point and the next point, and <math>p_{3}</math> indicates that the drawing is finished and subsequent points and the current point should not be drawn.<br />
<br />
=== Sketch-RNN ===<br />
[[File:sketchrnn.PNG]]<br />
<br />
The model is a Sequence-to-Sequence Variational Autoencoder (VAE). The encoder model is a symmetric and parallel set of two RNNs that individually process the sketch drawings in forward and reverse order, respectively. The hidden state produced by each encoder model is then concatenated into a single hidden state <math>h</math>. <br />
<br />
The concatenated hidden state <math>h</math> is then projected into two vectors <math>\mu</math> and <math>\hat{\sigma}</math> each of size <math>N_{z}</math> using a fully connected layer. <math>\hat{\sigma}</math> is then converted into a non-negative standard deviation parameter <math>\sigma</math> using an exponential operator. These two parameters <math>\mu</math> and <math>\sigma</math> are then used along with an IID Gaussian vector distributed as <math>\mathcal{N}(0, I)</math> of size <math>N_{z}</math> to construct a random vector <math>z \in ℝ^{N_{z}}</math>, similar to the method used for VAE:<br />
\begin{align}<br />
\mu = W_{\mu}h + b_{mu}\textrm{, }\hat{\sigma} = W_{\sigma}h + b_{\sigma}\textrm{, }\sigma = exp\bigg{(}\frac{\hat{\sigma}}{2}\bigg{)}\textrm{, }z = \mu + \sigma \odot \mathcal{N}(0,I)<br />
\end{align}<br />
<br />
The decoder model is another RNN that samples output sketches from the latent vector <math>z</math>. The initial hidden states of each recurrent neuron are determined using <math>[h_{0}, c_{0}] = tanh(W_{z}z + b_{z})</math>. Each step of the decoder RNN accepts the previous point <math>S_{i-1}</math> and the latent vector <math>z</math> as concatenated input. The initial point given is the origin point with pen state down. The output at each step are the parameters for a probability distribution of the next point <math>S_{i}</math>. Outputs <math>\Delta x</math> and <math>\Delta y</math> are modelled using a Gaussian Mixture Model (GMM) with M normal distributions and output pen states <math>(q_{1}, q_{2}, q_{3})</math> modelled as a categorical distribution with one-hot encoding.<br />
\begin{align}<br />
P(\Delta x, \Delta y) = \sum_{j=1}^{M}\Pi_{j}\mathcal{N}(\Delta x, \Delta y | \mu_{x, j}, \mu_{y, j}, \sigma_{x, j}, \sigma_{y, j}, \rho_{xy, j})\textrm{, where }\sum_{j=1}^{M}\Pi_{j} = 1<br />
\end{align}<br />
<br />
For each of the M distributions in the GMM, parameters <math>\mu</math> and <math>\sigma</math> are output for both the x and y locations signifying the mean location of the next point and the standard deviation, respectively. Also output from each model is parameter <math>\rho_{xy}</math> signifying correlation of each bivariate normal distribution. An additional vector <math>\Pi</math> is an output giving the mixture weights for the GMM. The output <math>S_{i}</math> is determined from each of the mixture models using softmax sampling from these distributions.<br />
<br />
One of the key difficulties in training this model is the highly imbalanced class distribution of pen states. In particular, the state that signifies a drawing is complete will only appear one time per each sketch and is difficult to incorporate into the model. In order to have the model stop drawing, the authors introduce a hyperparameter that limits the number of points per drawing to being no more than <math>N_{max}</math>, after which all output states form the model are set to (0, 0, 0, 0, 1) to force the drawing to stop.<br />
<br />
To sample from the model, the parameters required by the GMM and categorical distributions are generated at each time step and the model is sampled until a “stop drawing” state appears or the time state reaches time <math>N_{max}</math>. The authors also introduce a “temperature” parameter <math>\tau</math> that controls the randomness of the drawings by modifying the pen states, model standard deviations, and mixture weights as follows:<br />
<br />
\begin{align}<br />
\hat{q}_{k} \rightarrow \frac{\hat{q}_{k}}{\tau}\textrm{, }\hat{\Pi}_{k} \rightarrow \frac{\hat{\Pi}_{k}}{\tau}\textrm{, }\sigma^{2}_{x} \rightarrow \sigma^{2}_{x}\tau\textrm{, }\sigma^{2}_{y} \rightarrow \sigma^{2}_{y}\tau<br />
\end{align}<br />
<br />
This parameter <math>\tau</math> lies in the range (0, 1]. As the parameter approaches 0, the model becomes more deterministic and always produces the point locations with the maximum likelihood for a given timestep.<br />
<br />
=== Unconditional Generation ===<br />
The authors also explored unconditional generation of sketch drawings by only training the decoder RNN module. To do this, the initial hidden states of the RNN were set to 0, and only vectors from the drawing input are used as input without any conditional latent variable <math>z</math>. Different sketches are sampled from the network by only varying the temperature parameter <math>\tau</math> between 0.2 and 0.9<br />
<br />
=== Training ===<br />
The training procedure follows the same approach as training for VAE and uses a loss function that consists of the sum of Reconstruction Loss <math>L_{R}</math> and KL Divergence Loss <math>L_{KL}</math>. The reconstruction loss term is composed of two terms; <math>L_{s}</math>, which tries to maximize the log-likelihood of the generated probability distribution explaining the training data <math>S</math> and <math>L_{p}</math> which is the log loss of the pen state terms.<br />
\begin{align}<br />
L_{s} = -\frac{1}{N_{max}}\sum_{i=1}^{N_{S}}log\bigg{(}\sum_{j=1}^{M}\Pi_{j,i}\mathcal{N}(\Delta x_{i},\Delta y_{i} | \mu_{x,j,i},\mu_{y,j,i},\sigma_{x,j,i},\sigma_{y,j,i},\rho_{xy,j,i})\bigg{)}<br />
\end{align}<br />
\begin{align}<br />
L_{p} = -\frac{1}{N_{max}}\sum_{i=1}^{N_{max}} \sum_{k=1}^{3}p_{k,i}log(q_{k,i})<br />
\end{align}<br />
\begin{align}<br />
L_{R} = L_{s} + L{p}<br />
\end{align}<br />
<br />
The KL divergence loss <math>L_{KL}</math> measures the difference between the latent vector <math>z</math> and an IID Gaussian distribution with 0 mean and unit variance. This term, normalized by the number of dimensions <math>N_{z}</math> is calculated as:<br />
\begin{align}<br />
L_{KL} = -\frac{1}{2N_{z}}\big{(}1 + \hat{\sigma} - \mu^{2} – exp(\hat{\sigma})\big{)}<br />
\end{align}<br />
<br />
The loss for the entire model is thus the weighted sum:<br />
\begin{align}<br />
Loss = L_{R} + w_{KL}L_{KL}<br />
\end{align}<br />
<br />
The value of the weight parameter <math>w_{KL}</math> has the effect that as <math>w_{KL} \rightarrow 0</math>, there is a loss in ability to enforce a prior over the latent space and the model assumes the form of a pure autoencoder.<br />
<br />
= Experiments =<br />
The authors trained multiple conditional and unconditional models using varying values of <math>w_{KL}</math> and recorded the different <math>L_{R}</math> and <math>L_{KL}</math> values at convergence. The network used LSTM as it’s encoder RNN and HyperLSTM as the decoder network. The HyperLSTM model was used for decoding because it has a history of being useful in sequence generation tasks. (A HyperLSTM consists of two coupled LSTMS: an auxiliary LSTM and a main LSTM. At every time step, the auxiliary LSTM reads the previous hidden state and the current input vector, and computes an intermediate vector <math display="inline"> z </math>. The weights of the main LSTM used in the current time step are then a learned function of this intermediate vector <math display="inline"> z </math>. That is, the weights of the main LSTM are allowed to vary between time steps as a function of the output of the auxiliary LSTM. See Ha et al. (2016) for details)<br />
<br />
=== Conditional Reconstruction ===<br />
[[File:conditional_generation.PNG]]<br />
<br />
The authors qualitatively assessed the reconstructed images <math>S’</math> given input sketch <math>S</math> using different values for the temperature hyperparameter <math>\tau</math>. The figure above shows the results for different values of <math>\tau</math> starting with 0.01 at the far left and increasing to 1.0 on the far right. Interestingly, sketches with extra features like a cat with 3 eyes are reproduced as a sketch of a cat with two eyes and sketches of object of a different class such as a toothbrush are reproduced as a sketch of a cat that maintains several of the input toothbrush sketches features.<br />
<br />
=== Latent Space Interpolation ===<br />
[[File:latent_space_interp.PNG]]<br />
<br />
The latent space vectors <math>z</math> have few “gaps” between encoded latent space vectors due to the enforcement of a Guassian prior. This allowed the authors to do simple arithmetic on the latent vectors from different sketches and produce logical resulting images in the same style as latent space arithmetic on Word2Vec vectors.<br />
<br />
=== Sketch Drawing Analogies ===<br />
Given the latent space arithmetic possible, it was found that features of a sketch could be added after some sketch input was encoded. For example, a drawing of a cat with a body could be produced by providing the network with a drawing of a cat’s head, and then adding a latent vector to the embedding layer that represents “body”. As an example, this “body” vector might be produced by taking a drawing of a pig with a body and subtracting a vector representing the pigs head.<br />
<br />
=== Predicting Different Endings of Incomplete Sketches ===<br />
[[File:predicting_endings.PNG]]<br />
<br />
Using the decoder RNN only, it is possible to finish sketches by conditioning future vector line predictions on the previous points. To do this, the decoder RNN is first used to encode some existing points into the hidden state of the decoder network and then generating the remaining points of the sketch.<br />
<br />
= Applications and Future Work =<br />
Sketch-rnn may enable the production of several creative applications. These might include applications that help suggest ways an artist could finish a sketch, enable artists to explore latent space arithmetic to find interesting outputs given different sketch inputs, or allow the production of multiple different sketches of some object as a purely generative application. The authors suggest that providing some conditional sketch of an object to a model designed to produce output of a different class might be useful for producing sketches that morph the two different object classes into one sketch. For example, the image below was trained on drawing cats, but a chair was used as the input. This results in a chair looking cat.<br />
<br />
[[File:cat-chair.png]]<br />
<br />
Sketch-rnn may also be useful as a teaching tool to help people learn how to draw, especially if it were to be trained on higher quality images. Teaching tools might suggest to students how to procede to finish a sketch, or intake low fidelity sketches to produce a higher quality and “more coherent” output sketch.<br />
<br />
Finally the authors suggest that combining this model with another that produces photorealistic pixel-based images using sketch input, such as Pix2Pix may be an interesting direction for future research. In this case, the output from the sketch-rnn model would be used as input for Pix2Pix and could produce photorealistic images given some crude sketch from a user.<br />
<br />
= Conclusion =<br />
The authors presented sketch-rnn, a RNN model for modelling and generating vector-based sketch drawings. The VAE inspired architecture allows sampling the latent space to generate new drawings and also allows for applications that use latent space arithmetic in the style of Word2Vec to produce new drawings given operations on embedded sketch vectors. The authors also made available a large dataset of sketch drawings in the hope of encouraging more research in the area of vector-based image modelling.<br />
<br />
= Criticisms =<br />
The paper produces an interesting model that can effectively model vector-based images instead of traditional pixel-based images. This is an interesting problem because vector based images require producing a new way to encode the data. While the results from this paper are interesting, most of the techniques used are borrowed ideas from Variational Autoencoders and the main architecture is not terribly groundbreaking. <br />
<br />
One novel part about the architecture presented was the way the authors used GMMs in the decoder network. While this was interesting and seemed to allow the authors to produce different outputs given the same latent vector input <math>z</math> by manipulating the <math>\tau</math> hyperparameter, it was not that clear in the article why GMMs were used instead of a more simple architecture. Much time was spent explaining basics about GMM parameters like <math>\mu</math> and <math>\sigma</math>, but there was comparatively little explanation about how points were actually sampled from these mixture models.<br />
<br />
Finally, the authors gloss somewhat over how they were able to encode previous sketch points using only the decoder network into the hidden state of the decoder RNN to finish partially finished sketches. I can only assume that some kind of back-propagation was used to encode the expected sketch points into the hidden parameters of the decoder, but no explanation was given in the paper.<br />
<br />
= Source =<br />
<br />
Ha, D., & Eck, D. A neural representation of sketch drawings. In Proc. International Conference on Learning Representations (2018).</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Understanding_Image_Motion_with_Group_Representations&diff=33078Understanding Image Motion with Group Representations2018-03-09T15:51:23Z<p>Pa2forsy: </p>
<hr />
<div>== Introduction ==<br />
Motion perception is a key component of computer vision. It is critical to problems such as optical flow and visual odometry, where a sequence of images are used to calculate either the pixel level (local) motion or the motion of the entire scene (global). The smooth image transformation caused by camera motion is a subspace of all position image transformations. Here, we are interested in realistic transformation caused by motion, therefore unrealistic motion caused by say, face swapping, are not considered. <br />
<br />
Supervised learning of 3D motion is challenging since explicit motion labels are no trivial to obtain. The proposed learning method does not need label data. Instead, the method constraints learning by using the properties of motion space. The paper presents a general model of visual motion, and how the motion space properties of associativity and can be used to constrain the learning of a deep neural network. The results show evidence that the learned model captions motion in both 2D and 3D settings.<br />
<br />
[[File:paper13_fig1.png|650px|center|]]<br />
<br />
== Related Work ==<br />
The most common global representations of motion are from structure from motion (SfM) and simultaneous localization and mapping (SLAM), which represents poses in special Euclidean group <math> SE(3) </math> to represent a sequence of motions. However, these cannot be used to represent non-rigid or independent motions. Another approache to representing motion is spatiotemporal features (STFs), which are flexible enough to represent non-rigid motions.<br />
<br />
There are also works using CNN’s to learn optical flow using brightness constancy assumptions, and/or photometric local constraints. Works on stereo depth estimation using learning has also shown results. Regarding to image sequences, there are works on shuffling the order of images to learn representations of its contents, as well as learning representations equivariant to the egomotion of the camera. <br />
<br />
== Approach ==<br />
The proposed method is based on the observation that 3D motions, equipped with composition forms a group. By learning the underlying mapping that captures the motion transformations, we are approximating latent motion of the scene.The method is designed to capture group associativity and invertibility.<br />
<br />
Consider a latent structure space <math>S</math>, element of the structure space generates images via projection <math>\pi:S\rightarrow I</math>, latent motion space <math>M</math> which is some closed subgroup of the set of homeomorphism on <math>S</math>. For <math>s \in S</math>, a continuous motion sequence <math> \{m_t \in M | t \geq 0\} </math> generates continous image sequence <math> \{i_t \in I | t \geq 0\} </math> where <math> i_t=\pi(m_t(s)) </math>. Writing this as a hidden Markov model gives <math> i_t=\pi(m_{\Delta t}(s_{t-1}))) </math> where the current state is based on the change from the previous. Since <math> M </math> is a closed group on <math> S </math>, it is associative, has inverse, and contains idenity. <math> SE(3) </math> is an exmaple of this. To be more specific, the latent structure of a scene from rigid image motion could be modelled by a point cloud with a motion space <math>M=SE(3)</math>, where rigid image motion can be produced by a camera translating and rotating through a rigid scene in 3D. When a scene has N rigid bodies, the motion space can be represented as <math>M=[SE(3)]^N</math>.<br />
<br />
=== Learning Motion by Group Properties ===<br />
The goal is to learn a function <math> \Phi : I \times I \rightarrow \overline{M} </math>, <math> \overline{M} </math> indicating representation of <math> M </math>, as well as the composition operator <math> \diamond : \overline{M} \rightarrow \overline{M} </math> that represents composition in <math> M </math>. For all sequences, it is assumed <math> t_0 < t_1 < t_2 ... </math> <br />
# Associativity: <math> \Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_2}, I_{t_3}) = (\Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_1}, I_{t_2})) \diamond \Phi(I_{t_2}, I_{t_3}) = \Phi(I_{t_0}, I_{t_1}) \diamond (\Phi(I_{t_1}, I_{t_2}) \diamond \Phi(I_{t_2}, I_{t_3})) = \Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_1}, I_{t_3}) </math> <br />
# Has Identity: <math> \Phi(I_{t_0}, I_{t_1}) \diamond e = \Phi(I_{t_0}, I_{t_1}) = e \diamond \Phi(I_{t_0}, I_{t_1}) </math> and <math> e=\Phi(I_{t}, I_{t}) \forall t </math> <br />
# Invertibility: <math> \Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_1}, I_{t_0}) = e </math><br />
An embedding loss is used to approximately enforce associativity and invertibility among subsequences sampled from image sequence. Associativity is encouraged by pushing sequences with the same final motion but different transitions to the same representation. Invertibility is encouraged by pushing sequences corresponding to the same motion with but in opposite directions away from each other, as well as pushing all loops to the same representation. Uniqueness of the identity is encouraged by pushing loops away from non-identity representations. Loops from different sequences are also pushed to the same representation (the identity).<br />
<br />
These constraints are true to any type of transformation resulting from image motion. This puts little restriction on the learning problems and allows all features relevant to the motion structure to be captured. <br />
<br />
Also with this method, it is possible multiple representations <math> \overline{M} </math> can be learned from a single <math> M </math>, thus the learned representation is not necessary unique. In addition, the scenes are not expected to have rapid changing content, scene cuts, or long-term occlusions.<br />
<br />
=== Sequence Learning with Neural Networks ===<br />
The functions <math> \Phi </math> and <math> \diamond </math> are approximated by CNN and RNN, respectively. LSTM is used for RNN. The input to the network is a sequence of images <math> I_t = \{I_1,...,I_t\} </math>. The CNN processes pairs of images are intermediate representations, and the LSTM operates over the sequence of CNN outputs to produce and embedding sequence <math> R_t = \{R_{1,2},...,R_{t-1,t}\} </math>. Only the embedding at the final timstep is used for loss. The network is trained to minimize a hinge loss with respect to embeddings to pairs of sequences. The cost function is:<br />
<br />
<center><math>L(R^1,R^2) = \begin{cases} d(R^1,R^2), & \text{if positive pair} \\ max(0, m - d(R^1,R^2)), & \text{if negative pair} \end{cases}</math></center><br />
<center><math> d_{cosine}(R^1,R^2)=1-\frac{\langle R^1,R^2 \rangle}{\lVert R^1 \rVert \lVert R^2 \rVert} </math></center><br />
<br />
where <math>d(R^1,R^2)</math> measure the distance between the embeddings of two sequences used for training selected to be cosine distance, <math> m </math> is a fixed margin selected to be 0.5. Positive pair are training example where two sequences have the same final motion, negative pairs are training examples where two sequences have the exact opposite final motion. Using L2 distances yields similar results as cosine distances.<br />
<br />
Each training sequence is composed into 6 subsequences: two forward, two backward, and two identity. To prevent the network from only looking at static differences, subsequence pairs are sampled such that they have the same start and end frames but different motions in between. Sequences of varying lengths are also used to generalize motion on different temporal scale. Training the network with only one input images per timestep is also tried, but consistently yielded work results than image pairs.<br />
<br />
[[File:paper13_fig2.png|650px|center|]]<br />
<br />
Overall, training with image pairs resulted in lower error than training with just single images. This is demonstrated in the below table.<br />
<br />
<br />
[[File:table.png|700px|center|]]<br />
<br />
== Experimentation ==<br />
Trained network using rotated and translated MNIST dataset as well as KITTI dataset. <br />
* Used Torch<br />
* Used Adam for optimization, decay schedule of 30 epochs, learning rate chosen by random serach<br />
* 50-60 batch size for MNIST, 25-30 batch size for KITTI<br />
* Dilated convolution with Relu and batch normalization<br />
* Two LSTM cell per layer 256 hidden units each<br />
* Sequence length of 3-5 images<br />
<br />
=== Rigid Motion in 2D ===<br />
* MNIST data rotated <math>[0, 360)</math> degrees and translated <math>[-10, 10] </math> pixels, i.e. <math>SE(2)</math> transformations<br />
* Visualized the representation using t-SNE<br />
** Clear clustering by translation and rotation but not object classes<br />
** Suggests the representation captures the motion properties in the dataset, but is independent of image contents<br />
* Visualized the image-conditioned saliency maps<br />
** Take derivative of the network output respect to the map<br />
** The area that has the highest gradient means that part contributes the most to the output<br />
** The resulting salient map strongly resembles spatiotemporal energy filters of classical motion processing<br />
** Suggests the network is learn the right motion structure<br />
<br />
[[File:paper13_fig3.png|700px|center|]]<br />
<br />
=== Real World Motion in 3D ===<br />
* Uses KITTI dataset collected on a car driving through roads in Germany<br />
* On a separate dataset with ground truth camera pose, linearly regress the representation to the ground truth<br />
** The result is compared against self supervised flow algorithm Yu et al.(2016) after the output from the flow algorithm is downsampled, then feed through PCA, then regressed against the camera motion<br />
** The data shows it performs not as well as the supervised algorithm, but consistent better than chance (guessing the mean value)<br />
** Shows the method is able to capture dominant motion structure<br />
* Test performance on interpolation task<br />
** Check <math>R([I_1,I_T])</math> against <math>R([I_1, I_m, I_T])</math>, <math>R([I_1, I_{IN}, I_T])</math>, and <math>R([I_1, I_{OUT}, I_T])</math><br />
** Test how sensitive the network is to deviations from unnatural motion<br />
** High errors <math>\gg 1</math> means the network can distinguish between realistic and unrealistic motion<br />
** In order to do this, the distance between the embeddings of the frame sequences of the first and last frame <math>R([I_1,I_T])</math> and of the first, middle, and last frame <math>R([I_1, I_m, I_T])</math> is computed. This distance is compared with the distance when the middle frame of the second embedding is changed to a frame that is visually similar (inside sequence): <math>R([I_1, I_{IN}, I_T])</math> and one that is visually dissimilar (outside sequence): <math>R([I_1, I_{OUT}, I_T])</math>. The results are shown in Table 3. The embedding distance method is compared to the euclidean distance which is defined as the mean pixel distance between the test frame and <math>{I_1,I_T}</math>, whichever is smaller. It can be seen from the results that the embedding distance of the true frame is significantly lower than other frames. This means that the embedding distance used in the network is more sensitive to any atypical motions of the scenes. <br />
* Visualized saliency maps<br />
** Highs objects moving in the background, and motion of the car in the foreground<br />
** Suggests the method can be used for tracking as well<br />
<br />
[[File:paper13_tab2.png|700px|center|]]<br />
<br />
[[File:paper13_fig4.png|700px|center|]]<br />
<br />
[[File:paper13_fig5.png|700px|center|]]<br />
<br />
[[File:table3_motion.PNG|700px|center|]]<br />
<br />
== Conclusion ==<br />
The author presented a new model of motion and method for learning motion representations. It is shown that enforcing group properties can learn motion representations that is able to generalize between scenes with disparate content. The results can be useful for navigation, prediction, and other behavioral tasks relying on motion. Due to the fact that this method does not require labelled data, it can be applied to useful for large variety of tasks.<br />
<br />
== Criticism ==<br />
Although this method does not require any labelled data, it is still learning by supervision through defined constraints. The idea of training using unlabelled data is interesting and it does have meaningful practical application. Unfortunately, the author did not provide convincing experimental results. Results from motion estimation problems are typically compared against ground truth data for their accuracy. The author performed experiments on transformed MNIST data and KITTI data. The MNIST data is transformed by the author, thus the ground truth is readily available. However the author only claimed the validity of the results through indirect means of using t-SNE and saliency map visualization. For the KITTI dataset, the author regressed the representations against ground truth for some mapping from the network output to some physical motion representation. Again, the results again compared only indirectly against ground truth. Such experimentation made the method hardly convincing and applicable to real world applications. In addition, the network does not output motion representations with physical meanings, make the proposed method useless for any real world applications.<br />
<br />
Another criticism is that the group-properties constraint the authors impose is too weak. Any set consisting of functions, their inverses, and the identity forms a group. While physical motions are one example of such a group, there are many valid groups that do not represent any coherent physical motions.<br />
== References ==<br />
Jaegle, A. (2018). Understanding image motion with group representations . ICLR. Retrieved from https://openreview.net/pdf?id=SJLlmG-AZ.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Understanding_Image_Motion_with_Group_Representations&diff=33077Understanding Image Motion with Group Representations2018-03-09T15:19:55Z<p>Pa2forsy: </p>
<hr />
<div>== Introduction ==<br />
Motion perception is a key component of computer vision. It is critical to problems such as optical flow and visual odometry, where a sequence of images are used to calculate either the pixel level (local) motion or the motion of the entire scene (global). The smooth image transformation caused by camera motion is a subspace of all position image transformations. Here, we are interested in realistic transformation caused by motion, therefore unrealistic motion caused by say, face swapping, are not considered. <br />
<br />
Supervised learning of 3D motion is challenging since explicit motion labels are no trivial to obtain. The proposed learning method does not need label data. Instead, the method constraints learning by using the properties of motion space. The paper presents a general model of visual motion, and how the motion space properties of associativity and can be used to constrain the learning of a deep neural network. The results show evidence that the learned model captions motion in both 2D and 3D settings.<br />
<br />
[[File:paper13_fig1.png|650px|center|]]<br />
<br />
== Related Work ==<br />
The most common global representations of motion are from structure from motion (SfM) and simultaneous localization and mapping (SLAM), which represents poses in special Euclidean group <math> SE(3) </math> to represent a sequence of motions. However, these cannot be used to represent non-rigid or independent motions. Another approache to representing motion is spatiotemporal features (STFs), which are flexible enough to represent non-rigid motions.<br />
<br />
There are also works using CNN’s to learn optical flow using brightness constancy assumptions, and/or photometric local constraints. Works on stereo depth estimation using learning has also shown results. Regarding to image sequences, there are works on shuffling the order of images to learn representations of its contents, as well as learning representations equivariant to the egomotion of the camera. <br />
<br />
== Approach ==<br />
The proposed method is based on the observation that 3D motions, equipped with composition forms a group. By learning the underlying mapping that captures the motion transformations, we are approximating latent motion of the scene.The method is designed to capture group associativity and invertibility.<br />
<br />
Consider a latent structure space <math>S</math>, element of the structure space generates images via projection <math>\pi:S\rightarrow I</math>, latent motion space <math>M</math> which is some closed subgroup of the set of homeomorphism on <math>S</math>. For <math>s \in S</math>, a continuous motion sequence <math> \{m_t \in M | t \geq 0\} </math> generates continous image sequence <math> \{i_t \in I | t \geq 0\} </math> where <math> i_t=\pi(m_t(s)) </math>. Writing this as a hidden Markov model gives <math> i_t=\pi(m_{\Delta t}(s_{t-1}))) </math> where the current state is based on the change from the previous. Since <math> M </math> is a closed group on <math> S </math>, it is associative, has inverse, and contains idenity. <math> SE(3) </math> is an exmaple of this. To be more specific, the latent structure of a scene from rigid image motion could be modelled by a point cloud with a motion space <math>M=SE(3)</math>, where rigid image motion can be produced by a camera translating and rotating through a rigid scene in 3D. When a scene has N rigid bodies, the motion space can be represented as <math>M=[SE(3)]^N</math>.<br />
<br />
=== Learning Motion by Group Properties ===<br />
The goal is to learn a function <math> \Phi : I \times I \rightarrow \overline{M} </math>, <math> \overline{M} </math> indicating representation of <math> M </math>, as well as the composition operator <math> \diamond : \overline{M} \rightarrow \overline{M} </math> that represents composition in <math> M </math>. For all sequences, it is assumed <math> t_0 < t_1 < t_2 ... </math> <br />
# Associativity: <math> \Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_2}, I_{t_3}) = (\Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_1}, I_{t_2})) \diamond \Phi(I_{t_2}, I_{t_3}) = \Phi(I_{t_0}, I_{t_1}) \diamond (\Phi(I_{t_1}, I_{t_2}) \diamond \Phi(I_{t_2}, I_{t_3})) = \Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_1}, I_{t_3}) </math> <br />
# Has Identity: <math> \Phi(I_{t_0}, I_{t_1}) \diamond e = \Phi(I_{t_0}, I_{t_1}) = e \diamond \Phi(I_{t_0}, I_{t_1}) </math> and <math> e=\Phi(I_{t}, I_{t}) \forall t </math> <br />
# Invertibility: <math> \Phi(I_{t_0}, I_{t_1}) \diamond \Phi(I_{t_1}, I_{t_0}) = e </math><br />
An embedding loss is used to approximately enforce associativity and invertibility among subsequences sampled from image sequence. Associativity is encouraged by pushing sequences with the same final motion but different transitions to the same representation. Invertibility is encouraged by pushing sequences corresponding to the same motion with but in opposite directions away from each other, as well as pushing all loops to the same representation. Uniqueness of the identity is encouraged by pushing loops away from non-identity representations. Loops from different sequences are also pushed to the same representation (the identity).<br />
<br />
These constraints are true to any type of transformation resulting from image motion. This puts little restriction on the learning problems and allows all features relevant to the motion structure to be captured. <br />
<br />
Also with this method, it is possible multiple representations <math> \overline{M} </math> can be learned from a single <math> M </math>, thus the learned representation is not necessary unique. In addition, the scenes are not expected to have rapid changing content, scene cuts, or long-term occlusions.<br />
<br />
=== Sequence Learning with Neural Networks ===<br />
The functions <math> \Phi </math> and <math> \diamond </math> are approximated by CNN and RNN, respectively. LSTM is used for RNN. The input to the network is a sequence of images <math> I_t = \{I_1,...,I_t\} </math>. The CNN processes pairs of images are intermediate representations, and the LSTM operates over the sequence of CNN outputs to produce and embedding sequence <math> R_t = \{R_{1,2},...,R_{t-1,t}\} </math>. Only the embedding at the final timstep is used for loss. The network is trained to minimize a hinge loss with respect to embeddings to pairs of sequences. The cost function is:<br />
<br />
<center><math>L(R^1,R^2) = \begin{cases} d(R^1,R^2), & \text{if positive pair} \\ max(0, m - d(R^1,R^2)), & \text{if negative pair} \end{cases}</math></center><br />
<center><math> d_{cosine}(R^1,R^2)=1-\frac{\langle R^1,R^2 \rangle}{\lVert R^1 \rVert \lVert R^2 \rVert} </math></center><br />
<br />
where <math>d(R^1,R^2)</math> measure the distance between the embeddings of two sequences used for training selected to be cosine distance, <math> m </math> is a fixed margin selected to be 0.5. Positive pair are training example where two sequences have the same final motion, negative pairs are training examples where two sequences have the exact opposite final motion. Using L2 distances yields similar results as cosine distances.<br />
<br />
Each training sequence is composed into 6 subsequences: two forward, two backward, and two identity. To prevent the network from only looking at static differences, subsequence pairs are sampled such that they have the same start and end frames but different motions in between. Sequences of varying lengths are also used to generalize motion on different temporal scale. Training the network with only one input images per timestep is also tried, but consistently yielded work results than image pairs.<br />
<br />
[[File:paper13_fig2.png|650px|center|]]<br />
<br />
Overall, training with image pairs resulted in lower error than training with just single images. This is demonstrated in the below table.<br />
<br />
<br />
[[File:table.png|700px|center|]]<br />
<br />
== Experimentation ==<br />
Trained network using rotated and translated MNIST dataset as well as KITTI dataset. <br />
* Used Torch<br />
* Used Adam for optimization, decay schedule of 30 epochs, learning rate chosen by random serach<br />
* 50-60 batch size for MNIST, 25-30 batch size for KITTI<br />
* Dilated convolution with Relu and batch normalization<br />
* Two LSTM cell per layer 256 hidden units each<br />
* Sequence length of 3-5 images<br />
<br />
=== Rigid Motion in 2D ===<br />
* MNIST data rotated <math>[0, 360)</math> degrees and translated <math>[-10, 10] </math> pixels, i.e. <math>SE(2)</math> transformations<br />
* Visualized the representation using t-SNE<br />
** Clear clustering by translation and rotation but not object classes<br />
** Suggests the representation captures the motion properties in the dataset, but is independent of image contents<br />
* Visualized the image-conditioned saliency maps<br />
** Take derivative of the network output respect to the map<br />
** The area that has the highest gradient means that part contributes the most to the output<br />
** The resulting salient map strongly resembles spatiotemporal energy filters of classical motion processing<br />
** Suggests the network is learn the right motion structure<br />
<br />
[[File:paper13_fig3.png|700px|center|]]<br />
<br />
=== Real World Motion in 3D ===<br />
* Uses KITTI dataset collected on a car driving through roads in Germany<br />
* On a separate dataset with ground truth camera pose, linearly regress the representation to the ground truth<br />
** The result is compared against self supervised flow algorithm Yu et al.(2016) after the output from the flow algorithm is downsampled, then feed through PCA, then regressed against the camera motion<br />
** The data shows it performs not as well as the supervised algorithm, but consistent better than chance (guessing the mean value)<br />
** Shows the method is able to capture dominant motion structure<br />
* Test performance on interpolation task<br />
** Check <math>R([I_1,I_T])</math> against <math>R([I_1, I_m, I_T])</math>, <math>R([I_1, I_{IN}, I_T])</math>, and <math>R([I_1, I_{OUT}, I_T])</math><br />
** Test how sensitive the network is to deviations from unnatural motion<br />
** High errors <math>\gg 1</math> means the network can distinguish between realistic and unrealistic motion<br />
** In order to do this, the distance between the embeddings of the frame sequences of the first and last frame <math>R([I_1,I_T])</math> and of the first, middle, and last frame <math>R([I_1, I_m, I_T])</math> is computed. This distance is compared with the distance when the middle frame of the second embedding is changed to a frame that is visually similar (inside sequence): <math>R([I_1, I_{IN}, I_T])</math> and one that is visually dissimilar (outside sequence): <math>R([I_1, I_{OUT}, I_T])</math>. The results are shown in Table 3. The embedding distance method is compared to the euclidean distance which is defined as the mean pixel distance between the test frame and <math>{I_1,I_T}</math>, whichever is smaller. It can be seen from the results that the embedding distance of the true frame is significantly lower than other frames. This means that the embedding distance used in the network is more sensitive to any atypical motions of the scenes. <br />
* Visualized saliency maps<br />
** Highs objects moving in the background, and motion of the car in the foreground<br />
** Suggests the method can be used for tracking as well<br />
<br />
[[File:paper13_tab2.png|700px|center|]]<br />
<br />
[[File:paper13_fig4.png|700px|center|]]<br />
<br />
[[File:paper13_fig5.png|700px|center|]]<br />
<br />
[[File:table3_motion.PNG|700px|center|]]<br />
<br />
== Conclusion ==<br />
The author presented a new model of motion and method for learning motion representations. It is shown that enforcing group properties can learn motion representations that is able to generalize between scenes with disparate content. The results can be useful for navigation, prediction, and other behavioral tasks relying on motion. Due to the fact that this method does not require labelled data, it can be applied to useful for large variety of tasks.<br />
<br />
== Criticism ==<br />
Although this method does not require any labelled data, it is still learning by supervision through defined constraints. The idea of training using unlabelled data is interesting and it does have meaningful practical application. Unfortunately, the author did not provide convincing experimental results. Results from motion estimation problems are typically compared against ground truth data for their accuracy. The author performed experiments on transformed MNIST data and KITTI data. The MNIST data is transformed by the author, thus the ground truth is readily available. However the author only claimed the validity of the results through indirect means of using t-SNE and saliency map visualization. For the KITTI dataset, the author regressed the representations against ground truth for some mapping from the network output to some physical motion representation. Again, the results again compared only indirectly against ground truth. Such experimentation made the method hardly convincing and applicable to real world applications. In addition, the network does not output motion representations with physical meanings, make the proposed method useless for any real world applications.<br />
<br />
== References ==<br />
Jaegle, A. (2018). Understanding image motion with group representations . ICLR. Retrieved from https://openreview.net/pdf?id=SJLlmG-AZ.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Predicting_Floor-Level_for_911_Calls_with_Neural_Networks_and_Smartphone_Sensor_Data&diff=33022stat946w18/Predicting Floor-Level for 911 Calls with Neural Networks and Smartphone Sensor Data2018-03-08T17:55:14Z<p>Pa2forsy: </p>
<hr />
<div>= Introduction =<br />
During emergency 911 calls, knowing the exact position of the victim is crucial to a fast response and a successful rescue. Problems arise when the caller is unable to give their physical position accurately. This can happen for instance when the caller is disoriented, held hostage, or a child is calling on behalf of the victim. GPS sensors on smartphones can provide the rescuers with the geographic location. However GPS fails to give an accurate floor level inside a tall building. Previous works have explored using Wi-Fi signal or beacons placed inside the buildings, but these methods are not self-contained and require prior infrastructure knowledge.<br />
<br />
Fortunately, today’s smartphones are equipped with many more sensors including barometers and magnetometers. Deep learning can be applied to predict floor level based on these sensor readings. <br />
Firstly, an LSTM is trained to classify whether the caller is indoors or outdoors using GPS, RSSI (Received Signal Strength Indication), and magnetometer sensor readings. Next, an unsupervised clustering algorithm is used to predict the floor level depending on the barometric pressure difference. With these two parts working together, a self-contained floor level prediction system can achieve 100% accuracy, without any external prior knowledge.<br />
<br />
= Data Description =<br />
The authors developed an iOS app called Sensory and used it to collect data on an iPhone 6. The following sensor readings were recorded: indoors, created at, session id, floor, RSSI strength, GPS latitude, GPS longitude, GPS vertical accuracy, GPS horizontal accuracy, GPS course, GPS speed, barometric relative altitude, barometric pressure, environment context, environment mean building floors, environment activity, city name, country name, magnet x, magnet y, magnet z, magnet total.<br />
<br />
The indoor-outdoor data has to be manually entered as soon as the user enters or exits a building. To gather the data for floor level prediction, the authors conducted 63 trials among five different buildings throughout New York City. The actual floor level was recorded manually for validation purposes only, since unsupervised learning is being used.<br />
<br />
= Methods =<br />
The proposed method first determines if the user is indoors or outdoors, and detects the instances of transition between them. Then, vertical height is estimated by the barometer. Finally, the exact floor level is predicted through clustering techniques. Indoor/outdoor classification is critical to the working of this method. Once the user is detected to be outdoors, he is assumed to be at the ground level. The vertical height and floor estimation is applied only when the user is indoors. The indoor/outdoor transitions are used to save the barometer readings at the ground level for use as reference pressure.<br />
<br />
=== Indoor/Outdoor Classification === <br />
<br />
An LSTM network is used to solve the indoor-outdoor classification problem. Here is a diagram of the network architecture.<br />
<br />
[[File:lstm.jpg | 500px]]<br />
<br />
Figure 1: LSTM network architecture. A 3-layer LSTM. Inputs are sensor readings for d consecutive time-steps. Target is y = 1 if indoors and y = 0 if outdoors.<br />
<br />
<math> X_i</math> contains a set of <math>d</math> consecutive sensor readings, i.e. <math> X_i = [x_1, x_2,...,x_d] </math>. <math>Y</math> is labelled as 0 for outdoors and 1 for indoors. <math>d</math> is chosen to be 3 by random-search so that <math>X</math> has 3 points <math>X_i = [x_{j-1}, x_j, x_{j+1}]</math> and the middle <math>x_j</math> is used for the <math>y</math> label.<br />
The LSTM contains three layers. Layers one and two have 50 neurons followed by a dropout layer set to 0.2. Layer 3 has two neurons fed directly into a one-neuron feedforward layer with a sigmoid activation function. The input is the sensor readings, and the output is the indoor-outdoor label. The objective function is the cross-entropy between the true label and the prediction.<br />
<br />
\begin{equation}<br />
C(y_i, \hat{y}_i) = \frac{1}{n} \sum_{i=1}^{n} -(y_i log(\hat{y_i}) + (1 - y_i) log(1 - \hat{y_i}))<br />
\label{equation:binCE}<br />
\end{equation}<br />
<br />
The main reason why the neural network is able to predict whether the user is indoors or outdoors is that it learns a pattern of how the walls of buildings interfere with the GPS signals. The LSTM is able to find the pattern in the GPS signal strength in combination with other sensor readings to give an accurate prediction. However, the change in GPS signal does not happen instantaneously as the user walks indoor. Thus, a window of 20 seconds is allowed, and the minimum barometric pressure reading within that window is recorded as the ground floor.<br />
<br />
=== Indoor/Outdoor Transition === <br />
To determine the exact time the user makes an indoor-outdoor transition, two vector masks are convolved across the LSTM predictions.<br />
<br />
\begin{equation}<br />
V_1 = [1, 1, 1, 1, 1, 0, 0, 0, 0, 0]<br />
\end{equation} <br />
<br />
\begin{equation}<br />
V_2 = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]<br />
\end{equation}<br />
<br />
The Jaccard distances is then calculated by the following equation:<br />
<br />
\begin{equation}<br />
J_j = J(s_i, V_j) = \frac{|s_i \cap V_j|}{|s_i| + |V_j| - |s_i \cap V_j|} <br />
\label{equation:Jaccard}<br />
\end{equation}<br />
<br />
If the Jaccard distance to a sub-sequence <math> s_i </math> is greater than the threshold 0.4, it means the transition occurred in the vicinity of the range of the vector mask. These sets of transition windows are then merged if they are close together, and the center of the merged windows are marked as the transition points.<br />
<br />
[[File:FindIOIndexes.png | 700px]]<br />
<br />
=== Vertical Height Estimation === <br />
Once the barometric pressure of the ground floor is known, the user’s current relative altitude can be calculated by the international pressure equation, where <math>m_\Delta</math> is the estimated height, <math> p_1 </math> is the pressure reading of the device, and <math> p_0 </math> is the reference pressure at ground level while transitioning from outdoor to indoor.<br />
<br />
\begin{equation}<br />
m_\Delta = f_{floor}(p_0, p_1) = 44330 (1 - (\frac{p_1}{p_0})^{\frac{1}{5.255}})<br />
\label{equation:baroHeight}<br />
\end{equation}<br />
<br />
=== Floor Estimation === <br />
Given the user’s relative altitude, the floor level can be determined. However, this is not a straightforward task because different buildings have different floor heights, different floor labeling (E.g. not including the 13th floor), and floor heights within the same building can vary from floor to floor. To solve these problems, altitude data collected are clustered into groups. Each cluster represents the approximate altitude of a floor.<br />
<br />
Here is an example of altitude data collected across 41 trials in the Uris Hall building in New York City. Each dashed line represent the center of a cluster.<br />
<br />
[[File:clusters.png | 500px]]<br />
<br />
Figure 2: Distribution of measurements across 41 trials in the Uris Hall building in New York City. A clear size difference is specially noticeable at the lobby. Each dotted line corresponds to an actual floor in the building learned from clustered data-points.<br />
<br />
Here is the algorithm for the floor level prediction.<br />
<br />
[[File:PredictFloor.png | 700px]]<br />
<br />
= Experiments and Results =<br />
Here are the results for the indoor-outdoor classification problem using different machine learning techniques. LSTM has the best performance on the test set.<br />
The LSTM is trained for 24 epochs with a batch size of 128. All the hyper-parameters such as learning rate(0.006), number of layers, d size, number of hidden units and dropout rate were searched through random search algorithm.<br />
<br />
[[File:IOResults.png]]<br />
<br />
The following are the results for the floor level prediction from the 63 collected samples. Results are given as the percent which matched the floor exactly, off by one, or off by more than one. In each column, the left number is the accuracy using a fixed floor height, and the number on the right is the accuracy when clustering was used to calculate a variable floor height. It was found that using the clustering technique produced 100% accuracy on floor predictions. The conclusion from these results is that using building-specific floor heights produces significantly better results.<br />
<br />
[[File:FloorLevelResults.png]]<br />
<br />
Here is the comparison between the estimated floor height and the ground truth in the Uris Hall building.<br />
<br />
[[File:FloorComparison.png]]<br />
<br />
= Criticism =<br />
This paper is an interesting application of deep learning and achieves an outstanding result of 100% accuracy. However, it offers no new theoretical discoveries. The machine learning techniques used are fairly standard. The neural networks used in this paper only contains 3 layers, and the clustering is applied on one-dimensional data. This leads to the question whether deep learning is necessary and suitable for this task.<br />
<br />
It was explained in the paper that there are many cases where the system does not work. Some cases that were mentioned include: buildings with glass walls, delayed GPS signals, <br />
and pressure changes caused by air conditioning. Other examples I can think of are: uneven floors with some area higher than others, floors rarely visited, and tunnels from one building to another. These special cases are not mentioned in the paper.<br />
<br />
Another weakness of the method comes from the clustering technique. It requires a fair bit of training data. The author suggested two approaches. First, the data can be stored in the individual smartphone. This is not realistic as most people do not visit every single floor of every building, even if it is their own apartment buildings. The second approach is to let a central system (emergency department) to collect data from multiple users (which is what the paper’s results are based on). This is a serious violation of personal privacy and should not be legal.<br />
<br />
Aside from all the technical issues, would it be easier to let the rescuers carry a barometer with them and search for the floor with the transmitted pressure reading?</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Predicting_Floor-Level_for_911_Calls_with_Neural_Networks_and_Smartphone_Sensor_Data&diff=33021stat946w18/Predicting Floor-Level for 911 Calls with Neural Networks and Smartphone Sensor Data2018-03-08T17:46:58Z<p>Pa2forsy: </p>
<hr />
<div>= Introduction =<br />
During emergency 911 calls, knowing the exact position of the victim is crucial to a fast response and a successful rescue. Problems arise when the caller is unable to give their physical position accurately. This can happen for instance when the caller is disoriented, held hostage, or a child is calling on behalf of the victim. GPS sensors on smartphones can provide the rescuers with the geographic location. However GPS fails to give an accurate floor level inside a tall building. Previous works have explored using Wi-Fi signal or beacons placed inside the buildings, but these methods are not self-contained and require prior infrastructure knowledge.<br />
<br />
Fortunately, today’s smartphones are equipped with many more sensors including barometers and magnetometers. Deep learning can be applied to predict floor level based on these sensor readings. <br />
Firstly, an LSTM is trained to classify whether the caller is indoors or outdoors using GPS, RSSI (Received Signal Strength Indication), and magnetometer sensor readings. Next, an unsupervised clustering algorithm is used to predict the floor level depending on the barometric pressure difference. With these two parts working together, a self-contained floor level prediction system can achieve 100% accuracy, without any external prior knowledge.<br />
<br />
= Data Description =<br />
The authors developed an iOS app called Sensory and used it to collect data on an iPhone 6. The following sensor readings were recorded: indoors, created at, session id, floor, RSSI strength, GPS latitude, GPS longitude, GPS vertical accuracy, GPS horizontal accuracy, GPS course, GPS speed, barometric relative altitude, barometric pressure, environment context, environment mean building floors, environment activity, city name, country name, magnet x, magnet y, magnet z, magnet total.<br />
<br />
The indoor-outdoor data has to be manually entered as soon as the user enters or exits a building. To gather the data for floor level prediction, the authors conducted 63 trials among five different buildings throughout New York City. The actual floor level was recorded manually for validation purposes only, since unsupervised learning is being used.<br />
<br />
= Methods =<br />
The proposed method first determines if the user is indoors or outdoors, and detects the instances of transition between them. Then, vertical height is estimated by the barometer. Finally, the exact floor level is predicted through clustering techniques. Indoor/outdoor classification is critical to the working of this method. Once the user is detected to be outdoors, he is assumed to be at the ground level. The vertical height and floor estimation is applied only when the user is indoors. The indoor/outdoor transitions are used to save the barometer readings at the ground level for use as reference pressure.<br />
<br />
=== Indoor/Outdoor Classification === <br />
<br />
An LSTM network is used to solve the indoor-outdoor classification problem. Here is a diagram of the network architecture.<br />
<br />
[[File:lstm.jpg | 500px]]<br />
<br />
Figure 1: LSTM network architecture. A 3-layer LSTM. Inputs are sensor readings for d consecutive time-steps. Target is y = 1 if indoors and y = 0 if outdoors.<br />
<br />
<math> X_i</math> contains a set of <math>d</math> consecutive sensor readings, i.e. <math> X_i = [x_1, x_2,...,x_d] </math>. <math>Y</math> is labelled as 0 for outdoors and 1 for indoors. <math>d</math> is chosen to be 3 by random-search so that <math>X</math> has 3 points <math>X_i = [x_{j-1}, x_j, x_{j+1}]</math> and the middle <math>x_j</math> is used for the <math>y</math> label.<br />
The LSTM contains three layers. Layers one and two have 50 neurons followed by a dropout layer set to 0.2. Layer 3 has two neurons fed directly into a one-neuron feedforward layer with a sigmoid activation function. The input is the sensor readings, and the output is the indoor-outdoor label. The objective function is the cross-entropy between the true label and the prediction.<br />
<br />
\begin{equation}<br />
C(y_i, \hat{y}_i) = \frac{1}{n} \sum_{i=1}^{n} -(y_i log(\hat{y_i}) + (1 - y_i) log(1 - \hat{y_i}))<br />
\label{equation:binCE}<br />
\end{equation}<br />
<br />
The main reason why the neural network is able to predict whether the user is indoors or outdoors is that it learns a pattern of how the walls of buildings interfere with the GPS signals. The LSTM is able to find the pattern in the GPS signal strength in combination with other sensor readings to give an accurate prediction. However, the change in GPS signal does not happen instantaneously as the user walks indoor. Thus, a window of 20 seconds is allowed, and the minimum barometric pressure reading within that window is recorded as the ground floor.<br />
<br />
=== Indoor/Outdoor Transition === <br />
To determine the exact time the user makes an indoor-outdoor transition, two vector masks are convolved across the LSTM predictions.<br />
<br />
\begin{equation}<br />
V_1 = [1, 1, 1, 1, 1, 0, 0, 0, 0, 0]<br />
\end{equation} <br />
<br />
\begin{equation}<br />
V_2 = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]<br />
\end{equation}<br />
<br />
The Jaccard distances is then calculated by the following equation:<br />
<br />
\begin{equation}<br />
J_j = J(s_i, V_j) = \frac{|s_i \cap V_j|}{|s_i| + |V_j| - |s_i \cap V_j|} <br />
\label{equation:Jaccard}<br />
\end{equation}<br />
<br />
If the Jaccard distance to a sub-sequence <math> s_i </math> is greater than the threshold 0.4, it means the transition occurred in the vicinity of the range of the vector mask. These sets of transition windows are then merged if they are close together, and the center of the merged windows are marked as the transition points.<br />
<br />
[[File:FindIOIndexes.png | 700px]]<br />
<br />
=== Vertical Height Estimation === <br />
Once the barometric pressure of the ground floor is known, the user’s current relative altitude can be calculated by the international pressure equation, where <math>m_\Delta</math> is the estimated height, <math> p_1 </math> is the pressure reading of the device, and <math> p_0 </math> is the reference pressure at ground level while transitioning from outdoor to indoor.<br />
<br />
\begin{equation}<br />
m_\Delta = f_{floor}(p_0, p_1) = 44330 (1 - (\frac{p_1}{p_0})^{\frac{1}{5.255}})<br />
\label{equation:baroHeight}<br />
\end{equation}<br />
<br />
=== Floor Estimation === <br />
Given the user’s relative altitude, the floor level can be determined. However, this is not a straightforward task because different buildings have different floor heights, different floor labeling (E.g. not including the 13th floor), and floor heights within the same building can vary from floor to floor. To solve these problems, altitudes data collected are clustered into groups. Each cluster represents the approximate altitude of a floor.<br />
<br />
Here is an example of altitude data collected across 41 trials in the Uris Hall building in New York City. Each dashed line represent the center of a cluster.<br />
<br />
[[File:clusters.png | 500px]]<br />
<br />
Figure 2: Distribution of measurements across 41 trials in the Uris Hall building in New York City. A clear size difference is specially noticeable at the lobby. Each dotted line corresponds to an actual floor in the building learned from clustered data-points.<br />
<br />
Here is the algorithm for the floor level prediction.<br />
<br />
[[File:PredictFloor.png | 700px]]<br />
<br />
= Experiments and Results =<br />
Here are the results for the indoor-outdoor classification problem using different machine learning techniques. LSTM has the best performance on the test set.<br />
The LSTM is trained for 24 epochs with a batch size of 128. All the hyper-parameters such as learning rate(0.006), number of layers, d size, number of hidden units and dropout rate were searched through random search algorithm.<br />
<br />
[[File:IOResults.png]]<br />
<br />
The following are the results for the floor level prediction from the 63 collected samples. Results are given as the percent which matched the floor exactly, off by one, or off by more than one. In each column, the left number is the accuracy using a fixed floor height, and the number on the right is the accuracy when clustering was used to calculate a variable floor height. It was found that using the clustering technique produced 100% accuracy on floor predictions. The conclusion from these results is that using building-specific floor heights produces significantly better results.<br />
<br />
[[File:FloorLevelResults.png]]<br />
<br />
Here is the comparison between the estimated floor height and the ground truth in the Uris Hall building.<br />
<br />
[[File:FloorComparison.png]]<br />
<br />
= Criticism =<br />
This paper is an interesting application of deep learning and achieves an outstanding result of 100% accuracy. However, it offers no new theoretical discoveries. The machine learning techniques used are fairly standard. The neural networks used in this paper only contains 3 layers, and the clustering is applied on one-dimensional data. This leads to the question whether deep learning is necessary and suitable for this task.<br />
<br />
It was explained in the paper that there are many cases where the system does not work. Some cases that were mentioned include: buildings with glass walls, delayed GPS signals, <br />
and pressure changes caused by air conditioning. Other examples I can think of are: uneven floors with some area higher than others, floors rarely visited, and tunnels from one building to another. These special cases are not mentioned in the paper.<br />
<br />
Another weakness of the method comes from the clustering technique. It requires a fair bit of training data. The author suggested two approaches. First, the data can be stored in the individual smartphone. This is not realistic as most people do not visit every single floor of every building, even if it is their own apartment buildings. The second approach is to let a central system (emergency department) to collect data from multiple users (which is what the paper’s results are based on). This is a serious violation of personal privacy and should not be legal.<br />
<br />
Aside from all the technical issues, would it be easier to let the rescuers carry a barometer with them and search for the floor with the transmitted pressure reading?</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Predicting_Floor-Level_for_911_Calls_with_Neural_Networks_and_Smartphone_Sensor_Data&diff=33015stat946w18/Predicting Floor-Level for 911 Calls with Neural Networks and Smartphone Sensor Data2018-03-08T17:09:42Z<p>Pa2forsy: /* Methods */</p>
<hr />
<div>= Introduction =<br />
During emergency 911 calls, knowing the exact position of the victim is crucial to a fast response and a successful rescue. Problems arise when the caller is unable to give their position accurately. This can happen, for instance, when the caller is disoriented, held in hostage, or a child calling for the victim. GPS in our smartphones can give the rescuers the geographic location. However, in a tall building, GPS fails to give an accurate floor level. Previous works have explored using Wi-Fi signal or beacons placed inside the buildings, but these methods are not self-contained and require prior infrastructure knowledge.<br />
<br />
Fortunately, today’s smartphones are equipped with many more sensors including barometers and magnetometers. Deep learning can be applied to predict floor level based on these sensor readings. Firstly, an LSTM is trained to classify whether the caller is indoors or outdoors using GPS, RSSI (Received <br />
Signal Strength Indication), and magnetometer sensor readings. Next, an unsupervised clustering algorithm is used to predict the floor level depending on the barometric pressure difference. With these two parts working together, a self-contained floor level prediction system can achieve 100% accuracy, without any outside knowledge.<br />
<br />
= Data Description =<br />
The author developed an iOS app called Sensory and used it to collect data on an iPhone 6. The following sensor readings are recorded: indoors, created at, session id, floor, RSSI strength, GPS latitude, GPS longitude, GPS vertical accuracy, GPS horizontal accuracy, GPS course, GPS speed, barometric relative altitude, barometric pressure, environment context, environment mean building floors, environment activity, city name, country name, magnet x, magnet y, magnet z, magnet total.<br />
<br />
The indoor-outdoor data has to be manually entered as soon as the user enters or exits a building. To gather the data for floor level prediction, the authors conducted 63 trials among five different buildings throughout New York City. The actual floor level was recorded manually for validation purposes only, since unsupervised learning is being used.<br />
<br />
= Methods =<br />
The proposed method first determines if the user is indoors or outdoors, and detects the instances of transition between them. Then, vertical height is estimated from the barometer. Finally, the exact floor level is predicted through clustering techniques. Indoor/outdoor classification is critical to the working of this method. Once the user is detected to be outdoors, he is assumed to be at the ground level. The vertical height and floor estimations are applied only when the user is indoors. The indoor/outdoor transitions are used to save the barometer readings at the ground level for use as reference pressure.<br />
<br />
=== Indoor/Outdoor Classification === <br />
<br />
An LSTM network is used to solve the indoor-outdoor classification problem. Here is a diagram of the network architecture.<br />
<br />
[[File:lstm.jpg | 500px]]<br />
<br />
Figure 1: LSTM network architecture. A 3-layer LSTM. Inputs are sensor readings for d consecutive time-steps. Target is y = 1 if indoors and y = 0 if outdoors.<br />
<br />
<math> X_i</math> contains a set of <math>d</math> consecutive sensor readings, i.e. <math> X_i = [x_1, x_2,...,x_d] </math>. <math>Y</math> is labelled as 0 for outdoors and 1 for indoors. <math>d</math> is chosen to be 3 by random-search so that <math>X</math> has 3 points <math>X_i = [x_{j-1}, x_j, x_{j+1}]</math> and the middle <math>x_j</math> is used for the <math>y</math> label.<br />
The LSTM contains three layers. Layers one and two have 50 neurons followed by a dropout layer set to 0.2. Layer 3 has two neurons fed directly into a one-neuron feedforward layer with a sigmoid activation function. The input is the sensor readings, and the output is the indoor-outdoor label. The objective function is the cross-entropy between the true label and the prediction.<br />
<br />
\begin{equation}<br />
C(y_i, \hat{y}_i) = \frac{1}{n} \sum_{i=1}^{n} -(y_i log(\hat{y_i}) + (1 - y_i) log(1 - \hat{y_i}))<br />
\label{equation:binCE}<br />
\end{equation}<br />
<br />
The main reason why the neural network is able to predict whether the user is indoor or outdoor is that walls of buildings interfere with the GPS signals. The LSTM is able to find the pattern in the GPS signal strength in combination with other sensor readings to give an accurate prediction. However, the change in GPS signal does not happen instantaneously as the user walks indoor. Thus, a window of 20 seconds is allowed, and the minimum barometric pressure reading within that window is recorded as the ground floor.<br />
<br />
=== Indoor/Outdoor Transition === <br />
To determine the exact time the user makes an indoor-outdoor transition, two vector masks are convolved across the LSTM predictions.<br />
<br />
\begin{equation}<br />
V_1 = [1, 1, 1, 1, 1, 0, 0, 0, 0, 0]<br />
\end{equation} <br />
<br />
\begin{equation}<br />
V_2 = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]<br />
\end{equation}<br />
<br />
The Jaccard distances is then calculated by the following equation:<br />
<br />
\begin{equation}<br />
J_j = J(s_i, V_j) = \frac{|s_i \cap V_j|}{|s_i| + |V_j| - |s_i \cap V_j|} <br />
\label{equation:Jaccard}<br />
\end{equation}<br />
<br />
If the Jaccard distance is greater than the threshold 0.4, it means the transition occurred in the vicinity of the range of the vector mask. These sets of transition windows are then merged if they are close together, and the center of the merged windows are marked as the transition points.<br />
<br />
[[File:FindIOIndexes.png | 700px]]<br />
<br />
=== Vertical Height Estimation === <br />
Once the barometric pressure of the ground floor is known, the user’s current relative altitude can be calculated by the international pressure equation, where <math>m_\Delta</math> is the estimated height, <math> p_1 </math> is the pressure reading of the device, and <math> p_0 </math> is the reference pressure at ground level while transitioning from outdoor to indoor.<br />
<br />
\begin{equation}<br />
m_\Delta = f_{floor}(p_0, p_1) = 44330 (1 - (\frac{p_1}{p_0})^{\frac{1}{5.255}})<br />
\label{equation:baroHeight}<br />
\end{equation}<br />
<br />
=== Floor Estimation === <br />
Given the user’s relative altitude, the floor level can be determined. However, this is not a straightforward task because different buildings have different floor heights, different floor labeling (E.g. not including the 13th floor), and floor heights within the same building can vary from floor to floor. To solve these problems, altitudes data collected are clustered into groups. Each cluster represent the approximate altitude of a floor.<br />
<br />
Here is an example of altitude data collected across 41 trials in the Uris Hall building in New York City. Each dashed line represent the center of a cluster.<br />
<br />
[[File:clusters.png | 500px]]<br />
<br />
Figure 2: Distribution of measurements across 41 trials in the Uris Hall building in New York City. A clear size difference is specially noticeable at the lobby. Each dotted line corresponds to an actual floor in the building learned from clustered data-points.<br />
<br />
Here is the algorithm for the floor level prediction.<br />
<br />
[[File:PredictFloor.png | 700px]]<br />
<br />
= Experiments and Results =<br />
Here are the results for the indoor-outdoor classification problem using different machine learning techniques. LSTM has the best performance on the test set.<br />
The LSTM is trained for 24 epochs with a batch size of 128. All the hyper-parameters such as learning rate(0.006), number of layers, d size, number of hidden units and dropout rate were searched through random search algorithm.<br />
<br />
[[File:IOResults.png]]<br />
<br />
The following are the results for the floor level prediction from the 63 collected samples. Results are given as the percent which matched the floor exactly, off by one, or off by more than one. In each column, the left number is the accuracy using a fixed floor height, and the number on the right is the accuracy when clustering was used to calculate a variable floor height. It was found that using the clustering technique produced 100% accuracy on floor predictions. The conclusion from these results is that using building-specific floor heights produces significantly better results.<br />
<br />
[[File:FloorLevelResults.png]]<br />
<br />
Here is the comparison between the estimated floor height and the ground truth in the Uris Hall building.<br />
<br />
[[File:FloorComparison.png]]<br />
<br />
= Criticism =<br />
This paper is an interesting application of deep learning and achieved an outstanding result of 100% accuracy. However, it offers no new theoretical discoveries. The machine learning techniques used are fairly standard. The neural networks used only contains 3 layers, and the clustering is applied on a one-dimensional data. This makes me question whether deep learning is suitable for this task.<br />
<br />
It was explained in the paper that there are many cases where the system does not work. Some cases that were mentioned include: buildings with glass walls, delayed GPS signals, <br />
and pressure changes caused by air conditioning. Other examples I can think of are: uneven floors with some area higher than others, floors rarely visited, and tunnels from one building to another. These special cases are not mentioned in the paper.<br />
<br />
Another weakness of the method comes from the clustering technique. It requires a fair bit of training data. The author suggested two approaches. First, the data can be stored in the individual smartphone. This is not realistic as most people do not visit every single floor of every building, even if it is their own apartment buildings. The second approach is to let a central system (emergency department) to collect data from multiple users (which is what the paper’s results are based on). This is a serious violation of personal privacy and should not be legal.<br />
<br />
Aside from all the technical issues, would it be easier to let the rescuers carry a barometer with them and search for the floor with the transmitted pressure reading?</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Predicting_Floor-Level_for_911_Calls_with_Neural_Networks_and_Smartphone_Sensor_Data&diff=33013stat946w18/Predicting Floor-Level for 911 Calls with Neural Networks and Smartphone Sensor Data2018-03-08T17:05:00Z<p>Pa2forsy: </p>
<hr />
<div>= Introduction =<br />
During emergency 911 calls, knowing the exact position of the victim is crucial to a fast response and a successful rescue. Problems arise when the caller is unable to give their position accurately. This can happen, for instance, when the caller is disoriented, held in hostage, or a child calling for the victim. GPS in our smartphones can give the rescuers the geographic location. However, in a tall building, GPS fails to give an accurate floor level. Previous works have explored using Wi-Fi signal or beacons placed inside the buildings, but these methods are not self-contained and require prior infrastructure knowledge.<br />
<br />
Fortunately, today’s smartphones are equipped with many more sensors including barometers and magnetometers. Deep learning can be applied to predict floor level based on these sensor readings. Firstly, an LSTM is trained to classify whether the caller is indoors or outdoors using GPS, RSSI (Received <br />
Signal Strength Indication), and magnetometer sensor readings. Next, an unsupervised clustering algorithm is used to predict the floor level depending on the barometric pressure difference. With these two parts working together, a self-contained floor level prediction system can achieve 100% accuracy, without any outside knowledge.<br />
<br />
= Data Description =<br />
The author developed an iOS app called Sensory and used it to collect data on an iPhone 6. The following sensor readings are recorded: indoors, created at, session id, floor, RSSI strength, GPS latitude, GPS longitude, GPS vertical accuracy, GPS horizontal accuracy, GPS course, GPS speed, barometric relative altitude, barometric pressure, environment context, environment mean building floors, environment activity, city name, country name, magnet x, magnet y, magnet z, magnet total.<br />
<br />
The indoor-outdoor data has to be manually entered as soon as the user enters or exits a building. To gather the data for floor level prediction, the authors conducted 63 trials among five different buildings throughout New York City. The actual floor level was recorded manually for validation purposes only, since unsupervised learning is being used.<br />
<br />
= Methods =<br />
The proposed method first determines if the user is indoors or outdoors, and detect the instances of transition between them. Then, vertical height is estimated from the barometer. Finally, the exact floor level is predicted through clustering techniques. Indoor/outdoor classification is critical to the working of this method. Once the user is detected to be outdoors, he is assumed to be at the ground level. The vertical height and floor estimations are applied only when the user is indoors. The indoor/outdoor transitions are used to save the barometer readings at the ground level for use as reference pressure.<br />
<br />
=== Indoor/Outdoor Classification === <br />
<br />
An LSTM network is used to solve the indoor-outdoor classification problem. Here is a diagram of the network architecture.<br />
<br />
[[File:lstm.jpg | 500px]]<br />
<br />
Figure 1: LSTM network architecture. A 3-layer LSTM. Inputs are sensor readings for d consecutive time-steps. Target is y = 1 if indoors and y = 0 if outdoors.<br />
<br />
<math> X_i</math> contains a set of <math>d</math> consecutive sensor readings, i.e. <math> X_i = [x_1, x_2,...,x_d] </math>. <math>Y</math> is labelled as 0 for outdoors and 1 for indoors. <math>d</math> is chosen to be 3 by random-search so that <math>X</math> has 3 points <math>X_i = [x_{j-1}, x_j, x_{j+1}]</math> and the middle <math>x_j</math> is used for the <math>y</math> label.<br />
The LSTM contains three layers. Layers one and two have 50 neurons followed by a dropout layer set to 0.2. Layer 3 has two neurons fed directly into a one-neuron feedforward layer with a sigmoid activation function. The input is the sensor readings, and the output is the indoor-outdoor label. The objective function is the cross-entropy between the true label and the prediction.<br />
<br />
\begin{equation}<br />
C(y_i, \hat{y}_i) = \frac{1}{n} \sum_{i=1}^{n} -(y_i log(\hat{y_i}) + (1 - y_i) log(1 - \hat{y_i}))<br />
\label{equation:binCE}<br />
\end{equation}<br />
<br />
The main reason why the neural network is able to predict whether the user is indoor or outdoor is that walls of buildings interfere with the GPS signals. The LSTM is able to find the pattern in the GPS signal strength in combination with other sensor readings to give an accurate prediction. However, the change in GPS signal does not happen instantaneously as the user walks indoor. Thus, a window of 20 seconds is allowed, and the minimum barometric pressure reading within that window is recorded as the ground floor.<br />
<br />
=== Indoor/Outdoor Transition === <br />
To determine the exact time the user makes an indoor-outdoor transition, two vector masks are convolved across the LSTM predictions.<br />
<br />
\begin{equation}<br />
V_1 = [1, 1, 1, 1, 1, 0, 0, 0, 0, 0]<br />
\end{equation} <br />
<br />
\begin{equation}<br />
V_2 = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]<br />
\end{equation}<br />
<br />
The Jaccard distances is then calculated by the following equation:<br />
<br />
\begin{equation}<br />
J_j = J(s_i, V_j) = \frac{|s_i \cap V_j|}{|s_i| + |V_j| - |s_i \cap V_j|} <br />
\label{equation:Jaccard}<br />
\end{equation}<br />
<br />
If the Jaccard distance is greater than the threshold 0.4, it means the transition occurred in the vicinity of the range of the vector mask. These sets of transition windows are then merged if they are close together, and the center of the merged windows are marked as the transition points.<br />
<br />
[[File:FindIOIndexes.png | 700px]]<br />
<br />
=== Vertical Height Estimation === <br />
Once the barometric pressure of the ground floor is known, the user’s current relative altitude can be calculated by the international pressure equation, where <math>m_\Delta</math> is the estimated height, <math> p_1 </math> is the pressure reading of the device, and <math> p_0 </math> is the reference pressure at ground level while transitioning from outdoor to indoor.<br />
<br />
\begin{equation}<br />
m_\Delta = f_{floor}(p_0, p_1) = 44330 (1 - (\frac{p_1}{p_0})^{\frac{1}{5.255}})<br />
\label{equation:baroHeight}<br />
\end{equation}<br />
<br />
=== Floor Estimation === <br />
Given the user’s relative altitude, the floor level can be determined. However, this is not a straightforward task because different buildings have different floor heights, different floor labeling (E.g. not including the 13th floor), and floor heights within the same building can vary from floor to floor. To solve these problems, altitudes data collected are clustered into groups. Each cluster represent the approximate altitude of a floor.<br />
<br />
Here is an example of altitude data collected across 41 trials in the Uris Hall building in New York City. Each dashed line represent the center of a cluster.<br />
<br />
[[File:clusters.png | 500px]]<br />
<br />
Figure 2: Distribution of measurements across 41 trials in the Uris Hall building in New York City. A clear size difference is specially noticeable at the lobby. Each dotted line corresponds to an actual floor in the building learned from clustered data-points.<br />
<br />
Here is the algorithm for the floor level prediction.<br />
<br />
[[File:PredictFloor.png | 700px]]<br />
<br />
= Experiments and Results =<br />
Here are the results for the indoor-outdoor classification problem using different machine learning techniques. LSTM has the best performance on the test set.<br />
The LSTM is trained for 24 epochs with a batch size of 128. All the hyper-parameters such as learning rate(0.006), number of layers, d size, number of hidden units and dropout rate were searched through random search algorithm.<br />
<br />
[[File:IOResults.png]]<br />
<br />
The following are the results for the floor level prediction from the 63 collected samples. Results are given as the percent which matched the floor exactly, off by one, or off by more than one. In each column, the left number is the accuracy using a fixed floor height, and the number on the right is the accuracy when clustering was used to calculate a variable floor height. It was found that using the clustering technique produced 100% accuracy on floor predictions. The conclusion from these results is that using building-specific floor heights produces significantly better results.<br />
<br />
[[File:FloorLevelResults.png]]<br />
<br />
Here is the comparison between the estimated floor height and the ground truth in the Uris Hall building.<br />
<br />
[[File:FloorComparison.png]]<br />
<br />
= Criticism =<br />
This paper is an interesting application of deep learning and achieved an outstanding result of 100% accuracy. However, it offers no new theoretical discoveries. The machine learning techniques used are fairly standard. The neural networks used only contains 3 layers, and the clustering is applied on a one-dimensional data. This makes me question whether deep learning is suitable for this task.<br />
<br />
It was explained in the paper that there are many cases where the system does not work. Some cases that were mentioned include: buildings with glass walls, delayed GPS signals, <br />
and pressure changes caused by air conditioning. Other examples I can think of are: uneven floors with some area higher than others, floors rarely visited, and tunnels from one building to another. These special cases are not mentioned in the paper.<br />
<br />
Another weakness of the method comes from the clustering technique. It requires a fair bit of training data. The author suggested two approaches. First, the data can be stored in the individual smartphone. This is not realistic as most people do not visit every single floor of every building, even if it is their own apartment buildings. The second approach is to let a central system (emergency department) to collect data from multiple users (which is what the paper’s results are based on). This is a serious violation of personal privacy and should not be legal.<br />
<br />
Aside from all the technical issues, would it be easier to let the rescuers carry a barometer with them and search for the floor with the transmitted pressure reading?</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Predicting_Floor-Level_for_911_Calls_with_Neural_Networks_and_Smartphone_Sensor_Data&diff=33012stat946w18/Predicting Floor-Level for 911 Calls with Neural Networks and Smartphone Sensor Data2018-03-08T16:52:58Z<p>Pa2forsy: /* Introduction */</p>
<hr />
<div>= Introduction =<br />
During emergency 911 calls, knowing the exact position of the victim is crucial to a fast response and a successful rescue. Problems arise when the caller is unable to give their position accurately. This can happen, for instance, when the caller is disoriented, held in hostage, or a child calling for the victim. GPS in our smartphones can give the rescuers the geographic location. However, in a tall building, GPS fails to give an accurate floor level. Previous works have explored using Wi-Fi signal or beacons placed inside the buildings, but these methods are not self-contained and require prior infrastructure knowledge.<br />
<br />
Fortunately, today’s smartphones are equipped with many more sensors including barometers and magnetometers. Deep learning can be applied to predict floor level based on these sensor readings. Firstly, an LSTM is trained to classify whether the caller is indoors or outdoors using GPS, RSSI, and magnetometer sensor readings. Next, an unsupervised clustering algorithm is used to predict the floor level depending on the barometric pressure difference. With these two parts working together, a self-contained floor level prediction system can achieve 100% accuracy, without any outside knowledge.<br />
<br />
= Data Description =<br />
The author developed an iOS app called Sensory and used it to collect data on an iPhone 6. The following sensor readings are recorded: indoors, created at, session id, floor, RSSI strength, GPS latitude, GPS longitude, GPS vertical accuracy, GPS horizontal accuracy, GPS course, GPS speed, barometric relative altitude, barometric pressure, environment context, environment mean building floors, environment activity, city name, country name, magnet x, magnet y, magnet z, magnet total.<br />
<br />
The indoor-outdoor data has to be manually entered as soon as the user enters or exits a building. To gather the data for floor level prediction, the authors conducted 63 trials among five different buildings throughout New York City. The actual floor level was recorded manually for validation purposes only, since unsupervised learning is being used.<br />
<br />
= Methods =<br />
The proposed method first determines if the user is indoors or outdoors, and detect the instances of transition between them. Then, vertical height is estimated from the barometer. Finally, the exact floor level is predicted through clustering techniques. Indoor/outdoor classification is critical to the working of this method. Once the user is detected to be outdoors, he is assumed to be at the ground level. The vertical height and floor estimations are applied only when the user is indoors. The indoor/outdoor transitions are used to save the barometer readings at the ground level for use as reference pressure.<br />
<br />
=== Indoor/Outdoor Classification === <br />
<br />
An LSTM network is used to solve the indoor-outdoor classification problem. Here is a diagram of the network architecture.<br />
<br />
[[File:lstm.jpg | 500px]]<br />
<br />
Figure 1: LSTM network architecture. A 3-layer LSTM. Inputs are sensor readings for d consecutive time-steps. Target is y = 1 if indoors and y = 0 if outdoors.<br />
<br />
<math> X_i</math> contains a set of <math>d</math> consecutive sensor readings, i.e. <math> X_i = [x_1, x_2,...,x_d] </math>. <math>Y</math> is labelled as 0 for outdoors and 1 for indoors. <math>d</math> is chosen to be 3 by random-search so that <math>X</math> has 3 points <math>X_i = [x_{j-1}, x_j, x_{j+1}]</math> and the middle <math>x_j</math> is used for the <math>y</math> label.<br />
The LSTM contains three layers. Layers one and two have 50 neurons followed by a dropout layer set to 0.2. Layer 3 has two neurons fed directly into a one-neuron feedforward layer with a sigmoid activation function. The input is the sensor readings, and the output is the indoor-outdoor label. The objective function is the cross-entropy between the true label and the prediction.<br />
<br />
\begin{equation}<br />
C(y_i, \hat{y}_i) = \frac{1}{n} \sum_{i=1}^{n} -(y_i log(\hat{y_i}) + (1 - y_i) log(1 - \hat{y_i}))<br />
\label{equation:binCE}<br />
\end{equation}<br />
<br />
The main reason why the neural network is able to predict whether the user is indoor or outdoor is that walls of buildings interfere with the GPS signals. The LSTM is able to find the pattern in the GPS signal strength in combination with other sensor readings to give an accurate prediction. However, the change in GPS signal does not happen instantaneously as the user walks indoor. Thus, a window of 20 seconds is allowed, and the minimum barometric pressure reading within that window is recorded as the ground floor.<br />
<br />
=== Indoor/Outdoor Transition === <br />
To determine the exact time the user makes an indoor-outdoor transition, two vector masks are convolved across the LSTM predictions.<br />
<br />
\begin{equation}<br />
V_1 = [1, 1, 1, 1, 1, 0, 0, 0, 0, 0]<br />
\end{equation} <br />
<br />
\begin{equation}<br />
V_2 = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]<br />
\end{equation}<br />
<br />
The Jaccard distances is then calculated by the following equation:<br />
<br />
\begin{equation}<br />
J_j = J(s_i, V_j) = \frac{|s_i \cap V_j|}{|s_i| + |V_j| - |s_i \cap V_j|} <br />
\label{equation:Jaccard}<br />
\end{equation}<br />
<br />
If the Jaccard distance is greater than the threshold 0.4, it means the transition occurred in the vicinity of the range of the vector mask. These sets of transition windows are then merged if they are close together, and the center of the merged windows are marked as the transition points.<br />
<br />
[[File:FindIOIndexes.png | 700px]]<br />
<br />
=== Vertical Height Estimation === <br />
Once the barometric pressure of the ground floor is known, the user’s current relative altitude can be calculated by the international pressure equation, where <math>m_\Delta</math> is the estimated height, <math> p_1 </math> is the pressure reading of the device, and <math> p_0 </math> is the reference pressure at ground level while transitioning from outdoor to indoor.<br />
<br />
\begin{equation}<br />
m_\Delta = f_{floor}(p_0, p_1) = 44330 (1 - (\frac{p_1}{p_0})^{\frac{1}{5.255}})<br />
\label{equation:baroHeight}<br />
\end{equation}<br />
<br />
=== Floor Estimation === <br />
Given the user’s relative altitude, the floor level can be determined. However, this is not a straightforward task because different buildings have different floor heights, different floor labeling (E.g. not including the 13th floor), and floor heights within the same building can vary from floor to floor. To solve these problems, altitudes data collected are clustered into groups. Each cluster represent the approximate altitude of a floor.<br />
<br />
Here is an example of altitude data collected across 41 trials in the Uris Hall building in New York City. Each dashed line represent the center of a cluster.<br />
<br />
[[File:clusters.png | 500px]]<br />
<br />
Figure 2: Distribution of measurements across 41 trials in the Uris Hall building in New York City. A clear size difference is specially noticeable at the lobby. Each dotted line corresponds to an actual floor in the building learned from clustered data-points.<br />
<br />
Here is the algorithm for the floor level prediction.<br />
<br />
[[File:PredictFloor.png | 700px]]<br />
<br />
= Experiments and Results =<br />
Here are the results for the indoor-outdoor classification problem using different machine learning techniques. LSTM has the best performance on the test set.<br />
The LSTM is trained for 24 epochs with a batch size of 128. All the hyper-parameters such as learning rate(0.006), number of layers, d size, number of hidden units and dropout rate were searched through random search algorithm.<br />
<br />
[[File:IOResults.png]]<br />
<br />
The following are the results for the floor level prediction from the 63 collected samples. Results are given as the percent which matched the floor exactly, off by one, or off by more than one. In each column, the left number is the accuracy using a fixed floor height, and the number on the right is the accuracy when clustering was used to calculate a variable floor height. It was found that using the clustering technique produced 100% accuracy on floor predictions. The conclusion from these results is that using building-specific floor heights produces significantly better results.<br />
<br />
[[File:FloorLevelResults.png]]<br />
<br />
Here is the comparison between the estimated floor height and the ground truth in the Uris Hall building.<br />
<br />
[[File:FloorComparison.png]]<br />
<br />
= Criticism =<br />
This paper is an interesting application of deep learning and achieved an outstanding result of 100% accuracy. However, it offers no new theoretical discoveries. The machine learning techniques used are fairly standard. The neural networks used only contains 3 layers, and the clustering is applied on a one-dimensional data. This makes me question whether deep learning is suitable for this task.<br />
<br />
It was explained in the paper that there are many cases where the system does not work. Some cases that were mentioned include: buildings with glass walls, delayed GPS signals, <br />
and pressure changes caused by air conditioning. Other examples I can think of are: uneven floors with some area higher than others, floors rarely visited, and tunnels from one building to another. These special cases are not mentioned in the paper.<br />
<br />
Another weakness of the method comes from the clustering technique. It requires a fair bit of training data. The author suggested two approaches. First, the data can be stored in the individual smartphone. This is not realistic as most people do not visit every single floor of every building, even if it is their own apartment buildings. The second approach is to let a central system (emergency department) to collect data from multiple users (which is what the paper’s results are based on). This is a serious violation of personal privacy and should not be legal.<br />
<br />
Aside from all the technical issues, would it be easier to let the rescuers carry a barometer with them and search for the floor with the transmitted pressure reading?</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Self_Normalizing_Neural_Networks&diff=32919stat946w18/Self Normalizing Neural Networks2018-03-08T00:02:22Z<p>Pa2forsy: Added a few details to the computation of one of the moment-mapping integrals</p>
<hr />
<div>==Introduction and Motivation==<br />
<br />
While neural networks have been making a lot of headway in improving benchmark results and narrowing the gap with human-level performance, success has been fairly limited to visual and sequential processing tasks through advancements in convolutional network and recurrent network structures. Most data science competitions outside of those domains are still being won by algorithms such as gradient boosting and random forests. The traditional (densely connected) feed-forward neural networks (FNNs) are rarely used competitively, and when they do win on the rare occasions, they are won with very shallow networks with just up to four layers [10].<br />
<br />
The authors, Klambauer et al., believe that what prevents FNNs from becoming more useful is the inability to train a deeper FNN structure, which would allow the network to learn more levels of abstract representations. To have a deeper network, oscillations in the distribution of activations need to be kept under control so that stable gradients can be obtained during training. Several techniques are available to normalize activations, including batch normalization [6], layer normalization [1] and weight normalization [8]. These methods work well with CNNs and RNNs, but not so much with FNNs because backpropagating through normalization parameters introduces additional variance to the gradients, and regularization techniques like dropout further perturb the normalization effect. CNNs and RNNs are less sensitive to such perturbations, presumably due to their weight sharing architecture, but FNNs do not have such property, and thus suffer from high variance in training errors, which hinders learning. Furthermore, the aforementioned normalization techniques involving adding external layers to the model and can slow down computations. <br />
<br />
Therefore, the authors were motivated to develop a new FNN implementation that can achieve the intended effect of normalization techniques that works well with stochastic gradient descent and dropout. Self-normalizing neural networks (SNNs) are based on the idea of scaled exponential linear units (SELU), a new activation function introduced in this paper, whose output distribution is proved to converge to a fixed point, thus making it possible to train deeper networks. <br />
<br />
==Notations==<br />
<br />
As the paper (primarily in the supplementary materials) comes with lengthy proofs, important notations are listed first.<br />
<br />
Consider two fully-connected layers, let <math display="inline">x</math> denote the inputs to the second layer, then <math display="inline">z = Wx</math> represents the network inputs of the second layer, and <math display="inline">y = f(z)</math> represents the activations in the second layer.<br />
<br />
Assume that all <math display="inline">x_i</math>'s, <math display="inline">1 \leqslant i \leqslant n</math>, have mean <math display="inline">\mu := \mathrm{E}(x_i)</math> and variance <math display="inline">\nu := \mathrm{Var}(x_i)</math> and that each <math display="inline">y</math> has mean <math display="inline">\widetilde{\mu} := \mathrm{E}(y)</math> and variance <math display="inline">\widetilde{\nu} := \mathrm{Var}(y)</math>, then let <math display="inline">g</math> be the set of functions that maps <math display="inline">(\mu, \nu)</math> to <math display="inline">(\widetilde{\mu}, \widetilde{\nu})</math>. <br />
<br />
For the weight vector <math display="inline">w</math>, <math display="inline">n</math> times the mean of the weight vector is <math display="inline">\omega := \sum_{i = 1}^n \omega_i</math> and <math display="inline">n</math> times the second moment is <math display="inline">\tau := \sum_{i = 1}^{n} w_i^2</math>.<br />
<br />
==Key Concepts==<br />
<br />
===Self-Normalizing Neural-Net (SNN)===<br />
<br />
''A neural network is self-normalizing if it possesses a mapping <math display="inline">g: \Omega \rightarrow \Omega</math> for each activation <math display="inline">y</math> that maps mean and variance from one layer to the next and has a stable and attracting fixed point depending on <math display="inline">(\omega, \tau)</math> in <math display="inline">\Omega</math>. Furthermore, the mean and variance remain in the domain <math display="inline">\Omega</math>, that is <math display="inline">g(\Omega) \subseteq \Omega</math>, where <math display="inline">\Omega = \{ (\mu, \nu) | \mu \in [\mu_{min}, \mu_{max}], \nu \in [\nu_{min}, \nu_{max}] \}</math>. When iteratively applying the mapping <math display="inline">g</math>, each point within <math display="inline">\Omega</math> converges to this fixed point.''<br />
<br />
In other words, in SNNs, if the inputs from an earlier layer (<math display="inline">x</math>) already have their mean and variance within a predefined interval <math display="inline">\Omega</math>, then the activations to the next layer (<math display="inline">y = f(z = Wx)</math>) should remain within those intervals. This is true across all pairs of connecting layers as the normalizing effect gets propagated through the network, hence why the term self-normalizing. When the mapping is applied iteratively, it should draw the mean and variance values closer to a fixed point within <math display="inline">\Omega</math>, the value of which depends on <math display="inline">\omega</math> and <math display="inline">\tau</math> (recall that they are from the weight vector).<br />
<br />
The activation function that makes an SNN possible should meet the following four conditions:<br />
<br />
# It can take on both negative and positive values, so it can normalize the mean;<br />
# It has a saturation region, so it can dampen variances that are too large;<br />
# It has a slope larger than one, so it can increase variances that are too small; and<br />
# It is a continuous curve, which is necessary for the fixed point to exist (see the definition of Banach fixed point theorem to follow).<br />
<br />
Commonly used activation functions such as rectified linear units (ReLU), sigmoid, tanh, leaky ReLUs and exponential linear units (ELUs) do not meet all four criteria, therefore, a new activation function is needed.<br />
<br />
===Scaled Exponential Linear Units (SELUs)===<br />
<br />
One of the main ideas introduced in this paper is the SELU function. As the name suggests, it is closely related to ELU [3],<br />
<br />
\[ \mathrm{elu}(x) = \begin{cases} x & x > 0 \\<br />
\alpha e^x - \alpha & x \leqslant 0<br />
\end{cases} \]<br />
<br />
but further builds upon it by introducing a new scale parameter $\lambda$ and proving the exact values that $\alpha$ and $\lambda$ should take on to achieve self-normalization. SELU is defined as:<br />
<br />
\[ \mathrm{selu}(x) = \lambda \begin{cases} x & x > 0 \\<br />
\alpha e^x - \alpha & x \leqslant 0<br />
\end{cases} \]<br />
<br />
SELUs meet all four criteria listed above - it takes on positive values when <math display="inline">x > 0</math> and negative values when <math display="inline">x < 0</math>, it has a saturation region when <math display="inline">x</math> is a larger negative value, the value of <math display="inline">\lambda</math> can be set to greater than one to ensure a slope greater than one, and it is continuous at <math display="inline">x = 0</math>. <br />
<br />
Figure 1 below gives an intuition for how SELUs normalize activations across layers. As shown, a variance dampening effect occurs when inputs are negative and far away from zero, and a variance increasing effect occurs when inputs are close to zero.<br />
<br />
[[File:snnf1.png|500px]]<br />
<br />
Figure 2 below plots the progression of training error on the MNIST and CIFAR10 datasets when training with SNNs versus FNNs with batch normalization at varying model depths. As shown, FNNs that adopted the SELU activation function exhibited lower and less variable training loss compared to using batch normalization, even as the depth increased to 16 and 32 layers.<br />
<br />
[[File:snnf2.png|600px]]<br />
<br />
=== Banach Fixed Point Theorem and Contraction Mappings ===<br />
<br />
The underlying theory behind SNNs is the Banach fixed point theorem, which states the following: ''Let <math display="inline">(X, d)</math> be a non-empty complete metric space with a contraction mapping <math display="inline">f: X \rightarrow X</math>. Then <math display="inline">f</math> has a unique fixed point <math display="inline">x_f \subseteq X</math> with <math display="inline">f(x_f) = x_f</math>. Every sequence <math display="inline">x_n = f(x_{n-1})</math> with starting element <math display="inline">x_0 \subseteq X</math> converges to the fixed point: <math display="inline">x_n \underset{n \rightarrow \infty}\rightarrow x_f</math>.''<br />
<br />
A contraction mapping is a function <math display="inline">f: X \rightarrow X</math> on a metric space <math display="inline">X</math> with distance <math display="inline">d</math>, such that for all points <math display="inline">\mathbf{u}</math> and <math display="inline">\mathbf{v}</math> in <math display="inline">X</math>: <math display="inline">d(f(\mathbf{u}), f(\mathbf{v})) \leqslant \delta d(\mathbf{u}, \mathbf{v})</math>, for a <math display="inline">0 \leqslant \delta \leqslant 1</math>.<br />
<br />
The easiest way to prove a contraction mapping is usually to show that the spectral norm [12] of its Jacobian is less than 1 [13], as was done for this paper.<br />
<br />
==Proving the Self-Normalizing Property==<br />
<br />
===Mean and Variance Mapping Function===<br />
<br />
<math display="inline">g</math> is derived under the assumption that <math display="inline">x_i</math>'s are independent but not necessarily having the same mean and variance [[#Footnotes |(2)]]. Under this assumption (and recalling earlier notation of <math display="inline">\omega</math> and <math display="inline">\tau</math>),<br />
<br />
\begin{align}<br />
\mathrm{E}(z = \mathbf{w}^T \mathbf{x}) = \sum_{i = 1}^n w_i \mathrm{E}(x_i) = \mu \omega<br />
\end{align}<br />
<br />
\begin{align}<br />
\mathrm{Var}(z) = \mathrm{Var}(\sum_{i = 1}^n w_i x_i) = \sum_{i = 1}^n w_i^2 \mathrm{Var}(x_i) = \nu \sum_{i = 1}^n w_i^2 = \nu\tau \textrm{ .}<br />
\end{align}<br />
<br />
When the weight terms are normalized, <math display="inline">z</math> can be viewed as a weighted sum of <math display="inline">x_i</math>'s. Wide neural net layers with a large number of nodes is common, so <math display="inline">n</math> is usually large, and by the Central Limit Theorem, <math display="inline">z</math> approaches a normal distribution <math display="inline">\mathcal{N}(\mu\omega, \sqrt{\nu\tau})</math>. <br />
<br />
Using the above property, the exact form for <math display="inline">g</math> can be obtained using the definitions for mean and variance of continuous random variables: <br />
<br />
[[File:gmapping.png|600px|center]]<br />
<br />
Analytical solutions for the integrals can be obtained as follows: <br />
<br />
[[File:gintegral.png|600px|center]]<br />
<br />
The authors are interested in the fixed point <math display="inline">(\mu, \nu) = (0, 1)</math> as these are the parameters associated with the common standard normal distribution. The authors also proposed using normalized weights such that <math display="inline">\omega = \sum_{i = 1}^n = 0</math> and <math display="inline">\tau = \sum_{i = 1}^n w_i^2= 1</math> as it gives a simpler, cleaner expression for <math display="inline">\widetilde{\mu}</math> and <math display="inline">\widetilde{\nu}</math> in the calculations in the next steps. This weight scheme can be achieved in several ways, for example, by drawing from a normal distribution <math display="inline">\mathcal{N}(0, \frac{1}{n})</math> or from a uniform distribution <math display="inline">U(-\sqrt{3}, \sqrt{3})</math>.<br />
<br />
At <math display="inline">\widetilde{\mu} = \mu = 0</math>, <math display="inline">\widetilde{\nu} = \nu = 1</math>, <math display="inline">\omega = 0</math> and <math display="inline">\tau = 1</math>, the constants <math display="inline">\lambda</math> and <math display="inline">\alpha</math> from the SELU function can be solved for - <math display="inline">\lambda_{01} \approx 1.0507</math> and <math display="inline">\alpha_{01} \approx 1.6733</math>. These values are used throughout the rest of the paper whenever an expression calls for <math display="inline">\lambda</math> and <math display="inline">\alpha</math>.<br />
<br />
===Details of Moment-Mapping Integrals ===<br />
Consider the moment-mapping integrals:<br />
\begin{align}<br />
\widetilde{\mu} & = \int_{-\infty}^\infty \mathrm{selu} (z) p_N(z; \mu \omega, \sqrt{\nu \tau})dz\\<br />
\widetilde{\nu} & = \int_{-\infty}^\infty \mathrm{selu} (z)^2 p_N(z; \mu \omega, \sqrt{\nu \tau}) dz-\widetilde{\mu}^2.<br />
\end{align}<br />
<br />
The equation for <math display="inline">\widetilde{\mu}</math> can be expanded as <br />
\begin{align}<br />
\widetilde{\mu} & = \frac{\lambda}{2}\left( 2\alpha\int_{-\infty}^0 (\exp(z)-1) p_N(z; \mu \omega, \sqrt{\nu \tau})dz +2\int_{0}^\infty z p_N(z; \mu \omega, \sqrt{\nu \tau})dz \right)\\<br />
&= \frac{\lambda}{2}\left( 2 \alpha \frac{1}{\sqrt{2\pi\tau\nu}} \int_{-\infty}^0 (\exp(z)-1) \exp(\frac{-1}{2\tau \nu} (z-\mu \omega)^2 ) dz +2\frac{1}{\sqrt{2\pi\tau\nu}}\int_{0}^\infty z \exp(\frac{-1}{2\tau \nu} (z-\mu \omega)^2 dz \right)\\<br />
&= \frac{\lambda}{2}\left( 2 \alpha\frac{1}{\sqrt{2\pi\tau\nu}}\int_{-\infty}^0 \exp(z) \exp(\frac{-1}{2\tau \nu} (z-\mu \omega)^2 ) dz - 2 \alpha\frac{1}{\sqrt{2\pi\tau\nu}}\int_{-\infty}^0 \exp(\frac{-1}{2\tau \nu} (z-\mu \omega)^2 ) dz +2\frac{1}{\sqrt{2\pi\tau\nu}}\int_{0}^\infty z \exp(\frac{-1}{2\tau \nu} (z-\mu \omega)^2 dz \right)\\<br />
\end{align}<br />
<br />
The first integral can be simplified via the substituiton<br />
\begin{align}<br />
q:= \frac{1}{\sqrt{2\tau \nu}}(z-\mu \omega -\tau \nu).<br />
\end{align}<br />
While the second and third can be simplified via the substitution<br />
\begin{align}<br />
q:= \frac{1}{\sqrt{2\tau \nu}}(z-\mu \omega ).<br />
\end{align}<br />
Using the definitions of <math display="inline">\mathrm{erf}</math> and <math display="inline">\mathrm{erfc}</math> then yields the result of the previous section.<br />
<br />
===Self-Normalizing Property Under Normalized Weights===<br />
<br />
With the weights normalized, it is possible to calculate the exact value for the spectral norm [12] of <math display="inline">g</math>'s Jacobian around the fixed point <math display="inline">(\mu, \nu) = (0, 1)</math>, which turns out to be <math display="inline">0.7877</math>. Thus, at initialization, SNNs have a stable and attracting fixed point at <math display="inline">(0, 1)</math>, which means that when <math display="inline">g</math> is applied iteratively to a pair <math display="inline">(\mu_{new}, \nu_{new})</math>, it should draw the points closer to <math display="inline">(0, 1)</math>. The rate of convergence is determined by the spectral norm [12], whose value depends on <math display="inline">\mu</math>, <math display="inline">\nu</math>, <math display="inline">\omega</math> and <math display="inline">\tau</math>.<br />
<br />
===Self-Normalizing Property Under Unnormalized Weights===<br />
<br />
As weights are updated during training, there is no guarantee that they would remain normalized. The authors addressed this issue through the first key theorem presented in the paper, which states that a fixed point close to (0, 1) can still be obtained if <math display="inline">\mu</math>, <math display="inline">\nu</math>, <math display="inline">\omega</math> and <math display="inline">\tau</math> are restricted to a specified range. <br />
<br />
Additionally, there is no guarantee that the mean and variance of the inputs would stay within the range given by the first theorem, which led to the development of theorems #2 and #3. These two theorems established an upper and lower bound on the variance of inputs if the variance of activations from the previous layer are above or below the range specified, respectively. This ensures that the variance would not explode or vanish after being propagated through the network.<br />
<br />
The theorems come with lengthy proofs in the supplementary materials for the paper. High-level proof sketches are presented here.<br />
<br />
====Theorem 1: Stable and Attracting Fixed Points Close to (0, 1)====<br />
<br />
'''Definition:''' We assume <math display="inline">\alpha = \alpha_{01}</math> and <math display="inline">\lambda = \lambda_{01}</math>. We restrict the range of the variables to the domain <math display="inline">\mu \in [-0.1, 0.1]</math>, <math display="inline">\omega \in [-0.1, 0.1]</math>, <math display="inline">\nu \in [0.8, 1.5]</math>, and <math display="inline">\tau \in [0.9, 1.1]</math>. For <math display="inline">\omega = 0</math> and <math display="inline">\tau = 1</math>, the mapping has the stable fixed point <math display="inline">(\mu, \nu) = (0, 1</math>. For other <math display="inline">\omega</math> and <math display="inline">\tau</math>, g has a stable and attracting fixed point depending on <math display="inline">(\omega, \tau)</math> in the <math display="inline">(\mu, \nu)</math>-domain: <math display="inline">\mu \in [-0.03106, 0.06773]</math> and <math display="inline">\nu \in [0.80009, 1.48617]</math>. All points within the <math display="inline">(\mu, \nu)</math>-domain converge when iteratively applying the mapping to this fixed point.<br />
<br />
'''Proof:''' In order to show the the mapping <math display="inline">g</math> has a stable and attracting fixed point close to <math display="inline">(0, 1)</math>, the authors again applied Banach's fixed point theorem, which states that a contraction mapping on a nonempty complete metric space that does not map outside its domain has a unique fixed point, and that all points in the <math display="inline">(\mu, \nu)</math>-domain converge to the fixed point when <math display="inline">g</math> is iteratively applied. <br />
<br />
The two requirements are proven as follows:<br />
<br />
'''1. g is a contraction mapping.'''<br />
<br />
For <math display="inline">g</math> to be a contraction mapping in <math display="inline">\Omega</math> with distance <math display="inline">||\cdot||_2</math>, there must exist a Lipschitz constant <math display="inline">M < 1</math> such that: <br />
<br />
\begin{align} <br />
\forall \mu, \nu \in \Omega: ||g(\mu) - g(\nu)||_2 \leqslant M||\mu - \nu||_2 <br />
\end{align}<br />
<br />
As stated earlier, <math display="inline">g</math> is a contraction mapping if the spectral norm [12] of the Jacobian <math display="inline">\mathcal{H}</math> [[#Footnotes | (3)]] is below one, or equivalently, if the the largest singular value of <math display="inline">\mathcal{H}</math> is less than 1.<br />
<br />
To find the singular values of <math display="inline">\mathcal{H}</math>, the authors used an explicit formula derived by Blinn [2] for <math display="inline">2\times2</math> matrices, which states that the largest singular value of the matrix is <math display="inline">\frac{1}{2}(\sqrt{(a_{11} + a_{22}) ^ 2 + (a_{21} - a{12})^2} + \sqrt{(a_{11} - a_{22}) ^ 2 + (a_{21} + a{12})^2})</math>.<br />
<br />
For <math display="inline">\mathcal{H}</math>, an expression for the largest singular value of <math display="inline">\mathcal{H}</math>, made up of the first-order partial derivatives of the mapping <math display="inline">g</math> with respect to <math display="inline">\mu</math> and <math display="inline">\nu</math>, can be derived given the analytical solutions for <math display="inline">\widetilde{\mu}</math> and <math display="inline">\widetilde{\nu}</math> (and denoted <math display="inline">S(\mu, \omega, \nu, \tau, \lambda, \alpha)</math>).<br />
<br />
From the mean value theorem, we know that for a <math display="inline">t \in [0, 1]</math>, <br />
<br />
[[File:seq.png|600px|center]]<br />
<br />
Therefore, the distance of the singular value at <math display="inline">S(\mu, \omega, \nu, \tau, \lambda_{\mathrm{01}}, \alpha_{\mathrm{01}})</math> and at <math display="inline">S(\mu + \Delta\mu, \omega + \Delta\omega, \nu + \Delta\nu, \tau \Delta\tau, \lambda_{\mathrm{01}}, \alpha_{\mathrm{01}})</math> can be bounded above by <br />
<br />
[[File:seq2.png|600px|center]]<br />
<br />
An upper bound was obtained for each partial derivative term above, mainly through algebraic reformulations and by making use of the fact that many of the functions are monotonically increasing or decreasing on the variables they depend on in <math display="inline">\Omega</math> (see pages 17 - 25 in the supplementary materials).<br />
<br />
The <math display="inline">\Delta</math> terms were then set (rather arbitrarily) to be: <math display="inline">\Delta \mu=0.0068097371</math>,<br />
<math display="inline">\Delta \omega=0.0008292885</math>, <math display="inline">\Delta \nu=0.0009580840</math>, and <math display="inline">\Delta \tau=0.0007323095</math>. Plugging in the upper bounds on the absolute values of the derivative terms for <math display="inline">S</math> and the <math display="inline">\Delta</math> terms yields<br />
<br />
\[ S(\mu + \Delta \mu,\omega + \Delta \omega,\nu + \Delta \nu,\tau + \Delta \tau,\lambda_{\rm 01},\alpha_{\rm 01}) - S(\mu,\omega,\nu,\tau,\lambda_{\rm 01},\alpha_{\rm 01}) < 0.008747 \]<br />
<br />
Next, the largest singular value is found from a computer-assisted fine grid-search [[#Footnotes | (1)]] over the domain <math display="inline">\Omega</math>, with grid lengths <math display="inline">\Delta \mu=0.0068097371</math>, <math display="inline">\Delta \omega=0.0008292885</math>, <math display="inline">\Delta \nu=0.0009580840</math>, and <math display="inline">\Delta \tau=0.0007323095</math>, which turned out to be <math display="inline">0.9912524171058772</math>. Therefore, <br />
<br />
\[ S(\mu + \Delta \mu,\omega + \Delta \omega,\nu + \Delta \nu,\tau + \Delta \tau,\lambda_{\rm 01},\alpha_{\rm 01}) \leq 0.9912524171058772 + 0.008747 < 1 \]<br />
<br />
Since the largest singular value is smaller than 1, <math display="inline>g</math> is a contraction mapping.<br />
<br />
'''2. g does not map outside its domain.'''<br />
<br />
To prove that <math display="inline">g</math> does not map outside of the domain <math display="inline">\mu \in [-0.1, 0.1]</math> and <math display="inline">\nu \in [0.8, 1.5]</math>, lower and upper bounds on <math display="inline">\widetilde{\mu}</math> and <math display="inline">\widetilde{\nu}</math> were obtained to show that they stay within <math display="inline">\Omega</math>. <br />
<br />
First, it was shown that the derivatives of <math display="inline">\widetilde{\mu}</math> and <math display="inline">\widetilde{\xi}</math> with respect to <math display="inline">\mu</math> and <math display="inline">\nu</math> are either positive or have the sign of <math display="inline">\omega</math> in <math display="inline">\Omega</math>, so the minimum and maximum points are found at the borders. In <math display="inline">\Omega</math>, it then follows that<br />
<br />
\begin{align}<br />
-0.03106 <\widetilde{\mu}(-0.1,0.1, 0.8, 0.95, \lambda_{\rm 01}, \alpha_{\rm 01}) \leq & \widetilde{\mu} \leq \widetilde{\mu}(0.1,0.1,1.5, 1.1, \lambda_{\rm 01}, \alpha_{\rm 01}) < 0.06773<br />
\end{align}<br />
<br />
and <br />
<br />
\begin{align}<br />
0.80467 <\widetilde{\xi}(-0.1,0.1, 0.8, 0.95, \lambda_{\rm 01}, \alpha_{\rm 01}) \leq & \widetilde{\xi} \leq \widetilde{\xi}(0.1,0.1,1.5, 1.1, \lambda_{\rm 01}, \alpha_{\rm 01}) < 1.48617.<br />
\end{align}<br />
<br />
Since <math display="inline">\widetilde{\nu} = \widetilde{\xi} - \widetilde{\mu}^2</math>, <br />
<br />
\begin{align}<br />
0.80009 & \leqslant \widetilde{\nu} \leqslant 1.48617<br />
\end{align}<br />
<br />
The bounds on <math display="inline">\widetilde{\mu}</math> and <math display="inline">\widetilde{\nu}</math> are narrower than those for <math display="inline">\mu</math> and <math display="inline">\nu</math> set out in <math display="inline">\Omega</math>, therefore <math display="inline">g(\Omega) \subseteq \Omega</math>.<br />
<br />
==== Theorem 2: Decreasing Variance from Above ====<br />
<br />
'''Definition:''' For <math display="inline">\lambda = \lambda_{01}</math>, <math display="inline">\alpha = \alpha_{01}</math>, and the domain <math display="inline">\Omega^+: -1 \leqslant \mu \leqslant 1, -0.1 \leqslant \omega \leqslant 0.1, 3 \leqslant \nu \leqslant 16</math>, and <math display="inline">0.8 \leqslant \tau \leqslant 1.25</math>, we have for the mapping of the variance <math display="inline">\widetilde{\nu}(\mu, \omega, \nu, \tau, \lambda, \alpha)</math> under <math display="inline">g</math>: <math display="inline">\widetilde{\nu}(\mu, \omega, \nu, \tau, \lambda, \alpha) < \nu</math>.<br />
<br />
Theorem 2 states that when <math display="inline">\nu \in [3, 16]</math>, the mapping <math display="inline">g</math> draws it to below 3 when applied across layers, thereby establishing an upper bound of <math display="inline">\nu < 3</math> on variance.<br />
<br />
'''Proof:''' The authors proved the inequality by showing that <math display="inline">g(\mu, \omega, \xi, \tau, \lambda_{01}, \alpha_{01}) = \widetilde{\xi}(\mu, \omega, \xi, \tau, \lambda_{01}, \alpha_{01}) - \nu < 0</math>, since the second moment should be greater than or equal to variance <math display="inline">\widetilde{\nu}</math>. The behavior of <math display="inline">\frac{\partial }{\partial \mu } \widetilde{\xi}(\mu, \omega, \nu, \tau, \lambda, \alpha)</math>, <math display="inline">\frac{\partial }{\partial \omega } \widetilde{\xi}(\mu, \omega, \nu, \tau, \lambda, \alpha)</math>, <math display="inline">\frac{\partial }{\partial \nu } \widetilde{\xi}(\mu, \omega, \nu, \tau, \lambda, \alpha)</math>, and <math display="inline">\frac{\partial }{\partial \tau } \widetilde{\xi}(\mu, \omega, \nu, \tau, \lambda, \alpha)</math> are used to find the bounds on <math display="inline">g(\mu, \omega, \xi, \tau, \lambda_{01}, \alpha_{01})</math> (see pages 9 - 13 in the supplementary materials). Again, the partial derivative terms were monotonic, which made it possible to find the upper bound at the board values. It was shown that the maximum value of <math display="inline">g</math> does not exceed <math display="inline">-0.0180173</math>.<br />
<br />
==== Theorem 3: Increasing Variance from Below ====<br />
<br />
'''Definition''': We consider <math display="inline">\lambda = \lambda_{01}</math>, <math display="inline">\alpha = \alpha_{01}</math>, and the domain <math display="inline">\Omega^-: -0.1 \leqslant \mu \leqslant 0.1</math> and <math display="inline">-0.1 \leqslant \omega \leqslant 0.1</math>. For the domain <math display="inline">0.02 \leqslant \nu \leqslant 0.16</math> and <math display="inline">0.8 \leqslant \tau \leqslant 1.25</math> as well as for the domain <math display="inline">0.02 \leqslant \nu \leqslant 0.24</math> and <math display="inline">0.9 \leqslant \tau \leqslant 1.25</math>, the mapping of the variance <math display="inline">\widetilde{\nu}(\mu, \omega, \nu, \tau, \lambda, \alpha)</math> increases: <math display="inline">\widetilde{\nu}(\mu, \omega, \nu, \tau, \lambda, \alpha) > \nu</math>.<br />
<br />
Theorem 3 states that the variance <math display="inline">\widetilde{\nu}</math> increases when variance is smaller than in <math display="inline">Omega</math>. The lower bound on variance is <math display="inline">\widetilde{\nu} > 0.16</math> when <math display="inline">0.8 \leqslant \tau</math> and <math display="inline">\widetilde{\nu} > 0.24</math> when <math display="inline">0.9 \leqslant \tau</math> under the proposed mapping.<br />
<br />
'''Proof:''' According to the mean value theorem, for a <math display="inline">t \in [0, 1]</math>,<br />
<br />
[[File:th3.png|700px|center]]<br />
<br />
Similar to the proof for Theorem 2 (except we are interested in the smallest <math display="inline">\widetilde{\nu}</math> instead of the biggest), the lower bound for <math display="inline">\frac{\partial }{\partial \nu} \widetilde{\xi}(\mu,\omega,\nu+t(\nu_{\mathrm{min}}-\nu),\tau,\lambda_{\rm 01},\alpha_{\rm 01})</math> can be derived, and substituted into the relationship <math display="inline">\widetilde{\nu} = \widetilde{\xi}(\mu,\omega,\nu,\tau,\lambda_{\rm 01},\alpha_{\rm 01}) - (\widetilde{\mu}(\mu,\omega,\nu,\tau,\lambda_{\rm 01},\alpha_{\rm 01}))^2</math>. The lower bound depends on <math display="inline">\tau</math> and <math display="inline">\nu</math>, and in the <math display="inline">\Omega^{-1}</math> listed, it is slightly above <math display="inline">\nu</math>.<br />
<br />
== Implementation Details ==<br />
<br />
=== Initialization ===<br />
<br />
As previously explained, SNNs work best when inputs to the network are standardized, and the weights are initialized with mean of 0 and variance of <math display="inline">\frac{1}{n}</math> to help converge to the fixed point <math display="inline">(\mu, \nu) = (0, 1)</math>.<br />
<br />
=== Dropout Technique ===<br />
<br />
The authors reason that dropout, the act of randomly setting activations to 0 with probability <math display="inline">1 - q</math>, is not compatible with SELUs because the low variance region in SELUs is at <math display="inline">\lim_{x \rightarrow -\infty} = -\lambda \alpha</math>, not 0 (contrast this with ReLUs, which fits well with dropout and have <math display="inline">\lim_{x \rightarrow -\infty} = 0</math> as the saturation region). Additionally, activations need to be transformed (e.g. scaled) after dropout to maintain the same mean and variance. Therefore, a new dropout technique for SELUs was needed, termed ''alpha dropout''.<br />
<br />
With alpha dropout, activations are randomly set to <math display="inline">-\lambda\alpha = \alpha'</math>, or <math display="inline">1.7581</math>, by drawing from a Bernoulli distribution <math display="inline">d \sim B(1, q)</math>.<br />
<br />
The updated mean and variance of the activations are now:<br />
\[ \mathrm{E}(xd + \alpha'(1 - d)) = \mu q + \alpha'(1 - q) \] <br />
<br />
and<br />
<br />
\[ \mathrm{Var}(xd + \alpha'(1 - d)) = q((1-q)(\alpha' - \mu)^2 + \nu) \]<br />
<br />
To ensure that mean and variance are unchanged after dropout, the authors used an affine transformation <math display="inline">a(xd + \alpha'(1 - d) + b</math>, and solved for the values of <math display="inline">a</math> and <math display="inline">b</math> to give <math display="inline">a = (\frac{\nu}{q((1-q)(\alpha' - \mu)^2 + \nu)})^{\frac{1}{2}}</math> and <math display="inline">b = \mu - a(q\mu + (1-q)\alpha'))</math>. As the values for <math display="inline">\mu</math> and <math display="inline">\nu</math> are set to <math display="inline">0</math> and <math display="inline">1</math> throughout the paper, these expressions can be simplified into <math display="inline">a = (q + \alpha'^2 q(1-q))^{-\frac{1}{2}}</math> and <math display="inline">b = -(q + \alpha^2 q (1-q))^{-\frac{1}{2}}((1 - q)\alpha')</math>, where <math display="inline">\alpha' \approx 1.7581</math>.<br />
<br />
Empirically, the authors found that dropout rates of <math display="inline">0.05</math> or <math display="inline">0.10</math> worked well with SNNs.<br />
<br />
=== Optimizers ===<br />
<br />
Through experiments, the authors found that stochastic gradient descent, momentum, Adadelta and Adamax work well on SNNs. For Adam, configuration parameters <math display="inline">\beta_2 = 0.99</math> and <math display="inline">\epsilon = 0.01</math> were found to be more effective.<br />
<br />
==Experimental Results==<br />
<br />
Three sets of experiments were conducted to compare the performance of SNNs to six other FNN structures and to other machine learning algorithms, such as support vector machines and random forests. The experiments were carried out on (1) 121 UCI Machine Learning Repository datasets, (2) the Tox21 chemical compounds toxicity effects dataset (with 12,000 compounds and 270,000 features), and (3) the HTRU2 dataset of statistics on radio wave signals from pulsar candidates (with 18,000 observations and eight features). In each set of experiment, hyperparameter search was conducted on a validation set to select parameters such as the number of hidden units, number of hidden layers, learning rate, regularization parameter, and dropout rate (see pages 95 - 107 of the supplementary material for exact hyperparameters considered). Whenever models of different setups gave identical results on the validation data, preference was given to the structure with more layers, lower learning rate and higher dropout rate.<br />
<br />
The six FNN structures considered were: (1) FNNs with ReLU activations, no normalization and “Microsoft weight initialization” (MSRA) [5] to control the variance of input signals [5]; (2) FNNs with batch normalization [6], in which normalization is applied to activations of the same mini-batch; (3) FNNs with layer normalization [1], in which normalization is applied on a per layer basis for each training example; (4) FNNs with weight normalization [8], whereby each layer’s weights are normalized by learning the weight’s magnitude and direction instead of the weight vector itself; (5) highway networks, in which layers are not restricted to being sequentially connected [9]; and (6) an FNN-version of residual networks [4], with residual blocks made up of two or three densely connected layers.<br />
<br />
On the Tox21 dataset, the authors demonstrated the self-normalizing effect by comparing the distribution of neural inputs <math display="inline">z</math> at initialization and after 40 epochs of training to that of the standard normal. As Figure 3 show, the distribution of <math display="inline">z</math> remained similar to a normal distribution.<br />
<br />
[[File:snnf3.png|600px]]<br />
<br />
On all three sets of classification tasks, the authors demonstrated that SNN outperformed the other FNN counterparts on accuracy and AUC measures, came close to the state-of-the-art results on the Tox21 dataset with an 8-layer network, and produced a new state-of-the-art AUC on predicting pulsars for the HTRU2 dataset by a small margin (achieving an AUC 0.98, averaged over 10 cross-validation folds, versus the previous record of 0.976).<br />
<br />
On UCI datasets with fewer than 1,000 observations, SNNs did not outperform SVMs or random forests in terms of average rank in accuracy, but on datasets with at least 1,000 observations, SNNs showed the best overall performance (average rank of 5.8, compared to 6.1 for support vector machines and 6.6 for random forests). Through hyperparameter tuning, it was also discovered that the average depth of FNNs is 10.8 layers, more than the other FNN architectures tried.<br />
<br />
==Future Work==<br />
<br />
Although not the focus of this paper, the authors also briefly noted that their initial experiments with applying SELUs on relatively simple CNN structures showed promising results, which is not surprising given that ELUs, which do not have the self-normalizing property, has already been shown to work well with CNNs, demonstrating faster convergence than ReLU networks and even pushing the state-of-the-art error rates on CIFAR-100 at the time of publishing in 2015 [3].<br />
<br />
Since the paper was published, SELUs have been adopted by several researchers, not just with FNNs [https://github.com/bioinf-jku/SNNs see link], but also with CNNs, GANs, autoencoders, reinforcement learning and RNNs. In a few cases, researchers for those papers concluded that networks trained with SELUs converged faster than those trained with ReLUs, and that SELUs have the same convergence quality as batch normalization. There is potential for SELUs to be incorporated into more architectures in the future.<br />
<br />
==Critique==<br />
<br />
Overall, the authors presented a convincing case for using SELUs (along with proper initialization and alpha dropout) on FNNs. FNNs trained with SELU have more layers than those with other normalization techniques, so the work here provides a promising direction for making traditional FNNs more powerful. There are not as many well-established benchmark datasets to evaluate FNNs, but the experiments carried out, particularly on the larger Tox21 dataset, showed that SNNs can be very effective at classification tasks.<br />
<br />
The only question I have with the proofs is the lack of explanation for how the domains, <math display="inline">\Omega</math>, <math display="inline">\Omega^-</math> and <math display="inline">\Omega^+</math> are determined, which is an important consideration because they are used for deriving the upper and lower bounds on expressions needed for proving the three theorems. The ranges appear somewhat set through trial-and-error and heuristics to ensure the numbers work out (e.g. make the spectral norm [12] of <math display="inline">\mathcal{J}</math> as large as can be below 1 so as to ensure <math display="inline">g</math> is a contraction mapping), so it is not clear whether they are unique conditions, or that the parameters will remain within those prespecified ranges throughout training; and if the parameters can stray away from the ranges provided, then the issue of what will happen to the self-normalizing property was not addressed. Perhaps that is why the authors gave preference to models with a deeper structure and smaller learning rate during experiments to help the parameters stay within their domains. Further, in addition to the hyperparameters considered, it would be helpful to know the final values that went into the best-performing models, for a better understanding of what range of values work better for SNNs empirically.<br />
<br />
==Conclusion==<br />
<br />
The SNN structure proposed in this paper is built on the traditional FNN structure with a few modifications, including the use of SELUs as the activation function (with <math display="inline">\lambda \approx 1.0507</math> and <math display="inline">\alpha \approx 1.6733</math>), alpha dropout, network weights initialized with mean of zero and variance <math display="inline">\frac{1}{n}</math>, and inputs normalized to mean of zero and variance of one. It is simple to implement while being backed up by detailed theory. <br />
<br />
When properly initialized, SELUs will draw neural inputs towards a fixed point of zero mean and unit variance as the activations are propagated through the layers. The self-normalizing property is maintained even when weights deviate from their initial values during training (under mild conditions). When the variance of inputs goes beyond the prespecified range imposed, they are still bounded above and below so SNNs do not suffer from exploding and vanishing gradients. This self-normalizing property allows SNNs to be more robust to perturbations in stochastic gradient descent, so deeper structures with better prediction performance can be built. <br />
<br />
In the experiments conducted, the authors demonstrated that SNNs outperformed FNNs trained with other normalization techniques, such as batch, layer and weight normalization, and specialized architectures, such as highway or residual networks, on several classification tasks, including on the UCI Machine Learning Repository datasets. The adoption of SELUs by other researchers also lends credence to the potential for SELUs to be implemented in more neural network architectures.<br />
<br />
==References==<br />
<br />
# Ba, Kiros and Hinton. "Layer Normalization". arXiv:1607.06450. (2016).<br />
# Blinn. "Consider the Lowly 2X2 Matrix." IEEE Computer Graphics and Applications. (1996).<br />
# Clevert, Unterthiner, Hochreiter. "Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)." arXiv: 1511.07289. (2015).<br />
# He, Zhang, Ren and Sun. "Deep Residual Learning for Image Recognition." arXiv:1512.03385. (2015).<br />
# He, Zhang, Ren and Sun. "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification." arXiv:1502.01852. (2015). <br />
# Ioffe and Szegedy. "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariance Shift." arXiv:1502.03167. (2015).<br />
# Klambauer, Unterthiner, Mayr and Hochreiter. "Self-Normalizing Neural Networks." arXiv: 1706.02515. (2017).<br />
# Salimans and Kingma. "Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks." arXiv:1602.07868. (2016).<br />
# Srivastava, Greff and Schmidhuber. "Highway Networks." arXiv:1505.00387 (2015).<br />
# Unterthiner, Mayr, Klambauer and Hochreiter. "Toxicity Prediction Using Deep Learning." arXiv:1503.01445. (2015). <br />
# https://en.wikipedia.org/wiki/Central_limit_theorem <br />
# http://mathworld.wolfram.com/SpectralNorm.html <br />
# https://www.math.umd.edu/~petersd/466/fixedpoint.pdf<br />
<br />
==Online Resources==<br />
https://github.com/bioinf-jku/SNNs (GitHub repository maintained by some of the paper's authors)<br />
<br />
==Footnotes==<br />
<br />
1. Error propagation analysis: The authors performed an error analysis to quantify the potential numerical imprecisions propagated through the numerous operations performed. The potential imprecision <math display="inline">\epsilon</math> was quantified by applying the mean value theorem<br />
<br />
\[ |f(x + \Delta x - f(x)| \leqslant ||\triangledown f(x + t\Delta x|| ||\Delta x|| \textrm{ for } t \in [0, 1]\textrm{.} \] <br />
<br />
The error propagation rules, or <math display="inline">|f(x + \Delta x - f(x)|</math>, was first obtained for simple operations such as addition, subtraction, multiplication, division, square root, exponential function, error function and complementary error function. Them, the error bounds on the compound terms making up <math display="inline">\Delta (S(\mu, \omega, \nu, \tau, \lambda, \alpha)</math> were found by decomposing them into the simpler expressions. If each of the variables have a precision of <math display="inline">\epsilon</math>, then it turns out <math display="inline">S</math> has a precision better than <math display="inline">292\epsilon</math>. For a machine with a precision of <math display="inline">2^{-56}</math>, the rounding error is <math display="inline">\epsilon \approx 10^{-16}</math>, and <math display="inline">292\epsilon < 10^{-13}</math>. In addition, all computations are correct up to 3 ulps (“unit in last place”) for the hardware architectures and GNU C library used, with 1 ulp being the highest precision that can be achieved.<br />
<br />
2. Independence Assumption: The classic definition of central limit theorem requires <math display="inline">x_i</math>’s to be independent and identically distributed, which is not guaranteed to hold true in a neural network layer. However, according to the Lyapunov CLT, the <math display="inline">x_i</math>’s do not need to be identically distributed as long as the <math display="inline">(2 + \delta)</math>th moment exists for the variables and meet the Lyapunov condition for the rate of growth of the sum of the moments [11]. In addition, CLT has also shown to be valid under weak dependence under mixing conditions [11]. Therefore, the authors argue that the central limit theorem can be applied with network inputs.<br />
<br />
3. <math display="inline">\mathcal{H}</math> versus <math display="inline">\mathcal{J}</math> Jacobians: In solving for the largest singular value of the Jacobian <math display="inline">\mathcal{H}</math> for the mapping <math display="inline">g: (\mu, \nu)</math>, the authors first worked with the terms in the Jacobian <math display="inline">\mathcal{J}</math> for the mapping <math display="inline">h: (\mu, \nu) \rightarrow (\widetilde{\mu}, \widetilde{\xi})</math> instead, because the influence of <math display="inline">\widetilde{\mu}</math> on <math display="inline">\widetilde{\nu}</math> is small when <math display="inline">\widetilde{\mu}</math> is small in <math display="inline">\Omega</math> and <math display="inline">\mathcal{H}</math> can be easily expressed as terms in <math display="inline">\mathcal{J}</math>. <math display="inline">\mathcal{J}</math> was referenced in the paper, but I used <math display="inline">\mathcal{H}</math> in the summary here to avoid confusion.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Self_Normalizing_Neural_Networks&diff=32894stat946w18/Self Normalizing Neural Networks2018-03-07T18:13:03Z<p>Pa2forsy: /* Key Concepts */</p>
<hr />
<div>==Introduction and Motivation==<br />
<br />
While neural networks have been making a lot of headway in improving benchmark results and narrowing the gap with human-level performance, success has been fairly limited to visual and sequential processing tasks through advancements in convolutional network and recurrent network structures. Most data science competitions outside of those domains are still being won by algorithms such as gradient boosting and random forests. The traditional (densely connected) feed-forward neural networks (FNNs) are rarely used competitively, and when they do win on the rare occasions, they are won with very shallow networks with just up to four layers [10].<br />
<br />
The authors, Klambauer et al., believe that what prevents FNNs from becoming more useful is the inability to train a deeper FNN structure, which would allow the network to learn more levels of abstract representations. To have a deeper network, oscillations in the distribution of activations need to be kept under control so that stable gradients can be obtained during training. Several techniques are available to normalize activations, including batch normalization [6], layer normalization [1] and weight normalization [8]. These methods work well with CNNs and RNNs, but not so much with FNNs because backpropagating through normalization parameters introduces additional variance to the gradients, and regularization techniques like dropout further perturb the normalization effect. CNNs and RNNs are less sensitive to such perturbations, presumably due to their weight sharing architecture, but FNNs do not have such property, and thus suffer from high variance in training errors, which hinders learning. Furthermore, the aforementioned normalization techniques involving adding external layers to the model and can slow down computations. <br />
<br />
Therefore, the authors were motivated to develop a new FNN implementation that can achieve the intended effect of normalization techniques that works well with stochastic gradient descent and dropout. Self-normalizing neural networks (SNNs) are based on the idea of scaled exponential linear units (SELU), a new activation function introduced in this paper, whose output distribution is proved to converge to a fixed point, thus making it possible to train deeper networks. <br />
<br />
==Notations==<br />
<br />
As the paper (primarily in the supplementary materials) comes with lengthy proofs, important notations are listed first.<br />
<br />
Consider two fully-connected layers, let <math display="inline">x</math> denote the inputs to the second layer, then <math display="inline">z = Wx</math> represents the network inputs of the second layer, and <math display="inline">y = f(z)</math> represents the activations in the second layer.<br />
<br />
Assume that all <math display="inline">x_i</math>'s, <math display="inline">1 \leqslant i \leqslant n</math>, have mean <math display="inline">\mu := \mathrm{E}(x_i)</math> and variance <math display="inline">\nu := \mathrm{Var}(x_i)</math> and that each <math display="inline">y</math> has mean <math display="inline">\widetilde{\mu} := \mathrm{E}(y)</math> and variance <math display="inline">\widetilde{\nu} := \mathrm{Var}(y)</math>, then let <math display="inline">g</math> be the set of functions that maps <math display="inline">(\mu, \nu)</math> to <math display="inline">(\widetilde{\mu}, \widetilde{\nu})</math>. <br />
<br />
For the weight vector <math display="inline">w</math>, <math display="inline">n</math> times the mean of the weight vector is <math display="inline">\omega := \sum_{i = 1}^n \omega_i</math> and <math display="inline">n</math> times the second moment is <math display="inline">\tau := \sum_{i = 1}^{n} w_i^2</math>.<br />
<br />
==Key Concepts==<br />
<br />
===Self-Normalizing Neural-Net (SNN)===<br />
<br />
''A neural network is self-normalizing if it possesses a mapping <math display="inline">g: \Omega \rightarrow \Omega</math> for each activation <math display="inline">y</math> that maps mean and variance from one layer to the next and has a stable and attracting fixed point depending on <math display="inline">(\omega, \tau)</math> in <math display="inline">\Omega</math>. Furthermore, the mean and variance remain in the domain <math display="inline">\Omega</math>, that is <math display="inline">g(\Omega) \subseteq \Omega</math>, where <math display="inline">\Omega = \{ (\mu, \nu) | \mu \in [\mu_{min}, \mu_{max}], \nu \in [\nu_{min}, \nu_{max}] \}</math>. When iteratively applying the mapping <math display="inline">g</math>, each point within <math display="inline">\Omega</math> converges to this fixed point.''<br />
<br />
In other words, in SNNs, if the inputs from an earlier layer (<math display="inline">x</math>) already have their mean and variance within a predefined interval <math display="inline">\Omega</math>, then the activations to the next layer (<math display="inline">y = f(z = Wx)</math>) should remain within those intervals. This is true across all pairs of connecting layers as the normalizing effect gets propagated through the network, hence why the term self-normalizing. When the mapping is applied iteratively, it should draw the mean and variance values closer to a fixed point within <math display="inline">\Omega</math>, the value of which depends on <math display="inline">\omega</math> and <math display="inline">\tau</math> (recall that they are from the weight vector).<br />
<br />
The activation function that makes an SNN possible should meet the following four conditions:<br />
<br />
# It can take on both negative and positive values, so it can normalize the mean;<br />
# It has a saturation region, so it can dampen variances that are too large;<br />
# It has a slope larger than one, so it can increase variances that are too small; and<br />
# It is a continuous curve, which is necessary for the fixed point to exist (see the definition of Banach fixed point theorem to follow).<br />
<br />
Commonly used activation functions such as rectified linear units (ReLU), sigmoid, tanh, leaky ReLUs and exponential linear units (ELUs) do not meet all four criteria, therefore, a new activation function is needed.<br />
<br />
===Scaled Exponential Linear Units (SELUs)===<br />
<br />
One of the main ideas introduced in this paper is the SELU function. As the name suggests, it is closely related to ELU [3],<br />
<br />
\[ \mathrm{elu}(x) = \begin{cases} x & x > 0 \\<br />
\alpha e^x - \alpha & x \leqslant 0<br />
\end{cases} \]<br />
<br />
but further builds upon it by introducing a new scale parameter $\lambda$ and proving the exact values that $\alpha$ and $\lambda$ should take on to achieve self-normalization. SELU is defined as:<br />
<br />
\[ \mathrm{selu}(x) = \lambda \begin{cases} x & x > 0 \\<br />
\alpha e^x - \alpha & x \leqslant 0<br />
\end{cases} \]<br />
<br />
SELUs meet all four criteria listed above - it takes on positive values when <math display="inline">x > 0</math> and negative values when <math display="inline">x < 0</math>, it has a saturation region when <math display="inline">x</math> is a larger negative value, the value of <math display="inline">\lambda</math> can be set to greater than one to ensure a slope greater than one, and it is continuous at <math display="inline">x = 0</math>. <br />
<br />
Figure 1 below gives an intuition for how SELUs normalize activations across layers. As shown, a variance dampening effect occurs when inputs are negative and far away from zero, and a variance increasing effect occurs when inputs are close to zero.<br />
<br />
[[File:snnf1.png|500px]]<br />
<br />
Figure 2 below plots the progression of training error on the MNIST and CIFAR10 datasets when training with SNNs versus FNNs with batch normalization at varying model depths. As shown, FNNs that adopted the SELU activation function exhibited lower and less variable training loss compared to using batch normalization, even as the depth increased to 16 and 32 layers.<br />
<br />
[[File:snnf2.png|600px]]<br />
<br />
=== Banach Fixed Point Theorem and Contraction Mappings ===<br />
<br />
The underlying theory behind SNNs is the Banach fixed point theorem, which states the following: ''Let <math display="inline">(X, d)</math> be a non-empty complete metric space with a contraction mapping <math display="inline">f: X \rightarrow X</math>. Then <math display="inline">f</math> has a unique fixed point <math display="inline">x_f \subseteq X</math> with <math display="inline">f(x_f) = x_f</math>. Every sequence <math display="inline">x_n = f(x_{n-1})</math> with starting element <math display="inline">x_0 \subseteq X</math> converges to the fixed point: <math display="inline">x_n \underset{n \rightarrow \infty}\rightarrow x_f</math>.''<br />
<br />
A contraction mapping is a function <math display="inline">f: X \rightarrow X</math> on a metric space <math display="inline">X</math> with distance <math display="inline">d</math>, such that for all points <math display="inline">\mathbf{u}</math> and <math display="inline">\mathbf{v}</math> in <math display="inline">X</math>: <math display="inline">d(f(\mathbf{u}), f(\mathbf{v})) \leqslant \delta d(\mathbf{u}, \mathbf{v})</math>, for a <math display="inline">0 \leqslant \delta \leqslant 1</math>.<br />
<br />
The easiest way to prove a contraction mapping is usually to show that the spectral norm [12] of its Jacobian is less than 1 [13], as was done for this paper.<br />
<br />
==Proving the Self-Normalizing Property==<br />
<br />
===Mean and Variance Mapping Function===<br />
<br />
<math display="inline">g</math> is derived under the assumption that <math display="inline">x_i</math>'s are independent but not necessarily having the same mean and variance [[#Footnotes |(2)]]. Under this assumption (and recalling earlier notation of <math display="inline">\omega</math> and <math display="inline">\tau</math>),<br />
<br />
\begin{align}<br />
\mathrm{E}(z = \mathbf{w}^T \mathbf{x}) = \sum_{i = 1}^n w_i \mathrm{E}(x_i) = \mu \omega<br />
\end{align}<br />
<br />
\begin{align}<br />
\mathrm{Var}(z) = \mathrm{Var}(\sum_{i = 1}^n w_i x_i) = \sum_{i = 1}^n w_i^2 \mathrm{Var}(x_i) = \nu \sum_{i = 1}^n w_i^2 = \nu\tau \textrm{ .}<br />
\end{align}<br />
<br />
When the weight terms are normalized, <math display="inline">z</math> can be viewed as a weighted sum of <math display="inline">x_i</math>'s. Wide neural net layers with a large number of nodes is common, so <math display="inline">n</math> is usually large, and by the Central Limit Theorem, <math display="inline">z</math> approaches a normal distribution <math display="inline">\mathcal{N}(\mu\omega, \sqrt{\nu\tau})</math>. <br />
<br />
Using the above property, the exact form for <math display="inline">g</math> can be obtained using the definitions for mean and variance of continuous random variables: <br />
<br />
[[File:gmapping.png|600px|center]]<br />
<br />
Analytical solutions for the integrals can be obtained as follows: <br />
<br />
[[File:gintegral.png|600px|center]]<br />
<br />
The authors are interested in the fixed point <math display="inline">(\mu, \nu) = (0, 1)</math> as these are the parameters associated with the common standard normal distribution. The authors also proposed using normalized weights such that <math display="inline">\omega = \sum_{i = 1}^n = 0</math> and <math display="inline">\tau = \sum_{i = 1}^n w_i^2= 1</math> as it gives a simpler, cleaner expression for <math display="inline">\widetilde{\mu}</math> and <math display="inline">\widetilde{\nu}</math> in the calculations in the next steps. This weight scheme can be achieved in several ways, for example, by drawing from a normal distribution <math display="inline">\mathcal{N}(0, \frac{1}{n})</math> or from a uniform distribution <math display="inline">U(-\sqrt{3}, \sqrt{3})</math>.<br />
<br />
At <math display="inline">\widetilde{\mu} = \mu = 0</math>, <math display="inline">\widetilde{\nu} = \nu = 1</math>, <math display="inline">\omega = 0</math> and <math display="inline">\tau = 1</math>, the constants <math display="inline">\lambda</math> and <math display="inline">\alpha</math> from the SELU function can be solved for - <math display="inline">\lambda_{01} \approx 1.0507</math> and <math display="inline">\alpha_{01} \approx 1.6733</math>. These values are used throughout the rest of the paper whenever an expression calls for <math display="inline">\lambda</math> and <math display="inline">\alpha</math>.<br />
<br />
===Self-Normalizing Property Under Normalized Weights===<br />
<br />
With the weights normalized, it is possible to calculate the exact value for the spectral norm [12] of <math display="inline">g</math>'s Jacobian around the fixed point <math display="inline">(\mu, \nu) = (0, 1)</math>, which turns out to be <math display="inline">0.7877</math>. Thus, at initialization, SNNs have a stable and attracting fixed point at <math display="inline">(0, 1)</math>, which means that when <math display="inline">g</math> is applied iteratively to a pair <math display="inline">(\mu_{new}, \nu_{new})</math>, it should draw the points closer to <math display="inline">(0, 1)</math>. The rate of convergence is determined by the spectral norm [12], whose value depends on <math display="inline">\mu</math>, <math display="inline">\nu</math>, <math display="inline">\omega</math> and <math display="inline">\tau</math>.<br />
<br />
===Self-Normalizing Property Under Unnormalized Weights===<br />
<br />
As weights are updated during training, there is no guarantee that they would remain normalized. The authors addressed this issue through the first key theorem presented in the paper, which states that a fixed point close to (0, 1) can still be obtained if <math display="inline">\mu</math>, <math display="inline">\nu</math>, <math display="inline">\omega</math> and <math display="inline">\tau</math> are restricted to a specified range. <br />
<br />
Additionally, there is no guarantee that the mean and variance of the inputs would stay within the range given by the first theorem, which led to the development of theorems #2 and #3. These two theorems established an upper and lower bound on the variance of inputs if the variance of activations from the previous layer are above or below the range specified, respectively. This ensures that the variance would not explode or vanish after being propagated through the network.<br />
<br />
The theorems come with lengthy proofs in the supplementary materials for the paper. High-level proof sketches are presented here.<br />
<br />
====Theorem 1: Stable and Attracting Fixed Points Close to (0, 1)====<br />
<br />
'''Definition:''' We assume <math display="inline">\alpha = \alpha_{01}</math> and <math display="inline">\lambda = \lambda_{01}</math>. We restrict the range of the variables to the domain <math display="inline">\mu \in [-0.1, 0.1]</math>, <math display="inline">\omega \in [-0.1, 0.1]</math>, <math display="inline">\nu \in [0.8, 1.5]</math>, and <math display="inline">\tau \in [0.9, 1.1]</math>. For <math display="inline">\omega = 0</math> and <math display="inline">\tau = 1</math>, the mapping has the stable fixed point <math display="inline">(\mu, \nu) = (0, 1</math>. For other <math display="inline">\omega</math> and <math display="inline">\tau</math>, g has a stable and attracting fixed point depending on <math display="inline">(\omega, \tau)</math> in the <math display="inline">(\mu, \nu)</math>-domain: <math display="inline">\mu \in [-0.03106, 0.06773]</math> and <math display="inline">\nu \in [0.80009, 1.48617]</math>. All points within the <math display="inline">(\mu, \nu)</math>-domain converge when iteratively applying the mapping to this fixed point.<br />
<br />
'''Proof:''' In order to show the the mapping <math display="inline">g</math> has a stable and attracting fixed point close to <math display="inline">(0, 1)</math>, the authors again applied Banach's fixed point theorem, which states that a contraction mapping on a nonempty complete metric space that does not map outside its domain has a unique fixed point, and that all points in the <math display="inline">(\mu, \nu)</math>-domain converge to the fixed point when <math display="inline">g</math> is iteratively applied. <br />
<br />
The two requirements are proven as follows:<br />
<br />
'''1. g is a contraction mapping.'''<br />
<br />
For <math display="inline">g</math> to be a contraction mapping in <math display="inline">\Omega</math> with distance <math display="inline">||\cdot||_2</math>, there must exist a Lipschitz constant <math display="inline">M < 1</math> such that: <br />
<br />
\begin{align} <br />
\forall \mu, \nu \in \Omega: ||g(\mu) - g(\nu)||_2 \leqslant M||\mu - \nu||_2 <br />
\end{align}<br />
<br />
As stated earlier, <math display="inline">g</math> is a contraction mapping if the spectral norm [12] of the Jacobian <math display="inline">\mathcal{H}</math> [[#Footnotes | (3)]] is below one, or equivalently, if the the largest singular value of <math display="inline">\mathcal{H}</math> is less than 1.<br />
<br />
To find the singular values of <math display="inline">\mathcal{H}</math>, the authors used an explicit formula derived by Blinn [2] for <math display="inline">2\times2</math> matrices, which states that the largest singular value of the matrix is <math display="inline">\frac{1}{2}(\sqrt{(a_{11} + a_{22}) ^ 2 + (a_{21} - a{12})^2} + \sqrt{(a_{11} - a_{22}) ^ 2 + (a_{21} + a{12})^2})</math>.<br />
<br />
For <math display="inline">\mathcal{H}</math>, an expression for the largest singular value of <math display="inline">\mathcal{H}</math>, made up of the first-order partial derivatives of the mapping <math display="inline">g</math> with respect to <math display="inline">\mu</math> and <math display="inline">\nu</math>, can be derived given the analytical solutions for <math display="inline">\widetilde{\mu}</math> and <math display="inline">\widetilde{\nu}</math> (and denoted <math display="inline">S(\mu, \omega, \nu, \tau, \lambda, \alpha)</math>).<br />
<br />
From the mean value theorem, we know that for a <math display="inline">t \in [0, 1]</math>, <br />
<br />
[[File:seq.png|600px|center]]<br />
<br />
Therefore, the distance of the singular value at <math display="inline">S(\mu, \omega, \nu, \tau, \lambda_{\mathrm{01}}, \alpha_{\mathrm{01}})</math> and at <math display="inline">S(\mu + \Delta\mu, \omega + \Delta\omega, \nu + \Delta\nu, \tau \Delta\tau, \lambda_{\mathrm{01}}, \alpha_{\mathrm{01}})</math> can be bounded above by <br />
<br />
[[File:seq2.png|600px|center]]<br />
<br />
An upper bound was obtained for each partial derivative term above, mainly through algebraic reformulations and by making use of the fact that many of the functions are monotonically increasing or decreasing on the variables they depend on in <math display="inline">\Omega</math> (see pages 17 - 25 in the supplementary materials).<br />
<br />
The <math display="inline">\Delta</math> terms were then set (rather arbitrarily) to be: <math display="inline">\Delta \mu=0.0068097371</math>,<br />
<math display="inline">\Delta \omega=0.0008292885</math>, <math display="inline">\Delta \nu=0.0009580840</math>, and <math display="inline">\Delta \tau=0.0007323095</math>. Plugging in the upper bounds on the absolute values of the derivative terms for <math display="inline">S</math> and the <math display="inline">\Delta</math> terms yields<br />
<br />
\[ S(\mu + \Delta \mu,\omega + \Delta \omega,\nu + \Delta \nu,\tau + \Delta \tau,\lambda_{\rm 01},\alpha_{\rm 01}) - S(\mu,\omega,\nu,\tau,\lambda_{\rm 01},\alpha_{\rm 01}) < 0.008747 \]<br />
<br />
Next, the largest singular value is found from a computer-assisted fine grid-search [[#Footnotes | (1)]] over the domain <math display="inline">\Omega</math>, with grid lengths <math display="inline">\Delta \mu=0.0068097371</math>, <math display="inline">\Delta \omega=0.0008292885</math>, <math display="inline">\Delta \nu=0.0009580840</math>, and <math display="inline">\Delta \tau=0.0007323095</math>, which turned out to be <math display="inline">0.9912524171058772</math>. Therefore, <br />
<br />
\[ S(\mu + \Delta \mu,\omega + \Delta \omega,\nu + \Delta \nu,\tau + \Delta \tau,\lambda_{\rm 01},\alpha_{\rm 01}) \leq 0.9912524171058772 + 0.008747 < 1 \]<br />
<br />
Since the largest singular value is smaller than 1, <math display="inline>g</math> is a contraction mapping.<br />
<br />
'''2. g does not map outside its domain.'''<br />
<br />
To prove that <math display="inline">g</math> does not map outside of the domain <math display="inline">\mu \in [-0.1, 0.1]</math> and <math display="inline">\nu \in [0.8, 1.5]</math>, lower and upper bounds on <math display="inline">\widetilde{\mu}</math> and <math display="inline">\widetilde{\nu}</math> were obtained to show that they stay within <math display="inline">\Omega</math>. <br />
<br />
First, it was shown that the derivatives of <math display="inline">\widetilde{\mu}</math> and <math display="inline">\widetilde{\xi}</math> with respect to <math display="inline">\mu</math> and <math display="inline">\nu</math> are either positive or have the sign of <math display="inline">\omega</math> in <math display="inline">\Omega</math>, so the minimum and maximum points are found at the borders. In <math display="inline">\Omega</math>, it then follows that<br />
<br />
\begin{align}<br />
-0.03106 <\widetilde{\mu}(-0.1,0.1, 0.8, 0.95, \lambda_{\rm 01}, \alpha_{\rm 01}) \leq & \widetilde{\mu} \leq \widetilde{\mu}(0.1,0.1,1.5, 1.1, \lambda_{\rm 01}, \alpha_{\rm 01}) < 0.06773<br />
\end{align}<br />
<br />
and <br />
<br />
\begin{align}<br />
0.80467 <\widetilde{\xi}(-0.1,0.1, 0.8, 0.95, \lambda_{\rm 01}, \alpha_{\rm 01}) \leq & \widetilde{\xi} \leq \widetilde{\xi}(0.1,0.1,1.5, 1.1, \lambda_{\rm 01}, \alpha_{\rm 01}) < 1.48617.<br />
\end{align}<br />
<br />
Since <math display="inline">\widetilde{\nu} = \widetilde{\xi} - \widetilde{\mu}^2</math>, <br />
<br />
\begin{align}<br />
0.80009 & \leqslant \widetilde{\nu} \leqslant 1.48617<br />
\end{align}<br />
<br />
The bounds on <math display="inline">\widetilde{\mu}</math> and <math display="inline">\widetilde{\nu}</math> are narrower than those for <math display="inline">\mu</math> and <math display="inline">\nu</math> set out in <math display="inline">\Omega</math>, therefore <math display="inline">g(\Omega) \subseteq \Omega</math>.<br />
<br />
==== Theorem 2: Decreasing Variance from Above ====<br />
<br />
'''Definition:''' For <math display="inline">\lambda = \lambda_{01}</math>, <math display="inline">\alpha = \alpha_{01}</math>, and the domain <math display="inline">\Omega^+: -1 \leqslant \mu \leqslant 1, -0.1 \leqslant \omega \leqslant 0.1, 3 \leqslant \nu \leqslant 16</math>, and <math display="inline">0.8 \leqslant \tau \leqslant 1.25</math>, we have for the mapping of the variance <math display="inline">\widetilde{\nu}(\mu, \omega, \nu, \tau, \lambda, \alpha)</math> under <math display="inline">g</math>: <math display="inline">\widetilde{\nu}(\mu, \omega, \nu, \tau, \lambda, \alpha) < \nu</math>.<br />
<br />
Theorem 2 states that when <math display="inline">\nu \in [3, 16]</math>, the mapping <math display="inline">g</math> draws it to below 3 when applied across layers, thereby establishing an upper bound of <math display="inline">\nu < 3</math> on variance.<br />
<br />
'''Proof:''' The authors proved the inequality by showing that <math display="inline">g(\mu, \omega, \xi, \tau, \lambda_{01}, \alpha_{01}) = \widetilde{\xi}(\mu, \omega, \xi, \tau, \lambda_{01}, \alpha_{01}) - \nu < 0</math>, since the second moment should be greater than or equal to variance <math display="inline">\widetilde{\nu}</math>. The behavior of <math display="inline">\frac{\partial }{\partial \mu } \widetilde{\xi}(\mu, \omega, \nu, \tau, \lambda, \alpha)</math>, <math display="inline">\frac{\partial }{\partial \omega } \widetilde{\xi}(\mu, \omega, \nu, \tau, \lambda, \alpha)</math>, <math display="inline">\frac{\partial }{\partial \nu } \widetilde{\xi}(\mu, \omega, \nu, \tau, \lambda, \alpha)</math>, and <math display="inline">\frac{\partial }{\partial \tau } \widetilde{\xi}(\mu, \omega, \nu, \tau, \lambda, \alpha)</math> are used to find the bounds on <math display="inline">g(\mu, \omega, \xi, \tau, \lambda_{01}, \alpha_{01})</math> (see pages 9 - 13 in the supplementary materials). Again, the partial derivative terms were monotonic, which made it possible to find the upper bound at the board values. It was shown that the maximum value of <math display="inline">g</math> does not exceed <math display="inline">-0.0180173</math>.<br />
<br />
==== Theorem 3: Increasing Variance from Below ====<br />
<br />
'''Definition''': We consider <math display="inline">\lambda = \lambda_{01}</math>, <math display="inline">\alpha = \alpha_{01}</math>, and the domain <math display="inline">\Omega^-: -0.1 \leqslant \mu \leqslant 0.1</math> and <math display="inline">-0.1 \leqslant \omega \leqslant 0.1</math>. For the domain <math display="inline">0.02 \leqslant \nu \leqslant 0.16</math> and <math display="inline">0.8 \leqslant \tau \leqslant 1.25</math> as well as for the domain <math display="inline">0.02 \leqslant \nu \leqslant 0.24</math> and <math display="inline">0.9 \leqslant \tau \leqslant 1.25</math>, the mapping of the variance <math display="inline">\widetilde{\nu}(\mu, \omega, \nu, \tau, \lambda, \alpha)</math> increases: <math display="inline">\widetilde{\nu}(\mu, \omega, \nu, \tau, \lambda, \alpha) > \nu</math>.<br />
<br />
Theorem 3 states that the variance <math display="inline">\widetilde{\nu}</math> increases when variance is smaller than in <math display="inline">Omega</math>. The lower bound on variance is <math display="inline">\widetilde{\nu} > 0.16</math> when <math display="inline">0.8 \leqslant \tau</math> and <math display="inline">\widetilde{\nu} > 0.24</math> when <math display="inline">0.9 \leqslant \tau</math> under the proposed mapping.<br />
<br />
'''Proof:''' According to the mean value theorem, for a <math display="inline">t \in [0, 1]</math>,<br />
<br />
[[File:th3.png|700px|center]]<br />
<br />
Similar to the proof for Theorem 2 (except we are interested in the smallest <math display="inline">\widetilde{\nu}</math> instead of the biggest), the lower bound for <math display="inline">\frac{\partial }{\partial \nu} \widetilde{\xi}(\mu,\omega,\nu+t(\nu_{\mathrm{min}}-\nu),\tau,\lambda_{\rm 01},\alpha_{\rm 01})</math> can be derived, and substituted into the relationship <math display="inline">\widetilde{\nu} = \widetilde{\xi}(\mu,\omega,\nu,\tau,\lambda_{\rm 01},\alpha_{\rm 01}) - (\widetilde{\mu}(\mu,\omega,\nu,\tau,\lambda_{\rm 01},\alpha_{\rm 01}))^2</math>. The lower bound depends on <math display="inline">\tau</math> and <math display="inline">\nu</math>, and in the <math display="inline">\Omega^{-1}</math> listed, it is slightly above <math display="inline">\nu</math>.<br />
<br />
== Implementation Details ==<br />
<br />
=== Initialization ===<br />
<br />
As previously explained, SNNs work best when inputs to the network are standardized, and the weights are initialized with mean of 0 and variance of <math display="inline">\frac{1}{n}</math> to help converge to the fixed point <math display="inline">(\mu, \nu) = (0, 1)</math>.<br />
<br />
=== Dropout Technique ===<br />
<br />
The authors reason that dropout, the act of randomly setting activations to 0 with probability <math display="inline">1 - q</math>, is not compatible with SELUs because the low variance region in SELUs is at <math display="inline">\lim_{x \rightarrow -\infty} = -\lambda \alpha</math>, not 0 (contrast this with ReLUs, which fits well with dropout and have <math display="inline">\lim_{x \rightarrow -\infty} = 0</math> as the saturation region). Additionally, activations need to be transformed (e.g. scaled) after dropout to maintain the same mean and variance. Therefore, a new dropout technique for SELUs was needed, termed ''alpha dropout''.<br />
<br />
With alpha dropout, activations are randomly set to <math display="inline">-\lambda\alpha = \alpha'</math>, or <math display="inline">1.7581</math>, by drawing from a Bernoulli distribution <math display="inline">d \sim B(1, q)</math>.<br />
<br />
The updated mean and variance of the activations are now:<br />
\[ \mathrm{E}(xd + \alpha'(1 - d)) = \mu q + \alpha'(1 - q) \] <br />
<br />
and<br />
<br />
\[ \mathrm{Var}(xd + \alpha'(1 - d)) = q((1-q)(\alpha' - \mu)^2 + \nu) \]<br />
<br />
To ensure that mean and variance are unchanged after dropout, the authors used an affine transformation <math display="inline">a(xd + \alpha'(1 - d) + b</math>, and solved for the values of <math display="inline">a</math> and <math display="inline">b</math> to give <math display="inline">a = (\frac{\nu}{q((1-q)(\alpha' - \mu)^2 + \nu)})^{\frac{1}{2}}</math> and <math display="inline">b = \mu - a(q\mu + (1-q)\alpha'))</math>. As the values for <math display="inline">\mu</math> and <math display="inline">\nu</math> are set to <math display="inline">0</math> and <math display="inline">1</math> throughout the paper, these expressions can be simplified into <math display="inline">a = (q + \alpha'^2 q(1-q))^{-\frac{1}{2}}</math> and <math display="inline">b = -(q + \alpha^2 q (1-q))^{-\frac{1}{2}}((1 - q)\alpha')</math>, where <math display="inline">\alpha' \approx 1.7581</math>.<br />
<br />
Empirically, the authors found that dropout rates of <math display="inline">0.05</math> or <math display="inline">0.10</math> worked well with SNNs.<br />
<br />
=== Optimizers ===<br />
<br />
Through experiments, the authors found that stochastic gradient descent, momentum, Adadelta and Adamax work well on SNNs. For Adam, configuration parameters <math display="inline">\beta_2 = 0.99</math> and <math display="inline">\epsilon = 0.01</math> were found to be more effective.<br />
<br />
==Experimental Results==<br />
<br />
Three sets of experiments were conducted to compare the performance of SNNs to six other FNN structures and to other machine learning algorithms, such as support vector machines and random forests. The experiments were carried out on (1) 121 UCI Machine Learning Repository datasets, (2) the Tox21 chemical compounds toxicity effects dataset (with 12,000 compounds and 270,000 features), and (3) the HTRU2 dataset of statistics on radio wave signals from pulsar candidates (with 18,000 observations and eight features). In each set of experiment, hyperparameter search was conducted on a validation set to select parameters such as the number of hidden units, number of hidden layers, learning rate, regularization parameter, and dropout rate (see pages 95 - 107 of the supplementary material for exact hyperparameters considered). Whenever models of different setups gave identical results on the validation data, preference was given to the structure with more layers, lower learning rate and higher dropout rate.<br />
<br />
The six FNN structures considered were: (1) FNNs with ReLU activations, no normalization and “Microsoft weight initialization” (MSRA) [5] to control the variance of input signals [5]; (2) FNNs with batch normalization [6], in which normalization is applied to activations of the same mini-batch; (3) FNNs with layer normalization [1], in which normalization is applied on a per layer basis for each training example; (4) FNNs with weight normalization [8], whereby each layer’s weights are normalized by learning the weight’s magnitude and direction instead of the weight vector itself; (5) highway networks, in which layers are not restricted to being sequentially connected [9]; and (6) an FNN-version of residual networks [4], with residual blocks made up of two or three densely connected layers.<br />
<br />
On the Tox21 dataset, the authors demonstrated the self-normalizing effect by comparing the distribution of neural inputs <math display="inline">z</math> at initialization and after 40 epochs of training to that of the standard normal. As Figure 3 show, the distribution of <math display="inline">z</math> remained similar to a normal distribution.<br />
<br />
[[File:snnf3.png|600px]]<br />
<br />
On all three sets of classification tasks, the authors demonstrated that SNN outperformed the other FNN counterparts on accuracy and AUC measures, came close to the state-of-the-art results on the Tox21 dataset with an 8-layer network, and produced a new state-of-the-art AUC on predicting pulsars for the HTRU2 dataset by a small margin (achieving an AUC 0.98, averaged over 10 cross-validation folds, versus the previous record of 0.976).<br />
<br />
On UCI datasets with fewer than 1,000 observations, SNNs did not outperform SVMs or random forests in terms of average rank in accuracy, but on datasets with at least 1,000 observations, SNNs showed the best overall performance (average rank of 5.8, compared to 6.1 for support vector machines and 6.6 for random forests). Through hyperparameter tuning, it was also discovered that the average depth of FNNs is 10.8 layers, more than the other FNN architectures tried.<br />
<br />
==Future Work==<br />
<br />
Although not the focus of this paper, the authors also briefly noted that their initial experiments with applying SELUs on relatively simple CNN structures showed promising results, which is not surprising given that ELUs, which do not have the self-normalizing property, has already been shown to work well with CNNs, demonstrating faster convergence than ReLU networks and even pushing the state-of-the-art error rates on CIFAR-100 at the time of publishing in 2015 [3].<br />
<br />
Since the paper was published, SELUs have been adopted by several researchers, not just with FNNs [https://github.com/bioinf-jku/SNNs see link], but also with CNNs, GANs, autoencoders, reinforcement learning and RNNs. In a few cases, researchers for those papers concluded that networks trained with SELUs converged faster than those trained with ReLUs, and that SELUs have the same convergence quality as batch normalization. There is potential for SELUs to be incorporated into more architectures in the future.<br />
<br />
==Critique==<br />
<br />
Overall, the authors presented a convincing case for using SELUs (along with proper initialization and alpha dropout) on FNNs. FNNs trained with SELU have more layers than those with other normalization techniques, so the work here provides a promising direction for making traditional FNNs more powerful. There are not as many well-established benchmark datasets to evaluate FNNs, but the experiments carried out, particularly on the larger Tox21 dataset, showed that SNNs can be very effective at classification tasks.<br />
<br />
The only question I have with the proofs is the lack of explanation for how the domains, <math display="inline">\Omega</math>, <math display="inline">\Omega^-</math> and <math display="inline">\Omega^+</math> are determined, which is an important consideration because they are used for deriving the upper and lower bounds on expressions needed for proving the three theorems. The ranges appear somewhat set through trial-and-error and heuristics to ensure the numbers work out (e.g. make the spectral norm [12] of <math display="inline">\mathcal{J}</math> as large as can be below 1 so as to ensure <math display="inline">g</math> is a contraction mapping), so it is not clear whether they are unique conditions, or that the parameters will remain within those prespecified ranges throughout training; and if the parameters can stray away from the ranges provided, then the issue of what will happen to the self-normalizing property was not addressed. Perhaps that is why the authors gave preference to models with a deeper structure and smaller learning rate during experiments to help the parameters stay within their domains. Further, in addition to the hyperparameters considered, it would be helpful to know the final values that went into the best-performing models, for a better understanding of what range of values work better for SNNs empirically.<br />
<br />
==Conclusion==<br />
<br />
The SNN structure proposed in this paper is built on the traditional FNN structure with a few modifications, including the use of SELUs as the activation function (with <math display="inline">\lambda \approx 1.0507</math> and <math display="inline">\alpha \approx 1.6733</math>), alpha dropout, network weights initialized with mean of zero and variance <math display="inline">\frac{1}{n}</math>, and inputs normalized to mean of zero and variance of one. It is simple to implement while being backed up by detailed theory. <br />
<br />
When properly initialized, SELUs will draw neural inputs towards a fixed point of zero mean and unit variance as the activations are propagated through the layers. The self-normalizing property is maintained even when weights deviate from their initial values during training (under mild conditions). When the variance of inputs goes beyond the prespecified range imposed, they are still bounded above and below so SNNs do not suffer from exploding and vanishing gradients. This self-normalizing property allows SNNs to be more robust to perturbations in stochastic gradient descent, so deeper structures with better prediction performance can be built. <br />
<br />
In the experiments conducted, the authors demonstrated that SNNs outperformed FNNs trained with other normalization techniques, such as batch, layer and weight normalization, and specialized architectures, such as highway or residual networks, on several classification tasks, including on the UCI Machine Learning Repository datasets. The adoption of SELUs by other researchers also lends credence to the potential for SELUs to be implemented in more neural network architectures.<br />
<br />
==References==<br />
<br />
# Ba, Kiros and Hinton. "Layer Normalization". arXiv:1607.06450. (2016).<br />
# Blinn. "Consider the Lowly 2X2 Matrix." IEEE Computer Graphics and Applications. (1996).<br />
# Clevert, Unterthiner, Hochreiter. "Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)." arXiv: 1511.07289. (2015).<br />
# He, Zhang, Ren and Sun. "Deep Residual Learning for Image Recognition." arXiv:1512.03385. (2015).<br />
# He, Zhang, Ren and Sun. "Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification." arXiv:1502.01852. (2015). <br />
# Ioffe and Szegedy. "Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariance Shift." arXiv:1502.03167. (2015).<br />
# Klambauer, Unterthiner, Mayr and Hochreiter. "Self-Normalizing Neural Networks." arXiv: 1706.02515. (2017).<br />
# Salimans and Kingma. "Weight Normalization: A Simple Reparameterization to Accelerate Training of Deep Neural Networks." arXiv:1602.07868. (2016).<br />
# Srivastava, Greff and Schmidhuber. "Highway Networks." arXiv:1505.00387 (2015).<br />
# Unterthiner, Mayr, Klambauer and Hochreiter. "Toxicity Prediction Using Deep Learning." arXiv:1503.01445. (2015). <br />
# https://en.wikipedia.org/wiki/Central_limit_theorem <br />
# http://mathworld.wolfram.com/SpectralNorm.html <br />
# https://www.math.umd.edu/~petersd/466/fixedpoint.pdf<br />
<br />
==Online Resources==<br />
https://github.com/bioinf-jku/SNNs (GitHub repository maintained by some of the paper's authors)<br />
<br />
==Footnotes==<br />
<br />
1. Error propagation analysis: The authors performed an error analysis to quantify the potential numerical imprecisions propagated through the numerous operations performed. The potential imprecision <math display="inline">\epsilon</math> was quantified by applying the mean value theorem<br />
<br />
\[ |f(x + \Delta x - f(x)| \leqslant ||\triangledown f(x + t\Delta x|| ||\Delta x|| \textrm{ for } t \in [0, 1]\textrm{.} \] <br />
<br />
The error propagation rules, or <math display="inline">|f(x + \Delta x - f(x)|</math>, was first obtained for simple operations such as addition, subtraction, multiplication, division, square root, exponential function, error function and complementary error function. Them, the error bounds on the compound terms making up <math display="inline">\Delta (S(\mu, \omega, \nu, \tau, \lambda, \alpha)</math> were found by decomposing them into the simpler expressions. If each of the variables have a precision of <math display="inline">\epsilon</math>, then it turns out <math display="inline">S</math> has a precision better than <math display="inline">292\epsilon</math>. For a machine with a precision of <math display="inline">2^{-56}</math>, the rounding error is <math display="inline">\epsilon \approx 10^{-16}</math>, and <math display="inline">292\epsilon < 10^{-13}</math>. In addition, all computations are correct up to 3 ulps (“unit in last place”) for the hardware architectures and GNU C library used, with 1 ulp being the highest precision that can be achieved.<br />
<br />
2. Independence Assumption: The classic definition of central limit theorem requires <math display="inline">x_i</math>’s to be independent and identically distributed, which is not guaranteed to hold true in a neural network layer. However, according to the Lyapunov CLT, the <math display="inline">x_i</math>’s do not need to be identically distributed as long as the <math display="inline">(2 + \delta)</math>th moment exists for the variables and meet the Lyapunov condition for the rate of growth of the sum of the moments [11]. In addition, CLT has also shown to be valid under weak dependence under mixing conditions [11]. Therefore, the authors argue that the central limit theorem can be applied with network inputs.<br />
<br />
3. <math display="inline">\mathcal{H}</math> versus <math display="inline">\mathcal{J}</math> Jacobians: In solving for the largest singular value of the Jacobian <math display="inline">\mathcal{H}</math> for the mapping <math display="inline">g: (\mu, \nu)</math>, the authors first worked with the terms in the Jacobian <math display="inline">\mathcal{J}</math> for the mapping <math display="inline">h: (\mu, \nu) \rightarrow (\widetilde{\mu}, \widetilde{\xi})</math> instead, because the influence of <math display="inline">\widetilde{\mu}</math> on <math display="inline">\widetilde{\nu}</math> is small when <math display="inline">\widetilde{\mu}</math> is small in <math display="inline">\Omega</math> and <math display="inline">\mathcal{H}</math> can be easily expressed as terms in <math display="inline">\mathcal{J}</math>. <math display="inline">\mathcal{J}</math> was referenced in the paper, but I used <math display="inline">\mathcal{H}</math> in the summary here to avoid confusion.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data&diff=32783Word translation without parallel data2018-03-07T00:53:40Z<p>Pa2forsy: Added the proof I couldn't remember in class</p>
<hr />
<div>[[File:Toy_example.png]]<br />
<br />
= Presented by =<br />
<br />
Xia Fan<br />
<br />
= Introduction =<br />
<br />
Many successful methods for learning relationships between languages stem from the hypothesis that there is a relationship between the context of words and their meanings. This means that if an adequate representation of a language is found in a high dimensional space (this is called an embedding), then words similar to a given word are close to one another in this space (ex. some norm can be minimized to find a word with similar context). Historically, another significant hypothesis is that these embedding spaces show similar structures over different languages. That is to say that given an embedding space for English and one for Spanish, a mapping could be found that aligns the two spaces and such a mapping could be used as a tool for translation. Many papers exploit these hypotheses, but use large parallel datasets for training. Recently, to remove the need for supervised training, methods have been implemented that utilize identical character strings (ex. letters or digits) in order to try to align the embeddings. The downside of this approach is that the two languages need to be similar to begin with as they need to have some shared basic building block. The method proposed in this paper uses an adversarial method to find this mapping between the embedding spaces of two languages without the use of large parallel datasets.<br />
<br />
This paper introduces a model that either is on par, or outperforms supervised state-of-the-art methods, without employing any cross-lingual annotated data. This method uses an idea similar to GANs: it leverages adversarial training to learn a linear mapping from a source to distinguish between the mapped source embeddings and the target embeddings, while the mapping is jointly trained to fool the discriminator. Second, this paper extracts a synthetic dictionary from the resulting shared embedding space and fine-tunes the mapping with the closed-form Procrustes solution from Schonemann (1966). Third, this paper also introduces an unsupervised selection metric that is highly correlated with the mapping quality and that the authors use both as a stopping criterion and to select the best hyper-parameters.<br />
<br />
= Model =<br />
<br />
<br />
=== Estimation of Word Representations in Vector Space ===<br />
<br />
This model focuses on learning a mapping between the two sets such that translations are close in the shared space. Before talking about the model it used, a model which can exploit the similarities of monolingual embedding spaces should be introduced. Mikolov et al.(2013) use a known dictionary of n=5000 pairs of words <math> \{x_i,y_i\}_{i\in{1,n}} </math>. and learn a linear mapping W between the source and the target space such that <br />
<br />
\begin{align}<br />
W^*=argmin_{W{\in}M_d(R)}||WX-Y||_F \hspace{1cm} (1)<br />
\end{align}<br />
<br />
where d is the dimension of the embeddings, <math> M_d(R) </math> is the space of d*d matrices of real numbers, and X and Y are two aligned matrices of size d*n containing the embeddings of the words in the parallel vocabulary. <br />
<br />
Xing et al. (2015) showed that these results are improved by enforcing orthogonality constraint on W. In that case, equation (1) boils down to the Procrustes problem, which advantageously offers a closed form solution obtained from the singular value decomposition (SVD) of <math> YX^T </math> :<br />
<br />
\begin{align}<br />
W^*=argmin_{W{\in}M_d(R)}||WX-Y||_F=UV^T, with U\Sigma V^T=SVD(YX^T).<br />
\end{align}<br />
<br />
<br />
<br />
This can be proven as follows. First note that <br />
\begin{align}<br />
&||WX-Y||_F\\<br />
&= \langle WX, WX \rangle_F -2 \langle W X, Y \rangle_F + \langle Y, Y \rangle_F \\<br />
&= ||X||_F^2 -2 \langle W X, Y \rangle_F + || Y||_F^2, <br />
\end{align}<br />
<br />
where <math display="inline"> \langle \cdot, \cdot \rangle_F </math> denotes the Frobenius inner-product and we have used the orthogonality of <math display="inline"> W </math>. It follows that we need only maximize the inner-product above. Let <math display="inline"> u_1, \ldots, u_d </math> denote the columns of <math display="inline"> U </math>. Let <math display="inline"> v_1, \ldots , v_d </math> denote the columns of <math display="inline"> V </math>. Let <math display="inline"> \sigma_1, \ldots, \sigma_d </math> denote the diagonal entries of <math display="inline"> \Sigma </math>. We have<br />
\begin{align}<br />
&\langle W X, Y \rangle_F \\<br />
&= \text{Tr} (W^T Y X^T)\\<br />
&=\sum_i \sigma_i \text{Tr}(W^T u_i v_i^T)\\<br />
&=\sum_i \sigma_i ((Wv_i)^T u_i )\\<br />
&\le \sum_i \sigma_i ||Wv_i|| ||u_i||\\<br />
&= \sum_i \sigma_i<br />
\end{align}<br />
where we have used the invariance of trace under cyclic permutations, Cauchy-Schwarz, and the orthogonality of the columns of U and V. Note that choosing <br />
\begin{align}<br />
W=UV^T<br />
\end{align}<br />
achieves the bound. This completes the proof.<br />
<br />
=== Domain-adversarial setting ===<br />
<br />
This paper shows how to learn this mapping W without cross-lingual supervision. An illustration of the approach is given in Fig. 1. First, this model learn an initial proxy of W by using an adversarial criterion. Then, it use the words that match the best as anchor points for Procrustes. Finally, it improve performance over less frequent words by changing the metric of the space, which leads to spread more of those points in dense region. <br />
<br />
[[File:Toy_example.png |frame|none|alt=Alt text|Figure 1: Toy illustration of the method. (A) There are two distributions of word embeddings, English words in red denoted by X and Italian words in blue denoted by Y , which we want to align/translate. Each dot represents a word in that space. The size of the dot is proportional to the frequency of the words in the training corpus of that language. (B) Using adversarial learning, we learn a rotation matrix W which roughly aligns the two distributions. The green stars are randomly selected words that are fed to the discriminator to determine whether the two word embeddings come from the same distribution. (C) The mapping W is further refined via Procrustes. This method uses frequent words aligned by the previous step as anchor points, and minimizes an energy function that corresponds to a spring system between anchor points. The refined mapping is then used to map all words in the dictionary. (D) Finally, we translate by using the mapping W and a distance metric, dubbed CSLS, that expands the space where there is high density of points (like the area around the word “cat”), so that “hubs” (like the word “cat”) become less close to other word vectors than they would otherwise (compare to the same region in panel (A)).]]<br />
<br />
Let <math> X={x_1,...,x_n} </math> and <math> Y={y_1,...,y_m} </math> be two sets of n and m word embeddings coming from a source and a target language respectively. A model is trained is trained to discriminate between elements randomly sampled from <math> WX={Wx_1,...,Wx_n} </math> and Y, We call this model the discriminator. W is trained to prevent the discriminator from making accurate predictions. As a result, this is a two-player game, where the discriminator aims at maximizing its ability to identify the origin of an embedding, and W aims at preventing the discriminator from doing so by making WX and Y as similar as possible. This approach is in line with the work of Ganin et al.(2016), who proposed to learn latent representations invariant to the input domain, where in this case, a domain is represented by a language(source or target).<br />
<br />
1. Discriminator objective<br />
<br />
Refer to the discriminator parameters as <math> \theta_D </math>. Consider the probability <math> P_{\theta_D}(source = 1|z) </math> that a vector z is the mapping of a source embedding (as opposed to a target embedding) according to the discriminator. The discriminator loss can be written as:<br />
<br />
\begin{align}<br />
L_D(\theta_D|W)=-\frac{1}{n} \sum_{i=1}^n log P_{\theta_D}(source=1|Wx_i)-\frac{1}{m} \sum_{i=1}^m log P_{\theta_D}(source=0|y_i)<br />
\end{align}<br />
<br />
2. Mapping objective <br />
<br />
In the unsupervised setting, W is now trained so that the discriminator is unable to accurately predict the embedding origins: <br />
<br />
\begin{align}<br />
L_W(W|\theta_D)=-\frac{1}{n} \sum_{i=1}^n log P_{\theta_D}(source=0|Wx_i)-\frac{1}{m} \sum_{i=1}^m log P_{\theta_D}(source=1|y_i)<br />
\end{align}<br />
<br />
3. Learning algorithm <br />
To train the model, the authors follow the standard training procedure of deep adversarial networks of Goodfellow et al. (2014). For every input sample, the discriminator and the mapping matrix W are trained successively with stochastic gradient updates to respectively minimize <math> L_D </math> and <math> L_W </math><br />
<br />
=== Refinement procedre ===<br />
<br />
The matrix W obtained with adversarial training gives good performance (see Table 1), but the results are still not on par with the supervised approach. In fact, the adversarial approach tries to align all words irrespective of their frequencies. However, rare words have embeddings that are less updated and are more likely to appear in different contexts in each corpus, which makes them harder to align. Under the assumption that the mapping is linear, it is then better to infer the global mapping using only the most frequent words as anchors. Besides, the accuracy on the most frequent word pairs is high after adversarial training.<br />
To refine the mapping, this paper build a synthetic parallel vocabulary using the W just learned with adversarial training. Specifically, this paper consider the most frequent words and retain only mutual nearest neighbors to ensure a high-quality dictionary. Subsequently, this paper apply the Procrustes solution in (2) on this generated dictionary. Considering the improved solution generated with the Procrustes algorithm, it is possible to generate a more accurate dictionary and apply this method iteratively, similarly to Artetxe et al. (2017). However, given that the synthetic dictionary obtained using adversarial training is already strong, this paper only observe small improvements when doing more than one iteration, i.e., the improvements on the word translation task are usually below 1%.<br />
<br />
=== Cross-Domain Similarity Local Scaling (CSLS) ===<br />
<br />
This paper considers a bi-partite neighborhood graph, in which each word of a given dictionary is connected to its K nearest neighbors in the other language. <math> N_T(Wx_s) </math> is used to denote the neighborhood, on this bi-partite graph, associated with a mapped source word embedding <math> Wx_s </math>. All K elements of <math> N_T(Wx_s) </math> are words from the target language. Similarly we denote by <math> N_S(y_t) </math> the neighborhood associated with a word t of the target language. Consider the mean similarity of a source embedding <math> x_s </math> to its target neighborhood as<br />
<br />
\begin{align}<br />
r_T(Wx_s)=\frac{1}{K}\sum_{y\in N_T(Wx_s)}cos(Wx_s,y_t)<br />
\end{align}<br />
<br />
where cos(,) is the cosine similarity. Likewise, the mean similarity of a target word <math> y_t </math> to its neighborhood is denotes as <math> r_S(y_t) </math>. This is used to define similarity measure CSLS(.,.) between mapped source words and target words,as <br />
<br />
\begin{align}<br />
CSLS(Wx_s,y_t)=2cos(Wx_s,y_t)-r_T(Wx_s)-r_S(y_t)<br />
\end{align}<br />
<br />
This process increases the similarity associated with isolated word vectors, but decreases the similarity of vectors lying in dense areas.<br />
<br />
= Training and architectural choices =<br />
=== Architecture ===<br />
<br />
This paper use unsupervised word vectors that were trained using fastText2. These correspond to monolingual embeddings of dimension 300 trained on Wikipedia corpora; therefore, the mapping W has size 300 × 300. Words are lower-cased, and those that appear less than 5 times are discarded for training. As a post-processing step, only the first 200k most frequent words were selected in the experiments.<br />
For the discriminator, it use a multilayer perceptron with two hidden layers of size 2048, and Leaky-ReLU activation functions. The input to the discriminator is corrupted with dropout noise with a rate of 0.1. As suggested by Goodfellow (2016), a smoothing coefficient s = 0.2 is included in the discriminator predictions. This paper use stochastic gradient descent with a batch size of 32, a learning rate of 0.1 and a decay of 0.95 both for the discriminator and W . <br />
<br />
=== Discriminator inputs ===<br />
The embedding quality of rare words is generally not as good as the one of frequent words (Luong et al., 2013), and it is observed that feeding the discriminator with rare words had a small, but not negligible negative impact. As a result, this paper only feed the discriminator with the 50,000 most frequent words. At each training step, the word embeddings given to the discriminator are sampled uniformly. Sampling them according to the word frequency did not have any noticeable impact on the results.<br />
<br />
=== Orthogonality===<br />
In this work, it propose to use a simple update step to ensure that the matrix W stays close to an orthogonal matrix during training (Cisse et al. (2017)). Specifically, the following update rule on the matrix W is used :<br />
<br />
\begin{align}<br />
W \leftarrow (1+\beta)W-\beta(WW^T)W<br />
\end{align}<br />
<br />
where β = 0.01 is usually found to perform well. This method ensures that the matrix stays close to the manifold of orthogonal matrices after each update.<br />
<br />
This update rule can be justified as follows. Consider the function <br />
\begin{align}<br />
g: \mathbb{R}^{d\times d} \to \mathbb{R}^{d \times d}<br />
\end{align}<br />
defined by<br />
\begin{align}<br />
g(W)= W^T W -I.<br />
\end{align}<br />
<br />
The derivative of g at W is is the linear map<br />
\begin{align}<br />
Dg[W]: \mathbb{R}^{d \times d} \to \mathbb{R}^{d \times d}<br />
\end{align}<br />
defined by<br />
\begin{align}<br />
Dg[W](H)= H^T W + W^T H.<br />
\end{align}<br />
<br />
The adjoint of this linear map is<br />
<br />
\begin{align}<br />
D^\ast g[W](H)= WH^T +WH.<br />
\end{align}<br />
<br />
Now consider the function f<br />
\begin{align}<br />
f: \mathbb{R}^{d \times d} \to \mathbb{R}<br />
\end{align}<br />
<br />
defined by<br />
<br />
\begin{align}<br />
f(W)=||g(W) ||_F^2=||W^TW -I ||_F^2.<br />
\end{align}<br />
<br />
f has gradient:<br />
\begin{align}<br />
\nabla f (W) = 2D^\ast g[W] (g(W ) ) =2W(W^TW-I) +2W(W^TW-I)=4W W^TW-4W.<br />
\end{align}<br />
<br />
Thus the update<br />
\begin{align}<br />
W \leftarrow (1+\beta)W-\beta(WW^T)W<br />
\end{align}<br />
amounts to a step in the direction opposite the gradient of f. That is, a step toward the set of orthogonal matrices.<br />
<br />
=== Dictionary generation ===<br />
The refinement step requires to generate a new dictionary at each iteration. In order for the Procrustes solution to work well, it is best to apply it on correct word pairs. As a result, the CSLS method is used to select more accurate translation pairs in the dictionary. To increase even more the quality of the dictionary, and ensure that W is learned from correct translation pairs, only mutual nearest neighbors were considered, i.e. pairs of words that are mutually nearest neighbors of each other according to CSLS. This significantly decreases the size of the generated dictionary, but improves its accuracy, as well as the overall performance.<br />
<br />
=== Validation criterion for unsupervised model selection ===<br />
<br />
This paper consider the 10k most frequent source words, and use CSLS to generate a translation for each of them, then compute the average cosine similarity between these deemed translations, and use this average as a validation metric. Figure 2 shows the correlation between the evaluation score and this unsupervised criterion (without stabilization by learning rate shrinkage)<br />
<br />
<br />
<br />
[[File:fig2_fan.png |frame|none|alt=Alt text|Figure 2: Unsupervised model selection.<br />
Correlation between the unsupervised validation criterion (black line) and actual word translation accuracy (blue line). In this particular experiment, the selected model is at epoch 10. Observe how the criterion is well correlated with translation accuracy.]]<br />
<br />
= Results =<br />
<br />
In what follows, the results on word translation retrieval using the bilingual dictionaries were presented in Table 1 and the comparison to previous work in Table 2 where unsupervised model significantly outperform previous approaches. The results on the sentence translation retrieval task were also presented in Table 3 and the cross-lingual word similarity task in Table 4. Finally, the results on word-by-word translation for English-Esperanto was presented in Table 5.<br />
<br />
[[File:table1_fan.png |frame|none|alt=Alt text|Table 1: Word translation retrieval P@1 for the released vocabularies in various language pairs. The authors consider 1,500 source test queries, and 200k target words for each language pair. The authors use fastText embeddings trained on Wikipedia. NN: nearest neighbors. ISF: inverted softmax. (’en’ is English, ’fr’ is French, ’de’ is German, ’ru’ is Russian, ’zh’ is classical Chinese and ’eo’ is Esperanto)]]<br />
<br />
<br />
[[File:table2_fan.png |frame|none|alt=Alt text|English-Italian word translation average precisions (@1, @5, @10) from 1.5k source word queries using 200k target words. Results marked with the symbol † are from Smith et al. (2017). Wiki means the embeddings were trained on Wikipedia using fastText. Note that the method used by Artetxe et al. (2017) does not use the same supervision as other supervised methods, as they only use numbers in their ini- tial parallel dictionary.]]<br />
<br />
[[File:table3_fan.png |frame|none|alt=Alt text|Table 3: English-Italian sentence translation retrieval. The authors report the average P@k from 2,000 source queries using 200,000 target sentences. The authors use the same embeddings as in Smith et al. (2017). Their results are marked with the symbol †.]]<br />
<br />
[[File:table4_fan.png |frame|none|alt=Alt text|Table 4: Cross-lingual wordsim task. NASARI<br />
(Camacho-Collados et al. (2016)) refers to the official SemEval2017 baseline. The authors report Pearson correlation.]]<br />
<br />
[[File:table5_fan.png |frame|none|alt=Alt text|Table 5: BLEU score on English-Esperanto.<br />
Although being a naive approach, word-by- word translation is enough to get a rough idea of the input sentence. The quality of the gener- ated dictionary has a significant impact on the BLEU score.]]<br />
<br />
= Conclusion =<br />
This paper shows for the first time that one can align word embedding spaces without any cross-lingual supervision, i.e., solely based on unaligned datasets of each language, while reaching or outperforming the quality of previous supervised approaches in several cases. Using adversarial training, the model is able to initialize a linear mapping between a source and a target space, which is also used to produce a synthetic parallel dictionary. It is then possible to apply the same techniques proposed for supervised techniques, namely a Procrustean optimization.<br />
<br />
= Source =<br />
Lample, Guillaume; Denoyer, Ludovic; Ranzato, Marc'Aurelio <br />
| Unsupervised Machine Translation Using Monolingual Corpora Only<br />
| arVix: 1701.04087</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=One-Shot_Imitation_Learning&diff=32754One-Shot Imitation Learning2018-03-06T19:02:59Z<p>Pa2forsy: Added full citations for papers to which the body of the summary refers</p>
<hr />
<div>= Introduction =<br />
Robotic systems can be used for many applications, but to truly be useful for complex applications, they need to overcome 2 challenges: having the intent of the task at hand communicated to them, and being able to perform the manipulations necessary to complete this task. It is preferable to use demonstration to teach the robotic systems rather than natural language, as natural language may often fail to convey the details and intricacies required for the task. However, current work on learning from demonstrations is only successful with large amounts of feature engineering or a large number of demonstrations. The proposed model aims to achieve 'one-shot' imitation learning, ie. learning to complete a new task from just a single demonstration of it without any other supervision. As input, the proposed model takes the observation of the current instance a task, and a demonstration of successfully solving a different instance of the same task. Strong generalization was achieved by using a soft attention mechanism on both the sequence of actions and states that the demonstration consists of, as well as on the vector of element locations within the environment. The success of this proposed model at completing a series of block stacking tasks can be viewed at http://bit.ly/nips2017-oneshot.<br />
<br />
= Related Work =<br />
While one-shot imitation learning is a novel combination of ideas, each of the components has previously been studied.<br />
* Imitation Learning: <br />
** Behavioural learning uses supervised learning to map from observations to actions<br />
** Inverse reinforcement learning estimates a reward function that considers demonstrations as optimal behavior<br />
* One-Shot Learning:<br />
** Typically a form of meta-learning<br />
** Previously used for variety of tasks but all domain-specific<br />
** [https://arxiv.org/abs/1703.03400 (Finn et al. 2017)] proposed a generic solution but excluded imitation learning<br />
* Reinforcement Learning:<br />
** Demonstrated to work on variety of tasks and environments, in particular on games and robotic control<br />
** Requires large amount of trials and a user-specified reward function<br />
* Multi-task/Transfer Learning:<br />
** Shown to be particularly effective at computer vision tasks<br />
** Not meant for one-shot learning<br />
* Attention Modelling:<br />
** The proposed model makes use of the attention model from [https://arxiv.org/abs/1409.0473 (Bahdanau et al. 2016)]<br />
** The attention modelling over demonstration is similar in nature to the seq2seq models from the well known [https://papers.nips.cc/paper/5346-sequence-to-sequence-learning-with-neural-networks.pdf (Sutskever et al. 2014)]<br />
<br />
= One-Shot Imitation Learning =<br />
<br />
[[File:oneshot1.jpg|1000px]]<br />
<br />
The figure shoulds the differences between traditional and one-shot imitation learning. In a), the traditional method may require training different policies for performing similar tasks that are similar in nature. For example, stacking blocks in a height of 2 and in a height of 3. In b), the one-shot imitation learning allows the same policy to be used for these tasks given a single demonstration, achieving good performance without any additional system interactions. In c), the policy is trained by using a set of different training tasks, with enough examples so that the learned results can be generalized to other similar tasks. Each task has a set of successful demonstrations. Each iteration of training uses two demonstrations from a task, one is used as the input passing into the algorithm and the other is used at the output, the results from the two are then conditioned to produce the correct action.<br />
<br />
== Problem Formalization ==<br />
The problem is briefly formalized with the authors describing a distribution of tasks, an individual task, a distribution of demonstrations for this task, and a single demonstration respecitvely as \[T, t\sim T, D(t), d\sim D(t)\]<br />
In addition, an action, an observation, parameters, and a policy are respectively defined as \[a, o, \theta, \pi_\theta(a|o,d)\]<br />
In particular, a demonstration is a sequence of observation and action pairs \[d = [(o_1, a_1),(o_2, a_2), . . . ,(o_T , a_T )]\]<br />
Assuming that $$T$$ and some evaluation function $$R_t(d): R^T \rightarrow R$$ are given, and that succesful demonstrations are available for each task, then the objective is to maximize expectation of the policy performance over \[t\sim T, d\sim D(t)\].<br />
<br />
== Block Stacking Tasks ==<br />
The tasks that the authors focus on is block stacking. A user specifies in what final configuration cubic blocks should be stacked, and the goal is to use a 7-DOF Fetch robotic arm to arrange the blocks in this configuration. The number of blocks, and their desired configuration (ie. number of towers, the height of each tower, and order of blocks within each tower) can be varied and encoded as a string. For example, 'abc def' would signify 2 towers of height 3, with block A on block B on block C in one tower, and block D on block E on block F in a second tower. To add complexity, the initial configuration of the blocks can vary and is encoded as a set of 3-dimensional vectors describing the position of each block relative to the robotic arm.<br />
<br />
== Algorithm ==<br />
To avoid needing to specify a reward function, the authors use behavioral cloning and DAGGER, 2 imitation learning methods that require only demonstrations, for training. In each training step, a list of tasks is sampled, and for each, a demonstration with injected noise along with some observation-action pairs are sampled. Given the current observation and demonstration as input, the policy is trained against the sampled actions by minimizing L2 norm for continuous actions, and cross-entropy for discrete ones. Adamax is used as the optimizer with a learning rate of 0.001.<br />
<br />
= Architecture =<br />
The authors propose a novel architecture for imitation learning, consisting of 3 networks.<br />
<br />
[[File:oneshot2.jpg|1000px]]<br />
<br />
== Demonstration Network ==<br />
This network takes a demonstration as input and produces an embedding with size linearly proportional to the number of blocks and the size of the demonstration.<br />
=== Temporal Dropout: ===<br />
Since a demonstration for block stacking can be very long, the authors randomly discard 95% of the time steps, a process they call 'temporal dropout'. The reduced size of the demonstrations allows multiple trajectories to be explored during testing to calculate an ensemble estimate. Dilated temporal convolutions and neighborhood attention are then repeatedly applied to the downsampled demonstrations.<br />
<br />
=== Neighborhood Attention: ===<br />
Since demonstration sizes can vary, a mechanism is needed that is not restricted to fixed-length inputs. While soft attention is one such mechanism, the problem with it is that there may be increasingly large amounts of information lost if soft attention is used to map longer demonstrations to the same fixed length as shorter demonstrations. As a solution, the authors propose having the same number of outputs as inputs, but with attention performed on other inputs relative to the current input.<br />
<br />
A query, a list of context vectors, and a list of memory vectors are given as input to soft attention. Each attention weight is given by the product of a learned weight vector and a nonlinearity applied to the sum of the query and corresponding context vector. Softmaxed weights applied to the corresponding memory vector form the output of the soft attention.<br />
<br />
\[Inputs: q, \{c_j\}, \{m_j\}\]<br />
\[Weights: w_i \leftarrow v^Ttanh(q+c_i)\]<br />
\[Output: \sum_i{m_i\frac{\exp(w_i)}{\sum_j{\exp(w_j)}}}\]<br />
<br />
A list of same-length embeddings, coming from a previous neighbourhood attention layer or a projection from the list of block coordinates, is given as input to neighborhood attention. For each block, two separate linear layers produce a query vector and a context vector, while a memory vector is a list of tuples that describe the position of each block joined with the input embedding for that block. Soft attention is then performed on this query, context vector, and memory vector. The authors claim that the intuition behind this process is to allow each block to provide information about itself relative to the other blocks in the environment. Finally, for each block, a linear transformation is performed on the vector composed by concatenating the input embedding, the result of the soft attention for that block, and the robot's state.<br />
<br />
For an environment with B blocks:<br />
\[State: s\]<br />
\[Block_i: b_i \leftarrow (x_i, y_i, z_i)\]<br />
\[Embeddings: h_1^{in}, ..., h_B^{in}\] <br />
\[Query_i: q_i \leftarrow Linear(h_i^{in})\]<br />
\[Context_i: c_i \leftarrow Linear(h_i^{in})\]<br />
\[Memory_i: m_i \leftarrow (b_i, h_i^{in}) \]<br />
\[Result_i: result_i \leftarrow SoftAttn(q_i, \{c_j\}_{j=1}^B, \{m_k\}_{k=1}^B)\]<br />
\[Output_i: output_i \leftarrow Linear(concat(h_i^{in}, result_i, b_i, s))\]<br />
<br />
== Context network ==<br />
This network takes the current state and the embedding produced by the demonstration network as inputs and outputs a fixed-length "context embedding" which captures only the information relevant for the manipulation network at this particular step.<br />
=== Attention over demonstration: ===<br />
The current state is used to compute a query vector which is then used for attending over all the steps of the embedding. Since at each time step there are multiple blocks, the weights for each are summed together to produce a scalar for each time step. Neighbourhood attention is then applied several times, using an LSTM with untied weights, since the information at each time steps needs to be propagated to each block's embedding. <br />
<br />
Performing attention over the demonstration yields a vector whose size is independent of the demonstration size; however, it is still dependent on the number of blocks in the environment, so it is natural to now attend over the state in order to get a fixed-length vector.<br />
=== Attention over current state: ===<br />
The authors propose that in general, within each subtask, only a limited number of blocks are relevant for performing the subtask. If the subtask is to stack A on B, then intuitively, one would suppose that only block A and B are relevant, and perhaps any blocks that may be blocking access to either A or B. This is not enforced during training, but once soft attention is applied to the current state to produce a fixed-length context embedding, the authors believe that the model does indeed learn in this way.<br />
<br />
== Manipulation network ==<br />
Given the context embedding as input, this simple feedforward network decides on the particular action needed, to complete the subtask of stacking one particular 'source' block on top of another 'target' block.<br />
<br />
= Experiments = <br />
The proposed model was tested on the block stacking tasks. 140 training tasks and 43 testing tasks were collected, each with between 2 to 10 blocks and a different, desired final layout. Over 1000 demonstrations for each task were collected using a hard-coded policy rather than a human user. The authors compare 4 different architectures in these experiments:<br />
* Behavioural cloning used to train the proposed model<br />
* DAGGER used to train the proposed model<br />
* The proposed model, trained with DAGGER, but conditioned on the desired final state rather than an entire demonstration<br />
* The proposed model, trained with DAGGER, but conditioned on a 'snapshot' of the environment at the end of each subtask (ie every time a block is stacked on another block)<br />
<br />
== Performance Evaluation ==<br />
[[File:oneshot3.jpg|1000px]]<br />
<br />
The most confident action at each timestep is chosen in 100 different task configurations, and results are averaged over tasks that had the same number of blocks. The results suggest that the performance of each of the architectures is comparable to that of the hard-coded policy which they aim to imitate. Performance degrades similarly across all architectures and the hard-coded policy as the number of blocks increases. On the harder tasks, conditioning on the entire demonstration led to better performance than conditioning on snapshots or on the final state. The authors believe that this may be due to the lack of information when conditioning only on the final state as well as due to regularization caused by temporal dropout which leads to data augmentation when conditioning on the full demonstration but is omitted when conditioning only on the snapshots or final state. Both DAGGER and behavioral cloning performed comparably well. As mentioned above, noise injection was used in training to improve performance; in practice, additional noise can still be injected but some may already come from other sources.<br />
<br />
= Conclusions =<br />
The proposed model successfully learns to complete new instances of a new task from just a single demonstration. The model was demonstrated to work on a series of block stacking tasks. The authors propose several extensions including enabling few-shot learning when one demonstration is insufficient, using image data as the demonstrations, and attempting many other tasks aside from block stacking.<br />
<br />
= Criticisms =<br />
While the paper shows an incredibly impressive result: the ability to learn a new task from just a single demonstration, there are a few points that need clearing up.<br />
Firstly, the authors use a hard-coded policy in their experiments rather than a human. It is clear that the performance of this policy begins to degrade quickly as the complexity of the task increases. It would be useful to know what this hard-coded policy actually was, and if the proposed model could still have comparable performance if a more successful demonstration, perhaps one by a human user, were performed. Give the current popularity of adversarial examples, it would also be interesting to see the performance when conditioned on an "adversarial" demonstration, that achieves the correct final state, but intentionally performs complex or obfuscated steps to get there.<br />
Second, it would be useful to see the model's performance on a more complex family of tasks than block stacking, since although each block stacking task is slightly different, the differences may turn out be insignificant compared to other tasks that this model should work on if it is to be a general imitation learning architecture. Regardless, this work is a big step forward for imitation learning, permitting a wider range of tasks for which there is little training data and no reward function available, to still be successfully solved.<br />
<br />
= Illustrative Example: Particle Reaching =<br />
<br />
[[File:f1.png]]<br />
<br />
Figure 1: [Left] Agent, [Middle] Orange square is target, [Right] Green triangle is target [2].<br />
<br />
Another simple yet insightful example of the One-Shot Imitation Learning is the particle reaching problem which provides a relatively simple suite of tasks from which the network needs to solve an arbitrary one. The problem is formulated such that for each task: there is an agent which can move based on a 2D force vector, and n landmarks at varying 2D locations (n varies from task to task) with the goal of moving the agent to the specific landmark reached in the demonstration. This is illustrated in Figure 1. <br />
<br />
[[File:f2.png]]<br />
<br />
Figure 2: Experimental results [2].<br />
<br />
Some insight comes from the use of different network architectures to solve this problem. The three architectures to compare (described below) are plain LSTM, LSTM with attention, and final state with attention. The key insight is that the architectures go from generic to specific, with the best generalization performance achieved with the most specific architecture, final state with attention, as seen in Figure 2. It is important to note that this conclusion does not carry forward to more complicated tasks such as the block stacking task.<br />
*Plain LSTM: 512 hidden units, with the input being the demonstration trajectory (the position of the agent changes over time and approaches one of the targets). Output of the LSTM with the current state (from the task needed to be solved) is the input for a multi-layer perceptron (MLP) for finding the solution.<br />
*LSTM with attention: Output of LSTM is now a set of weights for the different targets during training. These weights and the test state are used in the test task. The, now, 2D output is the input for an MLP as before.<br />
*Final state with attention: Looks only at the final state of the demonstration since it can sufficiently provide the needed detail of which target to reach (trajectory is not required). Similar to previous architecture, produces weights used by MLP.<br />
<br />
= Source =<br />
# Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
# Duan, Yan, Marcin Andrychowicz, Bradly Stadie, OpenAI Jonathan Ho, Jonas Schneider, Ilya Sutskever, Pieter Abbeel, and Wojciech Zaremba. "One-shot imitation learning." In Advances in neural information processing systems, pp. 1087-1098. 2017.<br />
# Y. Duan, M. Andrychowicz, B. Stadie, J. Ho, J. Schneider, I. Sutskever, P. Abbeel, and W. Zaremba. One-shot imitation learning. arXiv preprint arXiv:1703.07326, 2017. (Newer revision)<br />
# Finn, Chelsea, Pieter Abbeel, and Sergey Levine. "Model-agnostic meta-learning for fast adaptation of deep networks." arXiv preprint arXiv:1703.03400 (2017).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Spectral_normalization_for_generative_adversial_network&diff=32748stat946w18/Spectral normalization for generative adversial network2018-03-06T18:11:31Z<p>Pa2forsy: </p>
<hr />
<div>= Presented by =<br />
<br />
1. liu, wenqing<br />
<br />
= Introduction =<br />
Generative adversarial networks (GANs) (Goodfellow et al., 2014) have been enjoying considerable success as a framework of generative models in recent years. The concept is to consecutively train the model distribution and the discriminator in turn, with the goal of reducing the difference between the model distribution and the target distribution measured by the best discriminator possible at each step of the training.<br />
<br />
A persisting challenge in the training of GANs is the performance control of the discriminator. When the support of the model distribution and the support of target distribution are disjoint, there exists a discriminator that can perfectly distinguish the model distribution from the target (Arjovsky & Bottou, 2017). One such discriminator is produced in this situation, the training of the generator comes to complete stop, because the derivative of the so-produced discriminator with respect to the input turns out to be 0. This motivates us to introduce some form of restriction to the choice of discriminator.<br />
<br />
In this paper, we propose a novel weight normalization method called ''spectral normalization'' that can stabilize the training of discriminator networks. Our normalization enjoys following favorable properties:<br />
<br />
* The only hyper-parameter that needs to be tuned is the Lipschitz constant, and the algorithm is not too sensitive to this constant's value<br />
* The additional computational needed to implement spectral normalization is small <br />
<br />
In this study, we provide explanations of the effectiveness of spectral normalization against other regularization or normalization techniques.<br />
<br />
= Model =<br />
<br />
Let us consider a simple discriminator made of a neural network of the following form, with the input x:<br />
<br />
\[f(x,\theta) = W^{L+1}a_L(W^L(a_{L-1}(W^{L-1}(\cdots a_1(W^1x)\cdots))))\]<br />
<br />
where <math> \theta:=W^1,\cdots,W^L, W^{L+1} </math> is the learning parameters set, <math>W^l\in R^{d_l*d_{l-1}}, W^{L+1}\in R^{1*d_L} </math>, and <math>a_l </math> is an element-wise non-linear activation function.The final output of the discriminator function is given by <math>D(x,\theta) = A(f(x,\theta)) </math>. The standard formulation of GANs is given by <math>\min_{G}\max_{D}V(G,D)</math> where min and max of G and D are taken over the set of generator and discriminator functions, respectively. <br />
<br />
The conventional form of <math>V(G,D) </math> is given by:<br />
<br />
\[E_{x\sim q_{data}}[\log D(x)] + E_{x'\sim p_G}[\log(1-D(x')\]<br />
<br />
where <math>q_{data}</math> is the data distribution and <math>p_G(x)</math> is the model generator distribution to be learned through the adversarial min-max optimization. It is known that, for a fixed generator G, the optimal discriminator for this form of <math>V(G,D) </math> is given by <br />
<br />
\[ D_G^{*}(x):=q_{data}(x)/(q_{data}(x)+p_G(x)) \]<br />
<br />
We search for the discriminator D from the set of K-lipshitz continuous functions, that is, <br />
<br />
\[ \arg\max_{||f||_{Lip}\le k}V(G,D)\]<br />
<br />
where we mean by <math> ||f||_{lip}</math> the smallest value M such that <math> ||f(x)-f(x')||/||x-x'||\le M </math> for any x,x', with the norm being the <math> l_2 </math> norm.<br />
<br />
Our spectral normalization controls the Lipschitz constant of the discriminator function <math> f </math> by literally constraining the spectral norm of each layer <math> g: h_{in}\rightarrow h_{out}</math>. By definition, Lipschitz norm <math> ||g||_{Lip} </math> is equal to <math> \sup_h\sigma(\nabla g(h)) </math>, where <math> \sigma(A) </math> is the spectral norm of the matrix A, which is equivalent to the largest singular value of A. Therefore, for a linear layer <math> g(h)=Wh </math>, the norm is given by <math> ||g||_{Lip}=\sigma(W) </math>. Observing the following bound:<br />
<br />
\[ ||f||_{Lip}\le ||(h_L\rightarrow W^{L+1}h_{L})||_{Lip}*||a_{L}||_{Lip}*||(h_{L-1}\rightarrow W^{L}h_{L-1})||_{Lip}\cdots ||a_1||_{Lip}*||(h_0\rightarrow W^1h_0)||_{Lip}=\prod_{l=1}^{L+1}\sigma(W^l) *\prod_{l=1}^{L} ||a_l||_{Lip} \]<br />
<br />
Our spectral normalization normalizes the spectral norm of the weight matrix W so that it satisfies the Lipschitz constraint <math> \sigma(W)=1 </math>:<br />
<br />
\[ \bar{W_{SN}}:= W/\sigma(W) \]<br />
<br />
In summary, just like what weight normalization does, we reparameterize weight matrix <math> \bar{W_{SN}} </math> as <math> W/\sigma(W) </math> to fix the singular value of weight matrix. Now we can calculate the gradient of new parameter W by chain rule:<br />
<br />
\[ \frac{\partial V(G,D)}{\partial W} = \frac{\partial V(G,D)}{\partial \bar{W_{SN}}}*\frac{\partial \bar{W_{SN}}}{\partial W} \]<br />
<br />
\[ \frac{\partial \bar{W_{SN}}}{\partial W_{ij}} = \frac{1}{\sigma(W)}E_{ij}-\frac{1}{\sigma(W)^2}*\frac{\partial \sigma(W)}{\partial(W_{ij})}W=\frac{1}{\sigma(W)}E_{ij}-\frac{[u_1v_1^T]_{ij}}{\sigma(W)^2}W=\frac{1}{\sigma(W)}(E_{ij}-[u_1v_1^T]_{ij}\bar{W_{SN}}) \]<br />
<br />
where <math> E_{ij} </math> is the matrix whose (i,j)-th entry is 1 and zero everywhere else, and <math> u_1, v_1</math> are respectively the first left and right singular vectors of W.<br />
<br />
To understand the above computation in more detail, note that <br />
\begin{align}<br />
\sigma(W)= \sup_{||u||=1, ||v||=1} \langle Wv, u \rangle = \sup_{||u||=1, ||v||=1} \text{trace} ( (uv^T)^T W).<br />
\end{align}<br />
By Theorem 4.4.2 in Lemaréchal and Hiriart-Urruty (1996), the sub-differential of a convex function defined as the the maximum of a set of differentiable convex functions over a compact index set is the convex hull of the gradients of the maximizing functions. Thus we have the sub-differential:<br />
<br />
\begin{align}<br />
\partial \sigma = \text{convex hull} \{ u v^T: u,v \text{ are left/right singular vectors associated with } \sigma(W) \}.<br />
\end{align}<br />
<br />
However, the authors assume that the maximum singular value of W has only one left and one right normalized singular vector. Thus <math> \sigma </math> is differentiable and <br />
\begin{align}<br />
\nabla_W \sigma(W) =u_1v_1^T,<br />
\end{align}<br />
which explains the above computation.<br />
<br />
= Spectral Normalization VS Other Regularization Techniques =<br />
<br />
The weight normalization introduced by Salimans & Kingma (2016) is a method that normalizes the <math> l_2 </math> norm of each row vector in the weight matrix. Mathematically it is equivalent to require the weight by the weight normalization <math> \bar{W_{WN}} </math>:<br />
<br />
<math> \sigma_1(\bar{W_{WN}})^2+\cdots+\sigma_T(\bar{W_{WN}})^2=d_0, \text{where } T=\min(d_i,d_0) </math> where <math> \sigma_t(A) </math> is a t-th singular value of matrix A. <br />
<br />
Note, if <math> \bar{W_{WN}} </math> is the weight normalized matrix of dimension <math> d_i*d_0 </math>, the norm <math> ||\bar{W_{WN}}h||_2 </math> for a fixed unit vector <math> h </math> is maximized at <math> ||\bar{W_{WN}}h||_2 \text{ when } \sigma_1(\bar{W_{WN}})=\sqrt{d_0} \text{ and } \sigma_t(\bar{W_{WN}})=0, t=2, \cdots, T </math> which means that <math> \bar{W_{WN}} </math> is of rank one. In order to retain as much norm of the input as possible and hence to make the discriminator more sensitive, one would hope to make the norm of <math> \bar{W_{WN}}h </math> large. For weight normalization, however, this comes at hte cost of reducing the rank and hence the number of features to be used for the discriminator. Thus, there is a conflict of interests between weight normalization and our desire to use as many features as possible to distinguish the generator distribution from the target distribution. The former interest often reigns over the other in many cases, inadvertently diminishing the number of features to be used by the discriminators. Consequently, the algorithm would produce a rather arbitrary model distribution that matches the target distribution only at select few features. <br />
<br />
Brock et al. (2016) introduced orthonormal regularization on each weight to stabilize the training of GANs. In their work, Brock et al. (2016) augmented the adversarial objective function by adding the following term:<br />
<br />
<math> ||W^TW-I||^2_F </math><br />
<br />
While this seems to serve the same purpose as spectral normalization, orthonormal regularization are mathematically quite different from our spectral normalization because the orthonormal regularization destroys the information about the spectrum by setting all the singular values to one. On the other hand, spectral normalization only scales the spectrum so that its maximum will be one. <br />
<br />
Gulrajani et al. (2017) used gradient penalty method in combination with WGAN. In their work, they placed K-Lipschitz constant on the discriminator by augmenting the objective function with the regularizer that rewards the function for having local 1-Lipschitz constant(i.e <math> ||\nabla_{\hat{x}} f ||_2 = 1 </math>) at discrete sets of points of the form <math> \hat{x}:=\epsilon \tilde{x} + (1-\epsilon)x </math> generated by interpolating a sample <math> \tilde{x} </math> from generative distribution and a sample <math> x </math> from the data distribution. This approach has an obvious weakness of being heavily dependent on the support of the current generative distribution. Moreover, WGAN-GP requires more computational cost than our spectral normalization with single-step power iteration, because the computation of <math> ||\nabla_{\hat{x}} f ||_2 </math> requires one whole round of forward and backward propagation.<br />
<br />
= Experimental settings and results = <br />
== Objective function ==<br />
For all methods other than WGAN-GP, we use <br />
<math> V(G,D) := E_{x\sim q_{data}(x)}[\log D(x)] + E_{z\sim p(z)}[\log (1-D(G(z)))]</math><br />
to update D, for the updates of G, use <math> -E_{z\sim p(z)}[\log(D(G(z)))] </math>. Alternatively, test performance of the algorithm with so-called hinge loss, which is given by <br />
<math> V_D(\hat{G},D)= E_{x\sim q_{data}(x)}[\min(0,-1+D(x))] + E_{z\sim p(z)}[\min(0,-1-D(\hat{G}(z)))] </math>, <math> V_G(G,\hat{D})=-E_{z\sim p(z)}[\hat{D}(G(z))] </math><br />
<br />
For WGAN-GP, we choose <br />
<math> V(G,D):=E_{x\sim q_{data}}[D(x)]-E_{z\sim p(z)}[D(G(z))]- \lambda E_{\hat{x}\sim p(\hat{x})}[(||\nabla_{\hat{x}}D(\hat{x}||-1)^2)]</math><br />
<br />
== Optimization ==<br />
Adam optimizer: 6 settings in total, related to <br />
* <math> n_{dis} </math>, the number of updates of the discriminator per one update of Adam. <br />
* learning rate <math> \alpha </math><br />
* the first and second momentum parameters <math> \beta_1, \beta_2 </math> of Adam<br />
<br />
[[File:inception score.png]]<br />
<br />
[[File:FID score.png]]<br />
<br />
The above image show the inception core and FID score of with settings A-F, and table show the inception scores of the different methods with optimal settings on CIFAR-10 and STL-10 dataset.<br />
<br />
== Singular values analysis on the weights of the discriminator D ==<br />
[[File:singular value.png]]<br />
<br />
In above figure, we show the squared singular values of the weight matrices in the final discriminator D produced by each method using the parameter that yielded the best inception score. As we predicted before, the singular values of the first fifth layers trained with weight clipping and weight normalization concentrate on a few components. On the other hand, the singular values of the weight matrices in those layers trained with spectral normalization is more broadly distributed.<br />
<br />
== Training time ==<br />
On CIFAR-10, SN-GANs is slightly slower than weight normalization, but significantly faster than WGAN-GP. As we mentioned in Section3, WGAN-GP is slower than other methods because WGAN-GP needs to calculate the gradient of gradient norm.<br />
<br />
== comparison between GN-GANs and orthonormal regularization ==<br />
[[File:comparison.png]]<br />
Above we explained in Section 3, orthonormal regularization is different from our method in that it destroys the spectral information and puts equal emphasis on all feature dimensions, including the ones that shall be weeded out in the training process. To see the extent of its possibly detrimental effect, we experimented by increasing the dimension of the feature space, especially at the final layer for which the training with our spectral normalization prefers relatively small feature space. Above figure shows the result of our experiments. As we predicted, the performance of the orthonormal regularization deteriorates as we increase the dimension of the feature maps at the final layer. SN-GANs, on the other hand, does not falter with this modification of the architecture.<br />
<br />
We also applied our method to the training of class conditional GANs on ILSVRC2012 dataset with 1000 classes, each consisting of approximately 1300 images, which we compressed to 128*128 pixels. GAN without normalization and GAN with layer normalization collapsed in the beginning of training and failed to produce any meaningful images. Above picture shows that the inception score of the orthonormal normalization plateaued around 20k iterations, while SN kept improving even afterward.<br />
<br />
= Algorithm of spectral normalization =<br />
To calculate the largest singular value of matrix <math> W </math> to implement spectral normalization, we appeal to power iterations. Algorithm is executed as follows:<br />
<br />
* Initialize <math>\tilde{u}_{l}\in R^{d_l} \text{for} l=1,\cdots,L </math> with a random vector (sampled from isotropic distribution) <br />
* For each update and each layer l:<br />
* Apply power iteration method to a unnormalized weight <math> W^l </math>:<br />
<br />
<math> \tilde{v_l}\leftarrow (W^l)^T\tilde{u_l}/||(W^l)^T\tilde{u_l}||_2 </math><br />
<br />
<math>\tilde{u_l}\leftarrow (W^l)^T\tilde{v_l}/||(W^l)^T\tilde{v_l}|| </math><br />
<br />
* Calculate <math> \bar{W_{SN}} </math> with the spectral norm :<br />
<br />
<math> \bar{W_{SN}}(W^l)=W^l/\sigma(W^l), \text{where} \sigma(W^l)=\tilde{u_l}^TW^l\tilde{v_l} </math><br />
<br />
* Update <math> W^l </math> with SGD on mini-batch dataset <math> D_M </math> with a learning rate <math> \alpha </math><br />
<br />
<math> W^l\leftarrow W^l-\alpha\nabla_{W^l}l(\bar{W_{SN}^l}(W^l),D_M) </math><br />
<br />
== Conclusions ==<br />
This paper proposes spectral normalization as a stabilizer of training of GANs. When we apply spectral normalization to the GANs on image generation tasks, the generated examples are more diverse than the conventional weight normalization and achieve better or comparative inception scores relative to previous studies. The method imposes global regularization on the discriminator as opposed to local regularization introduced by WGAN-GP, and can possibly used in combinations. In the future work, we would like to further investigate where our methods stand amongest other methods on more theoretical basis, and experiment our algorithm on larger and more complex datasets.<br />
<br />
== Critique(to be edited) ==<br />
<br />
== References ==<br />
# Lemaréchal, Claude, and J. B. Hiriart-Urruty. "Convex analysis and minimization algorithms I." Grundlehren der mathematischen Wissenschaften 305 (1996).</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/AmbientGAN:_Generative_Models_from_Lossy_Measurements&diff=32730stat946w18/AmbientGAN: Generative Models from Lossy Measurements2018-03-06T16:17:38Z<p>Pa2forsy: Explain inception score</p>
<hr />
<div>= Introduction =<br />
Generative Adversarial Networks operate by simulating complex distributions but training them requires access to large amounts of high quality data. Often times we only have access to noisy or partial observations, which will from here on be referred to as measurements of the true data. If we know the measurement function and would like to train a generative model for the true data, there are several ways to continue which have varying degrees of success. We will use noisy MNIST data as an illustrative example. Suppose we only see MNIST data that has been run through a Gaussian kernel (blurred) with some noise from a <math>N(0, 0.5^2)</math> distribution added to each pixel:<br />
<br />
<gallery mode="packed"><br />
File:mnist.png| True Data (Unobserved)<br />
File:mnistmeasured.png| Measured Data (Observed)<br />
</gallery><br />
<br />
<br />
=== Ignore the problem ===<br />
[[File:GANignore.png|500px]] [[File:mnistignore.png|300px]]<br />
<br />
Train a generative model directly on the measured data. This will obviously be unable to generate the true distribution before measurement has occurred. <br />
<br />
<br />
=== Try to recover the information lost ===<br />
[[File:GANrecovery.png|420px]] [[File:mnistrecover.png|300px]]<br />
<br />
Works better than ignoring the problem but depends on how easily the measurement function can be inverted.<br />
<br />
=== AmbientGAN ===<br />
[[File:GANambient.png|500px]] [[File:mnistambient.png|300px]]<br />
<br />
Ashish Bora, Eric Price and Alexandros G. Dimakis propose AmbientGAN as a way to recover the true underlying distribution from measurements of the true data. <br />
<br />
AmbientGAN works by training a generator which attempts to have the measurements of the output it generates fool the discriminator. The discriminator must distinguish between real and generated measurements.<br />
<br />
= Model =<br />
For the following variables superscript <math>r</math> represents the true distributions while superscript <math>g</math> represents the generated distributions. Let <math>x</math>, represent the underlying space and <math>y</math> for the measurement.<br />
<br />
Thus <math>p_x^r</math> is the real underlying distribution over <math>\mathbb{R}^n</math> that we are interested in. However if we assume our (known) measurement functions, <math>f_\theta: \mathbb{R}^n \to \mathbb{R}^m</math> are parameterized by <math>\theta \sim p_\theta</math>, we can only observe <math>y = f_\theta(x)</math>.<br />
<br />
Mirroring the standard GAN setup we let <math>Z \in \mathbb{R}^k, Z \sim p_z</math> and <math>\Theta \sim p_\theta</math> be random variables coming from a distribution that is easy to sample. <br />
<br />
If we have a generator <math>G: \mathbb{R}^k \to \mathbb{R}^n</math> then we can generate <math>X^g = G(Z)</math> which has distribution <math>p_x^g</math> a measurement <math>Y^g = f_\Theta(G(Z))</math> which has distribution <math>p_y^g</math>. <br />
<br />
Unfortunately we do not observe any <math>X^g \sim p_x</math> so we can use the discriminator directly on <math>G(Z)</math> to train the generator. Instead we will use the discriminator to distinguish between the <math>Y^g -<br />
f_\Theta(G(Z))</math> and <math>Y^r</math>. That is we train the discriminator, <math>D: \mathbb{R}^m \to \mathbb{R}</math> to detect if a measurement came from <math>p_y^r</math> or <math>p_y^g</math>.<br />
<br />
AmbientGAN has the objective function:<br />
<br />
<math>\min_G \max_D \mathbb{E}_{Y^r \sim p_y^r}[q(D(Y^r))] + \mathbb{E}_{Z \sim p_z, \Theta \sim p_\theta}[q(1 - D(f_\Theta(G(Z))))] </math><br />
<br />
where <math>q(.)</math> is the quality function; for the standard GAN <math>q(x) = log(x)</math> and for Wasserstein GAN <math>q(x) = x</math>.<br />
<br />
As a technical limitation we require <math>f_\theta</math> to be differentiable with the respect each input for all values of <math>\theta</math>.<br />
<br />
With this set up we sample <math>Z \sim p_z</math>, <math>\Theta \sim p_\theta</math>, and <math>Y^r \sim U\{y_1, \cdots, y_s\}</math> each iteration and use them to compute the stochastic gradients of the objective function. We alternate between updating <math>G</math> and updating <math>D</math>. <br />
<br />
= Empirical Results =<br />
<br />
The paper continues to present results of AmbientGAN under various measurement functions when compared to baseline models. We have already seen one example in the introduction: a comparison of AmbientGAN in the Convolve + Noise Measurement case compared to the ignore-baseline, and the unmeasure-baseline. <br />
<br />
=== Block-Pixels ===<br />
With the block-pixels measurement function each pixel is independently set to 0 with probability <math>p</math>.<br />
<br />
[[File:block-pixels.png]]<br />
<br />
Measurements from the celebA dataset with <math>p=0.95</math> (left). Images generated from GAN trained on unmeasured (via blurring) data (middle). Results generated from AmbientGAN (right).<br />
<br />
=== Block-Patch ===<br />
<br />
[[File:block-patch.png]]<br />
<br />
A random 14x14 patch is set to zero (left). Unmeasured using-navier-stoke inpainting (middle). AmbientGAN (right). <br />
<br />
=== Pad-Rotate-Project-<math>\theta</math> ===<br />
<br />
[[File:pad-rotate-project-theta.png]]<br />
<br />
Results generated by AmbientGAN where the measurement function 0 pads the images, rotates it by <math>\theta</math>, and projects it on to the x axis. For each measurement the value of <math>\theta</math> is known. <br />
<br />
The generated images only have the basic features of a face and is referred to as a failure case in the paper. However the measurement function performs relatively well given how lossy the measurement function is. <br />
<br />
=== Explanation of Inception Score ===<br />
To evaluate GAN performance, the authors make use of the inception score, a metric introduced by Salimans et al.(2016). To evaluate the inception score on a datapoint, a pre-trained inception classification model (Szegedy et al. 2016) is applied to that datapoint, and the KL divergence between its label distribution conditional on the datapoint and its marginal label distribution is computed. This KL divergence is the inception score. The idea is that meaningful images should be recognized by the inception model as belonging to some class, and so the conditional distribution should have low entropy, while the model should produce a variety of images, so the marginal should have high entropy. Thus an effective GAN should have a high inception score.<br />
<br />
=== MNIST Inception ===<br />
<br />
[[File:MNIST-inception.png]]<br />
<br />
<br />
AmbientGAN was compared with baselines through training several models with different probability <math>p</math> of blocking pixels. The plot on the left shows that the inception scores change as the block probability <math>p</math> changes. All four models are similar when no pixels are blocked <math>(p=0)</math>. By the increase of the blocking probability, AmbientGAN models present a relatively stable performance and perform better than the baseline models. Therefore, AmbientGAN is more robust than all other baseline models.<br />
<br />
The plot on the right reveals the changes in inception scores while the standard deviation of the additive Gaussian noise increased. Baselines perform better when the noise is small. By the increase of the variance, AmbientGAN models present a much better performance compare to the baseline models. Further AmbientGAN retains high inception scores as measurements become more and more lossy.<br />
<br />
=== CIFAR-10 Inception ===<br />
<br />
[[File:CIFAR-inception.png]]<br />
<br />
AmbientGAN is faster to train and more robust even on more complex distributions such as CIFAR-10. <br />
<br />
= Theoretical Results =<br />
<br />
The theoretical results in the paper prove the true underlying distribution of <math>p_x^r</math> can be recovered when we have data that comes from the Gaussian-Projection measurement, Fourier transform measurement and the block-pixels measurement. The do this by showing the distribution of the measurements <math>p_y^r</math> corresponds to a unique distribution <math>p_x^r</math>. Thus even when the measurement itself is non-invertible the effect of the measurement on the distribution <math>p_x^r</math> is invertible. Lemma 5.1 ensures this is sufficient to provide the AmbientGAN training process with a consistency guarantee. For full proofs of the results please see appendix A. <br />
<br />
=== Lemma 5.1 === <br />
Let <math>p_x^r</math> be the true data distribution, and <math>p_\theta</math> be the distributions over the parameters of the measurement function. Let <math>p_y^r</math> be the induced measurement distribution. <br />
<br />
Assume for <math>p_\theta</math> there is a unique probability distribution <math>p_x^r</math> that induces <math>p_y^r</math>. <br />
<br />
Then for the standard GAN model if the Discriminator is optimal, then a generator <math>G</math> is optimal if and only if <math>p_x^g = p_x^r</math>. <br />
<br />
=== Theorems 5.2===<br />
For the Gussian-Projection measurement model, there is a unique underlying distribution <math>p_x^{r} </math> that can induce the observed measurement distribution <math>p_y^{r} </math>.<br />
<br />
=== Theorems 5.3===<br />
Let <math> \mathcal{F} (\cdot) </math> denote the Fourier transform and let <math>supp (\cdot) </math> be the support of a function. Consider the Convolve+Noise measurement model with the convolution kernel <math> k </math>and additive noise distribution <math>p_\theta </math>. If <math> supp( \mathcal{F} (k))^{c}=\phi </math> and <math> supp( \mathcal{F} (p_\theta))^{c}=\phi </math>, then there is a unique distribution <math>p_x^{r} </math> that can induce the measurement distribution <math>p_y^{r} </math>.<br />
<br />
=== Theorems 5.4===<br />
Assume that each image pixel takes values in a finite set P. Thus <math>x \in P^n \subset \mathbb{R}^{n} </math>. Assume <math>0 \in P </math>, and consider the Block-Pixels measurement model with <math>p </math> being the probability of blocking a pixel. If <math>p <1</math>, then there is a unique distribution <math>p_x^{r} </math> that can induce the measurement distribution <math>p_y^{r} </math>. Further, for any <math> \epsilon > 0, \delta \in (0, 1] </math>, given a dataset of<br />
\begin{equation}<br />
s=\Omega \left( \frac{|P|^{2n}}{(1-p)^{2n} \epsilon^{2}} log \left( \frac{|P|^{n}}{\delta} \right) \right)<br />
\end{equation}<br />
IID measurement samples from pry , if the discriminator D is optimal, then with probability <math> \geq 1 - \delta </math> over the dataset, any optimal generator G must satisfy <math> d_{TV} \left( p^g_x , p^r_x \right) \leq \epsilon </math>, where <math> d_{TV} \left( \cdot, \cdot \right) </math> is the total variation distance.<br />
<br />
= Future Research =<br />
<br />
One critical weakness of AmbientGAN is the assumption that the measurement model is known. It would be nice to be able to train an AmbientGAN model when we have an unknown measurement model but also a small sample of unmeasured data. <br />
<br />
= References =<br />
# https://openreview.net/forum?id=Hy7fDog0b<br />
# Salimans, Tim, et al. "Improved techniques for training gans." Advances in Neural Information Processing Systems. 2016.<br />
# Szegedy, Christian, et al. "Rethinking the inception architecture for computer vision." Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. 2016.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=A_Neural_Representation_of_Sketch_Drawings&diff=32727A Neural Representation of Sketch Drawings2018-03-06T15:07:06Z<p>Pa2forsy: fix typo</p>
<hr />
<div>= Introduction =<br />
<br />
There have been many recent advances in neural generative models for low resolution pixel-based images. Humans however, do not see the world in a grid of pixels and more typically communicate drawings of the things we see using a series of pen strokes that represent components of objects. These pen strokes are similar to the way vector-based images store data. This paper proposes a new method for creating conditional and unconditional generative models for creating these kinds of vector sketch drawings based on recurrent neural networks (RNNs). The paper explores many applications of these kinds of models, especially creative applications and makes available their unique dataset of vector images.<br />
<br />
= Related Work =<br />
<br />
Previous work related to sketch drawing generation includes methods that focussed primarily on converting input photographs into equivalent vector line drawings. Image generating models using neural networks also exist but focussed more on generation of pixel-based imagery. Some recent work has focussed on handwritten character generation using RNNs and Mixture Density Networks to generate continuous data points. This work has been extended somewhat recently to conditionally and unconditionally generate handwritten vectorized Chinese Kanji characters by modelling them as a series of pen strokes. Furthermore, this paper builds on work that employed Sequence-to-Sequence models with Variational Autencoders to model English sentences in latent vector space.<br />
<br />
One of the limiting factors for creating models that operate on vector datasets has been the dearth of publicly available data. Previously available datasets include: Sketch, a set of 20K vector drawings; Sketchy, a set of 70K vector drawings; and ShadowDraw, a set of 30K raster images with extracted vector drawings.<br />
<br />
= Methodology =<br />
<br />
=== Dataset ===<br />
<br />
The “QuickDraw” dataset used in this research was assembled from 75K user drawings extracted from the game “Quick, Draw!” where users drew objects from one of hundreds of classes in 20 seconds or less. The dataset is split into 70K training samples and 2.5K validation and test samples each and represents each sketch a set of “pen stroke actions”. Each action is provided as a vector in the form <math>(\Delta x, \Delta y, p_{1}, p_{2}, p_{3})</math>. For each vector, <math>\Delta x</math> and <math>\Delta y</math> give the movement of the pen from the previous point, with the initial location being the origin. The last three vector elements are a one-hot representation of pen states; <math>p_{1}</math> indicates that the pen is down and a line should be drawn between the current point and the next point, <math>p_{2}</math> indicates that the pen is up and no line should be drawn between the current point and the next point, and <math>p_{3}</math> indicates that the drawing is finished and subsequent points and the current point should not be drawn.<br />
<br />
=== Sketch-RNN ===<br />
[[File:sketchrnn.PNG]]<br />
<br />
The model is a Sequence-to-Sequence Variational Autoencoder (VAE). The encoder model is a symmetric and parallel set of two RNNs that individually process the sketch drawings in forward and reverse order, respectively. The hidden state produced by each encoder model is then concatenated into a single hidden state <math>h</math>. <br />
<br />
The concatenated hidden state <math>h</math> is then projected into two vectors <math>\mu</math> and <math>\hat{\sigma}</math> each of size <math>N_{z}</math> using a fully connected layer. <math>\hat{\sigma}</math> is then converted into a non-negative standard deviation parameter <math>\sigma</math> using an exponential operator. These two parameters <math>\mu</math> and <math>\sigma</math> are then used along with an IID Gaussian vector distributed as <math>\mathcal{N}(0, I)</math> of size <math>N_{z}</math> to construct a random vector <math>z \in ℝ^{N_{z}}</math>, similar to the method used for VAE:<br />
\begin{align}<br />
\mu = W_{\mu}h + b_{mu}\textrm{, }\hat{\sigma} = W_{\sigma}h + b_{\sigma}\textrm{, }\sigma = exp\bigg{(}\frac{\hat{\sigma}}{2}\bigg{)}\textrm{, }z = \mu + \sigma \odot \mathcal{N}(0,I)<br />
\end{align}<br />
<br />
The decoder model is another RNN that samples output sketches from the latent vector <math>z</math>. The initial hidden states of each recurrent neuron are determined using <math>[h_{0}, c_{0}] = tanh(W_{z}z + b_{z})</math>. Each step of the decoder RNN accepts the previous point <math>S_{i-1}</math> and the latent vector <math>z</math> as concatenated input. The initial point given is the origin point with pen state down. The output at each step are the parameters for a probability distribution of the next point <math>S_{i}</math>. Outputs <math>\Delta x</math> and <math>\Delta y</math> are modelled using a Gaussian Mixture Model (GMM) with M normal distributions and output pen states <math>(q_{1}, q_{2}, q_{3})</math> modelled as a categorical distribution with one-hot encoding.<br />
\begin{align}<br />
P(\Delta x, \Delta y) = \sum_{j=1}^{M}\Pi_{j}\mathcal{N}(\Delta x, \Delta y | \mu_{x, j}, \mu_{y, j}, \sigma_{x, j}, \sigma_{y, j}, \rho_{xy, j})\textrm{, where }\sum_{j=1}^{M}\Pi_{j} = 1<br />
\end{align}<br />
<br />
For each of the M distributions in the GMM, parameters <math>\mu</math> and <math>\sigma</math> are output for both the x and y locations signifying the mean location of the next point and the standard deviation, respectively. Also output from each model is parameter <math>\rho_{xy}</math> signifying correlation of each bivariate normal distribution. An additional vector <math>\Pi</math> is output giving the mixture weights for the GMM. The output <math>S_{i}</math> is determined from each of the mixture models using softmax sampling from these distributions.<br />
<br />
One of the key difficulties in training this model is the highly imbalanced class distribution of pen states. In particular, the state that signifies a drawing is complete will only appear one time per each sketch and is difficult to incorporate into the model. In order to have the model stop drawing, the authors introduce a hyperparameter that limits the number of points per drawing to being no more than <math>N_{max}</math>, after which all output states form the model are set to (0, 0, 0, 0, 1) to force the drawing to stop.<br />
<br />
To sample from the model, the parameters required by the GMM and categorical distributions are generated at each time step and the model is sampled until a “stop drawing” state appears or the time state reaches time <math>N_{max}</math>. The authors also introduce a “temperature” parameter <math>\tau</math> that controls the randomness of the drawings by modifying the pen states, model standard deviations, and mixture weights as follows:<br />
<br />
\begin{align}<br />
\hat{q}_{k} \rightarrow \frac{\hat{q}_{k}}{\tau}\textrm{, }\hat{\Pi}_{k} \rightarrow \frac{\hat{\Pi}_{k}}{\tau}\textrm{, }\sigma^{2}_{x} \rightarrow \sigma^{2}_{x}\tau\textrm{, }\sigma^{2}_{y} \rightarrow \sigma^{2}_{y}\tau<br />
\end{align}<br />
<br />
This parameter <math>\tau</math> lies in the range (0, 1]. As the parameter approaches 0, the model becomes more deterministic and always produces the point locations with the maximum likelihood for a given timestep.<br />
<br />
=== Unconditional Generation ===<br />
The authors also explored unconditional generation of sketch drawings by only training the decoder RNN module. To do this, the initial hidden states of the RNN were set to 0, and only vectors from the drawing input are used as input without any conditional latent variable <math>z</math>. Different sketches are sampled from the network by only varying the temperature parameter <math>\tau</math> between 0.2 and 0.9<br />
<br />
=== Training ===<br />
The training procedure follows the same approach as training for VAE and uses a loss function that consists of the sum of Reconstruction Loss <math>L_{R}</math> and KL Divergence Loss <math>L_{KL}</math>. The reconstruction loss term is composed of two terms; <math>L_{s}</math>, which tries to maximize the log-likelihood of the generated probability distribution explaining the training data <math>S</math> and <math>L_{p}</math> which is the log loss of the pen state terms.<br />
\begin{align}<br />
L_{s} = -\frac{1}{N_{max}}\sum_{i=1}^{N_{S}}log\bigg{(}\sum_{j=1}^{M}\Pi_{j,i}\mathcal{N}(\Delta x_{i},\Delta y_{i} | \mu_{x,j,i},\mu_{y,j,i},\sigma_{x,j,i},\sigma_{y,j,i},\rho_{xy,j,i})\bigg{)}<br />
\end{align}<br />
\begin{align}<br />
L_{p} = -\frac{1}{N_{max}}\sum_{i=1}^{N_{max}} \sum_{k=1}^{3}p_{k,i}log(q_{k,i})<br />
\end{align}<br />
\begin{align}<br />
L_{R} = L_{s} + L{p}<br />
\end{align}<br />
<br />
The KL divergence loss <math>L_{KL}</math> measures the difference between the latent vector <math>z</math> and an IID Gaussian distribution with 0 mean and unit variance. This term, normalized by the number of dimensions <math>N_{z}</math> is calculated as:<br />
\begin{align}<br />
L_{KL} = -\frac{1}{2N_{z}}\big{(}1 + \hat{\sigma} - \mu^{2} – exp(\hat{\sigma})\big{)}<br />
\end{align}<br />
<br />
The loss for the entire model is thus the weighted sum:<br />
\begin{align}<br />
Loss = L_{R} + w_{KL}L_{KL}<br />
\end{align}<br />
<br />
The value of the weight parameter <math>w_{KL}</math> has the effect that as <math>w_{KL} \rightarrow 0</math>, there is a loss in ability to enforce a prior over the latent space and the model assumes the form of a pure autoencoder.<br />
<br />
= Experiments =<br />
The authors trained multiple conditional and unconditional models using varying values of <math>w_{KL}</math> and recorded the different <math>L_{R}</math> and <math>L_{KL}</math> values at convergence. The network used LSTM as it’s encoder RNN and HyperLSTM as the decoder network. The HyperLSTM model was used for decoding because it has a history of being useful in sequence generation tasks.<br />
<br />
=== Conditional Reconstruction ===<br />
[[File:conditional_generation.PNG]]<br />
<br />
The authors qualitatively assessed the reconstructed images <math>S’</math> given input sketch <math>S</math> using different values for the temperature hyperparameter <math>\tau</math>. The figure above shows the results for different values of <math>\tau</math> starting with 0.01 at the far left and increasing to 1.0 on the far right. Interestingly, sketches with extra features like a cat with 3 eyes is reproduced as a sketch of a cat with two eyes and sketches of object of a different class such as a toothbrush is reproduced as a sketch of a cat that maintains several of the input toothbrush sketches features.<br />
<br />
=== Latent Space Interpolation ===<br />
[[File:latent_space_interp.PNG]]<br />
<br />
The latent space vectors <math>z</math> have few “gaps” between encoded latent space vectors due to the enforcement of a Guassian prior. This allowed the authors to do simple arithmetic on the latent vectors from different sketches and produce logical resulting images in the same style as latent space arithmetic on Word2Vec vectors.<br />
<br />
=== Sketch Drawing Analogies ===<br />
Given the latent space arithmetic possible, it was found that features of a sketch could be added after some sketch input was encoded. For example, a drawing of a cat with a body could be produced by providing the network with a drawing of a cat’s head, and then adding a latent vector to the embedding layer that represents “body”. As an example, this “body” vector might be produced by taking a drawing of a pig with a body and subtracting a vector representing the pigs head.<br />
<br />
=== Predicting Different Endings of Incomplete Sketches ===<br />
[[File:predicting_endings.PNG]]<br />
<br />
Using the decoder RNN only, it is possible to finish sketches by conditioning future vector line predictions on the previous points. To do this, the decoder RNN is first used to encode some existing points into the hidden state of the decoder network and then generating the remaining points of the sketch.<br />
<br />
= Applications and Future Work =<br />
Sketch-rnn may enable the production of several creative applications. These might include applications that help suggest ways an artist could finish a sketch, enable artists to explore latent space arithmetic to find interesting outputs given different sketch inputs, or allow the production of multiple different sketches of some object as a purely generative application. The authors suggest that providing some conditional sketch of an object to a model designed to produce output of a different class might be useful for producing sketches that morph the two different object classes into one sketch. <br />
<br />
Sketch-rnn may also be useful as a teaching tool to help people learn how to draw, especially if it were to be trained on higher quality images. Teaching tools might suggest to students how to procede to finish a sketch, or intake low fidelity sketches to produce a higher quality and “more coherent” output sketch.<br />
<br />
Finally the authors suggest that combining this model with another that produces photorealistic pixel-based images using sketch input, such as Pix2Pix may be an interesting direction for future research. In this case, the output from the sketch-rnn model would be used as input for Pix2Pix and could produce photorealistic images given some crude sketch from a user.<br />
<br />
= Conclusion =<br />
The authors presented sketch-rnn, a RNN model for modelling and generating vector-based sketch drawings. The VAE inspired architecture allows sampling the latent space to generate new drawings and also allows for applications that use latent space arithmetic in the style of Word2Vec to produce new drawings given operations on embedded sketch vectors. The authors also made available a large dataset of sketch drawings in the hope of encouraging more research in the area of vector-based image modelling.<br />
<br />
= Criticisms =<br />
The paper produces an interesting model that can effectively model vector-based images instead of traditional pixel-based images. This is an interesting problem because vector based images require producing a new way to encode the data. While the results from this paper are interesting, most of the techniques used are borrowed ideas from Variational Autoencoders and the main architecture is not terribly groundbreaking. <br />
<br />
One novel part about the architecture presented was the way the authors used GMMs in the decoder network. While this was interesting and seemed to allow the authors to produce different outputs given the same latent vector input <math>z</math> by manipulating the <math>\tau</math> hyperparameter, it was not that clear in the article why GMMs were used instead of a more simple architecture. Much time was spent explaining basics about GMM parameters like <math>\mu</math> and <math>\sigma</math>, but there was comparatively little explanation about how points were actually sampled from these mixture models.<br />
<br />
Finally, the authors gloss somewhat over how they were able to encode previous sketch points using only the decoder network into the hidden state of the decoder RNN to finish partially finished sketches. I can only assume that some kind of back-propagation was used to encode the expected sketch points into the hidden parameters of the decoder, but no explanation was given in the paper.<br />
<br />
= Source =<br />
<br />
Ha, D., & Eck, D. A neural representation of sketch drawings. In Proc. International Conference on Learning Representations (2018).</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Synthetic_and_natural_noise_both_break_neural_machine_translation&diff=32725stat946w18/Synthetic and natural noise both break neural machine translation2018-03-06T14:56:18Z<p>Pa2forsy: Describe the MT systems in more detail</p>
<hr />
<div>== Introduction ==<br />
* Humans have surprisingly robust language processing systems which can easily overcome typos, e.g.<br />
<br />
Aoccdrnig to a rscheearch at Cmabrigde Uinervtisy, it deosn't mttaer in waht oredr the ltteers in a wrod are, the olny iprmoetnt tihng is taht the frist and lsat ltteer be at the rghit pclae.<br />
<br />
* A person's ability to read this text comes as no surprise to the Psychology literature<br />
*# Saberi \& Perrott (1999) found that this robustness extends to audio as well.<br />
*# Rayner et al. (2006) found that in noisier settings reading comprehension only slowed by 11 \%.<br />
*# McCusker et al. (1981) found that the common case of swapping letters could often go unnoticed by the reader.<br />
*# Mayall et al (1997) shows that we rely on word shape.<br />
*# Reicher, 1969; Pelli et al., (2003) found that we can switch between whole word recognition but the first and last letter positions are required to stay constant for comprehension<br />
<br />
However, NMT(neural machine translation) systems are brittle. i.e. The Arabic word<br />
[[File:Good_morning.PNG]] means a blessing for good morning, however [[File:Hunt.PNG]] means hunt or slaughter. <br />
<br />
Facebook's MT system mistakenly confused two words that only differ by one character, a situation that is challenging for a character-based NMT system.<br />
<br />
Figure 1 shows the performance translating German to English as a function of the percent of German words modified. Here we show two types of noise: (1) Random permutation of the word and (2) Swapping a pair of adjacent letters that does not include the first or last letter of the word. The important thing to note is that even small amounts of noise lead to substantial drops in performance.<br />
<br />
[[File:BLEU_plot.PNG]] <br />
<br />
BLEU (bilingual evaluation understudy) 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". BLEU is between 0 and 1.<br />
<br />
This paper explores two simple strategies for increasing model robustness:<br />
# using structure-invariant representations ( character CNN representation)<br />
# robust training on noisy data, a form of adversarial training.<br />
<br />
== Adversarial examples ==<br />
The growing literature on adversarial examples has demonstrated how dangerous it can be to have brittle machine learning systems being used so pervasively in the real world.<br />
<br />
The paper devises simple methods for generating adversarial examples for NMT. They do not assume any access to the NMT models' gradients, instead relying on cognitively-informed and naturally occurring language errors to generate noise.<br />
<br />
== MT system ==<br />
We experiment with three different NMT systems with access to character information at different levels.<br />
# Use <code>char2char</code>, the fully character-level model of (Lee et al. 2017). This model processes a sentence as a sequence of characters. The encoder works as follows: the characters are embedded as vectors, and then the sequence of vectors is fed to a convolutional layer. The sequence output by the convolutional layer is then shortened by max pooling in the time dimension. The output of the max-pooling layer is then fed to a four-layer highway network (Srivasta et al. 2015), and the output of the highway network is in turn fed to a bidirectional GRU, producing a sequence of hidden units. The sequence of hidden units is then processed by the decoder, a GRU with attention, to produce probabilities over sequences of output characters.<br />
# Use <code>Nematus</code> (Sennrich et al., 2017), a popular NMT toolkit. It is another sequence-to-sequence model with several architecture modifications, especially operating on sub-word units using byte-pair encoding. Byte-pair encoding (Sennich et al. 2015, Gage 1994) is an algorithm according to which we begin with a list of characters as our symbols, and repeatedly fuse common combinations to create new symbols. For example, if we begin with the letters a to z as our symbol list, and we find that "th" is the most common two-letter combination in a corpus, then we would add "th" to our symbol list in the first iteration. After we have used this algorithm to create a symbol list of the desired size, we apply a standard encoder-decoder with attention.<br />
# Use an attentional sequence-to-sequence model with a word representation based on a character convolutional neural network (<code>charCNN</code>). The <code>charCNN</code> model is similar to <code>char2char</code>, but uses a shallower highway network and, although it reads the input sentence as characters, it produces as output a probability distribution over words, not characters.<br />
<br />
== DATA ==<br />
=== MY DATA ===<br />
We use the TED talks parallel corpus prepared for IWSLT 2016 (Cettolo et al., 2012) for testing all of the NMT systems.<br />
[[File:Table1x.PNG]]<br />
<br />
=== NATURAL AND ARTIFICIAL NOISE ===<br />
==== NATURAL NOISE ====<br />
To three different languages French, German and Czech, they have their own frequent natural errors. <br />
<br />
The author harvest naturally occurring errors (typos, misspellings, etc.) corresponding to these three languages from available corpora of edits to build a look-up table of possible lexical replacements.<br />
<br />
==== Synthetic Noise ====<br />
In addition to naturally collected sources of error, we also experiment with four types of synthetic noise: Swap, Middle Random, Fully Random, and Key Typo. <br />
# <code>Swap</code>: The first and simplest source of noise is swapping two letters (do not alter the first or last letters).<br />
# <code>Middle Random</code>: Randomize the order of all the letters in a word except for the first and last.<br />
# <code>Fully Random</code> Completely randomized words.<br />
# <code>Keyboard Typo</code> Randomly replace one letter in each word with an adjacent key<br />
<br />
[[File:Table3x.PNG]]<br />
<br />
Table 3 shows BLEU scores of models trained on clean (Vanilla) texts and tested on clean and noisy<br />
texts. All models suffer a significant drop in BLEU when evaluated on noisy texts. This is true<br />
for both natural noise and all kinds of synthetic noise. The more noise in the text, the worse the<br />
translation quality, with random scrambling producing the lowest BLEU scores.<br />
<br />
== Dealing with noise ==<br />
=== STRUCTURE INVARIANT REPRESENTATIONS ===<br />
The three NMT models are all sensitive to word structure. The <code>char2char</code> and <code>charCNN</code> models both have convolutional layers on character sequences, designed to capture character n-grams. The model in <code>Nematus</code> is based on sub-word units obtained with BPE. It thus relies on character order.<br />
<br />
The simplest to improve such model is to take the average character embeddings as a word representation. This model, referred to as <code>meanChar</code>, first generates a word representation by averaging character embeddings, and then proceeds with a word-level encoder similar to the <code>charCNN</code> model.<br />
<br />
[[File:Table5x.PNG]]<br />
<br />
<code>meanChar</code> is good with the other three scrambling errors (Swap, Middle Random and Fully Random), but bad with Keyboard error and Natural errors.<br />
<br />
=== BLACK-BOX ADVERSARIAL TRAINING ===<br />
<br />
<code>charCNN</code> Performance<br />
[[File:Table6x.PNG]]<br />
<br />
== Analysis ==<br />
=== LEARNING MULTIPLE KINDS OF NOISE IN <code>charCNN</code> ===<br />
They analyze the weights learned by <code>charCNN</code> models trained on two kinds of input: completely scrambled words (Rand) without other kinds of noise, and a mix of Rand+Key+Nat kinds of noise.<br />
<br />
For each model, they compute the variance across the filter dimension for each one of the 1000 filters and for each one out of 25 character embedding dimensions. The we average the variances across the 1000 filters. <br />
<br />
[[File:Table7x.PNG]]<br />
<br />
== Conclusion ==<br />
In this work, they have shown that character-based NMT models are extremely brittle and tend to break when presented with both natural and synthetic kinds of noise. After models comparison, they found that a character-based CNN can learn to<br />
address multiple types of errors that are seen in training.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=Word_translation_without_parallel_data&diff=32671Word translation without parallel data2018-03-06T00:55:10Z<p>Pa2forsy: Added a proof of the closed-form solution to the orthogonal procrustes problem</p>
<hr />
<div>[[File:Toy_example.png]]<br />
<br />
= Presented by =<br />
<br />
Xia Fan<br />
<br />
= Introduction =<br />
<br />
Many successful methods for learning relationships between languages stem from the hypothesis that there is a relationship between the context of words and their meanings. This means that if an adequate representation of a language is found in a high dimensional space (this is called an embedding), then words similar to a given word are close to one another in this space (ex. some norm can be minimized to find a word with similar context). Historically, another significant hypothesis is that these embedding spaces show similar structures over different languages. That is to say that given an embedding space for English and one for Spanish, a mapping could be found that aligns the two spaces and such a mapping could be used as a tool for translation. Many papers exploit these hypothesis, but use large parallel datasets for training. Recently, to remove the need for supervised training, methods have been implemented that utilized identical character strings (ex. letters or digits) in order to try to align the embeddings. The downside of this approach is that the two languages need to be similar to begin with as they need to have some shared basic building block. The method proposed in this paper uses an adversarial method to find this mapping between the embedding spaces of two language without the use of large parallel datasets.<br />
<br />
This paper introduce a model that either is on par, or outperforms supervised state-of-the-art methods, without employing any cross-lingual annotated data. This method use a similar idea with GAN: it leverages adversarial training to learn a linear mapping from a source to a distinguish between the mapped source embeddings and the target embeddings, while the mapping is jointly trained to fool the discriminator. Second, this paper extract a synthetic dictionary from the resulting shared embedding space and fine-tune the mapping with the closed-form Procrustes solution from Schonemann (1966). Third, this paper also introduce an unsupervised selection metric that is highly correlated with the mapping quality and that we use both as a stopping criterion and to select the best hyper-parameters.<br />
<br />
= Model =<br />
<br />
<br />
=== Estimation of Word Representations in Vector Space ===<br />
<br />
This model focuses on learning a mapping between the two sets such that translations are close in the shared space. Before talking about the model it used, a model which can exploit the similarities of monolingual embedding spaces should be introduced. Mikolov et al.(2013) use a known dictionary of n=5000 pairs of words <math> \{x_i,y_i\}_{i\in{1,n}} </math>. and learn a linear mapping W between the source and the target space such that <br />
<br />
\begin{align}<br />
W^*=argmin_{W{\in}M_d(R)}||WX-Y||_F \hspace{1cm} (1)<br />
\end{align}<br />
<br />
where d is the dimension of the embeddings, <math> M_d(R) </math> is the space of d*d matrices of real numbers, and X and Y are two aligned matrices of size d*n containing the embeddings of the words in the parallel vocabulary. <br />
<br />
Xing et al. (2015) showed that these results are improved by enforcing orthogonality constraint on W. In that case, the equation (1) boils down to the Procrustes problem, which advantageously offers a closed form solution obtained from the singular value decomposition (SVD) of <math> YX^T </math> :<br />
<br />
\begin{align}<br />
W^*=argmin_{W{\in}M_d(R)}||WX-Y||_F=UV^T, with U\Sigma V^T=SVD(YX^T).<br />
\end{align}<br />
<br />
<br />
<br />
This can be proven as follows. First note that <br />
\begin{align}<br />
&||WX-Y||_F\\<br />
&= \langle WX, WX \rangle_F -2 \langle W X, Y \rangle_F + \langle Y, Y \rangle_F \\<br />
&= ||X||_F^2 -2 \langle W X, Y \rangle_F + || Y||_F^2, <br />
\end{align}<br />
<br />
where <math display="inline"> \langle \cdot, \cdot \rangle_F </math> denotes the Frobenius inner-product and we have used the orthogonality of <math display="inline"> W </math>. It follows that we need only maximize the inner-product above. Let <math display="inline"> u_1, \ldots, u_d </math> denote the columns of <math display="inline"> U </math>. Let <math display="inline"> v_1, \ldots , v_d </math> denote the columns of <math display="inline"> V </math>. Let <math display="inline"> \sigma_1, \ldots, \sigma_d </math> denote the diagonal entries of <math display="inline"> \Sigma </math>. We have<br />
\begin{align}<br />
&\langle W X, Y \rangle_F \\<br />
&= \text{Tr} (W^T Y X^T)\\<br />
&=\sum_i \sigma_i \text{Tr}(W^T u_i v_i^T)\\<br />
&=\sum_i \sigma_i ((Wv_i)^T u_i )\\<br />
&\le \sum_i \sigma_i ||Wv_i|| ||u_i||\\<br />
&= \sum_i \sigma_i<br />
\end{align}<br />
where we have used the invariance of trace under cyclic permutations, Cauchy-Schwarz, and the orthogonality of the columns of U and V. Note that choosing <br />
\begin{align}<br />
W=UV^T<br />
\end{align}<br />
achieves the bound. This completes the proof.<br />
<br />
=== Domain-adversarial setting ===<br />
<br />
This paper shows how to learn this mapping W without cross-lingual supervision. An illustration of the approach is given in Fig. 1. First, this model learn an initial proxy of W by using an adversarial criterion. Then, it use the words that match the best as anchor points for Procrustes. Finally, it improve performance over less frequent words by changing the metric of the space, which leads to spread more of those points in dense region. <br />
<br />
[[File:Toy_example.png |frame|none|alt=Alt text|Figure 1: Toy illustration of the method. (A) There are two distributions of word embeddings, English words in red denoted by X and Italian words in blue denoted by Y , which we want to align/translate. Each dot represents a word in that space. The size of the dot is proportional to the frequency of the words in the training corpus of that language. (B) Using adversarial learning, we learn a rotation matrix W which roughly aligns the two distributions. The green stars are randomly selected words that are fed to the discriminator to determine whether the two word embeddings come from the same distribution. (C) The mapping W is further refined via Procrustes. This method uses frequent words aligned by the previous step as anchor points, and minimizes an energy function that corresponds to a spring system between anchor points. The refined mapping is then used to map all words in the dictionary. (D) Finally, we translate by using the mapping W and a distance metric, dubbed CSLS, that expands the space where there is high density of points (like the area around the word “cat”), so that “hubs” (like the word “cat”) become less close to other word vectors than they would otherwise (compare to the same region in panel (A)).]]<br />
<br />
Let <math> X={x_1,...,x_n} </math> and <math> Y={y_1,...,y_m} </math> be two sets of n and m word embeddings coming from a source and a target language respectively. A model is trained is trained to discriminate between elements randomly sampled from <math> WX={Wx_1,...,Wx_n} </math> and Y, We call this model the discriminator. W is trained to prevent the discriminator from making accurate predictions. As a result, this is a two-player game, where the discriminator aims at maximizing its ability to identify the origin of an embedding, and W aims at preventing the discriminator from doing so by making WX and Y as similar as possible. This approach is in line with the work of Ganin et al.(2016), who proposed to learn latent representations invariant to the input domain, where in our case, a domain is represented by a language(source or target).<br />
<br />
1. Discriminator objective<br />
<br />
Refer to the discriminator parameters as <math> \theta_D </math>. Consider the probability <math> P_{\theta_D}(source = 1|z) </math> that a vector z is the mapping of a source embedding (as opposed to a target embedding) according to the discriminator. The discriminator loss can be written as:<br />
<br />
\begin{align}<br />
L_D(\theta_D|W)=-\frac{1}{n} \sum_{i=1}^n log P_{\theta_D}(source=1|Wx_i)-\frac{1}{m} \sum_{i=1}^m log P_{\theta_D}(source=0|Wx_i)<br />
\end{align}<br />
<br />
2. Mapping objective <br />
<br />
In the unsupervised setting, W is now trained so that the discriminator is unable to accurately predict the embedding origins: <br />
<br />
\begin{align}<br />
L_W(W|\theta_D)=-\frac{1}{n} \sum_{i=1}^n log P_{\theta_D}(source=0|Wx_i)-\frac{1}{m} \sum_{i=1}^m log P_{\theta_D}(source=1|Wx_i)<br />
\end{align}<br />
<br />
3. Learning algorithm <br />
To train our model, we follow the standard training procedure of deep adversarial networks of Goodfellow et al. (2014). For every input sample, the discriminator and the mapping matrix W are trained successively with stochastic gradient updates to respectively minimize <math> L_D </math> and <math> L_W </math><br />
<br />
=== Refinement procedre ===<br />
<br />
The matrix W obtained with adversarial training gives good performance (see Table 1), but the results are still not on par with the supervised approach. In fact, the adversarial approach tries to align all words irrespective of their frequencies. However, rare words have embeddings that are less updated and are more likely to appear in different contexts in each corpus, which makes them harder to align. Under the assumption that the mapping is linear, it is then better to infer the global mapping using only the most frequent words as anchors. Besides, the accuracy on the most frequent word pairs is high after adversarial training.<br />
To refine our mapping, this paper build a synthetic parallel vocabulary using the W just learned with ad- versarial training. Specifically, this paper consider the most frequent words and retain only mutual nearest neighbors to ensure a high-quality dictionary. Subsequently, this paper apply the Procrustes solution in (2) on this generated dictionary. Considering the improved solution generated with the Procrustes al- gorithm, it is possible to generate a more accurate dictionary and apply this method iteratively, similarly to Artetxe et al. (2017). However, given that the synthetic dictionary obtained using ad- versarial training is already strong, this paper only observe small improvements when doing more than one iteration, i.e., the improvements on the word translation task are usually below 1%.<br />
<br />
=== Cross-Domain similarity local scaling ===<br />
<br />
In this paper, it considers a bi-partite neighborhood graph, in which each word of a given dictionary is connected to its K nearest neighbors in the other language. <math> N_T(Wx_s) </math> is used to denote the neighborhood, on this bi-partite graph, associated with a mapped source word embedding <math> Wx_s </math>. All K elements of <math> N_T(Wx_s) </math> are words from the target language. Similarly we denote by <math> N_S(y_t) </math> the neighborhood associated with a word t of the target language. Consider the mean similarity of a source embedding <math> x_s </math> to its target neighborhood as<br />
<br />
\begin{align}<br />
r_T(Wx_s)=\frac{1}{K}\sum_{y\in N_T(Wx_s)}cos(Wx_s,y_t)<br />
\end{align}<br />
<br />
where cos(,) is the cosine similarity. This is used to define similarity measure CSLS(.,.) between mapped source words and target words,as <br />
<br />
\begin{align}<br />
CSLS(Wx_s,y_t)=2cos(Wx_s,y_t)-r_T(Wx_s)-r_S(y_t)<br />
\end{align}<br />
<br />
= Training and architectural choices =<br />
=== Architecture ===<br />
<br />
This paper use unsupervised word vectors that were trained using fastText2. These correspond to monolin- gual embeddings of dimension 300 trained on Wikipedia corpora; therefore, the mapping W has size 300 × 300. Words are lower-cased, and those that appear less than 5 times are discarded for training. As a post-processing step, only the first 200k most frequent words were selected in the experiments.<br />
For the discriminator, it use a multilayer perceptron with two hidden layers of size 2048, and Leaky-ReLU activation functions. The input to the discriminator is corrupted with dropout noise with a rate of 0.1. As suggested by Goodfellow (2016), a smoothing coefficient s = 0.2 is included in the discriminator predictions. This paper use stochastic gradient descent with a batch size of 32, a learning rate of 0.1 and a decay of 0.95 both for the discriminator and W . <br />
<br />
=== Discriminator inputs ===<br />
The embedding quality of rare words is generally not as good as the one of frequent words (Luong et al., 2013), and it is observed that feeding the discriminator with rare words had a small, but not negligible negative impact. As a result, this paper only feed the discriminator with the 50,000 most frequent words. At each training step, the word embeddings given to the discriminator are sampled uniformly. Sampling them according to the word frequency did not have any noticeable impact on the results.<br />
<br />
=== Orthogonality===<br />
In this work, it propose to use a simple update step to ensure that the matrix W stays close to an orthogonal matrix during training (Cisse et al. (2017)). Specifically, the following update rule on the matrix W is used :<br />
<br />
\begin{align}<br />
W \leftarrow (1+\beta)W-\beta(WW^T)W<br />
\end{align}<br />
<br />
where β = 0.01 is usually found to perform well. This method ensures that the matrix stays close to the manifold of orthogonal matrices after each update.<br />
<br />
=== Dictionary generation ===<br />
The refinement step requires to generate a new dictionary at each iteration. In order for the Procrustes solution to work well, it is best to apply it on correct word pairs. As a result, the CSLS method is used to select more accurate translation pairs in the dictionary. To increase even more the quality of the dictionary, and ensure that W is learned from correct translation pairs, only mutual nearest neighbors were considered, i.e. pairs of words that are mutually nearest neighbors of each other according to CSLS. This significantly decreases the size of the generated dictionary, but improves its accuracy, as well as the overall performance.<br />
<br />
=== Validation criterion for unsupervised model selection ===<br />
<br />
This paper consider the 10k most frequent source words, and use CSLS to generate a translation for each of them, then compute the average cosine similarity between these deemed translations, and use this average as a validation metric. Figure 2 shows the correlation between the evaluation score and this unsupervised criterion (without stabilization by learning rate shrinkage)<br />
<br />
<br />
<br />
[[File:fig2_fan.png |frame|none|alt=Alt text|Figure 2: Unsupervised model selection.<br />
Correlation between our unsupervised vali- dation criterion (black line) and actual word translation accuracy (blue line). In this par- ticular experiment, the selected model is at epoch 10. Observe how our criterion is well correlated with translation accuracy.]]<br />
<br />
= Results =<br />
<br />
In what follows, the results on word translation retrieval using our bilingual dictionaries were presented in Table 1 and the comparison to previous work in Table 2 where unsupervised model significantly outperform previous approaches. The results on the sentence translation retrieval task were also presented in Table 3 and the cross-lingual word similarity task in Table 4. Finally, the results on word-by-word translation for English-Esperanto was presented in Table 5.<br />
<br />
[[File:table1_fan.png |frame|none|alt=Alt text|Table 1: Word translation retrieval P@1 for our released vocabularies in various language pairs. We consider 1,500 source test queries, and 200k target words for each language pair. We use fastText embeddings trained on Wikipedia. NN: nearest neighbors. ISF: inverted softmax. (’en’ is English, ’fr’ is French, ’de’ is German, ’ru’ is Russian, ’zh’ is classical Chinese and ’eo’ is Esperanto)]]<br />
<br />
<br />
[[File:table2_fan.png |frame|none|alt=Alt text|English-Italian word translation average precisions (@1, @5, @10) from 1.5k source word queries using 200k target words. Re- sults marked with the symbol † are from Smith et al. (2017). Wiki means the embeddings were trained on Wikipedia using fastText. Note that the method used by Artetxe et al. (2017) does not use the same super- vision as other supervised methods, as they only use numbers in their ini- tial parallel dictionary.]]<br />
<br />
[[File:table3_fan.png |frame|none|alt=Alt text|Table 3: English-Italian sentence translation retrieval. We report the average P@k from 2,000 source queries using 200,000 target sen- tences. We use the same embeddings as in Smith et al. (2017). Their re- sults are marked with the symbol †.]]<br />
<br />
[[File:table4_fan.png |frame|none|alt=Alt text|Table 4: Cross-lingual wordsim task. NASARI<br />
(Camacho-Collados et al. (2016)) refers to the official SemEval2017 baseline. We report Pearson correlation.]]<br />
<br />
[[File:table5_fan.png |frame|none|alt=Alt text|Table 5: BLEU score on English-Esperanto.<br />
Although being a naive approach, word-by- word translation is enough to get a rough idea of the input sentence. The quality of the gener- ated dictionary has a significant impact on the BLEU score.]]<br />
<br />
= Conclusion =<br />
This paper shows for the first time that one can align word embedding spaces without any cross-lingual supervision, i.e., solely based on unaligned datasets of each language, while reaching or outperforming the quality of previous supervised approaches in several cases. Using adversarial training, the model is able to initialize a linear mapping between a source and a target space, which is also used to produce a synthetic parallel dictionary. It is then possible to apply the same techniques proposed for supervised techniques, namely a Procrustean optimization.<br />
<br />
= Source =<br />
Lample, Guillaume; Denoyer, Ludovic; Ranzato, Marc'Aurelio <br />
| Unsupervised Machine Translation Using Monolingual Corpora Only<br />
| arVix: 1701.04087</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Rethinking_the_Smaller-Norm-Less-Informative_Assumption_in_Channel_Pruning_of_Convolutional_Layers&diff=32670stat946w18/Rethinking the Smaller-Norm-Less-Informative Assumption in Channel Pruning of Convolutional Layers2018-03-05T23:57:06Z<p>Pa2forsy: /* Motivation */</p>
<hr />
<div>== Introduction ==<br />
<br />
With the recent and ongoing surge in low-power, intelligent agents (such as wearables, smartphones, and IoT devices), there exists a growing need for machine learning models to work well in memory- and cpu-constrained environments. Deep learning models have achieved state-of-the-art on a broad range of tasks; however, they are difficult to deploy in their original forms. For example, AlexNet (Krizhevsky et al., 2012), a model for image classification, contains 61 million parameters and requires 1.5 billion FLOPs in one inference pass. A more accurate model, ResNet-50 (He et al., 2016), has 25 million parameters but requires 4.08 billion FLOPs. Clearly, it would be difficult to deploy and run these models on low-power devices.<br />
<br />
In general, model compression can be accomplished using four main non-exclusive methods (Cheng et al., 2017): weight pruning, quantization, matrix transformations, and weight tying. By non-exclusive, we mean that these methods can be used in combination for pruning a single model; The use of one method does not exclude any of the other methods from being viable. <br />
<br />
Ye et al. (2018) explores pruning entire channels in a convolutional neural network. Past work has mostly focused on norm- or error-based heuristics to prune channels; instead, Ye et al. (2018) show that their approach is, "mathematically appealing from an optimization perspective and easy to reproduce" (Ye et al., 2018). In other words, they argue that the norm-based assumption is not as informative or theoretically justified as their approach, and provide strong empirical findings.<br />
<br />
== Motivation ==<br />
<br />
Some previous works on pruning channel filters (Li et al., 2016; Molchanov et al., 2016) have focused on using the L1 norm to determine the importance of a channel. Ye et al. (2018) show that, in the deep linear convolution case, penalizing the per-layer norm is coarse-grained; they argue that one cannot assign different coefficients to L1 penalties associated with different layers without risking the loss function being susceptible to trivial re-parameterizations. As an example, consider the following deep linear convolutional neural network with modified LASSO loss:<br />
<br />
$$\min \mathbb{E}_D \lVert W_{2n} * \dots * W_1 x - y\rVert^2 + \lambda \sum_{i=1}^n \lVert W_{2i} \rVert_1$$<br />
<br />
where W are the weights and * is convolution. Here we have chosen the coefficient 0 for the L1 penalty associated with odd-numbered layers and the coefficient 1 for the L1 penalty associated with even-numbered layers. This loss is susceptible to trivial re-paramterizations: without affecting the least-squares loss, we can always reduce the LASSO loss by halving the weights of all even-numbered layers and doubling the weights of all odd-numbered layers.<br />
<br />
Furthermore, batch normalization (Ioffe, 2015) is incompatible with this method of weight regularization. In other words, penalizing the norm of a filter in a deep convolutional network is hard to justify from a theoretical perspective.<br />
<br />
Thus, although not providing a complete theoretical guarantee on loss, Ye et al. (2018) develop a pruning technique that claims to be more justified than norm-based pruning is.<br />
<br />
== Method ==<br />
<br />
At a high level, Ye et al. (2018) propose that, instead of discovering sparsity via penalizing the per-filter or per-channel norm, penalize the batch normalization scale parameters ''gamma'' instead. The reasoning is that by having fewer parameters to constrain and working with normalized values, sparsity is easier to enforce, monitor, and learn. Having sparse batch normalization terms has the effect of pruning '''entire''' channels: if ''gamma'' is zero, then the output at that layer becomes constant (the bias term), and thus the preceding channels can be pruned.<br />
<br />
=== Summary ===<br />
<br />
The basic algorithm can be summarized as follows:<br />
<br />
1. Penalize the L1-norm of the batch normalization scaling parameters in the loss<br />
<br />
2. Train until loss plateaus<br />
<br />
3. Remove channels that correspond to a downstream zero in batch normalization<br />
<br />
4. Fine-tune the pruned model using regular learning<br />
<br />
=== Details ===<br />
<br />
There still exist a few problems that this summary has not addressed so far. Sub-gradient descent is known to have inverse square root convergence rate on subdifferentials (Gordon et al., 2012), so the sparsity gradient descent update may be suboptimal. Furthermore, the sparse penalty needs to be normalized with respect to previous channel sizes, since the penalty should be roughly equally distributed across all convolution layers.<br />
<br />
==== Slow Convergence ====<br />
To address the issue of slow convergence, Ye et al. (2018) use an iterative shrinking-thresholding algorithm (ISTA) (Beck & Teboulle, 2009) to update the batch normalization scale parameter. The intuition for ISTA is that the structure of the optimization objective can be taken advantage of; consider $$L(x) = f(x) + g(x)$$<br />
<br />
Let ''f'' be the model loss and ''g'' be the non-differentiable penalty (LASSO). ISTA is able to use the structure of the loss and converge in O(1/n), instead of O(1/sqrt(n)) when using subgradient descent, which assumes no structure about the loss. Even though ISTA is used in convex settings, Ye et. al (2018) argue that it still performs better than gradient descent.<br />
<br />
==== Penalty Normalization ====<br />
<br />
In the paper, Ye et al. (2018) normalize the per-layer sparse penalty with respect to the global input size, the current layer kernel areas, the previous layer kernel areas, and the local input feature map area.<br />
<br />
[[File:Screenshot_from_2018-02-28_17-06-41.png]] (Ye et al., 2018)<br />
<br />
To control the global penalty, a hyperparamter ''rho'' is multiplied with all the per-layer ''lambda'' in the final loss.<br />
<br />
=== Steps ===<br />
<br />
The final algorithm can be summarized as follows:<br />
<br />
1. Compute the per-layer normalized sparse penalty constant ''lambda''<br />
<br />
2. Compute the global LASSO loss with global scaling constant ''rho''<br />
<br />
3. Until convergence, train scaling parameters using ISTA and non-scaling parameters using regular gradient descent.<br />
<br />
4. Remove channels that correspond to a downstream zero in batch normalization<br />
<br />
5. Fine-tune the pruned model using regular learning<br />
<br />
== Results ==<br />
<br />
The authors show state-of-the-art performance, compared with other channel-pruning approaches. It is important to note that it would be unfair to compare against general pruning approaches; channel pruning specifically removes channels without introducing '''intra-kernel sparsity''', whereas other pruning approaches introduce irregular kernel sparsity and hence computational inefficiencies.<br />
<br />
Results on CIFAR-10:<br />
<br />
[[File:Screenshot_from_2018-02-28_17-24-25.png]]<br />
<br />
<br />
<br />
Results on ILSVRC2012:<br />
<br />
[[File:Screenshot_from_2018-02-28_17-24-36.png]]<br />
<br />
== Conclusion ==<br />
<br />
Pruning large neural architectures to fit on low-power devices is an important task. It would be interesting to conduct actual power measurements on the pruned models vs baseline for a real quantitative measure of efficiency; reduction in FLOPs doesn't necessarily correspond with vastly reduced power, since memory accesses dominate energy consumption (Han et al., 2015). However, the reduction in number of FLOPs and parameters is encouraging, so moderate power savings should be expected.<br />
<br />
It would also be interesting to combine multiple approaches, or "throw the whole kitchen sink" at this task. Han et al. (2015) sparked much recent interest by successfully combining weight pruning, quantization, and Huffman coding without loss in accuracy. However, their approach introduced irregular sparsity in the convolutional layers, so a direct comparison cannot be made.<br />
<br />
In conclusion, this novel, theoretically-motivated interpretation of channel pruning was successfully applied to several important tasks.<br />
<br />
== References ==<br />
<br />
* Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2012). Imagenet classification with deep convolutional neural networks. In Advances in neural information processing systems (pp. 1097-1105).<br />
* He, K., Zhang, X., Ren, S., & Sun, J. (2016). Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition (pp. 770-778).<br />
* Cheng, Y., Wang, D., Zhou, P., & Zhang, T. (2017). A Survey of Model Compression and Acceleration for Deep Neural Networks. arXiv preprint arXiv:1710.09282.<br />
* Ye, J., Lu, X., Lin, Z., & Wang, J. Z. (2018). Rethinking the Smaller-Norm-Less-Informative Assumption in Channel Pruning of Convolution Layers. arXiv preprint arXiv:1802.00124.<br />
* Li, H., Kadav, A., Durdanovic, I., Samet, H., & Graf, H. P. (2016). Pruning filters for efficient convnets. arXiv preprint arXiv:1608.08710.<br />
* Molchanov, P., Tyree, S., Karras, T., Aila, T., & Kautz, J. (2016). Pruning convolutional neural networks for resource efficient inference.<br />
* Ioffe, S., & Szegedy, C. (2015, June). Batch normalization: Accelerating deep network training by reducing internal covariate shift. In International conference on machine learning (pp. 448-456).<br />
* Gordon, G., & Tibshirani, R. (2012). Subgradient method. https://www.cs.cmu.edu/~ggordon/10725-F12/slides/06-sg-method.pdf<br />
* Beck, A., & Teboulle, M. (2009). A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM journal on imaging sciences, 2(1), 183-202.<br />
* Han, S., Mao, H., & Dally, W. J. (2015). Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. arXiv preprint arXiv:1510.00149</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only&diff=32182stat946w18/Unsupervised Machine Translation Using Monolingual Corpora Only2018-02-24T16:05:56Z<p>Pa2forsy: /* Commentary */</p>
<hr />
<div><br />
[[File:MC_Translation_Example.png]]<br />
== Introduction ==<br />
Neural machine translation systems must be trained on large corpora consisting of pairs of pre-translated sentences. The paper ''Unsupervised Machine Translation Using Monolingual Corpora Only'' by Guillaume Lample, Ludovic Denoyer, and Marc'Aurelio Ranzato proposes an unsupervised neural machine translation system, which can be trained without such parallel data.<br />
<br />
==Motivation==<br />
The authors offer two motivations for their work:<br />
# To translate between languages for which large parallel corpora do not exist.<br />
# To provide s strong baseline against which translation systems using parallel corpora can be compared.<br />
== Overview of unsupervised translation system ==<br />
The unsupervised translation scheme has the following outline:<br />
* The word-vector embeddings of the source and target languages are aligned in an unsupervised manner.<br />
* Sentences from the source and target language are mapped to a common latent vector space by an encoder, and then mapped to probability distributions over sentences in the target or source language by a decoder.<br />
* A de-noising auto-encoder loss encourages the latent-space representations to be insensitive to noise.<br />
* An adversarial loss encourages the latent-space representations of source and target sentences to be indistinguishable from each other. It is intended that the latent-space representation of a sentence should reflect its meaning, and not the particular language in which it is expressed.<br />
* A reconstruction loss encourages the model to improve on the translation model of the previous epoch.<br />
<br />
==Notation==<br />
Let <math display="inline">S</math> denote the set of words in the source language, and let <math display="inline">T</math> denote the set of words in the target language. Let <math display="inline">H \subset \mathbb{R}^{n_H}</math> denote the latent vector space. Moreover, let <math display="inline">S'</math> and <math display="inline">T'</math> denote the sets of finite sequences of words in the source and target language, and let <math display="inline">H'</math> denote the set of finite sequences of vectors in the latent space. For any set X, elide measure-theoretic details and let <math display="inline">\mathcal{P}(X)</math> denote the set of probability distributions over X.<br />
<br />
==Word vector alignment ==<br />
<br />
Conneau et al. (2017) describe an unsupervised method for aligning word vectors across languages. By "alignment", I mean that their method maps words with related meanings to nearby vectors, regardless of the language of the words. Moreover, if two words are one another's literal translations, their word vectors tend to be mutual nearest neighbors. <br />
<br />
The underlying idea of the alignment scheme can be summarized as follows: methods like word2vec or GLoVe generate vectors for which there is a correspondence between semantics and geometry. If <math display="inline">f</math> maps English words to their corresponding vectors, we have the approximate equation<br />
\begin{align}<br />
f(\text{king}) -f(\text{man}) +f(\text{woman})\approx f(\text{queen}).<br />
\end{align}<br />
Furthermore, if <math display="inline">g</math> maps French words to their corresponding vectors, then <br />
\begin{align}<br />
g(\text{roi}) -g(\text{homme}) +g(\text{femme})\approx g(\text{reine}).<br />
\end{align}<br />
<br />
Thus if <math display="inline">W</math> maps the word vectors of English words to the word vectors of their French translations, we should expect <math display="inline">W</math> to be linear. As was observed by Mikolov et al. (2013), the problem of word-vector alignment then becomes a problem of learning the linear transformation that best aligns two point clouds, one from the source language and one from the target language. For more on the history of the word-vector alignment problem, see my CS698 project ([https://uwaterloo.ca/scholar/sites/ca.scholar/files/pa2forsy/files/project_dec_3_0.pdf link]).<br />
<br />
Conneau et al. (2017)'s word vector alignment scheme is unique in that it requires no parallel data, and uses only the shapes of the two word-vector point clouds to be aligned. I will not go into detail, but the heart of the method is a special GAN, in which only the discriminator is a neural network, and the generator is the map corresponding to an orthogonal matrix.<br />
<br />
This unsupervised alignment method is crucial to the translation scheme of the current paper. From now on we denote by <br />
<math display="inline">A: S' \cup T' \to \mathcal{Z}'</math> the function that maps a source- or target- language word sequence to the corresponding aligned word vector sequence.<br />
<br />
==Encoder ==<br />
The encoder <math display="inline">E </math> reads a sequence of word vectors <math display="inline">(z_1,\ldots, z_m) \in \mathcal{Z}'</math> and outputs a sequence of hidden states <math display="inline">(h_1,\ldots, h_m) \in H'</math> in the latent space. Crucially, because the word vectors of the two languages have been aligned, the same encoder can be applied to both. That is, to map a source sentence <math display="inline">x=(x_1,\ldots, x_M)\in S'</math> to the latent space, we compute <math display="inline">E(A(x))</math>, and to map a target sentence <math display="inline">y=(y_1,\ldots, y_K)\in T'</math> to the latent space, we compute <math display="inline">E(A(y))</math>.<br />
<br />
The encoder consists of two LSTMs, one of which reads the word-vector sequence in the forward direction, and one of which reads it in the backward direction. The hidden state sequence is generated by concatenating the hidden states produced by the forward and backward LSTMs at each word vector.<br />
<br />
==Decoder==<br />
<br />
The decoder is a mono-directional LSTM that accepts a sequence of hidden states <math display="inline">h=(h_1,\ldots, h_m) \in H'</math> from the latent space and a language <math display="inline">L \in \{S,T \}</math> and outputs a probability distribution over sentences in that language. We have<br />
<br />
\begin{align}<br />
D: H' \times \{S,T \} \to \mathcal{P}(S') \cup \mathcal{P}(T').<br />
\end{align}<br />
<br />
The decoder makes use of the attention mechanism of Bahdanau et al. (2014). To compute the probability of a given sentence <math display="inline">y=(y_1,\ldots,y_K)</math> , the LSTM processes the sentence one word at a time, accepting at step <math display="inline">k</math> the aligned word vector of the previous word in the sentence <math display="inline">A(y_{k-1})</math> and a context vector <math display="inline">c_k\in H</math> computed from the hidden sequence <math display="inline">h\in H'</math>, and outputting a probability distribution over possible next words. The LSTM is initiated with a special, language-specific start-of-sequence token. Otherwise, the decoder is does not depend on the language of the sentence it is producing. The context vector is computed as described by Bahdanau et al. (2014), where we let <math display="inline">l_{k}</math> denote the hidden state of the LSTM at step <math display="inline">k</math>, and where <math display="inline">U,W</math> are learnable weight matrices, and <math display="inline">v</math> is a learnable weight vector:<br />
\begin{align}<br />
c_k&= \sum_{m=1}^M \alpha_{k,m} h_m\\<br />
\alpha_{k,m}&= \frac{\exp(e_{k,m})}{\sum_{m'=1}^M\exp(e_{k,m'}) },\\<br />
e_{k,m} &= v^T \tanh (Wl_{k-1} + U h_m ).<br />
\end{align}<br />
<br />
<br />
By learning <math display="inline">U,W</math> and <math display="inline">v</math>, the decoder can learn to decide which vectors in the sequence <math display="inline">h</math> are relevant to computing which words in the output sentence.<br />
<br />
At step <math display="inline">k</math>, after receiving the context vector <math display="inline">c_k\in H</math> and the aligned word vector of the previous word in the sequence,<math display="inline">A(y_{k-1})</math>, the LSTM outputs a probability distribution over words, which should be interpreted as the distribution of the next word according to the decoder. The probability the decoder assigns to a sentence is then the product of the probabilities computed for each word in this manner.<br />
<br />
==Overview of objective ==<br />
The objective function is the sum of:<br />
# The de-noising auto-encoder loss,<br />
# The translation loss,<br />
# The adversarial loss.<br />
I shall describe these in the following sections.<br />
<br />
==De-noising Auto-encoder Loss == <br />
A de-noising auto-encoder is a function optimized to map a corrupted sample from some dataset to the original un-corrupted sample. De-noising auto-encoders were introduced by Vincent et al. (2008), who provided numerous justifications, one of which is particularly illuminating. If we think of the dataset of interest as a thin manifold in a high-dimensional space, the corruption process is likely perturb a datapoint off the manifold. To learn to restore the corrupted datapoint, the de-noising auto-encoder must learn the shape of the manifold.<br />
<br />
Hill et al. (2016), used a de-noising auto-encoder to learn vectors representing sentences. They corrupted input sentences by randomly dropping and swapping words, and then trained a neural network to map the corrupted sentence to a vector, and then map the vector to the un-corrupted sentence. Interestingly, they found that sentence vectors learned this way were particularly effective when applied to tasks that involved generating paraphrases. This makes some sense: for a vector to be useful in restoring a corrupted sentence, it must capture something of the sentence's underlying meaning.<br />
<br />
The present paper uses the principal of de-noising auto-encoders to compute one of the terms in its loss function. In each iteration, a sentence is sampled from the source or target language, and a corruption process <math display="inline"> C</math> is applied to it. <math display="inline"> C</math> works by deleting each word in the sentence with probability <math display="inline">p_C</math> and applying to the sentence a permutation randomly selected from those that do not move words more than <math display="inline">k_C</math> spots from their original positions. The authors select <math display="inline">p_C=0.1</math> and <math display="inline">k_C=3</math>. The corrupted sentence is then mapped to the latent space using <math display="inline">E\circ A</math>. The loss is then the negative log probability of the original un-corrupted sentence according to the decoder <math display="inline">D</math> applied to the latent-space sequence.<br />
<br />
The explanation of Vincent et al. (2008) can help us understand this loss-function term: the de-noising auto-encoder loss forces the translation system to learn the shapes of the manifolds of the source and target languages.<br />
<br />
==Translation Loss==<br />
To compute the translation loss, we sample a sentence from one of the languages, translate it with the encoder and decoder of the previous epoch, and then corrupt its output with <math display="inline">C</math>. We then use the current encoder <math display="inline">E</math> to map the corrupted translation to a sequence <math display="inline">h \in H'</math> and the decoder <math display="inline">D</math> to map <math display="inline">h</math> to a probability distribution over sentences. The translation loss is the negative log probability the decoder assigns to the original uncorrupted sentence. <br />
<br />
It is interesting and useful to consider why this translation loss, which depends on the the translation model of the previous iteration, should promote an improved translation model in the current iteration. One loose way to understand this is to think of the translator as a de-noising translator. We are given a sentence perturbed from the manifold of possible sentences from a given language both by the corruption process and by the poor quality of the translation. The model must learn to both project and translate. The technique employed here resembles that used by Sennrich et al. (2014), who trained a neural machine translation system using both parallel and monolingual data. To make use of the monolingual target-language data, they used an auxiliary model to translate it to the source language, then trained their model to reconstruct the original target-language data from the source-language translation. Sennrich et al. argued that training the model to reconstruct true data from synthetic data was more robust than the opposite approach. The authors of the present paper use similar reasoning.<br />
<br />
==Adversarial Loss ==<br />
The intuition underlying the latent space is that it should encode the meaning of a sentence in a language-independent way. Accordingly, the authors introduce an adversarial loss, to encourage latent-space vectors mapped from the source and target languages to be indistinguishable. Central to this adversarial loss is the discriminator <math display="inline">R:H' \to [0,1]</math>, which makes use of <math display="inline">r: H\to [0,1]</math> a three-layer fully-connected neural network with 1024 hidden units per layer. Given a sequence of latent-space vectors <math display="inline">h=(h_1,\ldots,h_m)\in H'</math> the discriminator assigns probability <math display="inline">R(h)=\prod_{i=1}^m r(h_i)</math> that they originated in the target space. Each iteration, the discriminator is trained to maximize the objective function<br />
<br />
\begin{align}<br />
I_T(q) \log (R(E(q))) +(1-I_T(q) )\log(1-R(E(q)))<br />
\end{align}<br />
<br />
where <math display="inline">q</math> is a randomly selected sentence, and <math display="inline">I_T(q)</math> is 1 when <math display="inline">q\in I_T</math> is from the source language and 0 if <math display="inline">q\in I_S</math><br />
<br />
The same term is added to the primary objective function, which the encoder and decoder are trained to minimize. The result is that the encoder and decoder learn to fool the discriminator by mapping sentences from the source and target language to similar sequences of latent-space vectors.<br />
<br />
<br />
The authors note that they make use of label smoothing, a technique recommended by Goodfellow (2016) for regularizing GANs, in which the objective described above is replaced by <br />
<br />
\begin{align}<br />
I_T(q)( (1-\alpha)\log (R(E(q))) +\alpha\log(1-R(E(q))) )+(1-I_T(q) ) ( (1-\beta) \log(1-R(E(q))) +\beta\log (R(E(q)) ))<br />
\end{align}<br />
for some small nonnegative values of <math display="inline">\alpha, \beta</math>, the idea being to prevent the discriminator from making extreme predictions. While one-sided label smoothing (<math display="inline">\beta = 0</math>) is generally recommended, the present model differs from a standard GAN in that it is symmetric, and hence two-sided label smoothing would appear more reasonable.<br />
<br />
<br />
It is interesting to observe that while the intuition justifying the use of the latent space suggests that the latent space representation of a sentence should be language-independent, this is not actually true: if two sentences are translations of one another, but have different lengths, their latent-space representations will necessarily be different, since a a sentence's latent space representation has the same length as the sentence itself.<br />
<br />
==Objective Function==<br />
<br />
Combining the above-described terms, we can write the overall objective function. Let <math display="inline">Q_S</math> denote the monolingual dataset for the source language, and let <math display="inline">Q_T</math> denote the monolingual dataset for the target language. Let <math display="inline">D_S:= D(\cdot, S)</math> and<math display="inline">D_T= D(\cdot, T)</math> (i.e. <math display="inline">D_S, D_T</math>) be the decoder restricted to the source or target language, respectively. Let <math display="inline"> M_S </math> and <math display="inline"> M_T </math> denote the target-to-source and source-to-target translation models of the previous epoch. Then our objective function is<br />
<br />
\begin{align}<br />
\mathcal{L}(D,E,R)=\text{T Translation Loss}+\text{T De-noising Loss} +\text{T Adversarial Loss} +\text{S Translation Loss} +\text{S De-noising Loss} +\text{S Adversarial Loss}\\<br />
\end{align}<br />
\begin{align}<br />
=\sum_{q\in Q_T}\left( -\log D_T \circ E \circ C \circ M _S(q) (q) -\log D_T \circ E \circ C (q) (q)+(1-\alpha)\log (R\circ E(q)) +\alpha\log(1-R\circ E(q)) \right)+\sum_{q\in Q_S}\left( -\log D_S \circ E \circ C \circ M_T (q) (q) -\log D_S \circ E \circ C (q) (q)+(1-\beta) \log(1-R \circ E(q)) +\beta\log (R\circ E(q) \right).<br />
\end{align}<br />
<br />
They alternate between iterations minimizing <math display="inline">\mathcal{L} </math> with respect to <math display="inline">E, D</math> and iterations maximizing with respect to <math display="inline">R</math>. ADAM is used for minimization, while RMSprop is used for maximization. After each epoch, M is updated so that <math display="inline">M_S=D_S \circ E</math> and <math display="inline">M_T=D_T \circ E</math>, after which <math display="inline"> M </math> is frozen until the next epoch.<br />
<br />
==Validation==<br />
The authors' aim is for their method to be completely unsupervised, and so do not use parallel corpora even for the selection of hyper-parameters. Instead, they validate by translating sentences to the other language and back, and comparing the resulting sentence with the original according to BLEU, a similarity metric frequently used in translation (Papineni et al. 2002).<br />
<br />
==Experimental Procedure and Results==<br />
<br />
The authors test their method on four data sets. The first is from the English-French translation task of the Workshop on Machine Translation 2014 (WMT14). This data set consists of parallel data. The authors generate a monolingual English corpus by randomly sampling 15 million sentence pairs, and choosing only the English sentences. They then generate a French corpus by selecting the French sentences from those pairs that were not previous chosen. Importantly, this means that the monolingual data sets have no parallel sentences. The second data set is generated from the English-German translation task from WMT14 using he same procedure.<br />
<br />
The third and fourth data sets are generated from Multi30k data set, which consists of multilingual captions of various images. The images are discarded and the English, French, and German captions are used to generate monolingual data sets in the manner described above. These monolingual corpora are much smaller, consisting of 14500 sentence each.<br />
<br />
The unsupervised translation scheme performs well, though not as well as a supervised translation scheme. It converges after a small number of epochs. Besides supervised translation, the authors compare their method with three other baselines: "Word-by-Word" uses only the previously-discussed word-alignment scheme; "Word-Reordering" uses a simple LSTM based language model and a greedy algorithm to select a reordering of the words produced by "Word-by-Word". "Oracle Word Reordering" means the optimal reordering of the words produced by "Word-by-Word".<br />
<br />
==Result Figures==<br />
[[File:MC_Translation Results.png]]<br />
[[File:MC_Translation_Convergence.png]]<br />
<br />
==Commentary==<br />
This paper's results are impressive: that it is even possible to translate between languages without parallel data suggests that languages are more similar than we might initially suspect, and that the method the authors present has, at least in part, discovered some common deep structure. As the authors point out, using no parallel data at all, their method is able to produce results comparable to those produced by neural machine translation methods trained on hundreds of thousands of a parallel sentences on the WMT dataset. On the other hand, the results they offer come with a few significant caveats.<br />
<br />
The first caveat is that the workhorse of the method is the unsupervised word-vector alignment scheme presented in Conneau et al. (2017) (that paper shares 3 authors with this one). As the ablation study reveals, without word-vector alignment, this method preforms extremely poorly. Moreover, word-by-word translation using word-vector alignment alone performs well, albeit not as well as this method. This suggests that the method of this paper mainly learns to perform (sometimes significant) corrections to word-by-word translations by reordering and occasional word substitution. Presumably, it does this by learning something of the natural structure of sentences in each of the two languages, so that it can correct the errors made by word-by-word translation.<br />
<br />
The second caveat is that the best results are attained translating between English and French, two very closely related languages, and the quality of translation between English and German, a slightly-less related pair, is significantly worse ( according to the ''Shorter Oxford English Dictionary'', 28.3 percent of the English vocabulary is French-derived, 28.2 percent is Latin-derived, and 25 percent is derived from Germanic languages. This probably understates the degree of correspondence between the French and English vocabularies, since French likely derives from Latin many of the same words English does. ). The authors do not report results with more distantly-related pairs, but it is reasonable to expect that performance would degrade significantly, for two reasons. Firstly, Conneau et al. (2017) shows that the word-alignment scheme performs much worse on more distant language pairs. This may be because there are more one-to-one correspondences between the words of closely related languages than there are between more distant languages. Secondly, because the same encoder is used to read sentences of both languages, the encoder cannot adapt to the unique word-order properties of either language. This would become a problem for language pairs with very different grammar. The authors suggest that their scheme could be a useful tool for translating between language pairs for which their are few parallel corpora. However, language pairs lacking parallel corpora are often (though not always) distantly related, and it is for such pairs that the performance of the present method likely suffers.<br />
<br />
<br />
<br />
<br />
The proposed method always beats Oracle Word Reordering on the Multi30k data set, but sometimes does not on the WMT data set. This may be because the WMT sentences are much more syntactically complex than the simple image captions of the Multi30k data set.<br />
<br />
The ablation study also reveals the importance of the corruption process <math display="inline">C</math>: the absence of <math display="inline">C</math> significantly degrades translation quality, though not as much as the absence of word-vector alignment. We can understand this in two related ways. First of all, if we view the model as learning to correct structural errors in word-by-word translations, then the corruption process introduces more errors of this kind, and so provides additional data upon which the model can train. Second, as Vincent et al. (2008) point out, de-noising auto-encoder training encourages a model to learn the structure of the manifold from which the data is drawn. By learning the structure of the source and target languages, the model can better correct the errors of word-by-word translation.<br />
<br />
[[File:MC_Alignment_Results.png|frame|none|alt=Alt text|From Conneau et al. (2017). The final row shows the performance of alignment method used in the present paper. Note the degradation in performance for more distant languages.]]<br />
<br />
[[File:MC_Translation_Ablation.png|frame|none|alt=Alt text|From the present paper. Results of an ablation study. Of note are the first, third, and forth rows, which demonstrate that while the translation component of the loss is relatively unimportant, the word vector alignment scheme and de-noising auto-encoder matter a great deal.]]<br />
<br />
==Future Work==<br />
The principal of performing unsupervised translation by starting with a rough but reasonable guess, and then improving it using knowledge of the structure of target language seems promising. Word by word translation using word-vector alignment works well for closely related languages like English and French, but is unlikely to work as well for more distant languages. For those languages, a better method for getting an initial guess is required.<br />
<br />
==References==<br />
#Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
#Conneau, Alexis, Guillaume Lample, Marc’Aurelio Ranzato, Ludovic Denoyer, Hervé Jégou. "Word Translation without Parallel Data". arXiv:1710.04087, (2017)<br />
# Dictionary, Shorter Oxford English. "Shorter Oxford english dictionary." (2007).<br />
#Goodfellow, Ian. "NIPS 2016 tutorial: Generative adversarial networks." arXiv preprint arXiv:1701.00160 (2016).<br />
# Hill, Felix, Kyunghyun Cho, and Anna Korhonen. "Learning distributed representations of sentences from unlabelled data." arXiv preprint arXiv:1602.03483 (2016).<br />
# Lample, Guillaume, Ludovic Denoyer, and Marc'Aurelio Ranzato. "Unsupervised Machine Translation Using Monolingual Corpora Only." arXiv preprint arXiv:1711.00043 (2017).<br />
#Papineni, Kishore, et al. "BLEU: a method for automatic evaluation of machine translation." Proceedings of the 40th annual meeting on association for computational linguistics. Association for Computational Linguistics, 2002.<br />
# Mikolov, Tomas, Quoc V Le, and Ilya Sutskever. "Exploiting similarities among languages for machine translation." arXiv preprint arXiv:1309.4168. (2013).<br />
#Sennrich, Rico, Barry Haddow, and Alexandra Birch. "Improving neural machine translation models with monolingual data." arXiv preprint arXiv:1511.06709 (2015).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.<br />
# Vincent, Pascal, et al. "Extracting and composing robust features with denoising autoencoders." Proceedings of the 25th international conference on Machine learning. ACM, 2008.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only&diff=32148stat946w18/Unsupervised Machine Translation Using Monolingual Corpora Only2018-02-23T21:02:26Z<p>Pa2forsy: /* Future Work */</p>
<hr />
<div><br />
[[File:MC_Translation_Example.png]]<br />
== Introduction ==<br />
Neural machine translation systems must be trained on large corpora consisting of pairs of pre-translated sentences. The paper ''Unsupervised Machine Translation Using Monolingual Corpora Only'' by Guillaume Lample, Ludovic Denoyer, and Marc'Aurelio Ranzato proposes an unsupervised neural machine translation system, which can be trained without such parallel data.<br />
<br />
==Motivation==<br />
The authors offer two motivations for their work:<br />
# To translate between languages for which large parallel corpora do not exist.<br />
# To provide s strong baseline against which translation systems using parallel corpora can be compared.<br />
== Overview of unsupervised translation system ==<br />
The unsupervised translation scheme has the following outline:<br />
* The word-vector embeddings of the source and target languages are aligned in an unsupervised manner.<br />
* Sentences from the source and target language are mapped to a common latent vector space by an encoder, and then mapped to probability distributions over sentences in the target or source language by a decoder.<br />
* A de-noising auto-encoder loss encourages the latent-space representations to be insensitive to noise.<br />
* An adversarial loss encourages the latent-space representations of source and target sentences to be indistinguishable from each other. It is intended that the latent-space representation of a sentence should reflect its meaning, and not the particular language in which it is expressed.<br />
* A reconstruction loss encourages the model to improve on the translation model of the previous epoch.<br />
<br />
==Notation==<br />
Let <math display="inline">S</math> denote the set of words in the source language, and let <math display="inline">T</math> denote the set of words in the target language. Let <math display="inline">H \subset \mathbb{R}^{n_H}</math> denote the latent vector space. Moreover, let <math display="inline">S'</math> and <math display="inline">T'</math> denote the sets of finite sequences of words in the source and target language, and let <math display="inline">H'</math> denote the set of finite sequences of vectors in the latent space. For any set X, elide measure-theoretic details and let <math display="inline">\mathcal{P}(X)</math> denote the set of probability distributions over X.<br />
<br />
==Word vector alignment ==<br />
<br />
Conneau et al. (2017) describe an unsupervised method for aligning word vectors across languages. By "alignment", I mean that their method maps words with related meanings to nearby vectors, regardless of the language of the words. Moreover, if two words are one another's literal translations, their word vectors tend to be mutual nearest neighbors. <br />
<br />
The underlying idea of the alignment scheme can be summarized as follows: methods like word2vec or GLoVe generate vectors for which there is a correspondence between semantics and geometry. If <math display="inline">f</math> maps English words to their corresponding vectors, we have the approximate equation<br />
\begin{align}<br />
f(\text{king}) -f(\text{man}) +f(\text{woman})\approx f(\text{queen}).<br />
\end{align}<br />
Furthermore, if <math display="inline">g</math> maps French words to their corresponding vectors, then <br />
\begin{align}<br />
g(\text{roi}) -g(\text{homme}) +g(\text{femme})\approx g(\text{reine}).<br />
\end{align}<br />
<br />
Thus if <math display="inline">W</math> maps the word vectors of English words to the word vectors of their French translations, we should expect <math display="inline">W</math> to be linear. As was observed by Mikolov et al. (2013), the problem of word-vector alignment then becomes a problem of learning the linear transformation that best aligns two point clouds, one from the source language and one from the target language. For more on the history of the word-vector alignment problem, see my CS698 project ([https://uwaterloo.ca/scholar/sites/ca.scholar/files/pa2forsy/files/project_dec_3_0.pdf link]).<br />
<br />
Conneau et al. (2017)'s word vector alignment scheme is unique in that it requires no parallel data, and uses only the shapes of the two word-vector point clouds to be aligned. I will not go into detail, but the heart of the method is a special GAN, in which only the discriminator is a neural network, and the generator is the map corresponding to an orthogonal matrix.<br />
<br />
This unsupervised alignment method is crucial to the translation scheme of the current paper. From now on we denote by <br />
<math display="inline">A: S' \cup T' \to \mathcal{Z}'</math> the function that maps a source- or target- language word sequence to the corresponding aligned word vector sequence.<br />
<br />
==Encoder ==<br />
The encoder <math display="inline">E </math> reads a sequence of word vectors <math display="inline">(z_1,\ldots, z_m) \in \mathcal{Z}'</math> and outputs a sequence of hidden states <math display="inline">(h_1,\ldots, h_m) \in H'</math> in the latent space. Crucially, because the word vectors of the two languages have been aligned, the same encoder can be applied to both. That is, to map a source sentence <math display="inline">x=(x_1,\ldots, x_M)\in S'</math> to the latent space, we compute <math display="inline">E(A(x))</math>, and to map a target sentence <math display="inline">y=(y_1,\ldots, y_K)\in T'</math> to the latent space, we compute <math display="inline">E(A(y))</math>.<br />
<br />
The encoder consists of two LSTMs, one of which reads the word-vector sequence in the forward direction, and one of which reads it in the backward direction. The hidden state sequence is generated by concatenating the hidden states produced by the forward and backward LSTMs at each word vector.<br />
<br />
==Decoder==<br />
<br />
The decoder is a mono-directional LSTM that accepts a sequence of hidden states <math display="inline">h=(h_1,\ldots, h_m) \in H'</math> from the latent space and a language <math display="inline">L \in \{S,T \}</math> and outputs a probability distribution over sentences in that language. We have<br />
<br />
\begin{align}<br />
D: H' \times \{S,T \} \to \mathcal{P}(S') \cup \mathcal{P}(T').<br />
\end{align}<br />
<br />
The decoder makes use of the attention mechanism of Bahdanau et al. (2014). To compute the probability of a given sentence <math display="inline">y=(y_1,\ldots,y_K)</math> , the LSTM processes the sentence one word at a time, accepting at step <math display="inline">k</math> the aligned word vector of the previous word in the sentence <math display="inline">A(y_{k-1})</math> and a context vector <math display="inline">c_k\in H</math> computed from the hidden sequence <math display="inline">h\in H'</math>, and outputting a probability distribution over possible next words. The LSTM is initiated with a special, language-specific start-of-sequence token. Otherwise, the decoder is does not depend on the language of the sentence it is producing. The context vector is computed as described by Bahdanau et al. (2014), where we let <math display="inline">l_{k}</math> denote the hidden state of the LSTM at step <math display="inline">k</math>, and where <math display="inline">U,W</math> are learnable weight matrices, and <math display="inline">v</math> is a learnable weight vector:<br />
\begin{align}<br />
c_k&= \sum_{m=1}^M \alpha_{k,m} h_m\\<br />
\alpha_{k,m}&= \frac{\exp(e_{k,m})}{\sum_{m'=1}^M\exp(e_{k,m'}) },\\<br />
e_{k,m} &= v^T \tanh (Wl_{k-1} + U h_m ).<br />
\end{align}<br />
<br />
<br />
By learning <math display="inline">U,W</math> and <math display="inline">v</math>, the decoder can learn to decide which vectors in the sequence <math display="inline">h</math> are relevant to computing which words in the output sentence.<br />
<br />
At step <math display="inline">k</math>, after receiving the context vector <math display="inline">c_k\in H</math> and the aligned word vector of the previous word in the sequence,<math display="inline">A(y_{k-1})</math>, the LSTM outputs a probability distribution over words, which should be interpreted as the distribution of the next word according to the decoder. The probability the decoder assigns to a sentence is then the product of the probabilities computed for each word in this manner.<br />
<br />
==Overview of objective ==<br />
The objective function is the sum of:<br />
# The de-noising auto-encoder loss,<br />
# The translation loss,<br />
# The adversarial loss.<br />
I shall describe these in the following sections.<br />
<br />
==De-noising Auto-encoder Loss == <br />
A de-noising auto-encoder is a function optimized to map a corrupted sample from some dataset to the original un-corrupted sample. De-noising auto-encoders were introduced by Vincent et al. (2008), who provided numerous justifications, one of which is particularly illuminating. If we think of the dataset of interest as a thin manifold in a high-dimensional space, the corruption process is likely perturb a datapoint off the manifold. To learn to restore the corrupted datapoint, the de-noising auto-encoder must learn the shape of the manifold.<br />
<br />
Hill et al. (2016), used a de-noising auto-encoder to learn vectors representing sentences. They corrupted input sentences by randomly dropping and swapping words, and then trained a neural network to map the corrupted sentence to a vector, and then map the vector to the un-corrupted sentence. Interestingly, they found that sentence vectors learned this way were particularly effective when applied to tasks that involved generating paraphrases. This makes some sense: for a vector to be useful in restoring a corrupted sentence, it must capture something of the sentence's underlying meaning.<br />
<br />
The present paper uses the principal of de-noising auto-encoders to compute one of the terms in its loss function. In each iteration, a sentence is sampled from the source or target language, and a corruption process <math display="inline"> C</math> is applied to it. <math display="inline"> C</math> works by deleting each word in the sentence with probability <math display="inline">p_C</math> and applying to the sentence a permutation randomly selected from those that do not move words more than <math display="inline">k_C</math> spots from their original positions. The authors select <math display="inline">p_C=0.1</math> and <math display="inline">k_C=3</math>. The corrupted sentence is then mapped to the latent space using <math display="inline">E\circ A</math>. The loss is then the negative log probability of the original un-corrupted sentence according to the decoder <math display="inline">D</math> applied to the latent-space sequence.<br />
<br />
The explanation of Vincent et al. (2008) can help us understand this loss-function term: the de-noising auto-encoder loss forces the translation system to learn the shapes of the manifolds of the source and target languages.<br />
<br />
==Translation Loss==<br />
To compute the translation loss, we sample a sentence from one of the languages, translate it with the encoder and decoder of the previous epoch, and then corrupt its output with <math display="inline">C</math>. We then use the current encoder <math display="inline">E</math> to map the corrupted translation to a sequence <math display="inline">h \in H'</math> and the decoder <math display="inline">D</math> to map <math display="inline">h</math> to a probability distribution over sentences. The translation loss is the negative log probability the decoder assigns to the original uncorrupted sentence. <br />
<br />
It is interesting and useful to consider why this translation loss, which depends on the the translation model of the previous iteration, should promote an improved translation model in the current iteration. One loose way to understand this is to think of the translator as a de-noising translator. We are given a sentence perturbed from the manifold of possible sentences from a given language both by the corruption process and by the poor quality of the translation. The model must learn to both project and translate. The technique employed here resembles that used by Sennrich et al. (2014), who trained a neural machine translation system using both parallel and monolingual data. To make use of the monolingual target-language data, they used an auxiliary model to translate it to the source language, then trained their model to reconstruct the original target-language data from the source-language translation. Sennrich et al. argued that training the model to reconstruct true data from synthetic data was more robust than the opposite approach. The authors of the present paper use similar reasoning.<br />
<br />
==Adversarial Loss ==<br />
The intuition underlying the latent space is that it should encode the meaning of a sentence in a language-independent way. Accordingly, the authors introduce an adversarial loss, to encourage latent-space vectors mapped from the source and target languages to be indistinguishable. Central to this adversarial loss is the discriminator <math display="inline">R:H' \to [0,1]</math>, which makes use of <math display="inline">r: H\to [0,1]</math> a three-layer fully-connected neural network with 1024 hidden units per layer. Given a sequence of latent-space vectors <math display="inline">h=(h_1,\ldots,h_m)\in H'</math> the discriminator assigns probability <math display="inline">R(h)=\prod_{i=1}^m r(h_i)</math> that they originated in the target space. Each iteration, the discriminator is trained to maximize the objective function<br />
<br />
\begin{align}<br />
I_T(q) \log (R(E(q))) +(1-I_T(q) )\log(1-R(E(q)))<br />
\end{align}<br />
<br />
where <math display="inline">q</math> is a randomly selected sentence, and <math display="inline">I_T(q)</math> is 1 when <math display="inline">q\in I_T</math> is from the source language and 0 if <math display="inline">q\in I_S</math><br />
<br />
The same term is added to the primary objective function, which the encoder and decoder are trained to minimize. The result is that the encoder and decoder learn to fool the discriminator by mapping sentences from the source and target language to similar sequences of latent-space vectors.<br />
<br />
<br />
The authors note that they make use of label smoothing, a technique recommended by Goodfellow (2016) for regularizing GANs, in which the objective described above is replaced by <br />
<br />
\begin{align}<br />
I_T(q)( (1-\alpha)\log (R(E(q))) +\alpha\log(1-R(E(q))) )+(1-I_T(q) ) ( (1-\beta) \log(1-R(E(q))) +\beta\log (R(E(q)) ))<br />
\end{align}<br />
for some small nonnegative values of <math display="inline">\alpha, \beta</math>, the idea being to prevent the discriminator from making extreme predictions. While one-sided label smoothing (<math display="inline">\beta = 0</math>) is generally recommended, the present model differs from a standard GAN in that it is symmetric, and hence two-sided label smoothing would appear more reasonable.<br />
<br />
<br />
It is interesting to observe that while the intuition justifying the use of the latent space suggests that the latent space representation of a sentence should be language-independent, this is not actually true: if two sentences are translations of one another, but have different lengths, their latent-space representations will necessarily be different, since a a sentence's latent space representation has the same length as the sentence itself.<br />
<br />
==Objective Function==<br />
<br />
Combining the above-described terms, we can write the overall objective function. Let <math display="inline">Q_S</math> denote the monolingual dataset for the source language, and let <math display="inline">Q_T</math> denote the monolingual dataset for the target language. Let <math display="inline">D_S:= D(\cdot, S)</math> and<math display="inline">D_T= D(\cdot, T)</math> (i.e. <math display="inline">D_S, D_T</math>) be the decoder restricted to the source or target language, respectively. Let <math display="inline"> M_S </math> and <math display="inline"> M_T </math> denote the target-to-source and source-to-target translation models of the previous epoch. Then our objective function is<br />
<br />
\begin{align}<br />
\mathcal{L}(D,E,R)=\text{T Translation Loss}+\text{T De-noising Loss} +\text{T Adversarial Loss} +\text{S Translation Loss} +\text{S De-noising Loss} +\text{S Adversarial Loss}\\<br />
\end{align}<br />
\begin{align}<br />
=\sum_{q\in Q_T}\left( -\log D_T \circ E \circ C \circ M _S(q) (q) -\log D_T \circ E \circ C (q) (q)+(1-\alpha)\log (R\circ E(q)) +\alpha\log(1-R\circ E(q)) \right)+\sum_{q\in Q_S}\left( -\log D_S \circ E \circ C \circ M_T (q) (q) -\log D_S \circ E \circ C (q) (q)+(1-\beta) \log(1-R \circ E(q)) +\beta\log (R\circ E(q) \right).<br />
\end{align}<br />
<br />
They alternate between iterations minimizing <math display="inline">\mathcal{L} </math> with respect to <math display="inline">E, D</math> and iterations maximizing with respect to <math display="inline">R</math>. ADAM is used for minimization, while RMSprop is used for maximization. After each epoch, M is updated so that <math display="inline">M_S=D_S \circ E</math> and <math display="inline">M_T=D_T \circ E</math>, after which <math display="inline"> M </math> is frozen until the next epoch.<br />
<br />
==Validation==<br />
The authors' aim is for their method to be completely unsupervised, and so do not use parallel corpora even for the selection of hyper-parameters. Instead, they validate by translating sentences to the other language and back, and comparing the resulting sentence with the original according to BLEU, a similarity metric frequently used in translation (Papineni et al. 2002).<br />
<br />
==Experimental Procedure and Results==<br />
<br />
The authors test their method on four data sets. The first is from the English-French translation task of the Workshop on Machine Translation 2014 (WMT14). This data set consists of parallel data. The authors generate a monolingual English corpus by randomly sampling 15 million sentence pairs, and choosing only the English sentences. They then generate a French corpus by selecting the French sentences from those pairs that were not previous chosen. Importantly, this means that the monolingual data sets have no parallel sentences. The second data set is generated from the English-German translation task from WMT14 using he same procedure.<br />
<br />
The third and fourth data sets are generated from Multi30k data set, which consists of multilingual captions of various images. The images are discarded and the English, French, and German captions are used to generate monolingual data sets in the manner described above. These monolingual corpora are much smaller, consisting of 14500 sentence each.<br />
<br />
The unsupervised translation scheme performs well, though not as well as a supervised translation scheme. It converges after a small number of epochs. Besides supervised translation, the authors compare their method with three other baselines: "Word-by-Word" uses only the previously-discussed word-alignment scheme; "Word-Reordering" uses a simple LSTM based language model and a greedy algorithm to select a reordering of the words produced by "Word-by-Word". "Oracle Word Reordering" means the optimal reordering of the words produced by "Word-by-Word".<br />
<br />
==Result Figures==<br />
[[File:MC_Translation Results.png]]<br />
[[File:MC_Translation_Convergence.png]]<br />
<br />
==Commentary==<br />
This paper's results are impressive: that it is even possible to translate between languages without parallel data suggests that languages are more similar than we might initially suspect, and that the method the authors present has, at least in part, discovered some common deep structure. As the authors point out, using no parallel data at all, their method is able to produce results comparable to those produced by neural machine translation methods trained on hundreds of thousands of a parallel sentences on the WMT dataset. On the other hand, the results they offer come with a few significant caveats.<br />
<br />
The first caveat is that the workhorse of the method is the unsupervised word-vector alignment scheme presented in Conneau et al. (2017) (that paper shares 3 authors with this one). As the ablation study reveals, without word-vector alignment, this method preforms extremely poorly. Moreover, word-by-word translation using word-vector alignment alone performs well, albeit not as well as this method. This suggests that the method of this paper mainly learns to perform (sometimes significant) corrections to word-by-word translations by reordering and occasional word substitution. Presumably, it does this by learning something of the natural structure of sentences in each of the two languages, so that it can correct the errors made by word-by-word translation.<br />
<br />
The second caveat is that the best results are attained translating between English and French, two very closely related languages, and the quality of translation between English and German, a slightly-less related pair, is significantly worse ( according to the ''Shorter Oxford English Dictionary'', 28.3 percent of the English vocabulary is French-derived, 28.2 percent is Latin-derived, and 25 percent is derived from Germanic languages. This probably understates the degree of correspondence between the French and English vocabularies, since French likely derives from Latin many of the same words English does. ). The authors do not report results with more distantly-related pairs, but it is reasonable to expect that performance would degrade significantly, especially since Conneau et al. (2017) shows that the word-alignment scheme performs much worse on more distant language pairs. This may be because there are more one-to-one correspondences between the words of closely related languages than there are between more distant languages. The authors suggest that their scheme could be a useful tool for translating between language pairs for which their are few parallel corpora. However, language pairs lacking parallel corpora are often (though not always) distantly related, and it is for such pairs that the performance of the present method suffers.<br />
<br />
The proposed method always beats Oracle Word Reordering on the Multi30k data set, but sometimes does not on the WMT data set. This may be because the WMT sentences are much more syntactically complex than the simple image captions of the Multi30k data set.<br />
<br />
The ablation study also reveals the importance of the corruption process <math display="inline">C</math>: the absence of <math display="inline">C</math> significantly degrades translation quality, though not as much as the absence of word-vector alignment. We can understand this in two related ways. First of all, if we view the model as learning to correct structural errors in word-by-word translations, then the corruption process introduces more errors of this kind, and so provides additional data upon which the model can train. Second, as Vincent et al. (2008) point out, de-noising auto-encoder training encourages a model to learn the structure of the manifold from which the data is drawn. By learning the structure of the source and target languages, the model can better correct the errors of word-by-word translation.<br />
<br />
[[File:MC_Alignment_Results.png|frame|none|alt=Alt text|From Conneau et al. (2017). The final row shows the performance of alignment method used in the present paper. Note the degradation in performance for more distant languages.]]<br />
<br />
[[File:MC_Translation_Ablation.png|frame|none|alt=Alt text|From the present paper. Results of an ablation study. Of note are the first, third, and forth rows, which demonstrate that while the translation component of the loss is relatively unimportant, the word vector alignment scheme and de-noising auto-encoder matter a great deal.]]<br />
<br />
==Future Work==<br />
The principal of performing unsupervised translation by starting with a rough but reasonable guess, and then improving it using knowledge of the structure of target language seems promising. Word by word translation using word-vector alignment works well for closely related languages like English and French, but is unlikely to work as well for more distant languages. For those languages, a better method for getting an initial guess is required.<br />
<br />
==References==<br />
#Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
#Conneau, Alexis, Guillaume Lample, Marc’Aurelio Ranzato, Ludovic Denoyer, Hervé Jégou. "Word Translation without Parallel Data". arXiv:1710.04087, (2017)<br />
# Dictionary, Shorter Oxford English. "Shorter Oxford english dictionary." (2007).<br />
#Goodfellow, Ian. "NIPS 2016 tutorial: Generative adversarial networks." arXiv preprint arXiv:1701.00160 (2016).<br />
# Hill, Felix, Kyunghyun Cho, and Anna Korhonen. "Learning distributed representations of sentences from unlabelled data." arXiv preprint arXiv:1602.03483 (2016).<br />
# Lample, Guillaume, Ludovic Denoyer, and Marc'Aurelio Ranzato. "Unsupervised Machine Translation Using Monolingual Corpora Only." arXiv preprint arXiv:1711.00043 (2017).<br />
#Papineni, Kishore, et al. "BLEU: a method for automatic evaluation of machine translation." Proceedings of the 40th annual meeting on association for computational linguistics. Association for Computational Linguistics, 2002.<br />
# Mikolov, Tomas, Quoc V Le, and Ilya Sutskever. "Exploiting similarities among languages for machine translation." arXiv preprint arXiv:1309.4168. (2013).<br />
#Sennrich, Rico, Barry Haddow, and Alexandra Birch. "Improving neural machine translation models with monolingual data." arXiv preprint arXiv:1511.06709 (2015).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.<br />
# Vincent, Pascal, et al. "Extracting and composing robust features with denoising autoencoders." Proceedings of the 25th international conference on Machine learning. ACM, 2008.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only&diff=32147stat946w18/Unsupervised Machine Translation Using Monolingual Corpora Only2018-02-23T20:42:09Z<p>Pa2forsy: /* Adversarial Loss */</p>
<hr />
<div><br />
[[File:MC_Translation_Example.png]]<br />
== Introduction ==<br />
Neural machine translation systems must be trained on large corpora consisting of pairs of pre-translated sentences. The paper ''Unsupervised Machine Translation Using Monolingual Corpora Only'' by Guillaume Lample, Ludovic Denoyer, and Marc'Aurelio Ranzato proposes an unsupervised neural machine translation system, which can be trained without such parallel data.<br />
<br />
==Motivation==<br />
The authors offer two motivations for their work:<br />
# To translate between languages for which large parallel corpora do not exist.<br />
# To provide s strong baseline against which translation systems using parallel corpora can be compared.<br />
== Overview of unsupervised translation system ==<br />
The unsupervised translation scheme has the following outline:<br />
* The word-vector embeddings of the source and target languages are aligned in an unsupervised manner.<br />
* Sentences from the source and target language are mapped to a common latent vector space by an encoder, and then mapped to probability distributions over sentences in the target or source language by a decoder.<br />
* A de-noising auto-encoder loss encourages the latent-space representations to be insensitive to noise.<br />
* An adversarial loss encourages the latent-space representations of source and target sentences to be indistinguishable from each other. It is intended that the latent-space representation of a sentence should reflect its meaning, and not the particular language in which it is expressed.<br />
* A reconstruction loss encourages the model to improve on the translation model of the previous epoch.<br />
<br />
==Notation==<br />
Let <math display="inline">S</math> denote the set of words in the source language, and let <math display="inline">T</math> denote the set of words in the target language. Let <math display="inline">H \subset \mathbb{R}^{n_H}</math> denote the latent vector space. Moreover, let <math display="inline">S'</math> and <math display="inline">T'</math> denote the sets of finite sequences of words in the source and target language, and let <math display="inline">H'</math> denote the set of finite sequences of vectors in the latent space. For any set X, elide measure-theoretic details and let <math display="inline">\mathcal{P}(X)</math> denote the set of probability distributions over X.<br />
<br />
==Word vector alignment ==<br />
<br />
Conneau et al. (2017) describe an unsupervised method for aligning word vectors across languages. By "alignment", I mean that their method maps words with related meanings to nearby vectors, regardless of the language of the words. Moreover, if two words are one another's literal translations, their word vectors tend to be mutual nearest neighbors. <br />
<br />
The underlying idea of the alignment scheme can be summarized as follows: methods like word2vec or GLoVe generate vectors for which there is a correspondence between semantics and geometry. If <math display="inline">f</math> maps English words to their corresponding vectors, we have the approximate equation<br />
\begin{align}<br />
f(\text{king}) -f(\text{man}) +f(\text{woman})\approx f(\text{queen}).<br />
\end{align}<br />
Furthermore, if <math display="inline">g</math> maps French words to their corresponding vectors, then <br />
\begin{align}<br />
g(\text{roi}) -g(\text{homme}) +g(\text{femme})\approx g(\text{reine}).<br />
\end{align}<br />
<br />
Thus if <math display="inline">W</math> maps the word vectors of English words to the word vectors of their French translations, we should expect <math display="inline">W</math> to be linear. As was observed by Mikolov et al. (2013), the problem of word-vector alignment then becomes a problem of learning the linear transformation that best aligns two point clouds, one from the source language and one from the target language. For more on the history of the word-vector alignment problem, see my CS698 project ([https://uwaterloo.ca/scholar/sites/ca.scholar/files/pa2forsy/files/project_dec_3_0.pdf link]).<br />
<br />
Conneau et al. (2017)'s word vector alignment scheme is unique in that it requires no parallel data, and uses only the shapes of the two word-vector point clouds to be aligned. I will not go into detail, but the heart of the method is a special GAN, in which only the discriminator is a neural network, and the generator is the map corresponding to an orthogonal matrix.<br />
<br />
This unsupervised alignment method is crucial to the translation scheme of the current paper. From now on we denote by <br />
<math display="inline">A: S' \cup T' \to \mathcal{Z}'</math> the function that maps a source- or target- language word sequence to the corresponding aligned word vector sequence.<br />
<br />
==Encoder ==<br />
The encoder <math display="inline">E </math> reads a sequence of word vectors <math display="inline">(z_1,\ldots, z_m) \in \mathcal{Z}'</math> and outputs a sequence of hidden states <math display="inline">(h_1,\ldots, h_m) \in H'</math> in the latent space. Crucially, because the word vectors of the two languages have been aligned, the same encoder can be applied to both. That is, to map a source sentence <math display="inline">x=(x_1,\ldots, x_M)\in S'</math> to the latent space, we compute <math display="inline">E(A(x))</math>, and to map a target sentence <math display="inline">y=(y_1,\ldots, y_K)\in T'</math> to the latent space, we compute <math display="inline">E(A(y))</math>.<br />
<br />
The encoder consists of two LSTMs, one of which reads the word-vector sequence in the forward direction, and one of which reads it in the backward direction. The hidden state sequence is generated by concatenating the hidden states produced by the forward and backward LSTMs at each word vector.<br />
<br />
==Decoder==<br />
<br />
The decoder is a mono-directional LSTM that accepts a sequence of hidden states <math display="inline">h=(h_1,\ldots, h_m) \in H'</math> from the latent space and a language <math display="inline">L \in \{S,T \}</math> and outputs a probability distribution over sentences in that language. We have<br />
<br />
\begin{align}<br />
D: H' \times \{S,T \} \to \mathcal{P}(S') \cup \mathcal{P}(T').<br />
\end{align}<br />
<br />
The decoder makes use of the attention mechanism of Bahdanau et al. (2014). To compute the probability of a given sentence <math display="inline">y=(y_1,\ldots,y_K)</math> , the LSTM processes the sentence one word at a time, accepting at step <math display="inline">k</math> the aligned word vector of the previous word in the sentence <math display="inline">A(y_{k-1})</math> and a context vector <math display="inline">c_k\in H</math> computed from the hidden sequence <math display="inline">h\in H'</math>, and outputting a probability distribution over possible next words. The LSTM is initiated with a special, language-specific start-of-sequence token. Otherwise, the decoder is does not depend on the language of the sentence it is producing. The context vector is computed as described by Bahdanau et al. (2014), where we let <math display="inline">l_{k}</math> denote the hidden state of the LSTM at step <math display="inline">k</math>, and where <math display="inline">U,W</math> are learnable weight matrices, and <math display="inline">v</math> is a learnable weight vector:<br />
\begin{align}<br />
c_k&= \sum_{m=1}^M \alpha_{k,m} h_m\\<br />
\alpha_{k,m}&= \frac{\exp(e_{k,m})}{\sum_{m'=1}^M\exp(e_{k,m'}) },\\<br />
e_{k,m} &= v^T \tanh (Wl_{k-1} + U h_m ).<br />
\end{align}<br />
<br />
<br />
By learning <math display="inline">U,W</math> and <math display="inline">v</math>, the decoder can learn to decide which vectors in the sequence <math display="inline">h</math> are relevant to computing which words in the output sentence.<br />
<br />
At step <math display="inline">k</math>, after receiving the context vector <math display="inline">c_k\in H</math> and the aligned word vector of the previous word in the sequence,<math display="inline">A(y_{k-1})</math>, the LSTM outputs a probability distribution over words, which should be interpreted as the distribution of the next word according to the decoder. The probability the decoder assigns to a sentence is then the product of the probabilities computed for each word in this manner.<br />
<br />
==Overview of objective ==<br />
The objective function is the sum of:<br />
# The de-noising auto-encoder loss,<br />
# The translation loss,<br />
# The adversarial loss.<br />
I shall describe these in the following sections.<br />
<br />
==De-noising Auto-encoder Loss == <br />
A de-noising auto-encoder is a function optimized to map a corrupted sample from some dataset to the original un-corrupted sample. De-noising auto-encoders were introduced by Vincent et al. (2008), who provided numerous justifications, one of which is particularly illuminating. If we think of the dataset of interest as a thin manifold in a high-dimensional space, the corruption process is likely perturb a datapoint off the manifold. To learn to restore the corrupted datapoint, the de-noising auto-encoder must learn the shape of the manifold.<br />
<br />
Hill et al. (2016), used a de-noising auto-encoder to learn vectors representing sentences. They corrupted input sentences by randomly dropping and swapping words, and then trained a neural network to map the corrupted sentence to a vector, and then map the vector to the un-corrupted sentence. Interestingly, they found that sentence vectors learned this way were particularly effective when applied to tasks that involved generating paraphrases. This makes some sense: for a vector to be useful in restoring a corrupted sentence, it must capture something of the sentence's underlying meaning.<br />
<br />
The present paper uses the principal of de-noising auto-encoders to compute one of the terms in its loss function. In each iteration, a sentence is sampled from the source or target language, and a corruption process <math display="inline"> C</math> is applied to it. <math display="inline"> C</math> works by deleting each word in the sentence with probability <math display="inline">p_C</math> and applying to the sentence a permutation randomly selected from those that do not move words more than <math display="inline">k_C</math> spots from their original positions. The authors select <math display="inline">p_C=0.1</math> and <math display="inline">k_C=3</math>. The corrupted sentence is then mapped to the latent space using <math display="inline">E\circ A</math>. The loss is then the negative log probability of the original un-corrupted sentence according to the decoder <math display="inline">D</math> applied to the latent-space sequence.<br />
<br />
The explanation of Vincent et al. (2008) can help us understand this loss-function term: the de-noising auto-encoder loss forces the translation system to learn the shapes of the manifolds of the source and target languages.<br />
<br />
==Translation Loss==<br />
To compute the translation loss, we sample a sentence from one of the languages, translate it with the encoder and decoder of the previous epoch, and then corrupt its output with <math display="inline">C</math>. We then use the current encoder <math display="inline">E</math> to map the corrupted translation to a sequence <math display="inline">h \in H'</math> and the decoder <math display="inline">D</math> to map <math display="inline">h</math> to a probability distribution over sentences. The translation loss is the negative log probability the decoder assigns to the original uncorrupted sentence. <br />
<br />
It is interesting and useful to consider why this translation loss, which depends on the the translation model of the previous iteration, should promote an improved translation model in the current iteration. One loose way to understand this is to think of the translator as a de-noising translator. We are given a sentence perturbed from the manifold of possible sentences from a given language both by the corruption process and by the poor quality of the translation. The model must learn to both project and translate. The technique employed here resembles that used by Sennrich et al. (2014), who trained a neural machine translation system using both parallel and monolingual data. To make use of the monolingual target-language data, they used an auxiliary model to translate it to the source language, then trained their model to reconstruct the original target-language data from the source-language translation. Sennrich et al. argued that training the model to reconstruct true data from synthetic data was more robust than the opposite approach. The authors of the present paper use similar reasoning.<br />
<br />
==Adversarial Loss ==<br />
The intuition underlying the latent space is that it should encode the meaning of a sentence in a language-independent way. Accordingly, the authors introduce an adversarial loss, to encourage latent-space vectors mapped from the source and target languages to be indistinguishable. Central to this adversarial loss is the discriminator <math display="inline">R:H' \to [0,1]</math>, which makes use of <math display="inline">r: H\to [0,1]</math> a three-layer fully-connected neural network with 1024 hidden units per layer. Given a sequence of latent-space vectors <math display="inline">h=(h_1,\ldots,h_m)\in H'</math> the discriminator assigns probability <math display="inline">R(h)=\prod_{i=1}^m r(h_i)</math> that they originated in the target space. Each iteration, the discriminator is trained to maximize the objective function<br />
<br />
\begin{align}<br />
I_T(q) \log (R(E(q))) +(1-I_T(q) )\log(1-R(E(q)))<br />
\end{align}<br />
<br />
where <math display="inline">q</math> is a randomly selected sentence, and <math display="inline">I_T(q)</math> is 1 when <math display="inline">q\in I_T</math> is from the source language and 0 if <math display="inline">q\in I_S</math><br />
<br />
The same term is added to the primary objective function, which the encoder and decoder are trained to minimize. The result is that the encoder and decoder learn to fool the discriminator by mapping sentences from the source and target language to similar sequences of latent-space vectors.<br />
<br />
<br />
The authors note that they make use of label smoothing, a technique recommended by Goodfellow (2016) for regularizing GANs, in which the objective described above is replaced by <br />
<br />
\begin{align}<br />
I_T(q)( (1-\alpha)\log (R(E(q))) +\alpha\log(1-R(E(q))) )+(1-I_T(q) ) ( (1-\beta) \log(1-R(E(q))) +\beta\log (R(E(q)) ))<br />
\end{align}<br />
for some small nonnegative values of <math display="inline">\alpha, \beta</math>, the idea being to prevent the discriminator from making extreme predictions. While one-sided label smoothing (<math display="inline">\beta = 0</math>) is generally recommended, the present model differs from a standard GAN in that it is symmetric, and hence two-sided label smoothing would appear more reasonable.<br />
<br />
<br />
It is interesting to observe that while the intuition justifying the use of the latent space suggests that the latent space representation of a sentence should be language-independent, this is not actually true: if two sentences are translations of one another, but have different lengths, their latent-space representations will necessarily be different, since a a sentence's latent space representation has the same length as the sentence itself.<br />
<br />
==Objective Function==<br />
<br />
Combining the above-described terms, we can write the overall objective function. Let <math display="inline">Q_S</math> denote the monolingual dataset for the source language, and let <math display="inline">Q_T</math> denote the monolingual dataset for the target language. Let <math display="inline">D_S:= D(\cdot, S)</math> and<math display="inline">D_T= D(\cdot, T)</math> (i.e. <math display="inline">D_S, D_T</math>) be the decoder restricted to the source or target language, respectively. Let <math display="inline"> M_S </math> and <math display="inline"> M_T </math> denote the target-to-source and source-to-target translation models of the previous epoch. Then our objective function is<br />
<br />
\begin{align}<br />
\mathcal{L}(D,E,R)=\text{T Translation Loss}+\text{T De-noising Loss} +\text{T Adversarial Loss} +\text{S Translation Loss} +\text{S De-noising Loss} +\text{S Adversarial Loss}\\<br />
\end{align}<br />
\begin{align}<br />
=\sum_{q\in Q_T}\left( -\log D_T \circ E \circ C \circ M _S(q) (q) -\log D_T \circ E \circ C (q) (q)+(1-\alpha)\log (R\circ E(q)) +\alpha\log(1-R\circ E(q)) \right)+\sum_{q\in Q_S}\left( -\log D_S \circ E \circ C \circ M_T (q) (q) -\log D_S \circ E \circ C (q) (q)+(1-\beta) \log(1-R \circ E(q)) +\beta\log (R\circ E(q) \right).<br />
\end{align}<br />
<br />
They alternate between iterations minimizing <math display="inline">\mathcal{L} </math> with respect to <math display="inline">E, D</math> and iterations maximizing with respect to <math display="inline">R</math>. ADAM is used for minimization, while RMSprop is used for maximization. After each epoch, M is updated so that <math display="inline">M_S=D_S \circ E</math> and <math display="inline">M_T=D_T \circ E</math>, after which <math display="inline"> M </math> is frozen until the next epoch.<br />
<br />
==Validation==<br />
The authors' aim is for their method to be completely unsupervised, and so do not use parallel corpora even for the selection of hyper-parameters. Instead, they validate by translating sentences to the other language and back, and comparing the resulting sentence with the original according to BLEU, a similarity metric frequently used in translation (Papineni et al. 2002).<br />
<br />
==Experimental Procedure and Results==<br />
<br />
The authors test their method on four data sets. The first is from the English-French translation task of the Workshop on Machine Translation 2014 (WMT14). This data set consists of parallel data. The authors generate a monolingual English corpus by randomly sampling 15 million sentence pairs, and choosing only the English sentences. They then generate a French corpus by selecting the French sentences from those pairs that were not previous chosen. Importantly, this means that the monolingual data sets have no parallel sentences. The second data set is generated from the English-German translation task from WMT14 using he same procedure.<br />
<br />
The third and fourth data sets are generated from Multi30k data set, which consists of multilingual captions of various images. The images are discarded and the English, French, and German captions are used to generate monolingual data sets in the manner described above. These monolingual corpora are much smaller, consisting of 14500 sentence each.<br />
<br />
The unsupervised translation scheme performs well, though not as well as a supervised translation scheme. It converges after a small number of epochs. Besides supervised translation, the authors compare their method with three other baselines: "Word-by-Word" uses only the previously-discussed word-alignment scheme; "Word-Reordering" uses a simple LSTM based language model and a greedy algorithm to select a reordering of the words produced by "Word-by-Word". "Oracle Word Reordering" means the optimal reordering of the words produced by "Word-by-Word".<br />
<br />
==Result Figures==<br />
[[File:MC_Translation Results.png]]<br />
[[File:MC_Translation_Convergence.png]]<br />
<br />
==Commentary==<br />
This paper's results are impressive: that it is even possible to translate between languages without parallel data suggests that languages are more similar than we might initially suspect, and that the method the authors present has, at least in part, discovered some common deep structure. As the authors point out, using no parallel data at all, their method is able to produce results comparable to those produced by neural machine translation methods trained on hundreds of thousands of a parallel sentences on the WMT dataset. On the other hand, the results they offer come with a few significant caveats.<br />
<br />
The first caveat is that the workhorse of the method is the unsupervised word-vector alignment scheme presented in Conneau et al. (2017) (that paper shares 3 authors with this one). As the ablation study reveals, without word-vector alignment, this method preforms extremely poorly. Moreover, word-by-word translation using word-vector alignment alone performs well, albeit not as well as this method. This suggests that the method of this paper mainly learns to perform (sometimes significant) corrections to word-by-word translations by reordering and occasional word substitution. Presumably, it does this by learning something of the natural structure of sentences in each of the two languages, so that it can correct the errors made by word-by-word translation.<br />
<br />
The second caveat is that the best results are attained translating between English and French, two very closely related languages, and the quality of translation between English and German, a slightly-less related pair, is significantly worse ( according to the ''Shorter Oxford English Dictionary'', 28.3 percent of the English vocabulary is French-derived, 28.2 percent is Latin-derived, and 25 percent is derived from Germanic languages. This probably understates the degree of correspondence between the French and English vocabularies, since French likely derives from Latin many of the same words English does. ). The authors do not report results with more distantly-related pairs, but it is reasonable to expect that performance would degrade significantly, especially since Conneau et al. (2017) shows that the word-alignment scheme performs much worse on more distant language pairs. This may be because there are more one-to-one correspondences between the words of closely related languages than there are between more distant languages. The authors suggest that their scheme could be a useful tool for translating between language pairs for which their are few parallel corpora. However, language pairs lacking parallel corpora are often (though not always) distantly related, and it is for such pairs that the performance of the present method suffers.<br />
<br />
The proposed method always beats Oracle Word Reordering on the Multi30k data set, but sometimes does not on the WMT data set. This may be because the WMT sentences are much more syntactically complex than the simple image captions of the Multi30k data set.<br />
<br />
The ablation study also reveals the importance of the corruption process <math display="inline">C</math>: the absence of <math display="inline">C</math> significantly degrades translation quality, though not as much as the absence of word-vector alignment. We can understand this in two related ways. First of all, if we view the model as learning to correct structural errors in word-by-word translations, then the corruption process introduces more errors of this kind, and so provides additional data upon which the model can train. Second, as Vincent et al. (2008) point out, de-noising auto-encoder training encourages a model to learn the structure of the manifold from which the data is drawn. By learning the structure of the source and target languages, the model can better correct the errors of word-by-word translation.<br />
<br />
[[File:MC_Alignment_Results.png|frame|none|alt=Alt text|From Conneau et al. (2017). The final row shows the performance of alignment method used in the present paper. Note the degradation in performance for more distant languages.]]<br />
<br />
[[File:MC_Translation_Ablation.png|frame|none|alt=Alt text|From the present paper. Results of an ablation study. Of note are the first, third, and forth rows, which demonstrate that while the translation component of the loss is relatively unimportant, the word vector alignment scheme and de-noising auto-encoder matter a great deal.]]<br />
<br />
==Future Work==<br />
The principal of performing unsupervised translation by starting with a rough but reasonable guess, and then improving it using knowledge of the structure of target language seem promising. Word by word translation using word-vector alignment works well for closely related languages like English and French, but is unlikely to work as well for more distant languages. For those languages, a better method for getting an initial guess is required.<br />
<br />
==References==<br />
#Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
#Conneau, Alexis, Guillaume Lample, Marc’Aurelio Ranzato, Ludovic Denoyer, Hervé Jégou. "Word Translation without Parallel Data". arXiv:1710.04087, (2017)<br />
# Dictionary, Shorter Oxford English. "Shorter Oxford english dictionary." (2007).<br />
#Goodfellow, Ian. "NIPS 2016 tutorial: Generative adversarial networks." arXiv preprint arXiv:1701.00160 (2016).<br />
# Hill, Felix, Kyunghyun Cho, and Anna Korhonen. "Learning distributed representations of sentences from unlabelled data." arXiv preprint arXiv:1602.03483 (2016).<br />
# Lample, Guillaume, Ludovic Denoyer, and Marc'Aurelio Ranzato. "Unsupervised Machine Translation Using Monolingual Corpora Only." arXiv preprint arXiv:1711.00043 (2017).<br />
#Papineni, Kishore, et al. "BLEU: a method for automatic evaluation of machine translation." Proceedings of the 40th annual meeting on association for computational linguistics. Association for Computational Linguistics, 2002.<br />
# Mikolov, Tomas, Quoc V Le, and Ilya Sutskever. "Exploiting similarities among languages for machine translation." arXiv preprint arXiv:1309.4168. (2013).<br />
#Sennrich, Rico, Barry Haddow, and Alexandra Birch. "Improving neural machine translation models with monolingual data." arXiv preprint arXiv:1511.06709 (2015).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.<br />
# Vincent, Pascal, et al. "Extracting and composing robust features with denoising autoencoders." Proceedings of the 25th international conference on Machine learning. ACM, 2008.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only&diff=32143stat946w18/Unsupervised Machine Translation Using Monolingual Corpora Only2018-02-23T18:51:46Z<p>Pa2forsy: /* Word vector alignment */</p>
<hr />
<div><br />
[[File:MC_Translation_Example.png]]<br />
== Introduction ==<br />
Neural machine translation systems must be trained on large corpora consisting of pairs of pre-translated sentences. The paper ''Unsupervised Machine Translation Using Monolingual Corpora Only'' by Guillaume Lample, Ludovic Denoyer, and Marc'Aurelio Ranzato proposes an unsupervised neural machine translation system, which can be trained without such parallel data.<br />
<br />
==Motivation==<br />
The authors offer two motivations for their work:<br />
# To translate between languages for which large parallel corpora do not exist.<br />
# To provide s strong baseline against which translation systems using parallel corpora can be compared.<br />
== Overview of unsupervised translation system ==<br />
The unsupervised translation scheme has the following outline:<br />
* The word-vector embeddings of the source and target languages are aligned in an unsupervised manner.<br />
* Sentences from the source and target language are mapped to a common latent vector space by an encoder, and then mapped to probability distributions over sentences in the target or source language by a decoder.<br />
* A de-noising auto-encoder loss encourages the latent-space representations to be insensitive to noise.<br />
* An adversarial loss encourages the latent-space representations of source and target sentences to be indistinguishable from each other. It is intended that the latent-space representation of a sentence should reflect its meaning, and not the particular language in which it is expressed.<br />
* A reconstruction loss encourages the model to improve on the translation model of the previous epoch.<br />
<br />
==Notation==<br />
Let <math display="inline">S</math> denote the set of words in the source language, and let <math display="inline">T</math> denote the set of words in the target language. Let <math display="inline">H \subset \mathbb{R}^{n_H}</math> denote the latent vector space. Moreover, let <math display="inline">S'</math> and <math display="inline">T'</math> denote the sets of finite sequences of words in the source and target language, and let <math display="inline">H'</math> denote the set of finite sequences of vectors in the latent space. For any set X, elide measure-theoretic details and let <math display="inline">\mathcal{P}(X)</math> denote the set of probability distributions over X.<br />
<br />
==Word vector alignment ==<br />
<br />
Conneau et al. (2017) describe an unsupervised method for aligning word vectors across languages. By "alignment", I mean that their method maps words with related meanings to nearby vectors, regardless of the language of the words. Moreover, if two words are one another's literal translations, their word vectors tend to be mutual nearest neighbors. <br />
<br />
The underlying idea of the alignment scheme can be summarized as follows: methods like word2vec or GLoVe generate vectors for which there is a correspondence between semantics and geometry. If <math display="inline">f</math> maps English words to their corresponding vectors, we have the approximate equation<br />
\begin{align}<br />
f(\text{king}) -f(\text{man}) +f(\text{woman})\approx f(\text{queen}).<br />
\end{align}<br />
Furthermore, if <math display="inline">g</math> maps French words to their corresponding vectors, then <br />
\begin{align}<br />
g(\text{roi}) -g(\text{homme}) +g(\text{femme})\approx g(\text{reine}).<br />
\end{align}<br />
<br />
Thus if <math display="inline">W</math> maps the word vectors of English words to the word vectors of their French translations, we should expect <math display="inline">W</math> to be linear. As was observed by Mikolov et al. (2013), the problem of word-vector alignment then becomes a problem of learning the linear transformation that best aligns two point clouds, one from the source language and one from the target language. For more on the history of the word-vector alignment problem, see my CS698 project ([https://uwaterloo.ca/scholar/sites/ca.scholar/files/pa2forsy/files/project_dec_3_0.pdf link]).<br />
<br />
Conneau et al. (2017)'s word vector alignment scheme is unique in that it requires no parallel data, and uses only the shapes of the two word-vector point clouds to be aligned. I will not go into detail, but the heart of the method is a special GAN, in which only the discriminator is a neural network, and the generator is the map corresponding to an orthogonal matrix.<br />
<br />
This unsupervised alignment method is crucial to the translation scheme of the current paper. From now on we denote by <br />
<math display="inline">A: S' \cup T' \to \mathcal{Z}'</math> the function that maps a source- or target- language word sequence to the corresponding aligned word vector sequence.<br />
<br />
==Encoder ==<br />
The encoder <math display="inline">E </math> reads a sequence of word vectors <math display="inline">(z_1,\ldots, z_m) \in \mathcal{Z}'</math> and outputs a sequence of hidden states <math display="inline">(h_1,\ldots, h_m) \in H'</math> in the latent space. Crucially, because the word vectors of the two languages have been aligned, the same encoder can be applied to both. That is, to map a source sentence <math display="inline">x=(x_1,\ldots, x_M)\in S'</math> to the latent space, we compute <math display="inline">E(A(x))</math>, and to map a target sentence <math display="inline">y=(y_1,\ldots, y_K)\in T'</math> to the latent space, we compute <math display="inline">E(A(y))</math>.<br />
<br />
The encoder consists of two LSTMs, one of which reads the word-vector sequence in the forward direction, and one of which reads it in the backward direction. The hidden state sequence is generated by concatenating the hidden states produced by the forward and backward LSTMs at each word vector.<br />
<br />
==Decoder==<br />
<br />
The decoder is a mono-directional LSTM that accepts a sequence of hidden states <math display="inline">h=(h_1,\ldots, h_m) \in H'</math> from the latent space and a language <math display="inline">L \in \{S,T \}</math> and outputs a probability distribution over sentences in that language. We have<br />
<br />
\begin{align}<br />
D: H' \times \{S,T \} \to \mathcal{P}(S') \cup \mathcal{P}(T').<br />
\end{align}<br />
<br />
The decoder makes use of the attention mechanism of Bahdanau et al. (2014). To compute the probability of a given sentence <math display="inline">y=(y_1,\ldots,y_K)</math> , the LSTM processes the sentence one word at a time, accepting at step <math display="inline">k</math> the aligned word vector of the previous word in the sentence <math display="inline">A(y_{k-1})</math> and a context vector <math display="inline">c_k\in H</math> computed from the hidden sequence <math display="inline">h\in H'</math>, and outputting a probability distribution over possible next words. The LSTM is initiated with a special, language-specific start-of-sequence token. Otherwise, the decoder is does not depend on the language of the sentence it is producing. The context vector is computed as described by Bahdanau et al. (2014), where we let <math display="inline">l_{k}</math> denote the hidden state of the LSTM at step <math display="inline">k</math>, and where <math display="inline">U,W</math> are learnable weight matrices, and <math display="inline">v</math> is a learnable weight vector:<br />
\begin{align}<br />
c_k&= \sum_{m=1}^M \alpha_{k,m} h_m\\<br />
\alpha_{k,m}&= \frac{\exp(e_{k,m})}{\sum_{m'=1}^M\exp(e_{k,m'}) },\\<br />
e_{k,m} &= v^T \tanh (Wl_{k-1} + U h_m ).<br />
\end{align}<br />
<br />
<br />
By learning <math display="inline">U,W</math> and <math display="inline">v</math>, the decoder can learn to decide which vectors in the sequence <math display="inline">h</math> are relevant to computing which words in the output sentence.<br />
<br />
At step <math display="inline">k</math>, after receiving the context vector <math display="inline">c_k\in H</math> and the aligned word vector of the previous word in the sequence,<math display="inline">A(y_{k-1})</math>, the LSTM outputs a probability distribution over words, which should be interpreted as the distribution of the next word according to the decoder. The probability the decoder assigns to a sentence is then the product of the probabilities computed for each word in this manner.<br />
<br />
==Overview of objective ==<br />
The objective function is the sum of:<br />
# The de-noising auto-encoder loss,<br />
# The translation loss,<br />
# The adversarial loss.<br />
I shall describe these in the following sections.<br />
<br />
==De-noising Auto-encoder Loss == <br />
A de-noising auto-encoder is a function optimized to map a corrupted sample from some dataset to the original un-corrupted sample. De-noising auto-encoders were introduced by Vincent et al. (2008), who provided numerous justifications, one of which is particularly illuminating. If we think of the dataset of interest as a thin manifold in a high-dimensional space, the corruption process is likely perturb a datapoint off the manifold. To learn to restore the corrupted datapoint, the de-noising auto-encoder must learn the shape of the manifold.<br />
<br />
Hill et al. (2016), used a de-noising auto-encoder to learn vectors representing sentences. They corrupted input sentences by randomly dropping and swapping words, and then trained a neural network to map the corrupted sentence to a vector, and then map the vector to the un-corrupted sentence. Interestingly, they found that sentence vectors learned this way were particularly effective when applied to tasks that involved generating paraphrases. This makes some sense: for a vector to be useful in restoring a corrupted sentence, it must capture something of the sentence's underlying meaning.<br />
<br />
The present paper uses the principal of de-noising auto-encoders to compute one of the terms in its loss function. In each iteration, a sentence is sampled from the source or target language, and a corruption process <math display="inline"> C</math> is applied to it. <math display="inline"> C</math> works by deleting each word in the sentence with probability <math display="inline">p_C</math> and applying to the sentence a permutation randomly selected from those that do not move words more than <math display="inline">k_C</math> spots from their original positions. The authors select <math display="inline">p_C=0.1</math> and <math display="inline">k_C=3</math>. The corrupted sentence is then mapped to the latent space using <math display="inline">E\circ A</math>. The loss is then the negative log probability of the original un-corrupted sentence according to the decoder <math display="inline">D</math> applied to the latent-space sequence.<br />
<br />
The explanation of Vincent et al. (2008) can help us understand this loss-function term: the de-noising auto-encoder loss forces the translation system to learn the shapes of the manifolds of the source and target languages.<br />
<br />
==Translation Loss==<br />
To compute the translation loss, we sample a sentence from one of the languages, translate it with the encoder and decoder of the previous epoch, and then corrupt its output with <math display="inline">C</math>. We then use the current encoder <math display="inline">E</math> to map the corrupted translation to a sequence <math display="inline">h \in H'</math> and the decoder <math display="inline">D</math> to map <math display="inline">h</math> to a probability distribution over sentences. The translation loss is the negative log probability the decoder assigns to the original uncorrupted sentence. <br />
<br />
It is interesting and useful to consider why this translation loss, which depends on the the translation model of the previous iteration, should promote an improved translation model in the current iteration. One loose way to understand this is to think of the translator as a de-noising translator. We are given a sentence perturbed from the manifold of possible sentences from a given language both by the corruption process and by the poor quality of the translation. The model must learn to both project and translate. The technique employed here resembles that used by Sennrich et al. (2014), who trained a neural machine translation system using both parallel and monolingual data. To make use of the monolingual target-language data, they used an auxiliary model to translate it to the source language, then trained their model to reconstruct the original target-language data from the source-language translation. Sennrich et al. argued that training the model to reconstruct true data from synthetic data was more robust than the opposite approach. The authors of the present paper use similar reasoning.<br />
<br />
==Adversarial Loss ==<br />
The intuition underlying the latent space is that it should encode the meaning of a sentence in a language-independent way. Accordingly, the authors introduce an adversarial loss, to encourage latent-space vectors mapped from the source and target languages to be indistinguishable. Central to this adversarial loss is the discriminator <math display="inline">R:H' \to [0,1]</math>, which makes use of <math display="inline">r: H\to [0,1]</math> a three-layer fully-connected neural network with 1024 hidden units per layer. Given a sequence of latent-space vectors <math display="inline">h=(h_1,\ldots,h_m)\in H'</math> the discriminator assigns probability <math display="inline">R(h)=\prod_{i=1}^m r(h_i)</math> that they originated in the target space. Each iteration, the discriminator is trained to maximize the objective function<br />
<br />
\begin{align}<br />
I_T(q) \log (R(E(q))) +(1-I_T(q) )\log(1-R(E(q)))<br />
\end{align}<br />
<br />
where <math display="inline">q</math> is a randomly selected sentence, and <math display="inline">I_T(q)</math> is 1 when <math display="inline">q\in I_T</math> is from the source language and 0 if <math display="inline">q\in I_S</math><br />
<br />
The same term is added to the primary objective function, which the encoder and decoder are trained to minimize. The result is that the encoder and decoder learn to fool the discriminator by mapping sentences from the source and target language to similar sequences of latent-space vectors.<br />
<br />
<br />
The authors note that they make use of label smoothing, a technique recommended by Goodfellow (2016) for regularizing GANs, in which the objective described above is replaced by <br />
<br />
\begin{align}<br />
I_T(q)( (1-\alpha)\log (R(E(q))) +\alpha\log(1-R(E(q))) )+(1-I_T(q) ) ( (1-\beta) \log(1-R(E(q))) +\beta\log (R(E(q)) ))<br />
\end{align}<br />
for some small nonnegative values of <math display="inline">\alpha, \beta</math>, the idea being to prevent the discriminator from making extreme predictions. While one-sided label smoothing (<math display="inline">\beta = 0</math>) is generally recommended, the present model differs from a standard GAN in that it is symmetric, and hence two-sided label smoothing would appear more reasonable.<br />
<br />
==Objective Function==<br />
<br />
Combining the above-described terms, we can write the overall objective function. Let <math display="inline">Q_S</math> denote the monolingual dataset for the source language, and let <math display="inline">Q_T</math> denote the monolingual dataset for the target language. Let <math display="inline">D_S:= D(\cdot, S)</math> and<math display="inline">D_T= D(\cdot, T)</math> (i.e. <math display="inline">D_S, D_T</math>) be the decoder restricted to the source or target language, respectively. Let <math display="inline"> M_S </math> and <math display="inline"> M_T </math> denote the target-to-source and source-to-target translation models of the previous epoch. Then our objective function is<br />
<br />
\begin{align}<br />
\mathcal{L}(D,E,R)=\text{T Translation Loss}+\text{T De-noising Loss} +\text{T Adversarial Loss} +\text{S Translation Loss} +\text{S De-noising Loss} +\text{S Adversarial Loss}\\<br />
\end{align}<br />
\begin{align}<br />
=\sum_{q\in Q_T}\left( -\log D_T \circ E \circ C \circ M _S(q) (q) -\log D_T \circ E \circ C (q) (q)+(1-\alpha)\log (R\circ E(q)) +\alpha\log(1-R\circ E(q)) \right)+\sum_{q\in Q_S}\left( -\log D_S \circ E \circ C \circ M_T (q) (q) -\log D_S \circ E \circ C (q) (q)+(1-\beta) \log(1-R \circ E(q)) +\beta\log (R\circ E(q) \right).<br />
\end{align}<br />
<br />
They alternate between iterations minimizing <math display="inline">\mathcal{L} </math> with respect to <math display="inline">E, D</math> and iterations maximizing with respect to <math display="inline">R</math>. ADAM is used for minimization, while RMSprop is used for maximization. After each epoch, M is updated so that <math display="inline">M_S=D_S \circ E</math> and <math display="inline">M_T=D_T \circ E</math>, after which <math display="inline"> M </math> is frozen until the next epoch.<br />
<br />
==Validation==<br />
The authors' aim is for their method to be completely unsupervised, and so do not use parallel corpora even for the selection of hyper-parameters. Instead, they validate by translating sentences to the other language and back, and comparing the resulting sentence with the original according to BLEU, a similarity metric frequently used in translation (Papineni et al. 2002).<br />
<br />
==Experimental Procedure and Results==<br />
<br />
The authors test their method on four data sets. The first is from the English-French translation task of the Workshop on Machine Translation 2014 (WMT14). This data set consists of parallel data. The authors generate a monolingual English corpus by randomly sampling 15 million sentence pairs, and choosing only the English sentences. They then generate a French corpus by selecting the French sentences from those pairs that were not previous chosen. Importantly, this means that the monolingual data sets have no parallel sentences. The second data set is generated from the English-German translation task from WMT14 using he same procedure.<br />
<br />
The third and fourth data sets are generated from Multi30k data set, which consists of multilingual captions of various images. The images are discarded and the English, French, and German captions are used to generate monolingual data sets in the manner described above. These monolingual corpora are much smaller, consisting of 14500 sentence each.<br />
<br />
The unsupervised translation scheme performs well, though not as well as a supervised translation scheme. It converges after a small number of epochs. Besides supervised translation, the authors compare their method with three other baselines: "Word-by-Word" uses only the previously-discussed word-alignment scheme; "Word-Reordering" uses a simple LSTM based language model and a greedy algorithm to select a reordering of the words produced by "Word-by-Word". "Oracle Word Reordering" means the optimal reordering of the words produced by "Word-by-Word".<br />
<br />
==Result Figures==<br />
[[File:MC_Translation Results.png]]<br />
[[File:MC_Translation_Convergence.png]]<br />
<br />
==Commentary==<br />
This paper's results are impressive: that it is even possible to translate between languages without parallel data suggests that languages are more similar than we might initially suspect, and that the method the authors present has, at least in part, discovered some common deep structure. As the authors point out, using no parallel data at all, their method is able to produce results comparable to those produced by neural machine translation methods trained on hundreds of thousands of a parallel sentences on the WMT dataset. On the other hand, the results they offer come with a few significant caveats.<br />
<br />
The first caveat is that the workhorse of the method is the unsupervised word-vector alignment scheme presented in Conneau et al. (2017) (that paper shares 3 authors with this one). As the ablation study reveals, without word-vector alignment, this method preforms extremely poorly. Moreover, word-by-word translation using word-vector alignment alone performs well, albeit not as well as this method. This suggests that the method of this paper mainly learns to perform (sometimes significant) corrections to word-by-word translations by reordering and occasional word substitution. Presumably, it does this by learning something of the natural structure of sentences in each of the two languages, so that it can correct the errors made by word-by-word translation.<br />
<br />
The second caveat is that the best results are attained translating between English and French, two very closely related languages, and the quality of translation between English and German, a slightly-less related pair, is significantly worse ( according to the ''Shorter Oxford English Dictionary'', 28.3 percent of the English vocabulary is French-derived, 28.2 percent is Latin-derived, and 25 percent is derived from Germanic languages. This probably understates the degree of correspondence between the French and English vocabularies, since French likely derives from Latin many of the same words English does. ). The authors do not report results with more distantly-related pairs, but it is reasonable to expect that performance would degrade significantly, especially since Conneau et al. (2017) shows that the word-alignment scheme performs much worse on more distant language pairs. This may be because there are more one-to-one correspondences between the words of closely related languages than there are between more distant languages. The authors suggest that their scheme could be a useful tool for translating between language pairs for which their are few parallel corpora. However, language pairs lacking parallel corpora are often (though not always) distantly related, and it is for such pairs that the performance of the present method suffers.<br />
<br />
The proposed method always beats Oracle Word Reordering on the Multi30k data set, but sometimes does not on the WMT data set. This may be because the WMT sentences are much more syntactically complex than the simple image captions of the Multi30k data set.<br />
<br />
The ablation study also reveals the importance of the corruption process <math display="inline">C</math>: the absence of <math display="inline">C</math> significantly degrades translation quality, though not as much as the absence of word-vector alignment. We can understand this in two related ways. First of all, if we view the model as learning to correct structural errors in word-by-word translations, then the corruption process introduces more errors of this kind, and so provides additional data upon which the model can train. Second, as Vincent et al. (2008) point out, de-noising auto-encoder training encourages a model to learn the structure of the manifold from which the data is drawn. By learning the structure of the source and target languages, the model can better correct the errors of word-by-word translation.<br />
<br />
[[File:MC_Alignment_Results.png|frame|none|alt=Alt text|From Conneau et al. (2017). The final row shows the performance of alignment method used in the present paper. Note the degradation in performance for more distant languages.]]<br />
<br />
[[File:MC_Translation_Ablation.png|frame|none|alt=Alt text|From the present paper. Results of an ablation study. Of note are the first, third, and forth rows, which demonstrate that while the translation component of the loss is relatively unimportant, the word vector alignment scheme and de-noising auto-encoder matter a great deal.]]<br />
<br />
==Future Work==<br />
The principal of performing unsupervised translation by starting with a rough but reasonable guess, and then improving it using knowledge of the structure of target language seem promising. Word by word translation using word-vector alignment works well for closely related languages like English and French, but is unlikely to work as well for more distant languages. For those languages, a better method for getting an initial guess is required.<br />
<br />
==References==<br />
#Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
#Conneau, Alexis, Guillaume Lample, Marc’Aurelio Ranzato, Ludovic Denoyer, Hervé Jégou. "Word Translation without Parallel Data". arXiv:1710.04087, (2017)<br />
# Dictionary, Shorter Oxford English. "Shorter Oxford english dictionary." (2007).<br />
#Goodfellow, Ian. "NIPS 2016 tutorial: Generative adversarial networks." arXiv preprint arXiv:1701.00160 (2016).<br />
# Hill, Felix, Kyunghyun Cho, and Anna Korhonen. "Learning distributed representations of sentences from unlabelled data." arXiv preprint arXiv:1602.03483 (2016).<br />
# Lample, Guillaume, Ludovic Denoyer, and Marc'Aurelio Ranzato. "Unsupervised Machine Translation Using Monolingual Corpora Only." arXiv preprint arXiv:1711.00043 (2017).<br />
#Papineni, Kishore, et al. "BLEU: a method for automatic evaluation of machine translation." Proceedings of the 40th annual meeting on association for computational linguistics. Association for Computational Linguistics, 2002.<br />
# Mikolov, Tomas, Quoc V Le, and Ilya Sutskever. "Exploiting similarities among languages for machine translation." arXiv preprint arXiv:1309.4168. (2013).<br />
#Sennrich, Rico, Barry Haddow, and Alexandra Birch. "Improving neural machine translation models with monolingual data." arXiv preprint arXiv:1511.06709 (2015).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.<br />
# Vincent, Pascal, et al. "Extracting and composing robust features with denoising autoencoders." Proceedings of the 25th international conference on Machine learning. ACM, 2008.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only&diff=32142stat946w18/Unsupervised Machine Translation Using Monolingual Corpora Only2018-02-23T18:45:18Z<p>Pa2forsy: </p>
<hr />
<div><br />
[[File:MC_Translation_Example.png]]<br />
== Introduction ==<br />
Neural machine translation systems must be trained on large corpora consisting of pairs of pre-translated sentences. The paper ''Unsupervised Machine Translation Using Monolingual Corpora Only'' by Guillaume Lample, Ludovic Denoyer, and Marc'Aurelio Ranzato proposes an unsupervised neural machine translation system, which can be trained without such parallel data.<br />
<br />
==Motivation==<br />
The authors offer two motivations for their work:<br />
# To translate between languages for which large parallel corpora do not exist.<br />
# To provide s strong baseline against which translation systems using parallel corpora can be compared.<br />
== Overview of unsupervised translation system ==<br />
The unsupervised translation scheme has the following outline:<br />
* The word-vector embeddings of the source and target languages are aligned in an unsupervised manner.<br />
* Sentences from the source and target language are mapped to a common latent vector space by an encoder, and then mapped to probability distributions over sentences in the target or source language by a decoder.<br />
* A de-noising auto-encoder loss encourages the latent-space representations to be insensitive to noise.<br />
* An adversarial loss encourages the latent-space representations of source and target sentences to be indistinguishable from each other. It is intended that the latent-space representation of a sentence should reflect its meaning, and not the particular language in which it is expressed.<br />
* A reconstruction loss encourages the model to improve on the translation model of the previous epoch.<br />
<br />
==Notation==<br />
Let <math display="inline">S</math> denote the set of words in the source language, and let <math display="inline">T</math> denote the set of words in the target language. Let <math display="inline">H \subset \mathbb{R}^{n_H}</math> denote the latent vector space. Moreover, let <math display="inline">S'</math> and <math display="inline">T'</math> denote the sets of finite sequences of words in the source and target language, and let <math display="inline">H'</math> denote the set of finite sequences of vectors in the latent space. For any set X, elide measure-theoretic details and let <math display="inline">\mathcal{P}(X)</math> denote the set of probability distributions over X.<br />
<br />
==Word vector alignment ==<br />
<br />
Conneau et al. (2017) describe an unsupervised method for aligning word vectors across languages. By "alignment", I mean that their method maps words with related meanings to nearby vectors, regardless of the language of the words. Moreover, if two words are one another's literal translations, their word vectors tend to be mutual nearest neighbors. <br />
<br />
The underlying idea of the alignment scheme can be summarized as follows: methods like word2vec or GLoVe generate vectors for which there is a correspondence between semantics and geometry. If <math display="inline">f</math> maps English words to their corresponding vectors, we have the approximate equation<br />
\begin{align}<br />
f(\text{king}) -f(\text{man}) +f(\text{woman})\approx f(\text{queen}).<br />
\end{align}<br />
Furthermore, if <math display="inline">g</math> maps French words to their corresponding vectors, then <br />
\begin{align}<br />
g(\text{roi}) -f(\text{homme}) +f(\text{femme})\approx f(\text{reine}).<br />
\end{align}<br />
<br />
Thus if <math display="inline">W</math> maps the word vectors of English words to the word vectors of their French translations, we should expect <math display="inline">W</math> to be linear. As was observed by Mikolov et al. (2013), the problem of word-vector alignment then becomes a problem of learning the linear transformation that best aligns two point clouds, one from the source language and one from the target language. For more on the history of the word-vector alignment problem, see my CS698 project ([https://uwaterloo.ca/scholar/sites/ca.scholar/files/pa2forsy/files/project_dec_3_0.pdf link]).<br />
<br />
Conneau et al. (2017)'s word vector alignment scheme is unique in that it requires no parallel data, and uses only the shapes of the two word-vector point clouds to be aligned. I will not go into detail, but the heart of the method is a special GAN, in which only the discriminator is a neural network, and the generator is the map corresponding to an orthogonal matrix.<br />
<br />
This unsupervised alignment method is crucial to the translation scheme of the current paper. From now on we denote by <br />
<math display="inline">A: S' \cup T' \to \mathcal{Z}'</math> the function that maps a source- or target- language word sequence to the corresponding aligned word vector sequence.<br />
<br />
==Encoder ==<br />
The encoder <math display="inline">E </math> reads a sequence of word vectors <math display="inline">(z_1,\ldots, z_m) \in \mathcal{Z}'</math> and outputs a sequence of hidden states <math display="inline">(h_1,\ldots, h_m) \in H'</math> in the latent space. Crucially, because the word vectors of the two languages have been aligned, the same encoder can be applied to both. That is, to map a source sentence <math display="inline">x=(x_1,\ldots, x_M)\in S'</math> to the latent space, we compute <math display="inline">E(A(x))</math>, and to map a target sentence <math display="inline">y=(y_1,\ldots, y_K)\in T'</math> to the latent space, we compute <math display="inline">E(A(y))</math>.<br />
<br />
The encoder consists of two LSTMs, one of which reads the word-vector sequence in the forward direction, and one of which reads it in the backward direction. The hidden state sequence is generated by concatenating the hidden states produced by the forward and backward LSTMs at each word vector.<br />
<br />
==Decoder==<br />
<br />
The decoder is a mono-directional LSTM that accepts a sequence of hidden states <math display="inline">h=(h_1,\ldots, h_m) \in H'</math> from the latent space and a language <math display="inline">L \in \{S,T \}</math> and outputs a probability distribution over sentences in that language. We have<br />
<br />
\begin{align}<br />
D: H' \times \{S,T \} \to \mathcal{P}(S') \cup \mathcal{P}(T').<br />
\end{align}<br />
<br />
The decoder makes use of the attention mechanism of Bahdanau et al. (2014). To compute the probability of a given sentence <math display="inline">y=(y_1,\ldots,y_K)</math> , the LSTM processes the sentence one word at a time, accepting at step <math display="inline">k</math> the aligned word vector of the previous word in the sentence <math display="inline">A(y_{k-1})</math> and a context vector <math display="inline">c_k\in H</math> computed from the hidden sequence <math display="inline">h\in H'</math>, and outputting a probability distribution over possible next words. The LSTM is initiated with a special, language-specific start-of-sequence token. Otherwise, the decoder is does not depend on the language of the sentence it is producing. The context vector is computed as described by Bahdanau et al. (2014), where we let <math display="inline">l_{k}</math> denote the hidden state of the LSTM at step <math display="inline">k</math>, and where <math display="inline">U,W</math> are learnable weight matrices, and <math display="inline">v</math> is a learnable weight vector:<br />
\begin{align}<br />
c_k&= \sum_{m=1}^M \alpha_{k,m} h_m\\<br />
\alpha_{k,m}&= \frac{\exp(e_{k,m})}{\sum_{m'=1}^M\exp(e_{k,m'}) },\\<br />
e_{k,m} &= v^T \tanh (Wl_{k-1} + U h_m ).<br />
\end{align}<br />
<br />
<br />
By learning <math display="inline">U,W</math> and <math display="inline">v</math>, the decoder can learn to decide which vectors in the sequence <math display="inline">h</math> are relevant to computing which words in the output sentence.<br />
<br />
At step <math display="inline">k</math>, after receiving the context vector <math display="inline">c_k\in H</math> and the aligned word vector of the previous word in the sequence,<math display="inline">A(y_{k-1})</math>, the LSTM outputs a probability distribution over words, which should be interpreted as the distribution of the next word according to the decoder. The probability the decoder assigns to a sentence is then the product of the probabilities computed for each word in this manner.<br />
<br />
==Overview of objective ==<br />
The objective function is the sum of:<br />
# The de-noising auto-encoder loss,<br />
# The translation loss,<br />
# The adversarial loss.<br />
I shall describe these in the following sections.<br />
<br />
==De-noising Auto-encoder Loss == <br />
A de-noising auto-encoder is a function optimized to map a corrupted sample from some dataset to the original un-corrupted sample. De-noising auto-encoders were introduced by Vincent et al. (2008), who provided numerous justifications, one of which is particularly illuminating. If we think of the dataset of interest as a thin manifold in a high-dimensional space, the corruption process is likely perturb a datapoint off the manifold. To learn to restore the corrupted datapoint, the de-noising auto-encoder must learn the shape of the manifold.<br />
<br />
Hill et al. (2016), used a de-noising auto-encoder to learn vectors representing sentences. They corrupted input sentences by randomly dropping and swapping words, and then trained a neural network to map the corrupted sentence to a vector, and then map the vector to the un-corrupted sentence. Interestingly, they found that sentence vectors learned this way were particularly effective when applied to tasks that involved generating paraphrases. This makes some sense: for a vector to be useful in restoring a corrupted sentence, it must capture something of the sentence's underlying meaning.<br />
<br />
The present paper uses the principal of de-noising auto-encoders to compute one of the terms in its loss function. In each iteration, a sentence is sampled from the source or target language, and a corruption process <math display="inline"> C</math> is applied to it. <math display="inline"> C</math> works by deleting each word in the sentence with probability <math display="inline">p_C</math> and applying to the sentence a permutation randomly selected from those that do not move words more than <math display="inline">k_C</math> spots from their original positions. The authors select <math display="inline">p_C=0.1</math> and <math display="inline">k_C=3</math>. The corrupted sentence is then mapped to the latent space using <math display="inline">E\circ A</math>. The loss is then the negative log probability of the original un-corrupted sentence according to the decoder <math display="inline">D</math> applied to the latent-space sequence.<br />
<br />
The explanation of Vincent et al. (2008) can help us understand this loss-function term: the de-noising auto-encoder loss forces the translation system to learn the shapes of the manifolds of the source and target languages.<br />
<br />
==Translation Loss==<br />
To compute the translation loss, we sample a sentence from one of the languages, translate it with the encoder and decoder of the previous epoch, and then corrupt its output with <math display="inline">C</math>. We then use the current encoder <math display="inline">E</math> to map the corrupted translation to a sequence <math display="inline">h \in H'</math> and the decoder <math display="inline">D</math> to map <math display="inline">h</math> to a probability distribution over sentences. The translation loss is the negative log probability the decoder assigns to the original uncorrupted sentence. <br />
<br />
It is interesting and useful to consider why this translation loss, which depends on the the translation model of the previous iteration, should promote an improved translation model in the current iteration. One loose way to understand this is to think of the translator as a de-noising translator. We are given a sentence perturbed from the manifold of possible sentences from a given language both by the corruption process and by the poor quality of the translation. The model must learn to both project and translate. The technique employed here resembles that used by Sennrich et al. (2014), who trained a neural machine translation system using both parallel and monolingual data. To make use of the monolingual target-language data, they used an auxiliary model to translate it to the source language, then trained their model to reconstruct the original target-language data from the source-language translation. Sennrich et al. argued that training the model to reconstruct true data from synthetic data was more robust than the opposite approach. The authors of the present paper use similar reasoning.<br />
<br />
==Adversarial Loss ==<br />
The intuition underlying the latent space is that it should encode the meaning of a sentence in a language-independent way. Accordingly, the authors introduce an adversarial loss, to encourage latent-space vectors mapped from the source and target languages to be indistinguishable. Central to this adversarial loss is the discriminator <math display="inline">R:H' \to [0,1]</math>, which makes use of <math display="inline">r: H\to [0,1]</math> a three-layer fully-connected neural network with 1024 hidden units per layer. Given a sequence of latent-space vectors <math display="inline">h=(h_1,\ldots,h_m)\in H'</math> the discriminator assigns probability <math display="inline">R(h)=\prod_{i=1}^m r(h_i)</math> that they originated in the target space. Each iteration, the discriminator is trained to maximize the objective function<br />
<br />
\begin{align}<br />
I_T(q) \log (R(E(q))) +(1-I_T(q) )\log(1-R(E(q)))<br />
\end{align}<br />
<br />
where <math display="inline">q</math> is a randomly selected sentence, and <math display="inline">I_T(q)</math> is 1 when <math display="inline">q\in I_T</math> is from the source language and 0 if <math display="inline">q\in I_S</math><br />
<br />
The same term is added to the primary objective function, which the encoder and decoder are trained to minimize. The result is that the encoder and decoder learn to fool the discriminator by mapping sentences from the source and target language to similar sequences of latent-space vectors.<br />
<br />
<br />
The authors note that they make use of label smoothing, a technique recommended by Goodfellow (2016) for regularizing GANs, in which the objective described above is replaced by <br />
<br />
\begin{align}<br />
I_T(q)( (1-\alpha)\log (R(E(q))) +\alpha\log(1-R(E(q))) )+(1-I_T(q) ) ( (1-\beta) \log(1-R(E(q))) +\beta\log (R(E(q)) ))<br />
\end{align}<br />
for some small nonnegative values of <math display="inline">\alpha, \beta</math>, the idea being to prevent the discriminator from making extreme predictions. While one-sided label smoothing (<math display="inline">\beta = 0</math>) is generally recommended, the present model differs from a standard GAN in that it is symmetric, and hence two-sided label smoothing would appear more reasonable.<br />
<br />
==Objective Function==<br />
<br />
Combining the above-described terms, we can write the overall objective function. Let <math display="inline">Q_S</math> denote the monolingual dataset for the source language, and let <math display="inline">Q_T</math> denote the monolingual dataset for the target language. Let <math display="inline">D_S:= D(\cdot, S)</math> and<math display="inline">D_T= D(\cdot, T)</math> (i.e. <math display="inline">D_S, D_T</math>) be the decoder restricted to the source or target language, respectively. Let <math display="inline"> M_S </math> and <math display="inline"> M_T </math> denote the target-to-source and source-to-target translation models of the previous epoch. Then our objective function is<br />
<br />
\begin{align}<br />
\mathcal{L}(D,E,R)=\text{T Translation Loss}+\text{T De-noising Loss} +\text{T Adversarial Loss} +\text{S Translation Loss} +\text{S De-noising Loss} +\text{S Adversarial Loss}\\<br />
\end{align}<br />
\begin{align}<br />
=\sum_{q\in Q_T}\left( -\log D_T \circ E \circ C \circ M _S(q) (q) -\log D_T \circ E \circ C (q) (q)+(1-\alpha)\log (R\circ E(q)) +\alpha\log(1-R\circ E(q)) \right)+\sum_{q\in Q_S}\left( -\log D_S \circ E \circ C \circ M_T (q) (q) -\log D_S \circ E \circ C (q) (q)+(1-\beta) \log(1-R \circ E(q)) +\beta\log (R\circ E(q) \right).<br />
\end{align}<br />
<br />
They alternate between iterations minimizing <math display="inline">\mathcal{L} </math> with respect to <math display="inline">E, D</math> and iterations maximizing with respect to <math display="inline">R</math>. ADAM is used for minimization, while RMSprop is used for maximization. After each epoch, M is updated so that <math display="inline">M_S=D_S \circ E</math> and <math display="inline">M_T=D_T \circ E</math>, after which <math display="inline"> M </math> is frozen until the next epoch.<br />
<br />
==Validation==<br />
The authors' aim is for their method to be completely unsupervised, and so do not use parallel corpora even for the selection of hyper-parameters. Instead, they validate by translating sentences to the other language and back, and comparing the resulting sentence with the original according to BLEU, a similarity metric frequently used in translation (Papineni et al. 2002).<br />
<br />
==Experimental Procedure and Results==<br />
<br />
The authors test their method on four data sets. The first is from the English-French translation task of the Workshop on Machine Translation 2014 (WMT14). This data set consists of parallel data. The authors generate a monolingual English corpus by randomly sampling 15 million sentence pairs, and choosing only the English sentences. They then generate a French corpus by selecting the French sentences from those pairs that were not previous chosen. Importantly, this means that the monolingual data sets have no parallel sentences. The second data set is generated from the English-German translation task from WMT14 using he same procedure.<br />
<br />
The third and fourth data sets are generated from Multi30k data set, which consists of multilingual captions of various images. The images are discarded and the English, French, and German captions are used to generate monolingual data sets in the manner described above. These monolingual corpora are much smaller, consisting of 14500 sentence each.<br />
<br />
The unsupervised translation scheme performs well, though not as well as a supervised translation scheme. It converges after a small number of epochs. Besides supervised translation, the authors compare their method with three other baselines: "Word-by-Word" uses only the previously-discussed word-alignment scheme; "Word-Reordering" uses a simple LSTM based language model and a greedy algorithm to select a reordering of the words produced by "Word-by-Word". "Oracle Word Reordering" means the optimal reordering of the words produced by "Word-by-Word".<br />
<br />
==Result Figures==<br />
[[File:MC_Translation Results.png]]<br />
[[File:MC_Translation_Convergence.png]]<br />
<br />
==Commentary==<br />
This paper's results are impressive: that it is even possible to translate between languages without parallel data suggests that languages are more similar than we might initially suspect, and that the method the authors present has, at least in part, discovered some common deep structure. As the authors point out, using no parallel data at all, their method is able to produce results comparable to those produced by neural machine translation methods trained on hundreds of thousands of a parallel sentences on the WMT dataset. On the other hand, the results they offer come with a few significant caveats.<br />
<br />
The first caveat is that the workhorse of the method is the unsupervised word-vector alignment scheme presented in Conneau et al. (2017) (that paper shares 3 authors with this one). As the ablation study reveals, without word-vector alignment, this method preforms extremely poorly. Moreover, word-by-word translation using word-vector alignment alone performs well, albeit not as well as this method. This suggests that the method of this paper mainly learns to perform (sometimes significant) corrections to word-by-word translations by reordering and occasional word substitution. Presumably, it does this by learning something of the natural structure of sentences in each of the two languages, so that it can correct the errors made by word-by-word translation.<br />
<br />
The second caveat is that the best results are attained translating between English and French, two very closely related languages, and the quality of translation between English and German, a slightly-less related pair, is significantly worse ( according to the ''Shorter Oxford English Dictionary'', 28.3 percent of the English vocabulary is French-derived, 28.2 percent is Latin-derived, and 25 percent is derived from Germanic languages. This probably understates the degree of correspondence between the French and English vocabularies, since French likely derives from Latin many of the same words English does. ). The authors do not report results with more distantly-related pairs, but it is reasonable to expect that performance would degrade significantly, especially since Conneau et al. (2017) shows that the word-alignment scheme performs much worse on more distant language pairs. This may be because there are more one-to-one correspondences between the words of closely related languages than there are between more distant languages. The authors suggest that their scheme could be a useful tool for translating between language pairs for which their are few parallel corpora. However, language pairs lacking parallel corpora are often (though not always) distantly related, and it is for such pairs that the performance of the present method suffers.<br />
<br />
The proposed method always beats Oracle Word Reordering on the Multi30k data set, but sometimes does not on the WMT data set. This may be because the WMT sentences are much more syntactically complex than the simple image captions of the Multi30k data set.<br />
<br />
The ablation study also reveals the importance of the corruption process <math display="inline">C</math>: the absence of <math display="inline">C</math> significantly degrades translation quality, though not as much as the absence of word-vector alignment. We can understand this in two related ways. First of all, if we view the model as learning to correct structural errors in word-by-word translations, then the corruption process introduces more errors of this kind, and so provides additional data upon which the model can train. Second, as Vincent et al. (2008) point out, de-noising auto-encoder training encourages a model to learn the structure of the manifold from which the data is drawn. By learning the structure of the source and target languages, the model can better correct the errors of word-by-word translation.<br />
<br />
[[File:MC_Alignment_Results.png|frame|none|alt=Alt text|From Conneau et al. (2017). The final row shows the performance of alignment method used in the present paper. Note the degradation in performance for more distant languages.]]<br />
<br />
[[File:MC_Translation_Ablation.png|frame|none|alt=Alt text|From the present paper. Results of an ablation study. Of note are the first, third, and forth rows, which demonstrate that while the translation component of the loss is relatively unimportant, the word vector alignment scheme and de-noising auto-encoder matter a great deal.]]<br />
<br />
==Future Work==<br />
The principal of performing unsupervised translation by starting with a rough but reasonable guess, and then improving it using knowledge of the structure of target language seem promising. Word by word translation using word-vector alignment works well for closely related languages like English and French, but is unlikely to work as well for more distant languages. For those languages, a better method for getting an initial guess is required.<br />
<br />
==References==<br />
#Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
#Conneau, Alexis, Guillaume Lample, Marc’Aurelio Ranzato, Ludovic Denoyer, Hervé Jégou. "Word Translation without Parallel Data". arXiv:1710.04087, (2017)<br />
# Dictionary, Shorter Oxford English. "Shorter Oxford english dictionary." (2007).<br />
#Goodfellow, Ian. "NIPS 2016 tutorial: Generative adversarial networks." arXiv preprint arXiv:1701.00160 (2016).<br />
# Hill, Felix, Kyunghyun Cho, and Anna Korhonen. "Learning distributed representations of sentences from unlabelled data." arXiv preprint arXiv:1602.03483 (2016).<br />
# Lample, Guillaume, Ludovic Denoyer, and Marc'Aurelio Ranzato. "Unsupervised Machine Translation Using Monolingual Corpora Only." arXiv preprint arXiv:1711.00043 (2017).<br />
#Papineni, Kishore, et al. "BLEU: a method for automatic evaluation of machine translation." Proceedings of the 40th annual meeting on association for computational linguistics. Association for Computational Linguistics, 2002.<br />
# Mikolov, Tomas, Quoc V Le, and Ilya Sutskever. "Exploiting similarities among languages for machine translation." arXiv preprint arXiv:1309.4168. (2013).<br />
#Sennrich, Rico, Barry Haddow, and Alexandra Birch. "Improving neural machine translation models with monolingual data." arXiv preprint arXiv:1511.06709 (2015).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.<br />
# Vincent, Pascal, et al. "Extracting and composing robust features with denoising autoencoders." Proceedings of the 25th international conference on Machine learning. ACM, 2008.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only&diff=32129stat946w18/Unsupervised Machine Translation Using Monolingual Corpora Only2018-02-23T16:17:55Z<p>Pa2forsy: /* Introduction */</p>
<hr />
<div><br />
[[File:MC_Translation_Example.png]]<br />
== Introduction ==<br />
Neural machine translation systems must be trained on large corpora consisting of pairs of pre-translated sentences. The paper ''Unsupervised Machine Translation Using Monolingual Corpora Only'' by Guillaume Lample, Ludovic Denoyer, and Marc'Aurelio Ranzato proposes an unsupervised neural machine translation system, which can be trained without such parallel data.<br />
<br />
== Overview of unsupervised translation system ==<br />
The unsupervised translation scheme has the following outline:<br />
* The word-vector embeddings of the source and target languages are aligned in an unsupervised manner.<br />
* Sentences from the source and target language are mapped to a common latent vector space by an encoder, and then mapped to probability distributions over sentences in the target or source language by a decoder.<br />
* A de-noising auto-encoder loss encourages the latent-space representations to be insensitive to noise.<br />
* An adversarial loss encourages the latent-space representations of source and target sentences to be indistinguishable from each other. It is intended that the latent-space representation of a sentence should reflect its meaning, and not the particular language in which it is expressed.<br />
* A reconstruction loss encourages the model to improve on the translation model of the previous epoch.<br />
<br />
==Notation==<br />
Let <math display="inline">S</math> denote the set of words in the source language, and let <math display="inline">T</math> denote the set of words in the target language. Let <math display="inline">H \subset \mathbb{R}^{n_H}</math> denote the latent vector space. Moreover, let <math display="inline">S'</math> and <math display="inline">T'</math> denote the sets of finite sequences of words in the source and target language, and let <math display="inline">H'</math> denote the set of finite sequences of vectors in the latent space. For any set X, elide measure-theoretic details and let <math display="inline">\mathcal{P}(X)</math> denote the set of probability distributions over X.<br />
<br />
==Word vector alignment ==<br />
<br />
Conneau et al. (2017) describe an unsupervised method for aligning word vectors across languages. By "alignment", I mean that their method maps words with related meanings to nearby vectors, regardless of the language of the words. Moreover, if two words are one another's literal translations, their word vectors tend to be mutual nearest neighbors. <br />
<br />
The underlying idea of the alignment scheme can be summarized as follows: methods like word2vec or GLoVe generate vectors for which there is a correspondence between semantics and geometry. If <math display="inline">f</math> maps English words to their corresponding vectors, we have the approximate equation<br />
\begin{align}<br />
f(\text{king}) -f(\text{man}) +f(\text{woman})\approx f(\text{queen}).<br />
\end{align}<br />
Furthermore, if <math display="inline">g</math> maps French words to their corresponding vectors, then <br />
\begin{align}<br />
g(\text{roi}) -f(\text{homme}) +f(\text{femme})\approx f(\text{reine}).<br />
\end{align}<br />
<br />
Thus if <math display="inline">W</math> maps the word vectors of English words to the word vectors of their French translations, we should expect <math display="inline">W</math> to be linear. As was observed by Mikolov et al. (2013), the problem of word-vector alignment then becomes a problem of learning the linear transformation that best aligns two point clouds, one from the source language and one from the target language. For more on the history of the word-vector alignment problem, see my CS698 project ([https://uwaterloo.ca/scholar/sites/ca.scholar/files/pa2forsy/files/project_dec_3_0.pdf link]).<br />
<br />
Conneau et al. (2017)'s word vector alignment scheme is unique in that it requires no parallel data, and uses only the shapes of the two word-vector point clouds to be aligned. I will not go into detail, but the heart of the method is a special GAN, in which only the discriminator is a neural network, and the generator is the map corresponding to an orthogonal matrix.<br />
<br />
This unsupervised alignment method is crucial to the translation scheme of the current paper. From now on we denote by <br />
<math display="inline">A: S' \cup T' \to \mathcal{Z}'</math> the function that maps a source- or target- language word sequence to the corresponding aligned word vector sequence.<br />
<br />
==Encoder ==<br />
The encoder <math display="inline">E </math> reads a sequence of word vectors <math display="inline">(z_1,\ldots, z_m) \in \mathcal{Z}'</math> and outputs a sequence of hidden states <math display="inline">(h_1,\ldots, h_m) \in H'</math> in the latent space. Crucially, because the word vectors of the two languages have been aligned, the same encoder can be applied to both. That is, to map a source sentence <math display="inline">x=(x_1,\ldots, x_M)\in S'</math> to the latent space, we compute <math display="inline">E(A(x))</math>, and to map a target sentence <math display="inline">y=(y_1,\ldots, y_K)\in T'</math> to the latent space, we compute <math display="inline">E(A(y))</math>.<br />
<br />
The encoder consists of two LSTMs, one of which reads the word-vector sequence in the forward direction, and one of which reads it in the backward direction. The hidden state sequence is generated by concatenating the hidden states produced by the forward and backward LSTMs at each word vector.<br />
<br />
==Decoder==<br />
<br />
The decoder is a mono-directional LSTM that accepts a sequence of hidden states <math display="inline">h=(h_1,\ldots, h_m) \in H'</math> from the latent space and a language <math display="inline">L \in \{S,T \}</math> and outputs a probability distribution over sentences in that language. We have<br />
<br />
\begin{align}<br />
D: H' \times \{S,T \} \to \mathcal{P}(S') \cup \mathcal{P}(T').<br />
\end{align}<br />
<br />
The decoder makes use of the attention mechanism of Bahdanau et al. (2014). To compute the probability of a given sentence <math display="inline">y=(y_1,\ldots,y_K)</math> , the LSTM processes the sentence one word at a time, accepting at step <math display="inline">k</math> the aligned word vector of the previous word in the sentence <math display="inline">A(y_{k-1})</math> and a context vector <math display="inline">c_k\in H</math> computed from the hidden sequence <math display="inline">h\in H'</math>, and outputting a probability distribution over possible next words. The LSTM is initiated with a special, language-specific start-of-sequence token. Otherwise, the decoder is does not depend on the language of the sentence it is producing. The context vector is computed as described by Bahdanau et al. (2014), where we let <math display="inline">l_{k}</math> denote the hidden state of the LSTM at step <math display="inline">k</math>, and where <math display="inline">U,W</math> are learnable weight matrices, and <math display="inline">v</math> is a learnable weight vector:<br />
\begin{align}<br />
c_k&= \sum_{m=1}^M \alpha_{k,m} h_m\\<br />
\alpha_{k,m}&= \frac{\exp(e_{k,m})}{\sum_{m'=1}^M\exp(e_{k,m'}) },\\<br />
e_{k,m} &= v^T \tanh (Wl_{k-1} + U h_m ).<br />
\end{align}<br />
<br />
<br />
By learning <math display="inline">U,W</math> and <math display="inline">v</math>, the decoder can learn to decide which vectors in the sequence <math display="inline">h</math> are relevant to computing which words in the output sentence.<br />
<br />
At step <math display="inline">k</math>, after receiving the context vector <math display="inline">c_k\in H</math> and the aligned word vector of the previous word in the sequence,<math display="inline">A(y_{k-1})</math>, the LSTM outputs a probability distribution over words, which should be interpreted as the distribution of the next word according to the decoder. The probability the decoder assigns to a sentence is then the product of the probabilities computed for each word in this manner.<br />
<br />
==Overview of objective ==<br />
The objective function is the sum of:<br />
# The de-noising auto-encoder loss,<br />
# The translation loss,<br />
# The adversarial loss.<br />
I shall describe these in the following sections.<br />
<br />
==De-noising Auto-encoder Loss == <br />
A de-noising auto-encoder is a function optimized to map a corrupted sample from some dataset to the original un-corrupted sample. De-noising auto-encoders were introduced by Vincent et al. (2008), who provided numerous justifications, one of which is particularly illuminating. If we think of the dataset of interest as a thin manifold in a high-dimensional space, the corruption process is likely perturb a datapoint off the manifold. To learn to restore the corrupted datapoint, the de-noising auto-encoder must learn the shape of the manifold.<br />
<br />
Hill et al. (2016), used a de-noising auto-encoder to learn vectors representing sentences. They corrupted input sentences by randomly dropping and swapping words, and then trained a neural network to map the corrupted sentence to a vector, and then map the vector to the un-corrupted sentence. Interestingly, they found that sentence vectors learned this way were particularly effective when applied to tasks that involved generating paraphrases. This makes some sense: for a vector to be useful in restoring a corrupted sentence, it must capture something of the sentence's underlying meaning.<br />
<br />
The present paper uses the principal of de-noising auto-encoders to compute one of the terms in its loss function. In each iteration, a sentence is sampled from the source or target language, and a corruption process <math display="inline"> C</math> is applied to it. <math display="inline"> C</math> works by deleting each word in the sentence with probability <math display="inline">p_C</math> and applying to the sentence a permutation randomly selected from those that do not move words more than <math display="inline">k_C</math> spots from their original positions. The authors select <math display="inline">p_C=0.1</math> and <math display="inline">k_C=3</math>. The corrupted sentence is then mapped to the latent space using <math display="inline">E\circ A</math>. The loss is then the negative log probability of the original un-corrupted sentence according to the decoder <math display="inline">D</math> applied to the latent-space sequence.<br />
<br />
The explanation of Vincent et al. (2008) can help us understand this loss-function term: the de-noising auto-encoder loss forces the translation system to learn the shapes of the manifolds of the source and target languages.<br />
<br />
==Translation Loss==<br />
To compute the translation loss, we sample a sentence from one of the languages, translate it with the encoder and decoder of the previous epoch, and then corrupt its output with <math display="inline">C</math>. We then use the current encoder <math display="inline">E</math> to map the corrupted translation to a sequence <math display="inline">h \in H'</math> and the decoder <math display="inline">D</math> to map <math display="inline">h</math> to a probability distribution over sentences. The translation loss is the negative log probability the decoder assigns to the original uncorrupted sentence. <br />
<br />
It is interesting and useful to consider why this translation loss, which depends on the the translation model of the previous iteration, should promote an improved translation model in the current iteration. One loose way to understand this is to think of the translator as a de-noising translator. We are given a sentence perturbed from the manifold of possible sentences from a given language both by the corruption process and by the poor quality of the translation. The model must learn to both project and translate. The technique employed here resembles that used by Sennrich et al. (2014), who trained a neural machine translation system using both parallel and monolingual data. To make use of the monolingual target-language data, they used an auxiliary model to translate it to the source language, then trained their model to reconstruct the original target-language data from the source-language translation. Sennrich et al. argued that training the model to reconstruct true data from synthetic data was more robust than the opposite approach. The authors of the present paper use similar reasoning.<br />
<br />
==Adversarial Loss ==<br />
The intuition underlying the latent space is that it should encode the meaning of a sentence in a language-independent way. Accordingly, the authors introduce an adversarial loss, to encourage latent-space vectors mapped from the source and target languages to be indistinguishable. Central to this adversarial loss is the discriminator <math display="inline">R:H' \to [0,1]</math>, which makes use of <math display="inline">r: H\to [0,1]</math> a three-layer fully-connected neural network with 1024 hidden units per layer. Given a sequence of latent-space vectors <math display="inline">h=(h_1,\ldots,h_m)\in H'</math> the discriminator assigns probability <math display="inline">R(h)=\prod_{i=1}^m r(h_i)</math> that they originated in the target space. Each iteration, the discriminator is trained to maximize the objective function<br />
<br />
\begin{align}<br />
I_T(q) \log (R(E(q))) +(1-I_T(q) )\log(1-R(E(q)))<br />
\end{align}<br />
<br />
where <math display="inline">q</math> is a randomly selected sentence, and <math display="inline">I_T(q)</math> is 1 when <math display="inline">q\in I_T</math> is from the source language and 0 if <math display="inline">q\in I_S</math><br />
<br />
The same term is added to the primary objective function, which the encoder and decoder are trained to minimize. The result is that the encoder and decoder learn to fool the discriminator by mapping sentences from the source and target language to similar sequences of latent-space vectors.<br />
<br />
<br />
The authors note that they make use of label smoothing, a technique recommended by Goodfellow (2016) for regularizing GANs, in which the objective described above is replaced by <br />
<br />
\begin{align}<br />
I_T(q)( (1-\alpha)\log (R(E(q))) +\alpha\log(1-R(E(q))) )+(1-I_T(q) ) ( (1-\beta) \log(1-R(E(q))) +\beta\log (R(E(q)) ))<br />
\end{align}<br />
for some small nonnegative values of <math display="inline">\alpha, \beta</math>, the idea being to prevent the discriminator from making extreme predictions. While one-sided label smoothing (<math display="inline">\beta = 0</math>) is generally recommended, the present model differs from a standard GAN in that it is symmetric, and hence two-sided label smoothing would appear more reasonable.<br />
<br />
==Objective Function==<br />
<br />
Combining the above-described terms, we can write the overall objective function. Let <math display="inline">Q_S</math> denote the monolingual dataset for the source language, and let <math display="inline">Q_T</math> denote the monolingual dataset for the target language. Let <math display="inline">D_S:= D(\cdot, S)</math> and<math display="inline">D_T= D(\cdot, T)</math> (i.e. <math display="inline">D_S, D_T</math>) be the decoder restricted to the source or target language, respectively. Let <math display="inline"> M_S </math> and <math display="inline"> M_T </math> denote the target-to-source and source-to-target translation models of the previous epoch. Then our objective function is<br />
<br />
\begin{align}<br />
\mathcal{L}(D,E,R)=\text{T Translation Loss}+\text{T De-noising Loss} +\text{T Adversarial Loss} +\text{S Translation Loss} +\text{S De-noising Loss} +\text{S Adversarial Loss}\\<br />
\end{align}<br />
\begin{align}<br />
=\sum_{q\in Q_T}\left( -\log D_T \circ E \circ C \circ M _S(q) (q) -\log D_T \circ E \circ C (q) (q)+(1-\alpha)\log (R\circ E(q)) +\alpha\log(1-R\circ E(q)) \right)+\sum_{q\in Q_S}\left( -\log D_S \circ E \circ C \circ M_T (q) (q) -\log D_S \circ E \circ C (q) (q)+(1-\beta) \log(1-R \circ E(q)) +\beta\log (R\circ E(q) \right).<br />
\end{align}<br />
<br />
They alternate between iterations minimizing <math display="inline">\mathcal{L} </math> with respect to <math display="inline">E, D</math> and iterations maximizing with respect to <math display="inline">R</math>. ADAM is used for minimization, while RMSprop is used for maximization. After each epoch, M is updated so that <math display="inline">M_S=D_S \circ E</math> and <math display="inline">M_T=D_T \circ E</math>, after which <math display="inline"> M </math> is frozen until the next epoch.<br />
<br />
==Validation==<br />
The authors' aim is for their method to be completely unsupervised, and so do not use parallel corpora even for the selection of hyper-parameters. Instead, they validate by translating sentences to the other language and back, and comparing the resulting sentence with the original according to BLEU, a similarity metric frequently used in translation (Papineni et al. 2002).<br />
<br />
==Experimental Procedure and Results==<br />
<br />
The authors test their method on four data sets. The first is from the English-French translation task of the Workshop on Machine Translation 2014 (WMT14). This data set consists of parallel data. The authors generate a monolingual English corpus by randomly sampling 15 million sentence pairs, and choosing only the English sentences. They then generate a French corpus by selecting the French sentences from those pairs that were not previous chosen. Importantly, this means that the monolingual data sets have no parallel sentences. The second data set is generated from the English-German translation task from WMT14 using he same procedure.<br />
<br />
The third and fourth data sets are generated from Multi30k data set, which consists of multilingual captions of various images. The images are discarded and the English, French, and German captions are used to generate monolingual data sets in the manner described above. These monolingual corpora are much smaller, consisting of 14500 sentence each.<br />
<br />
The unsupervised translation scheme performs well, though not as well as a supervised translation scheme. It converges after a small number of epochs. Besides supervised translation, the authors compare their method with three other baselines: "Word-by-Word" uses only the previously-discussed word-alignment scheme; "Word-Reordering" uses a simple LSTM based language model and a greedy algorithm to select a reordering of the words produced by "Word-by-Word". "Oracle Word Reordering" means the optimal reordering of the words produced by "Word-by-Word".<br />
<br />
==Result Figures==<br />
[[File:MC_Translation Results.png]]<br />
[[File:MC_Translation_Convergence.png]]<br />
<br />
==Commentary==<br />
This paper's results are impressive: that it is even possible to translate between languages without parallel data suggests that languages are more similar than we might initially suspect, and that the method the authors present has, at least in part, discovered some common deep structure. As the authors point out, using no parallel data at all, their method is able to produce results comparable to those produced by neural machine translation methods trained on hundreds of thousands of a parallel sentences on the WMT dataset. On the other hand, the results they offer come with a few significant caveats.<br />
<br />
The first caveat is that the workhorse of the method is the unsupervised word-vector alignment scheme presented in Conneau et al. (2017) (that paper shares 3 authors with this one). As the ablation study reveals, without word-vector alignment, this method preforms extremely poorly. Moreover, word-by-word translation using word-vector alignment alone performs well, albeit not as well as this method. This suggests that the method of this paper mainly learns to perform (sometimes significant) corrections to word-by-word translations by reordering and occasional word substitution. Presumably, it does this by learning something of the natural structure of sentences in each of the two languages, so that it can correct the errors made by word-by-word translation.<br />
<br />
The second caveat is that the best results are attained translating between English and French, two very closely related languages, and the quality of translation between English and German, a slightly-less related pair, is significantly worse ( according to the ''Shorter Oxford English Dictionary'', 28.3 percent of the English vocabulary is French-derived, 28.2 percent is Latin-derived, and 25 percent is derived from Germanic languages. This probably understates the degree of correspondence between the French and English vocabularies, since French likely derives from Latin many of the same words English does. ). The authors do not report results with more distantly-related pairs, but it is reasonable to expect that performance would degrade significantly, especially since Conneau et al. (2017) shows that the word-alignment scheme performs much worse on more distant language pairs. This may be because there are more one-to-one correspondences between the words of closely related languages than there are between more distant languages. The authors suggest that their scheme could be a useful tool for translating between language pairs for which their are few parallel corpora. However, language pairs lacking parallel corpora are often (though not always) distantly related, and it is for such pairs that the performance of the present method suffers.<br />
<br />
The proposed method always beats Oracle Word Reordering on the Multi30k data set, but sometimes does not on the WMT data set. This may be because the WMT sentences are much more syntactically complex than the simple image captions of the Multi30k data set.<br />
<br />
The ablation study also reveals the importance of the corruption process <math display="inline">C</math>: the absence of <math display="inline">C</math> significantly degrades translation quality, though not as much as the absence of word-vector alignment. We can understand this in two related ways. First of all, if we view the model as learning to correct structural errors in word-by-word translations, then the corruption process introduces more errors of this kind, and so provides additional data upon which the model can train. Second, as Vincent et al. (2008) point out, de-noising auto-encoder training encourages a model to learn the structure of the manifold from which the data is drawn. By learning the structure of the source and target languages, the model can better correct the errors of word-by-word translation.<br />
<br />
[[File:MC_Alignment_Results.png|frame|none|alt=Alt text|From Conneau et al. (2017). The final row shows the performance of alignment method used in the present paper. Note the degradation in performance for more distant languages.]]<br />
<br />
[[File:MC_Translation_Ablation.png|frame|none|alt=Alt text|From the present paper. Results of an ablation study. Of note are the first, third, and forth rows, which demonstrate that while the translation component of the loss is relatively unimportant, the word vector alignment scheme and de-noising auto-encoder matter a great deal.]]<br />
<br />
==Future Work==<br />
The principal of performing unsupervised translation by starting with a rough but reasonable guess, and then improving it using knowledge of the structure of target language seem promising. Word by word translation using word-vector alignment works well for closely related languages like English and French, but is unlikely to work as well for more distant languages. For those languages, a better method for getting an initial guess is required.<br />
<br />
==References==<br />
#Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
#Conneau, Alexis, Guillaume Lample, Marc’Aurelio Ranzato, Ludovic Denoyer, Hervé Jégou. "Word Translation without Parallel Data". arXiv:1710.04087, (2017)<br />
# Dictionary, Shorter Oxford English. "Shorter Oxford english dictionary." (2007).<br />
#Goodfellow, Ian. "NIPS 2016 tutorial: Generative adversarial networks." arXiv preprint arXiv:1701.00160 (2016).<br />
# Hill, Felix, Kyunghyun Cho, and Anna Korhonen. "Learning distributed representations of sentences from unlabelled data." arXiv preprint arXiv:1602.03483 (2016).<br />
# Lample, Guillaume, Ludovic Denoyer, and Marc'Aurelio Ranzato. "Unsupervised Machine Translation Using Monolingual Corpora Only." arXiv preprint arXiv:1711.00043 (2017).<br />
#Papineni, Kishore, et al. "BLEU: a method for automatic evaluation of machine translation." Proceedings of the 40th annual meeting on association for computational linguistics. Association for Computational Linguistics, 2002.<br />
# Mikolov, Tomas, Quoc V Le, and Ilya Sutskever. "Exploiting similarities among languages for machine translation." arXiv preprint arXiv:1309.4168. (2013).<br />
#Sennrich, Rico, Barry Haddow, and Alexandra Birch. "Improving neural machine translation models with monolingual data." arXiv preprint arXiv:1511.06709 (2015).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.<br />
# Vincent, Pascal, et al. "Extracting and composing robust features with denoising autoencoders." Proceedings of the 25th international conference on Machine learning. ACM, 2008.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only&diff=32128stat946w18/Unsupervised Machine Translation Using Monolingual Corpora Only2018-02-23T16:10:32Z<p>Pa2forsy: /* Overview of unsupervised translation system */</p>
<hr />
<div><br />
[[File:MC_Translation_Example.png]]<br />
== Introduction ==<br />
Neural machine translation systems must be trained on large corpora consisting of pairs of pre-translated sentences. This paper (Lample et al. 2017) proposes an unsupervised neural machine translation system, which can be trained without such parallel data.<br />
<br />
== Overview of unsupervised translation system ==<br />
The unsupervised translation scheme has the following outline:<br />
* The word-vector embeddings of the source and target languages are aligned in an unsupervised manner.<br />
* Sentences from the source and target language are mapped to a common latent vector space by an encoder, and then mapped to probability distributions over sentences in the target or source language by a decoder.<br />
* A de-noising auto-encoder loss encourages the latent-space representations to be insensitive to noise.<br />
* An adversarial loss encourages the latent-space representations of source and target sentences to be indistinguishable from each other. It is intended that the latent-space representation of a sentence should reflect its meaning, and not the particular language in which it is expressed.<br />
* A reconstruction loss encourages the model to improve on the translation model of the previous epoch.<br />
<br />
==Notation==<br />
Let <math display="inline">S</math> denote the set of words in the source language, and let <math display="inline">T</math> denote the set of words in the target language. Let <math display="inline">H \subset \mathbb{R}^{n_H}</math> denote the latent vector space. Moreover, let <math display="inline">S'</math> and <math display="inline">T'</math> denote the sets of finite sequences of words in the source and target language, and let <math display="inline">H'</math> denote the set of finite sequences of vectors in the latent space. For any set X, elide measure-theoretic details and let <math display="inline">\mathcal{P}(X)</math> denote the set of probability distributions over X.<br />
<br />
==Word vector alignment ==<br />
<br />
Conneau et al. (2017) describe an unsupervised method for aligning word vectors across languages. By "alignment", I mean that their method maps words with related meanings to nearby vectors, regardless of the language of the words. Moreover, if two words are one another's literal translations, their word vectors tend to be mutual nearest neighbors. <br />
<br />
The underlying idea of the alignment scheme can be summarized as follows: methods like word2vec or GLoVe generate vectors for which there is a correspondence between semantics and geometry. If <math display="inline">f</math> maps English words to their corresponding vectors, we have the approximate equation<br />
\begin{align}<br />
f(\text{king}) -f(\text{man}) +f(\text{woman})\approx f(\text{queen}).<br />
\end{align}<br />
Furthermore, if <math display="inline">g</math> maps French words to their corresponding vectors, then <br />
\begin{align}<br />
g(\text{roi}) -f(\text{homme}) +f(\text{femme})\approx f(\text{reine}).<br />
\end{align}<br />
<br />
Thus if <math display="inline">W</math> maps the word vectors of English words to the word vectors of their French translations, we should expect <math display="inline">W</math> to be linear. As was observed by Mikolov et al. (2013), the problem of word-vector alignment then becomes a problem of learning the linear transformation that best aligns two point clouds, one from the source language and one from the target language. For more on the history of the word-vector alignment problem, see my CS698 project ([https://uwaterloo.ca/scholar/sites/ca.scholar/files/pa2forsy/files/project_dec_3_0.pdf link]).<br />
<br />
Conneau et al. (2017)'s word vector alignment scheme is unique in that it requires no parallel data, and uses only the shapes of the two word-vector point clouds to be aligned. I will not go into detail, but the heart of the method is a special GAN, in which only the discriminator is a neural network, and the generator is the map corresponding to an orthogonal matrix.<br />
<br />
This unsupervised alignment method is crucial to the translation scheme of the current paper. From now on we denote by <br />
<math display="inline">A: S' \cup T' \to \mathcal{Z}'</math> the function that maps a source- or target- language word sequence to the corresponding aligned word vector sequence.<br />
<br />
==Encoder ==<br />
The encoder <math display="inline">E </math> reads a sequence of word vectors <math display="inline">(z_1,\ldots, z_m) \in \mathcal{Z}'</math> and outputs a sequence of hidden states <math display="inline">(h_1,\ldots, h_m) \in H'</math> in the latent space. Crucially, because the word vectors of the two languages have been aligned, the same encoder can be applied to both. That is, to map a source sentence <math display="inline">x=(x_1,\ldots, x_M)\in S'</math> to the latent space, we compute <math display="inline">E(A(x))</math>, and to map a target sentence <math display="inline">y=(y_1,\ldots, y_K)\in T'</math> to the latent space, we compute <math display="inline">E(A(y))</math>.<br />
<br />
The encoder consists of two LSTMs, one of which reads the word-vector sequence in the forward direction, and one of which reads it in the backward direction. The hidden state sequence is generated by concatenating the hidden states produced by the forward and backward LSTMs at each word vector.<br />
<br />
==Decoder==<br />
<br />
The decoder is a mono-directional LSTM that accepts a sequence of hidden states <math display="inline">h=(h_1,\ldots, h_m) \in H'</math> from the latent space and a language <math display="inline">L \in \{S,T \}</math> and outputs a probability distribution over sentences in that language. We have<br />
<br />
\begin{align}<br />
D: H' \times \{S,T \} \to \mathcal{P}(S') \cup \mathcal{P}(T').<br />
\end{align}<br />
<br />
The decoder makes use of the attention mechanism of Bahdanau et al. (2014). To compute the probability of a given sentence <math display="inline">y=(y_1,\ldots,y_K)</math> , the LSTM processes the sentence one word at a time, accepting at step <math display="inline">k</math> the aligned word vector of the previous word in the sentence <math display="inline">A(y_{k-1})</math> and a context vector <math display="inline">c_k\in H</math> computed from the hidden sequence <math display="inline">h\in H'</math>, and outputting a probability distribution over possible next words. The LSTM is initiated with a special, language-specific start-of-sequence token. Otherwise, the decoder is does not depend on the language of the sentence it is producing. The context vector is computed as described by Bahdanau et al. (2014), where we let <math display="inline">l_{k}</math> denote the hidden state of the LSTM at step <math display="inline">k</math>, and where <math display="inline">U,W</math> are learnable weight matrices, and <math display="inline">v</math> is a learnable weight vector:<br />
\begin{align}<br />
c_k&= \sum_{m=1}^M \alpha_{k,m} h_m\\<br />
\alpha_{k,m}&= \frac{\exp(e_{k,m})}{\sum_{m'=1}^M\exp(e_{k,m'}) },\\<br />
e_{k,m} &= v^T \tanh (Wl_{k-1} + U h_m ).<br />
\end{align}<br />
<br />
<br />
By learning <math display="inline">U,W</math> and <math display="inline">v</math>, the decoder can learn to decide which vectors in the sequence <math display="inline">h</math> are relevant to computing which words in the output sentence.<br />
<br />
At step <math display="inline">k</math>, after receiving the context vector <math display="inline">c_k\in H</math> and the aligned word vector of the previous word in the sequence,<math display="inline">A(y_{k-1})</math>, the LSTM outputs a probability distribution over words, which should be interpreted as the distribution of the next word according to the decoder. The probability the decoder assigns to a sentence is then the product of the probabilities computed for each word in this manner.<br />
<br />
==Overview of objective ==<br />
The objective function is the sum of:<br />
# The de-noising auto-encoder loss,<br />
# The translation loss,<br />
# The adversarial loss.<br />
I shall describe these in the following sections.<br />
<br />
==De-noising Auto-encoder Loss == <br />
A de-noising auto-encoder is a function optimized to map a corrupted sample from some dataset to the original un-corrupted sample. De-noising auto-encoders were introduced by Vincent et al. (2008), who provided numerous justifications, one of which is particularly illuminating. If we think of the dataset of interest as a thin manifold in a high-dimensional space, the corruption process is likely perturb a datapoint off the manifold. To learn to restore the corrupted datapoint, the de-noising auto-encoder must learn the shape of the manifold.<br />
<br />
Hill et al. (2016), used a de-noising auto-encoder to learn vectors representing sentences. They corrupted input sentences by randomly dropping and swapping words, and then trained a neural network to map the corrupted sentence to a vector, and then map the vector to the un-corrupted sentence. Interestingly, they found that sentence vectors learned this way were particularly effective when applied to tasks that involved generating paraphrases. This makes some sense: for a vector to be useful in restoring a corrupted sentence, it must capture something of the sentence's underlying meaning.<br />
<br />
The present paper uses the principal of de-noising auto-encoders to compute one of the terms in its loss function. In each iteration, a sentence is sampled from the source or target language, and a corruption process <math display="inline"> C</math> is applied to it. <math display="inline"> C</math> works by deleting each word in the sentence with probability <math display="inline">p_C</math> and applying to the sentence a permutation randomly selected from those that do not move words more than <math display="inline">k_C</math> spots from their original positions. The authors select <math display="inline">p_C=0.1</math> and <math display="inline">k_C=3</math>. The corrupted sentence is then mapped to the latent space using <math display="inline">E\circ A</math>. The loss is then the negative log probability of the original un-corrupted sentence according to the decoder <math display="inline">D</math> applied to the latent-space sequence.<br />
<br />
The explanation of Vincent et al. (2008) can help us understand this loss-function term: the de-noising auto-encoder loss forces the translation system to learn the shapes of the manifolds of the source and target languages.<br />
<br />
==Translation Loss==<br />
To compute the translation loss, we sample a sentence from one of the languages, translate it with the encoder and decoder of the previous epoch, and then corrupt its output with <math display="inline">C</math>. We then use the current encoder <math display="inline">E</math> to map the corrupted translation to a sequence <math display="inline">h \in H'</math> and the decoder <math display="inline">D</math> to map <math display="inline">h</math> to a probability distribution over sentences. The translation loss is the negative log probability the decoder assigns to the original uncorrupted sentence. <br />
<br />
It is interesting and useful to consider why this translation loss, which depends on the the translation model of the previous iteration, should promote an improved translation model in the current iteration. One loose way to understand this is to think of the translator as a de-noising translator. We are given a sentence perturbed from the manifold of possible sentences from a given language both by the corruption process and by the poor quality of the translation. The model must learn to both project and translate. The technique employed here resembles that used by Sennrich et al. (2014), who trained a neural machine translation system using both parallel and monolingual data. To make use of the monolingual target-language data, they used an auxiliary model to translate it to the source language, then trained their model to reconstruct the original target-language data from the source-language translation. Sennrich et al. argued that training the model to reconstruct true data from synthetic data was more robust than the opposite approach. The authors of the present paper use similar reasoning.<br />
<br />
==Adversarial Loss ==<br />
The intuition underlying the latent space is that it should encode the meaning of a sentence in a language-independent way. Accordingly, the authors introduce an adversarial loss, to encourage latent-space vectors mapped from the source and target languages to be indistinguishable. Central to this adversarial loss is the discriminator <math display="inline">R:H' \to [0,1]</math>, which makes use of <math display="inline">r: H\to [0,1]</math> a three-layer fully-connected neural network with 1024 hidden units per layer. Given a sequence of latent-space vectors <math display="inline">h=(h_1,\ldots,h_m)\in H'</math> the discriminator assigns probability <math display="inline">R(h)=\prod_{i=1}^m r(h_i)</math> that they originated in the target space. Each iteration, the discriminator is trained to maximize the objective function<br />
<br />
\begin{align}<br />
I_T(q) \log (R(E(q))) +(1-I_T(q) )\log(1-R(E(q)))<br />
\end{align}<br />
<br />
where <math display="inline">q</math> is a randomly selected sentence, and <math display="inline">I_T(q)</math> is 1 when <math display="inline">q\in I_T</math> is from the source language and 0 if <math display="inline">q\in I_S</math><br />
<br />
The same term is added to the primary objective function, which the encoder and decoder are trained to minimize. The result is that the encoder and decoder learn to fool the discriminator by mapping sentences from the source and target language to similar sequences of latent-space vectors.<br />
<br />
<br />
The authors note that they make use of label smoothing, a technique recommended by Goodfellow (2016) for regularizing GANs, in which the objective described above is replaced by <br />
<br />
\begin{align}<br />
I_T(q)( (1-\alpha)\log (R(E(q))) +\alpha\log(1-R(E(q))) )+(1-I_T(q) ) ( (1-\beta) \log(1-R(E(q))) +\beta\log (R(E(q)) ))<br />
\end{align}<br />
for some small nonnegative values of <math display="inline">\alpha, \beta</math>, the idea being to prevent the discriminator from making extreme predictions. While one-sided label smoothing (<math display="inline">\beta = 0</math>) is generally recommended, the present model differs from a standard GAN in that it is symmetric, and hence two-sided label smoothing would appear more reasonable.<br />
<br />
==Objective Function==<br />
<br />
Combining the above-described terms, we can write the overall objective function. Let <math display="inline">Q_S</math> denote the monolingual dataset for the source language, and let <math display="inline">Q_T</math> denote the monolingual dataset for the target language. Let <math display="inline">D_S:= D(\cdot, S)</math> and<math display="inline">D_T= D(\cdot, T)</math> (i.e. <math display="inline">D_S, D_T</math>) be the decoder restricted to the source or target language, respectively. Let <math display="inline"> M_S </math> and <math display="inline"> M_T </math> denote the target-to-source and source-to-target translation models of the previous epoch. Then our objective function is<br />
<br />
\begin{align}<br />
\mathcal{L}(D,E,R)=\text{T Translation Loss}+\text{T De-noising Loss} +\text{T Adversarial Loss} +\text{S Translation Loss} +\text{S De-noising Loss} +\text{S Adversarial Loss}\\<br />
\end{align}<br />
\begin{align}<br />
=\sum_{q\in Q_T}\left( -\log D_T \circ E \circ C \circ M _S(q) (q) -\log D_T \circ E \circ C (q) (q)+(1-\alpha)\log (R\circ E(q)) +\alpha\log(1-R\circ E(q)) \right)+\sum_{q\in Q_S}\left( -\log D_S \circ E \circ C \circ M_T (q) (q) -\log D_S \circ E \circ C (q) (q)+(1-\beta) \log(1-R \circ E(q)) +\beta\log (R\circ E(q) \right).<br />
\end{align}<br />
<br />
They alternate between iterations minimizing <math display="inline">\mathcal{L} </math> with respect to <math display="inline">E, D</math> and iterations maximizing with respect to <math display="inline">R</math>. ADAM is used for minimization, while RMSprop is used for maximization. After each epoch, M is updated so that <math display="inline">M_S=D_S \circ E</math> and <math display="inline">M_T=D_T \circ E</math>, after which <math display="inline"> M </math> is frozen until the next epoch.<br />
<br />
==Validation==<br />
The authors' aim is for their method to be completely unsupervised, and so do not use parallel corpora even for the selection of hyper-parameters. Instead, they validate by translating sentences to the other language and back, and comparing the resulting sentence with the original according to BLEU, a similarity metric frequently used in translation (Papineni et al. 2002).<br />
<br />
==Experimental Procedure and Results==<br />
<br />
The authors test their method on four data sets. The first is from the English-French translation task of the Workshop on Machine Translation 2014 (WMT14). This data set consists of parallel data. The authors generate a monolingual English corpus by randomly sampling 15 million sentence pairs, and choosing only the English sentences. They then generate a French corpus by selecting the French sentences from those pairs that were not previous chosen. Importantly, this means that the monolingual data sets have no parallel sentences. The second data set is generated from the English-German translation task from WMT14 using he same procedure.<br />
<br />
The third and fourth data sets are generated from Multi30k data set, which consists of multilingual captions of various images. The images are discarded and the English, French, and German captions are used to generate monolingual data sets in the manner described above. These monolingual corpora are much smaller, consisting of 14500 sentence each.<br />
<br />
The unsupervised translation scheme performs well, though not as well as a supervised translation scheme. It converges after a small number of epochs. Besides supervised translation, the authors compare their method with three other baselines: "Word-by-Word" uses only the previously-discussed word-alignment scheme; "Word-Reordering" uses a simple LSTM based language model and a greedy algorithm to select a reordering of the words produced by "Word-by-Word". "Oracle Word Reordering" means the optimal reordering of the words produced by "Word-by-Word".<br />
<br />
==Result Figures==<br />
[[File:MC_Translation Results.png]]<br />
[[File:MC_Translation_Convergence.png]]<br />
<br />
==Commentary==<br />
This paper's results are impressive: that it is even possible to translate between languages without parallel data suggests that languages are more similar than we might initially suspect, and that the method the authors present has, at least in part, discovered some common deep structure. As the authors point out, using no parallel data at all, their method is able to produce results comparable to those produced by neural machine translation methods trained on hundreds of thousands of a parallel sentences on the WMT dataset. On the other hand, the results they offer come with a few significant caveats.<br />
<br />
The first caveat is that the workhorse of the method is the unsupervised word-vector alignment scheme presented in Conneau et al. (2017) (that paper shares 3 authors with this one). As the ablation study reveals, without word-vector alignment, this method preforms extremely poorly. Moreover, word-by-word translation using word-vector alignment alone performs well, albeit not as well as this method. This suggests that the method of this paper mainly learns to perform (sometimes significant) corrections to word-by-word translations by reordering and occasional word substitution. Presumably, it does this by learning something of the natural structure of sentences in each of the two languages, so that it can correct the errors made by word-by-word translation.<br />
<br />
The second caveat is that the best results are attained translating between English and French, two very closely related languages, and the quality of translation between English and German, a slightly-less related pair, is significantly worse ( according to the ''Shorter Oxford English Dictionary'', 28.3 percent of the English vocabulary is French-derived, 28.2 percent is Latin-derived, and 25 percent is derived from Germanic languages. This probably understates the degree of correspondence between the French and English vocabularies, since French likely derives from Latin many of the same words English does. ). The authors do not report results with more distantly-related pairs, but it is reasonable to expect that performance would degrade significantly, especially since Conneau et al. (2017) shows that the word-alignment scheme performs much worse on more distant language pairs. This may be because there are more one-to-one correspondences between the words of closely related languages than there are between more distant languages. The authors suggest that their scheme could be a useful tool for translating between language pairs for which their are few parallel corpora. However, language pairs lacking parallel corpora are often (though not always) distantly related, and it is for such pairs that the performance of the present method suffers.<br />
<br />
The proposed method always beats Oracle Word Reordering on the Multi30k data set, but sometimes does not on the WMT data set. This may be because the WMT sentences are much more syntactically complex than the simple image captions of the Multi30k data set.<br />
<br />
The ablation study also reveals the importance of the corruption process <math display="inline">C</math>: the absence of <math display="inline">C</math> significantly degrades translation quality, though not as much as the absence of word-vector alignment. We can understand this in two related ways. First of all, if we view the model as learning to correct structural errors in word-by-word translations, then the corruption process introduces more errors of this kind, and so provides additional data upon which the model can train. Second, as Vincent et al. (2008) point out, de-noising auto-encoder training encourages a model to learn the structure of the manifold from which the data is drawn. By learning the structure of the source and target languages, the model can better correct the errors of word-by-word translation.<br />
<br />
[[File:MC_Alignment_Results.png|frame|none|alt=Alt text|From Conneau et al. (2017). The final row shows the performance of alignment method used in the present paper. Note the degradation in performance for more distant languages.]]<br />
<br />
[[File:MC_Translation_Ablation.png|frame|none|alt=Alt text|From the present paper. Results of an ablation study. Of note are the first, third, and forth rows, which demonstrate that while the translation component of the loss is relatively unimportant, the word vector alignment scheme and de-noising auto-encoder matter a great deal.]]<br />
<br />
==Future Work==<br />
The principal of performing unsupervised translation by starting with a rough but reasonable guess, and then improving it using knowledge of the structure of target language seem promising. Word by word translation using word-vector alignment works well for closely related languages like English and French, but is unlikely to work as well for more distant languages. For those languages, a better method for getting an initial guess is required.<br />
<br />
==References==<br />
#Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
#Conneau, Alexis, Guillaume Lample, Marc’Aurelio Ranzato, Ludovic Denoyer, Hervé Jégou. "Word Translation without Parallel Data". arXiv:1710.04087, (2017)<br />
# Dictionary, Shorter Oxford English. "Shorter Oxford english dictionary." (2007).<br />
#Goodfellow, Ian. "NIPS 2016 tutorial: Generative adversarial networks." arXiv preprint arXiv:1701.00160 (2016).<br />
# Hill, Felix, Kyunghyun Cho, and Anna Korhonen. "Learning distributed representations of sentences from unlabelled data." arXiv preprint arXiv:1602.03483 (2016).<br />
# Lample, Guillaume, Ludovic Denoyer, and Marc'Aurelio Ranzato. "Unsupervised Machine Translation Using Monolingual Corpora Only." arXiv preprint arXiv:1711.00043 (2017).<br />
#Papineni, Kishore, et al. "BLEU: a method for automatic evaluation of machine translation." Proceedings of the 40th annual meeting on association for computational linguistics. Association for Computational Linguistics, 2002.<br />
# Mikolov, Tomas, Quoc V Le, and Ilya Sutskever. "Exploiting similarities among languages for machine translation." arXiv preprint arXiv:1309.4168. (2013).<br />
#Sennrich, Rico, Barry Haddow, and Alexandra Birch. "Improving neural machine translation models with monolingual data." arXiv preprint arXiv:1511.06709 (2015).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.<br />
# Vincent, Pascal, et al. "Extracting and composing robust features with denoising autoencoders." Proceedings of the 25th international conference on Machine learning. ACM, 2008.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only&diff=32065stat946w18/Unsupervised Machine Translation Using Monolingual Corpora Only2018-02-22T18:54:01Z<p>Pa2forsy: /* Overview of objective */</p>
<hr />
<div><br />
[[File:MC_Translation_Example.png]]<br />
== Introduction ==<br />
Neural machine translation systems must be trained on large corpora consisting of pairs of pre-translated sentences. This paper (Lample et al. 2017) proposes an unsupervised neural machine translation system, which can be trained without such parallel data.<br />
<br />
== Overview of unsupervised translation system ==<br />
The unsupervised translation scheme has the following outline:<br />
* The word-vector embedding of the source and target languages are aligned in an unsupervised manner.<br />
* Sentences from the source and target language are mapped to a common latent vector space by an encoder, and then mapped to probability distributions over sentences in the target or source language by a decoder.<br />
* A de-noising auto-encoder loss encourages the latent-space representations to be insensitive to noise.<br />
* An adversarial loss encourages the latent-space representations of source and target sentences to be indistinguishable from each other. It is intended that the latent-space representation of a sentence should reflect its meaning, and not the particular language in which it is expressed.<br />
* A reconstruction loss encourages the model to improve on the translation model of the previous epoch. <br />
I will next discuss this scheme in more detail.<br />
<br />
==Notation==<br />
Let <math display="inline">S</math> denote the set of words in the source language, and let <math display="inline">T</math> denote the set of words in the target language. Let <math display="inline">H \subset \mathbb{R}^{n_H}</math> denote the latent vector space. Moreover, let <math display="inline">S'</math> and <math display="inline">T'</math> denote the sets of finite sequences of words in the source and target language, and let <math display="inline">H'</math> denote the set of finite sequences of vectors in the latent space. For any set X, elide measure-theoretic details and let <math display="inline">\mathcal{P}(X)</math> denote the set of probability distributions over X.<br />
<br />
==Word vector alignment ==<br />
<br />
Conneau et al. (2017) describe an unsupervised method for aligning word vectors across languages. By "alignment", I mean that their method maps words with related meanings to nearby vectors, regardless of the language of the words. Moreover, if two words are one another's literal translations, their word vectors tend to be mutual nearest neighbors. <br />
<br />
The underlying idea of the alignment scheme can be summarized as follows: methods like word2vec or GLoVe generate vectors for which there is a correspondence between semantics and geometry. If <math display="inline">f</math> maps English words to their corresponding vectors, we have the approximate equation<br />
\begin{align}<br />
f(\text{king}) -f(\text{man}) +f(\text{woman})\approx f(\text{queen}).<br />
\end{align}<br />
Furthermore, if <math display="inline">g</math> maps French words to their corresponding vectors, then <br />
\begin{align}<br />
g(\text{roi}) -f(\text{homme}) +f(\text{femme})\approx f(\text{reine}).<br />
\end{align}<br />
<br />
Thus if <math display="inline">W</math> maps the word vectors of English words to the word vectors of their French translations, we should expect <math display="inline">W</math> to be linear. As was observed by Mikolov et al. (2013), the problem of word-vector alignment then becomes a problem of learning the linear transformation that best aligns two point clouds, one from the source language and one from the target language. For more on the history of the word-vector alignment problem, see my CS698 project ([https://uwaterloo.ca/scholar/sites/ca.scholar/files/pa2forsy/files/project_dec_3_0.pdf link]).<br />
<br />
Conneau et al. (2017)'s word vector alignment scheme is unique in that it requires no parallel data, and uses only the shapes of the two word-vector point clouds to be aligned. I will not go into detail, but the heart of the method is a special GAN, in which only the discriminator is a neural network, and the generator is the map corresponding to an orthogonal matrix.<br />
<br />
This unsupervised alignment method is crucial to the translation scheme of the current paper. From now on we denote by <br />
<math display="inline">A: S' \cup T' \to \mathcal{Z}'</math> the function that maps a source- or target- language word sequence to the corresponding aligned word vector sequence.<br />
<br />
==Encoder ==<br />
The encoder <math display="inline">E </math> reads a sequence of word vectors <math display="inline">(z_1,\ldots, z_m) \in \mathcal{Z}'</math> and outputs a sequence of hidden states <math display="inline">(h_1,\ldots, h_m) \in H'</math> in the latent space. Crucially, because the word vectors of the two languages have been aligned, the same encoder can be applied to both. That is, to map a source sentence <math display="inline">x=(x_1,\ldots, x_M)\in S'</math> to the latent space, we compute <math display="inline">E(A(x))</math>, and to map a target sentence <math display="inline">y=(y_1,\ldots, y_K)\in T'</math> to the latent space, we compute <math display="inline">E(A(y))</math>.<br />
<br />
The encoder consists of two LSTMs, one of which reads the word-vector sequence in the forward direction, and one of which reads it in the backward direction. The hidden state sequence is generated by concatenating the hidden states produced by the forward and backward LSTMs at each word vector.<br />
<br />
==Decoder==<br />
<br />
The decoder is a mono-directional LSTM that accepts a sequence of hidden states <math display="inline">h=(h_1,\ldots, h_m) \in H'</math> from the latent space and a language <math display="inline">L \in \{S,T \}</math> and outputs a probability distribution over sentences in that language. We have<br />
<br />
\begin{align}<br />
D: H' \times \{S,T \} \to \mathcal{P}(S') \cup \mathcal{P}(T').<br />
\end{align}<br />
<br />
The decoder makes use of the attention mechanism of Bahdanau et al. (2014). To compute the probability of a given sentence <math display="inline">y=(y_1,\ldots,y_K)</math> , the LSTM processes the sentence one word at a time, accepting at step <math display="inline">k</math> the aligned word vector of the previous word in the sentence <math display="inline">A(y_{k-1})</math> and a context vector <math display="inline">c_k\in H</math> computed from the hidden sequence <math display="inline">h\in H'</math>, and outputting a probability distribution over possible next words. The LSTM is initiated with a special, language-specific start-of-sequence token. Otherwise, the decoder is does not depend on the language of the sentence it is producing. The context vector is computed as described by Bahdanau et al. (2014), where we let <math display="inline">l_{k}</math> denote the hidden state of the LSTM at step <math display="inline">k</math>, and where <math display="inline">U,W</math> are learnable weight matrices, and <math display="inline">v</math> is a learnable weight vector:<br />
\begin{align}<br />
c_k&= \sum_{m=1}^M \alpha_{k,m} h_m\\<br />
\alpha_{k,m}&= \frac{\exp(e_{k,m})}{\sum_{m'=1}^M\exp(e_{k,m'}) },\\<br />
e_{k,m} &= v^T \tanh (Wl_{k-1} + U h_m ).<br />
\end{align}<br />
<br />
<br />
By learning <math display="inline">U,W</math> and <math display="inline">v</math>, the decoder can learn to decide which vectors in the sequence <math display="inline">h</math> are relevant to computing which words in the output sentence.<br />
<br />
At step <math display="inline">k</math>, after receiving the context vector <math display="inline">c_k\in H</math> and the aligned word vector of the previous word in the sequence,<math display="inline">A(y_{k-1})</math>, the LSTM outputs a probability distribution over words, which should be interpreted as the distribution of the next word according to the decoder. The probability the decoder assigns to a sentence is then the product of the probabilities computed for each word in this manner.<br />
<br />
==Overview of objective ==<br />
The objective function is the sum of:<br />
# The de-noising auto-encoder loss,<br />
# The translation loss,<br />
# The adversarial loss.<br />
I shall describe these in the following sections.<br />
<br />
==De-noising Auto-encoder Loss == <br />
A de-noising auto-encoder is a function optimized to map a corrupted sample from some dataset to the original un-corrupted sample. De-noising auto-encoders were introduced by Vincent et al. (2008), who provided numerous justifications, one of which is particularly illuminating. If we think of the dataset of interest as a thin manifold in a high-dimensional space, the corruption process is likely perturb a datapoint off the manifold. To learn to restore the corrupted datapoint, the de-noising auto-encoder must learn the shape of the manifold.<br />
<br />
Hill et al. (2016), used a de-noising auto-encoder to learn vectors representing sentences. They corrupted input sentences by randomly dropping and swapping words, and then trained a neural network to map the corrupted sentence to a vector, and then map the vector to the un-corrupted sentence. Interestingly, they found that sentence vectors learned this way were particularly effective when applied to tasks that involved generating paraphrases. This makes some sense: for a vector to be useful in restoring a corrupted sentence, it must capture something of the sentence's underlying meaning.<br />
<br />
The present paper uses the principal of de-noising auto-encoders to compute one of the terms in its loss function. In each iteration, a sentence is sampled from the source or target language, and a corruption process <math display="inline"> C</math> is applied to it. <math display="inline"> C</math> works by deleting each word in the sentence with probability <math display="inline">p_C</math> and applying to the sentence a permutation randomly selected from those that do not move words more than <math display="inline">k_C</math> spots from their original positions. The authors select <math display="inline">p_C=0.1</math> and <math display="inline">k_C=3</math>. The corrupted sentence is then mapped to the latent space using <math display="inline">E\circ A</math>. The loss is then the negative log probability of the original un-corrupted sentence according to the decoder <math display="inline">D</math> applied to the latent-space sequence.<br />
<br />
The explanation of Vincent et al. (2008) can help us understand this loss-function term: the de-noising auto-encoder loss forces the translation system to learn the shapes of the manifolds of the source and target languages.<br />
<br />
==Translation Loss==<br />
To compute the translation loss, we sample a sentence from one of the languages, translate it with the encoder and decoder of the previous epoch, and then corrupt its output with <math display="inline">C</math>. We then use the current encoder <math display="inline">E</math> to map the corrupted translation to a sequence <math display="inline">h \in H'</math> and the decoder <math display="inline">D</math> to map <math display="inline">h</math> to a probability distribution over sentences. The translation loss is the negative log probability the decoder assigns to the original uncorrupted sentence. <br />
<br />
It is interesting and useful to consider why this translation loss, which depends on the the translation model of the previous iteration, should promote an improved translation model in the current iteration. One loose way to understand this is to think of the translator as a de-noising translator. We are given a sentence perturbed from the manifold of possible sentences from a given language both by the corruption process and by the poor quality of the translation. The model must learn to both project and translate. The technique employed here resembles that used by Sennrich et al. (2014), who trained a neural machine translation system using both parallel and monolingual data. To make use of the monolingual target-language data, they used an auxiliary model to translate it to the source language, then trained their model to reconstruct the original target-language data from the source-language translation. Sennrich et al. argued that training the model to reconstruct true data from synthetic data was more robust than the opposite approach. The authors of the present paper use similar reasoning.<br />
<br />
==Adversarial Loss ==<br />
The intuition underlying the latent space is that it should encode the meaning of a sentence in a language-independent way. Accordingly, the authors introduce an adversarial loss, to encourage latent-space vectors mapped from the source and target languages to be indistinguishable. Central to this adversarial loss is the discriminator <math display="inline">R:H' \to [0,1]</math>, which makes use of <math display="inline">r: H\to [0,1]</math> a three-layer fully-connected neural network with 1024 hidden units per layer. Given a sequence of latent-space vectors <math display="inline">h=(h_1,\ldots,h_m)\in H'</math> the discriminator assigns probability <math display="inline">R(h)=\prod_{i=1}^m r(h_i)</math> that they originated in the target space. Each iteration, the discriminator is trained to maximize the objective function<br />
<br />
\begin{align}<br />
I_T(q) \log (R(E(q))) +(1-I_T(q) )\log(1-R(E(q)))<br />
\end{align}<br />
<br />
where <math display="inline">q</math> is a randomly selected sentence, and <math display="inline">I_T(q)</math> is 1 when <math display="inline">q\in I_T</math> is from the source language and 0 if <math display="inline">q\in I_S</math><br />
<br />
The same term is added to the primary objective function, which the encoder and decoder are trained to minimize. The result is that the encoder and decoder learn to fool the discriminator by mapping sentences from the source and target language to similar sequences of latent-space vectors.<br />
<br />
<br />
The authors note that they make use of label smoothing, a technique recommended by Goodfellow (2016) for regularizing GANs, in which the objective described above is replaced by <br />
<br />
\begin{align}<br />
I_T(q)( (1-\alpha)\log (R(E(q))) +\alpha\log(1-R(E(q))) )+(1-I_T(q) ) ( (1-\beta) \log(1-R(E(q))) +\beta\log (R(E(q)) ))<br />
\end{align}<br />
for some small nonnegative values of <math display="inline">\alpha, \beta</math>, the idea being to prevent the discriminator from making extreme predictions. While one-sided label smoothing (<math display="inline">\beta = 0</math>) is generally recommended, the present model differs from a standard GAN in that it is symmetric, and hence two-sided label smoothing would appear more reasonable.<br />
<br />
==Objective Function==<br />
<br />
Combining the above-described terms, we can write the overall objective function. Let <math display="inline">Q_S</math> denote the monolingual dataset for the source language, and let <math display="inline">Q_T</math> denote the monolingual dataset for the target language. Let <math display="inline">D_S:= D(\cdot, S)</math> and<math display="inline">D_T= D(\cdot, T)</math> (i.e. <math display="inline">D_S, D_T</math>) be the decoder restricted to the source or target language, respectively. Let <math display="inline"> M_S </math> and <math display="inline"> M_T </math> denote the target-to-source and source-to-target translation models of the previous epoch. Then our objective function is<br />
<br />
\begin{align}<br />
\mathcal{L}(D,E,R)=\text{T Translation Loss}+\text{T De-noising Loss} +\text{T Adversarial Loss} +\text{S Translation Loss} +\text{S De-noising Loss} +\text{S Adversarial Loss}\\<br />
\end{align}<br />
\begin{align}<br />
=\sum_{q\in Q_T}\left( -\log D_T \circ E \circ C \circ M _S(q) (q) -\log D_T \circ E \circ C (q) (q)+(1-\alpha)\log (R\circ E(q)) +\alpha\log(1-R\circ E(q)) \right)+\sum_{q\in Q_S}\left( -\log D_S \circ E \circ C \circ M_T (q) (q) -\log D_S \circ E \circ C (q) (q)+(1-\beta) \log(1-R \circ E(q)) +\beta\log (R\circ E(q) \right).<br />
\end{align}<br />
<br />
They alternate between iterations minimizing <math display="inline">\mathcal{L} </math> with respect to <math display="inline">E, D</math> and iterations maximizing with respect to <math display="inline">R</math>. ADAM is used for minimization, while RMSprop is used for maximization. After each epoch, M is updated so that <math display="inline">M_S=D_S \circ E</math> and <math display="inline">M_T=D_T \circ E</math>, after which <math display="inline"> M </math> is frozen until the next epoch.<br />
<br />
==Validation==<br />
The authors' aim is for their method to be completely unsupervised, and so do not use parallel corpora even for the selection of hyper-parameters. Instead, they validate by translating sentences to the other language and back, and comparing the resulting sentence with the original according to BLEU, a similarity metric frequently used in translation (Papineni et al. 2002).<br />
<br />
==Experimental Procedure and Results==<br />
<br />
The authors test their method on four data sets. The first is from the English-French translation task of the Workshop on Machine Translation 2014 (WMT14). This data set consists of parallel data. The authors generate a monolingual English corpus by randomly sampling 15 million sentence pairs, and choosing only the English sentences. They then generate a French corpus by selecting the French sentences from those pairs that were not previous chosen. Importantly, this means that the monolingual data sets have no parallel sentences. The second data set is generated from the English-German translation task from WMT14 using he same procedure.<br />
<br />
The third and fourth data sets are generated from Multi30k data set, which consists of multilingual captions of various images. The images are discarded and the English, French, and German captions are used to generate monolingual data sets in the manner described above. These monolingual corpora are much smaller, consisting of 14500 sentence each.<br />
<br />
The unsupervised translation scheme performs well, though not as well as a supervised translation scheme. It converges after a small number of epochs. Besides supervised translation, the authors compare their method with three other baselines: "Word-by-Word" uses only the previously-discussed word-alignment scheme; "Word-Reordering" uses a simple LSTM based language model and a greedy algorithm to select a reordering of the words produced by "Word-by-Word". "Oracle Word Reordering" means the optimal reordering of the words produced by "Word-by-Word".<br />
<br />
==Result Figures==<br />
[[File:MC_Translation Results.png]]<br />
[[File:MC_Translation_Convergence.png]]<br />
<br />
==Commentary==<br />
This paper's results are impressive: that it is even possible to translate between languages without parallel data suggests that languages are more similar than we might initially suspect, and that the method the authors present has, at least in part, discovered some common deep structure. As the authors point out, using no parallel data at all, their method is able to produce results comparable to those produced by neural machine translation methods trained on hundreds of thousands of a parallel sentences on the WMT dataset. On the other hand, the results they offer come with a few significant caveats.<br />
<br />
The first caveat is that the workhorse of the method is the unsupervised word-vector alignment scheme presented in Conneau et al. (2017) (that paper shares 3 authors with this one). As the ablation study reveals, without word-vector alignment, this method preforms extremely poorly. Moreover, word-by-word translation using word-vector alignment alone performs well, albeit not as well as this method. This suggests that the method of this paper mainly learns to perform (sometimes significant) corrections to word-by-word translations by reordering and occasional word substitution. Presumably, it does this by learning something of the natural structure of sentences in each of the two languages, so that it can correct the errors made by word-by-word translation.<br />
<br />
The second caveat is that the best results are attained translating between English and French, two very closely related languages, and the quality of translation between English and German, a slightly-less related pair, is significantly worse ( according to the ''Shorter Oxford English Dictionary'', 28.3 percent of the English vocabulary is French-derived, 28.2 percent is Latin-derived, and 25 percent is derived from Germanic languages. This probably understates the degree of correspondence between the French and English vocabularies, since French likely derives from Latin many of the same words English does. ). The authors do not report results with more distantly-related pairs, but it is reasonable to expect that performance would degrade significantly, especially since Conneau et al. (2017) shows that the word-alignment scheme performs much worse on more distant language pairs. This may be because there are more one-to-one correspondences between the words of closely related languages than there are between more distant languages. The authors suggest that their scheme could be a useful tool for translating between language pairs for which their are few parallel corpora. However, language pairs lacking parallel corpora are often (though not always) distantly related, and it is for such pairs that the performance of the present method suffers.<br />
<br />
The proposed method always beats Oracle Word Reordering on the Multi30k data set, but sometimes does not on the WMT data set. This may be because the WMT sentences are much more syntactically complex than the simple image captions of the Multi30k data set.<br />
<br />
The ablation study also reveals the importance of the corruption process <math display="inline">C</math>: the absence of <math display="inline">C</math> significantly degrades translation quality, though not as much as the absence of word-vector alignment. We can understand this in two related ways. First of all, if we view the model as learning to correct structural errors in word-by-word translations, then the corruption process introduces more errors of this kind, and so provides additional data upon which the model can train. Second, as Vincent et al. (2008) point out, de-noising auto-encoder training encourages a model to learn the structure of the manifold from which the data is drawn. By learning the structure of the source and target languages, the model can better correct the errors of word-by-word translation.<br />
<br />
[[File:MC_Alignment_Results.png|frame|none|alt=Alt text|From Conneau et al. (2017). The final row shows the performance of alignment method used in the present paper. Note the degradation in performance for more distant languages.]]<br />
<br />
[[File:MC_Translation_Ablation.png|frame|none|alt=Alt text|From the present paper. Results of an ablation study. Of note are the first, third, and forth rows, which demonstrate that while the translation component of the loss is relatively unimportant, the word vector alignment scheme and de-noising auto-encoder matter a great deal.]]<br />
<br />
==Future Work==<br />
The principal of performing unsupervised translation by starting with a rough but reasonable guess, and then improving it using knowledge of the structure of target language seem promising. Word by word translation using word-vector alignment works well for closely related languages like English and French, but is unlikely to work as well for more distant languages. For those languages, a better method for getting an initial guess is required.<br />
<br />
==References==<br />
#Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
#Conneau, Alexis, Guillaume Lample, Marc’Aurelio Ranzato, Ludovic Denoyer, Hervé Jégou. "Word Translation without Parallel Data". arXiv:1710.04087, (2017)<br />
# Dictionary, Shorter Oxford English. "Shorter Oxford english dictionary." (2007).<br />
#Goodfellow, Ian. "NIPS 2016 tutorial: Generative adversarial networks." arXiv preprint arXiv:1701.00160 (2016).<br />
# Hill, Felix, Kyunghyun Cho, and Anna Korhonen. "Learning distributed representations of sentences from unlabelled data." arXiv preprint arXiv:1602.03483 (2016).<br />
# Lample, Guillaume, Ludovic Denoyer, and Marc'Aurelio Ranzato. "Unsupervised Machine Translation Using Monolingual Corpora Only." arXiv preprint arXiv:1711.00043 (2017).<br />
#Papineni, Kishore, et al. "BLEU: a method for automatic evaluation of machine translation." Proceedings of the 40th annual meeting on association for computational linguistics. Association for Computational Linguistics, 2002.<br />
# Mikolov, Tomas, Quoc V Le, and Ilya Sutskever. "Exploiting similarities among languages for machine translation." arXiv preprint arXiv:1309.4168. (2013).<br />
#Sennrich, Rico, Barry Haddow, and Alexandra Birch. "Improving neural machine translation models with monolingual data." arXiv preprint arXiv:1511.06709 (2015).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.<br />
# Vincent, Pascal, et al. "Extracting and composing robust features with denoising autoencoders." Proceedings of the 25th international conference on Machine learning. ACM, 2008.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only&diff=32064stat946w18/Unsupervised Machine Translation Using Monolingual Corpora Only2018-02-22T18:46:01Z<p>Pa2forsy: /* Experimental Procedure and Results */</p>
<hr />
<div><br />
[[File:MC_Translation_Example.png]]<br />
== Introduction ==<br />
Neural machine translation systems must be trained on large corpora consisting of pairs of pre-translated sentences. This paper (Lample et al. 2017) proposes an unsupervised neural machine translation system, which can be trained without such parallel data.<br />
<br />
== Overview of unsupervised translation system ==<br />
The unsupervised translation scheme has the following outline:<br />
* The word-vector embedding of the source and target languages are aligned in an unsupervised manner.<br />
* Sentences from the source and target language are mapped to a common latent vector space by an encoder, and then mapped to probability distributions over sentences in the target or source language by a decoder.<br />
* A de-noising auto-encoder loss encourages the latent-space representations to be insensitive to noise.<br />
* An adversarial loss encourages the latent-space representations of source and target sentences to be indistinguishable from each other. It is intended that the latent-space representation of a sentence should reflect its meaning, and not the particular language in which it is expressed.<br />
* A reconstruction loss encourages the model to improve on the translation model of the previous epoch. <br />
I will next discuss this scheme in more detail.<br />
<br />
==Notation==<br />
Let <math display="inline">S</math> denote the set of words in the source language, and let <math display="inline">T</math> denote the set of words in the target language. Let <math display="inline">H \subset \mathbb{R}^{n_H}</math> denote the latent vector space. Moreover, let <math display="inline">S'</math> and <math display="inline">T'</math> denote the sets of finite sequences of words in the source and target language, and let <math display="inline">H'</math> denote the set of finite sequences of vectors in the latent space. For any set X, elide measure-theoretic details and let <math display="inline">\mathcal{P}(X)</math> denote the set of probability distributions over X.<br />
<br />
==Word vector alignment ==<br />
<br />
Conneau et al. (2017) describe an unsupervised method for aligning word vectors across languages. By "alignment", I mean that their method maps words with related meanings to nearby vectors, regardless of the language of the words. Moreover, if two words are one another's literal translations, their word vectors tend to be mutual nearest neighbors. <br />
<br />
The underlying idea of the alignment scheme can be summarized as follows: methods like word2vec or GLoVe generate vectors for which there is a correspondence between semantics and geometry. If <math display="inline">f</math> maps English words to their corresponding vectors, we have the approximate equation<br />
\begin{align}<br />
f(\text{king}) -f(\text{man}) +f(\text{woman})\approx f(\text{queen}).<br />
\end{align}<br />
Furthermore, if <math display="inline">g</math> maps French words to their corresponding vectors, then <br />
\begin{align}<br />
g(\text{roi}) -f(\text{homme}) +f(\text{femme})\approx f(\text{reine}).<br />
\end{align}<br />
<br />
Thus if <math display="inline">W</math> maps the word vectors of English words to the word vectors of their French translations, we should expect <math display="inline">W</math> to be linear. As was observed by Mikolov et al. (2013), the problem of word-vector alignment then becomes a problem of learning the linear transformation that best aligns two point clouds, one from the source language and one from the target language. For more on the history of the word-vector alignment problem, see my CS698 project ([https://uwaterloo.ca/scholar/sites/ca.scholar/files/pa2forsy/files/project_dec_3_0.pdf link]).<br />
<br />
Conneau et al. (2017)'s word vector alignment scheme is unique in that it requires no parallel data, and uses only the shapes of the two word-vector point clouds to be aligned. I will not go into detail, but the heart of the method is a special GAN, in which only the discriminator is a neural network, and the generator is the map corresponding to an orthogonal matrix.<br />
<br />
This unsupervised alignment method is crucial to the translation scheme of the current paper. From now on we denote by <br />
<math display="inline">A: S' \cup T' \to \mathcal{Z}'</math> the function that maps a source- or target- language word sequence to the corresponding aligned word vector sequence.<br />
<br />
==Encoder ==<br />
The encoder <math display="inline">E </math> reads a sequence of word vectors <math display="inline">(z_1,\ldots, z_m) \in \mathcal{Z}'</math> and outputs a sequence of hidden states <math display="inline">(h_1,\ldots, h_m) \in H'</math> in the latent space. Crucially, because the word vectors of the two languages have been aligned, the same encoder can be applied to both. That is, to map a source sentence <math display="inline">x=(x_1,\ldots, x_M)\in S'</math> to the latent space, we compute <math display="inline">E(A(x))</math>, and to map a target sentence <math display="inline">y=(y_1,\ldots, y_K)\in T'</math> to the latent space, we compute <math display="inline">E(A(y))</math>.<br />
<br />
The encoder consists of two LSTMs, one of which reads the word-vector sequence in the forward direction, and one of which reads it in the backward direction. The hidden state sequence is generated by concatenating the hidden states produced by the forward and backward LSTMs at each word vector.<br />
<br />
==Decoder==<br />
<br />
The decoder is a mono-directional LSTM that accepts a sequence of hidden states <math display="inline">h=(h_1,\ldots, h_m) \in H'</math> from the latent space and a language <math display="inline">L \in \{S,T \}</math> and outputs a probability distribution over sentences in that language. We have<br />
<br />
\begin{align}<br />
D: H' \times \{S,T \} \to \mathcal{P}(S') \cup \mathcal{P}(T').<br />
\end{align}<br />
<br />
The decoder makes use of the attention mechanism of Bahdanau et al. (2014). To compute the probability of a given sentence <math display="inline">y=(y_1,\ldots,y_K)</math> , the LSTM processes the sentence one word at a time, accepting at step <math display="inline">k</math> the aligned word vector of the previous word in the sentence <math display="inline">A(y_{k-1})</math> and a context vector <math display="inline">c_k\in H</math> computed from the hidden sequence <math display="inline">h\in H'</math>, and outputting a probability distribution over possible next words. The LSTM is initiated with a special, language-specific start-of-sequence token. Otherwise, the decoder is does not depend on the language of the sentence it is producing. The context vector is computed as described by Bahdanau et al. (2014), where we let <math display="inline">l_{k}</math> denote the hidden state of the LSTM at step <math display="inline">k</math>, and where <math display="inline">U,W</math> are learnable weight matrices, and <math display="inline">v</math> is a learnable weight vector:<br />
\begin{align}<br />
c_k&= \sum_{m=1}^M \alpha_{k,m} h_m\\<br />
\alpha_{k,m}&= \frac{\exp(e_{k,m})}{\sum_{m'=1}^M\exp(e_{k,m'}) },\\<br />
e_{k,m} &= v^T \tanh (Wl_{k-1} + U h_m ).<br />
\end{align}<br />
<br />
<br />
By learning <math display="inline">U,W</math> and <math display="inline">v</math>, the decoder can learn to decide which vectors in the sequence <math display="inline">h</math> are relevant to computing which words in the output sentence.<br />
<br />
At step <math display="inline">k</math>, after receiving the context vector <math display="inline">c_k\in H</math> and the aligned word vector of the previous word in the sequence,<math display="inline">A(y_{k-1})</math>, the LSTM outputs a probability distribution over words, which should be interpreted as the distribution of the next word according to the decoder. The probability the decoder assigns to a sentence is then the product of the probabilities computed for each word in this manner.<br />
<br />
==Overview of objective ==<br />
The objective function is the sum of three terms:<br />
# The de-noising auto-encoder loss,<br />
# The translation loss,<br />
# The adversarial loss.<br />
I shall describe these in the following sections.<br />
<br />
==De-noising Auto-encoder Loss == <br />
A de-noising auto-encoder is a function optimized to map a corrupted sample from some dataset to the original un-corrupted sample. De-noising auto-encoders were introduced by Vincent et al. (2008), who provided numerous justifications, one of which is particularly illuminating. If we think of the dataset of interest as a thin manifold in a high-dimensional space, the corruption process is likely perturb a datapoint off the manifold. To learn to restore the corrupted datapoint, the de-noising auto-encoder must learn the shape of the manifold.<br />
<br />
Hill et al. (2016), used a de-noising auto-encoder to learn vectors representing sentences. They corrupted input sentences by randomly dropping and swapping words, and then trained a neural network to map the corrupted sentence to a vector, and then map the vector to the un-corrupted sentence. Interestingly, they found that sentence vectors learned this way were particularly effective when applied to tasks that involved generating paraphrases. This makes some sense: for a vector to be useful in restoring a corrupted sentence, it must capture something of the sentence's underlying meaning.<br />
<br />
The present paper uses the principal of de-noising auto-encoders to compute one of the terms in its loss function. In each iteration, a sentence is sampled from the source or target language, and a corruption process <math display="inline"> C</math> is applied to it. <math display="inline"> C</math> works by deleting each word in the sentence with probability <math display="inline">p_C</math> and applying to the sentence a permutation randomly selected from those that do not move words more than <math display="inline">k_C</math> spots from their original positions. The authors select <math display="inline">p_C=0.1</math> and <math display="inline">k_C=3</math>. The corrupted sentence is then mapped to the latent space using <math display="inline">E\circ A</math>. The loss is then the negative log probability of the original un-corrupted sentence according to the decoder <math display="inline">D</math> applied to the latent-space sequence.<br />
<br />
The explanation of Vincent et al. (2008) can help us understand this loss-function term: the de-noising auto-encoder loss forces the translation system to learn the shapes of the manifolds of the source and target languages.<br />
<br />
==Translation Loss==<br />
To compute the translation loss, we sample a sentence from one of the languages, translate it with the encoder and decoder of the previous epoch, and then corrupt its output with <math display="inline">C</math>. We then use the current encoder <math display="inline">E</math> to map the corrupted translation to a sequence <math display="inline">h \in H'</math> and the decoder <math display="inline">D</math> to map <math display="inline">h</math> to a probability distribution over sentences. The translation loss is the negative log probability the decoder assigns to the original uncorrupted sentence. <br />
<br />
It is interesting and useful to consider why this translation loss, which depends on the the translation model of the previous iteration, should promote an improved translation model in the current iteration. One loose way to understand this is to think of the translator as a de-noising translator. We are given a sentence perturbed from the manifold of possible sentences from a given language both by the corruption process and by the poor quality of the translation. The model must learn to both project and translate. The technique employed here resembles that used by Sennrich et al. (2014), who trained a neural machine translation system using both parallel and monolingual data. To make use of the monolingual target-language data, they used an auxiliary model to translate it to the source language, then trained their model to reconstruct the original target-language data from the source-language translation. Sennrich et al. argued that training the model to reconstruct true data from synthetic data was more robust than the opposite approach. The authors of the present paper use similar reasoning.<br />
<br />
==Adversarial Loss ==<br />
The intuition underlying the latent space is that it should encode the meaning of a sentence in a language-independent way. Accordingly, the authors introduce an adversarial loss, to encourage latent-space vectors mapped from the source and target languages to be indistinguishable. Central to this adversarial loss is the discriminator <math display="inline">R:H' \to [0,1]</math>, which makes use of <math display="inline">r: H\to [0,1]</math> a three-layer fully-connected neural network with 1024 hidden units per layer. Given a sequence of latent-space vectors <math display="inline">h=(h_1,\ldots,h_m)\in H'</math> the discriminator assigns probability <math display="inline">R(h)=\prod_{i=1}^m r(h_i)</math> that they originated in the target space. Each iteration, the discriminator is trained to maximize the objective function<br />
<br />
\begin{align}<br />
I_T(q) \log (R(E(q))) +(1-I_T(q) )\log(1-R(E(q)))<br />
\end{align}<br />
<br />
where <math display="inline">q</math> is a randomly selected sentence, and <math display="inline">I_T(q)</math> is 1 when <math display="inline">q\in I_T</math> is from the source language and 0 if <math display="inline">q\in I_S</math><br />
<br />
The same term is added to the primary objective function, which the encoder and decoder are trained to minimize. The result is that the encoder and decoder learn to fool the discriminator by mapping sentences from the source and target language to similar sequences of latent-space vectors.<br />
<br />
<br />
The authors note that they make use of label smoothing, a technique recommended by Goodfellow (2016) for regularizing GANs, in which the objective described above is replaced by <br />
<br />
\begin{align}<br />
I_T(q)( (1-\alpha)\log (R(E(q))) +\alpha\log(1-R(E(q))) )+(1-I_T(q) ) ( (1-\beta) \log(1-R(E(q))) +\beta\log (R(E(q)) ))<br />
\end{align}<br />
for some small nonnegative values of <math display="inline">\alpha, \beta</math>, the idea being to prevent the discriminator from making extreme predictions. While one-sided label smoothing (<math display="inline">\beta = 0</math>) is generally recommended, the present model differs from a standard GAN in that it is symmetric, and hence two-sided label smoothing would appear more reasonable.<br />
<br />
==Objective Function==<br />
<br />
Combining the above-described terms, we can write the overall objective function. Let <math display="inline">Q_S</math> denote the monolingual dataset for the source language, and let <math display="inline">Q_T</math> denote the monolingual dataset for the target language. Let <math display="inline">D_S:= D(\cdot, S)</math> and<math display="inline">D_T= D(\cdot, T)</math> (i.e. <math display="inline">D_S, D_T</math>) be the decoder restricted to the source or target language, respectively. Let <math display="inline"> M_S </math> and <math display="inline"> M_T </math> denote the target-to-source and source-to-target translation models of the previous epoch. Then our objective function is<br />
<br />
\begin{align}<br />
\mathcal{L}(D,E,R)=\text{T Translation Loss}+\text{T De-noising Loss} +\text{T Adversarial Loss} +\text{S Translation Loss} +\text{S De-noising Loss} +\text{S Adversarial Loss}\\<br />
\end{align}<br />
\begin{align}<br />
=\sum_{q\in Q_T}\left( -\log D_T \circ E \circ C \circ M _S(q) (q) -\log D_T \circ E \circ C (q) (q)+(1-\alpha)\log (R\circ E(q)) +\alpha\log(1-R\circ E(q)) \right)+\sum_{q\in Q_S}\left( -\log D_S \circ E \circ C \circ M_T (q) (q) -\log D_S \circ E \circ C (q) (q)+(1-\beta) \log(1-R \circ E(q)) +\beta\log (R\circ E(q) \right).<br />
\end{align}<br />
<br />
They alternate between iterations minimizing <math display="inline">\mathcal{L} </math> with respect to <math display="inline">E, D</math> and iterations maximizing with respect to <math display="inline">R</math>. ADAM is used for minimization, while RMSprop is used for maximization. After each epoch, M is updated so that <math display="inline">M_S=D_S \circ E</math> and <math display="inline">M_T=D_T \circ E</math>, after which <math display="inline"> M </math> is frozen until the next epoch.<br />
<br />
==Validation==<br />
The authors' aim is for their method to be completely unsupervised, and so do not use parallel corpora even for the selection of hyper-parameters. Instead, they validate by translating sentences to the other language and back, and comparing the resulting sentence with the original according to BLEU, a similarity metric frequently used in translation (Papineni et al. 2002).<br />
<br />
==Experimental Procedure and Results==<br />
<br />
The authors test their method on four data sets. The first is from the English-French translation task of the Workshop on Machine Translation 2014 (WMT14). This data set consists of parallel data. The authors generate a monolingual English corpus by randomly sampling 15 million sentence pairs, and choosing only the English sentences. They then generate a French corpus by selecting the French sentences from those pairs that were not previous chosen. Importantly, this means that the monolingual data sets have no parallel sentences. The second data set is generated from the English-German translation task from WMT14 using he same procedure.<br />
<br />
The third and fourth data sets are generated from Multi30k data set, which consists of multilingual captions of various images. The images are discarded and the English, French, and German captions are used to generate monolingual data sets in the manner described above. These monolingual corpora are much smaller, consisting of 14500 sentence each.<br />
<br />
The unsupervised translation scheme performs well, though not as well as a supervised translation scheme. It converges after a small number of epochs. Besides supervised translation, the authors compare their method with three other baselines: "Word-by-Word" uses only the previously-discussed word-alignment scheme; "Word-Reordering" uses a simple LSTM based language model and a greedy algorithm to select a reordering of the words produced by "Word-by-Word". "Oracle Word Reordering" means the optimal reordering of the words produced by "Word-by-Word".<br />
<br />
==Result Figures==<br />
[[File:MC_Translation Results.png]]<br />
[[File:MC_Translation_Convergence.png]]<br />
<br />
==Commentary==<br />
This paper's results are impressive: that it is even possible to translate between languages without parallel data suggests that languages are more similar than we might initially suspect, and that the method the authors present has, at least in part, discovered some common deep structure. As the authors point out, using no parallel data at all, their method is able to produce results comparable to those produced by neural machine translation methods trained on hundreds of thousands of a parallel sentences on the WMT dataset. On the other hand, the results they offer come with a few significant caveats.<br />
<br />
The first caveat is that the workhorse of the method is the unsupervised word-vector alignment scheme presented in Conneau et al. (2017) (that paper shares 3 authors with this one). As the ablation study reveals, without word-vector alignment, this method preforms extremely poorly. Moreover, word-by-word translation using word-vector alignment alone performs well, albeit not as well as this method. This suggests that the method of this paper mainly learns to perform (sometimes significant) corrections to word-by-word translations by reordering and occasional word substitution. Presumably, it does this by learning something of the natural structure of sentences in each of the two languages, so that it can correct the errors made by word-by-word translation.<br />
<br />
The second caveat is that the best results are attained translating between English and French, two very closely related languages, and the quality of translation between English and German, a slightly-less related pair, is significantly worse ( according to the ''Shorter Oxford English Dictionary'', 28.3 percent of the English vocabulary is French-derived, 28.2 percent is Latin-derived, and 25 percent is derived from Germanic languages. This probably understates the degree of correspondence between the French and English vocabularies, since French likely derives from Latin many of the same words English does. ). The authors do not report results with more distantly-related pairs, but it is reasonable to expect that performance would degrade significantly, especially since Conneau et al. (2017) shows that the word-alignment scheme performs much worse on more distant language pairs. This may be because there are more one-to-one correspondences between the words of closely related languages than there are between more distant languages. The authors suggest that their scheme could be a useful tool for translating between language pairs for which their are few parallel corpora. However, language pairs lacking parallel corpora are often (though not always) distantly related, and it is for such pairs that the performance of the present method suffers.<br />
<br />
The proposed method always beats Oracle Word Reordering on the Multi30k data set, but sometimes does not on the WMT data set. This may be because the WMT sentences are much more syntactically complex than the simple image captions of the Multi30k data set.<br />
<br />
The ablation study also reveals the importance of the corruption process <math display="inline">C</math>: the absence of <math display="inline">C</math> significantly degrades translation quality, though not as much as the absence of word-vector alignment. We can understand this in two related ways. First of all, if we view the model as learning to correct structural errors in word-by-word translations, then the corruption process introduces more errors of this kind, and so provides additional data upon which the model can train. Second, as Vincent et al. (2008) point out, de-noising auto-encoder training encourages a model to learn the structure of the manifold from which the data is drawn. By learning the structure of the source and target languages, the model can better correct the errors of word-by-word translation.<br />
<br />
[[File:MC_Alignment_Results.png|frame|none|alt=Alt text|From Conneau et al. (2017). The final row shows the performance of alignment method used in the present paper. Note the degradation in performance for more distant languages.]]<br />
<br />
[[File:MC_Translation_Ablation.png|frame|none|alt=Alt text|From the present paper. Results of an ablation study. Of note are the first, third, and forth rows, which demonstrate that while the translation component of the loss is relatively unimportant, the word vector alignment scheme and de-noising auto-encoder matter a great deal.]]<br />
<br />
==Future Work==<br />
The principal of performing unsupervised translation by starting with a rough but reasonable guess, and then improving it using knowledge of the structure of target language seem promising. Word by word translation using word-vector alignment works well for closely related languages like English and French, but is unlikely to work as well for more distant languages. For those languages, a better method for getting an initial guess is required.<br />
<br />
==References==<br />
#Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
#Conneau, Alexis, Guillaume Lample, Marc’Aurelio Ranzato, Ludovic Denoyer, Hervé Jégou. "Word Translation without Parallel Data". arXiv:1710.04087, (2017)<br />
# Dictionary, Shorter Oxford English. "Shorter Oxford english dictionary." (2007).<br />
#Goodfellow, Ian. "NIPS 2016 tutorial: Generative adversarial networks." arXiv preprint arXiv:1701.00160 (2016).<br />
# Hill, Felix, Kyunghyun Cho, and Anna Korhonen. "Learning distributed representations of sentences from unlabelled data." arXiv preprint arXiv:1602.03483 (2016).<br />
# Lample, Guillaume, Ludovic Denoyer, and Marc'Aurelio Ranzato. "Unsupervised Machine Translation Using Monolingual Corpora Only." arXiv preprint arXiv:1711.00043 (2017).<br />
#Papineni, Kishore, et al. "BLEU: a method for automatic evaluation of machine translation." Proceedings of the 40th annual meeting on association for computational linguistics. Association for Computational Linguistics, 2002.<br />
# Mikolov, Tomas, Quoc V Le, and Ilya Sutskever. "Exploiting similarities among languages for machine translation." arXiv preprint arXiv:1309.4168. (2013).<br />
#Sennrich, Rico, Barry Haddow, and Alexandra Birch. "Improving neural machine translation models with monolingual data." arXiv preprint arXiv:1511.06709 (2015).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.<br />
# Vincent, Pascal, et al. "Extracting and composing robust features with denoising autoencoders." Proceedings of the 25th international conference on Machine learning. ACM, 2008.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only&diff=32063stat946w18/Unsupervised Machine Translation Using Monolingual Corpora Only2018-02-22T18:42:37Z<p>Pa2forsy: /* Objective Function */</p>
<hr />
<div><br />
[[File:MC_Translation_Example.png]]<br />
== Introduction ==<br />
Neural machine translation systems must be trained on large corpora consisting of pairs of pre-translated sentences. This paper (Lample et al. 2017) proposes an unsupervised neural machine translation system, which can be trained without such parallel data.<br />
<br />
== Overview of unsupervised translation system ==<br />
The unsupervised translation scheme has the following outline:<br />
* The word-vector embedding of the source and target languages are aligned in an unsupervised manner.<br />
* Sentences from the source and target language are mapped to a common latent vector space by an encoder, and then mapped to probability distributions over sentences in the target or source language by a decoder.<br />
* A de-noising auto-encoder loss encourages the latent-space representations to be insensitive to noise.<br />
* An adversarial loss encourages the latent-space representations of source and target sentences to be indistinguishable from each other. It is intended that the latent-space representation of a sentence should reflect its meaning, and not the particular language in which it is expressed.<br />
* A reconstruction loss encourages the model to improve on the translation model of the previous epoch. <br />
I will next discuss this scheme in more detail.<br />
<br />
==Notation==<br />
Let <math display="inline">S</math> denote the set of words in the source language, and let <math display="inline">T</math> denote the set of words in the target language. Let <math display="inline">H \subset \mathbb{R}^{n_H}</math> denote the latent vector space. Moreover, let <math display="inline">S'</math> and <math display="inline">T'</math> denote the sets of finite sequences of words in the source and target language, and let <math display="inline">H'</math> denote the set of finite sequences of vectors in the latent space. For any set X, elide measure-theoretic details and let <math display="inline">\mathcal{P}(X)</math> denote the set of probability distributions over X.<br />
<br />
==Word vector alignment ==<br />
<br />
Conneau et al. (2017) describe an unsupervised method for aligning word vectors across languages. By "alignment", I mean that their method maps words with related meanings to nearby vectors, regardless of the language of the words. Moreover, if two words are one another's literal translations, their word vectors tend to be mutual nearest neighbors. <br />
<br />
The underlying idea of the alignment scheme can be summarized as follows: methods like word2vec or GLoVe generate vectors for which there is a correspondence between semantics and geometry. If <math display="inline">f</math> maps English words to their corresponding vectors, we have the approximate equation<br />
\begin{align}<br />
f(\text{king}) -f(\text{man}) +f(\text{woman})\approx f(\text{queen}).<br />
\end{align}<br />
Furthermore, if <math display="inline">g</math> maps French words to their corresponding vectors, then <br />
\begin{align}<br />
g(\text{roi}) -f(\text{homme}) +f(\text{femme})\approx f(\text{reine}).<br />
\end{align}<br />
<br />
Thus if <math display="inline">W</math> maps the word vectors of English words to the word vectors of their French translations, we should expect <math display="inline">W</math> to be linear. As was observed by Mikolov et al. (2013), the problem of word-vector alignment then becomes a problem of learning the linear transformation that best aligns two point clouds, one from the source language and one from the target language. For more on the history of the word-vector alignment problem, see my CS698 project ([https://uwaterloo.ca/scholar/sites/ca.scholar/files/pa2forsy/files/project_dec_3_0.pdf link]).<br />
<br />
Conneau et al. (2017)'s word vector alignment scheme is unique in that it requires no parallel data, and uses only the shapes of the two word-vector point clouds to be aligned. I will not go into detail, but the heart of the method is a special GAN, in which only the discriminator is a neural network, and the generator is the map corresponding to an orthogonal matrix.<br />
<br />
This unsupervised alignment method is crucial to the translation scheme of the current paper. From now on we denote by <br />
<math display="inline">A: S' \cup T' \to \mathcal{Z}'</math> the function that maps a source- or target- language word sequence to the corresponding aligned word vector sequence.<br />
<br />
==Encoder ==<br />
The encoder <math display="inline">E </math> reads a sequence of word vectors <math display="inline">(z_1,\ldots, z_m) \in \mathcal{Z}'</math> and outputs a sequence of hidden states <math display="inline">(h_1,\ldots, h_m) \in H'</math> in the latent space. Crucially, because the word vectors of the two languages have been aligned, the same encoder can be applied to both. That is, to map a source sentence <math display="inline">x=(x_1,\ldots, x_M)\in S'</math> to the latent space, we compute <math display="inline">E(A(x))</math>, and to map a target sentence <math display="inline">y=(y_1,\ldots, y_K)\in T'</math> to the latent space, we compute <math display="inline">E(A(y))</math>.<br />
<br />
The encoder consists of two LSTMs, one of which reads the word-vector sequence in the forward direction, and one of which reads it in the backward direction. The hidden state sequence is generated by concatenating the hidden states produced by the forward and backward LSTMs at each word vector.<br />
<br />
==Decoder==<br />
<br />
The decoder is a mono-directional LSTM that accepts a sequence of hidden states <math display="inline">h=(h_1,\ldots, h_m) \in H'</math> from the latent space and a language <math display="inline">L \in \{S,T \}</math> and outputs a probability distribution over sentences in that language. We have<br />
<br />
\begin{align}<br />
D: H' \times \{S,T \} \to \mathcal{P}(S') \cup \mathcal{P}(T').<br />
\end{align}<br />
<br />
The decoder makes use of the attention mechanism of Bahdanau et al. (2014). To compute the probability of a given sentence <math display="inline">y=(y_1,\ldots,y_K)</math> , the LSTM processes the sentence one word at a time, accepting at step <math display="inline">k</math> the aligned word vector of the previous word in the sentence <math display="inline">A(y_{k-1})</math> and a context vector <math display="inline">c_k\in H</math> computed from the hidden sequence <math display="inline">h\in H'</math>, and outputting a probability distribution over possible next words. The LSTM is initiated with a special, language-specific start-of-sequence token. Otherwise, the decoder is does not depend on the language of the sentence it is producing. The context vector is computed as described by Bahdanau et al. (2014), where we let <math display="inline">l_{k}</math> denote the hidden state of the LSTM at step <math display="inline">k</math>, and where <math display="inline">U,W</math> are learnable weight matrices, and <math display="inline">v</math> is a learnable weight vector:<br />
\begin{align}<br />
c_k&= \sum_{m=1}^M \alpha_{k,m} h_m\\<br />
\alpha_{k,m}&= \frac{\exp(e_{k,m})}{\sum_{m'=1}^M\exp(e_{k,m'}) },\\<br />
e_{k,m} &= v^T \tanh (Wl_{k-1} + U h_m ).<br />
\end{align}<br />
<br />
<br />
By learning <math display="inline">U,W</math> and <math display="inline">v</math>, the decoder can learn to decide which vectors in the sequence <math display="inline">h</math> are relevant to computing which words in the output sentence.<br />
<br />
At step <math display="inline">k</math>, after receiving the context vector <math display="inline">c_k\in H</math> and the aligned word vector of the previous word in the sequence,<math display="inline">A(y_{k-1})</math>, the LSTM outputs a probability distribution over words, which should be interpreted as the distribution of the next word according to the decoder. The probability the decoder assigns to a sentence is then the product of the probabilities computed for each word in this manner.<br />
<br />
==Overview of objective ==<br />
The objective function is the sum of three terms:<br />
# The de-noising auto-encoder loss,<br />
# The translation loss,<br />
# The adversarial loss.<br />
I shall describe these in the following sections.<br />
<br />
==De-noising Auto-encoder Loss == <br />
A de-noising auto-encoder is a function optimized to map a corrupted sample from some dataset to the original un-corrupted sample. De-noising auto-encoders were introduced by Vincent et al. (2008), who provided numerous justifications, one of which is particularly illuminating. If we think of the dataset of interest as a thin manifold in a high-dimensional space, the corruption process is likely perturb a datapoint off the manifold. To learn to restore the corrupted datapoint, the de-noising auto-encoder must learn the shape of the manifold.<br />
<br />
Hill et al. (2016), used a de-noising auto-encoder to learn vectors representing sentences. They corrupted input sentences by randomly dropping and swapping words, and then trained a neural network to map the corrupted sentence to a vector, and then map the vector to the un-corrupted sentence. Interestingly, they found that sentence vectors learned this way were particularly effective when applied to tasks that involved generating paraphrases. This makes some sense: for a vector to be useful in restoring a corrupted sentence, it must capture something of the sentence's underlying meaning.<br />
<br />
The present paper uses the principal of de-noising auto-encoders to compute one of the terms in its loss function. In each iteration, a sentence is sampled from the source or target language, and a corruption process <math display="inline"> C</math> is applied to it. <math display="inline"> C</math> works by deleting each word in the sentence with probability <math display="inline">p_C</math> and applying to the sentence a permutation randomly selected from those that do not move words more than <math display="inline">k_C</math> spots from their original positions. The authors select <math display="inline">p_C=0.1</math> and <math display="inline">k_C=3</math>. The corrupted sentence is then mapped to the latent space using <math display="inline">E\circ A</math>. The loss is then the negative log probability of the original un-corrupted sentence according to the decoder <math display="inline">D</math> applied to the latent-space sequence.<br />
<br />
The explanation of Vincent et al. (2008) can help us understand this loss-function term: the de-noising auto-encoder loss forces the translation system to learn the shapes of the manifolds of the source and target languages.<br />
<br />
==Translation Loss==<br />
To compute the translation loss, we sample a sentence from one of the languages, translate it with the encoder and decoder of the previous epoch, and then corrupt its output with <math display="inline">C</math>. We then use the current encoder <math display="inline">E</math> to map the corrupted translation to a sequence <math display="inline">h \in H'</math> and the decoder <math display="inline">D</math> to map <math display="inline">h</math> to a probability distribution over sentences. The translation loss is the negative log probability the decoder assigns to the original uncorrupted sentence. <br />
<br />
It is interesting and useful to consider why this translation loss, which depends on the the translation model of the previous iteration, should promote an improved translation model in the current iteration. One loose way to understand this is to think of the translator as a de-noising translator. We are given a sentence perturbed from the manifold of possible sentences from a given language both by the corruption process and by the poor quality of the translation. The model must learn to both project and translate. The technique employed here resembles that used by Sennrich et al. (2014), who trained a neural machine translation system using both parallel and monolingual data. To make use of the monolingual target-language data, they used an auxiliary model to translate it to the source language, then trained their model to reconstruct the original target-language data from the source-language translation. Sennrich et al. argued that training the model to reconstruct true data from synthetic data was more robust than the opposite approach. The authors of the present paper use similar reasoning.<br />
<br />
==Adversarial Loss ==<br />
The intuition underlying the latent space is that it should encode the meaning of a sentence in a language-independent way. Accordingly, the authors introduce an adversarial loss, to encourage latent-space vectors mapped from the source and target languages to be indistinguishable. Central to this adversarial loss is the discriminator <math display="inline">R:H' \to [0,1]</math>, which makes use of <math display="inline">r: H\to [0,1]</math> a three-layer fully-connected neural network with 1024 hidden units per layer. Given a sequence of latent-space vectors <math display="inline">h=(h_1,\ldots,h_m)\in H'</math> the discriminator assigns probability <math display="inline">R(h)=\prod_{i=1}^m r(h_i)</math> that they originated in the target space. Each iteration, the discriminator is trained to maximize the objective function<br />
<br />
\begin{align}<br />
I_T(q) \log (R(E(q))) +(1-I_T(q) )\log(1-R(E(q)))<br />
\end{align}<br />
<br />
where <math display="inline">q</math> is a randomly selected sentence, and <math display="inline">I_T(q)</math> is 1 when <math display="inline">q\in I_T</math> is from the source language and 0 if <math display="inline">q\in I_S</math><br />
<br />
The same term is added to the primary objective function, which the encoder and decoder are trained to minimize. The result is that the encoder and decoder learn to fool the discriminator by mapping sentences from the source and target language to similar sequences of latent-space vectors.<br />
<br />
<br />
The authors note that they make use of label smoothing, a technique recommended by Goodfellow (2016) for regularizing GANs, in which the objective described above is replaced by <br />
<br />
\begin{align}<br />
I_T(q)( (1-\alpha)\log (R(E(q))) +\alpha\log(1-R(E(q))) )+(1-I_T(q) ) ( (1-\beta) \log(1-R(E(q))) +\beta\log (R(E(q)) ))<br />
\end{align}<br />
for some small nonnegative values of <math display="inline">\alpha, \beta</math>, the idea being to prevent the discriminator from making extreme predictions. While one-sided label smoothing (<math display="inline">\beta = 0</math>) is generally recommended, the present model differs from a standard GAN in that it is symmetric, and hence two-sided label smoothing would appear more reasonable.<br />
<br />
==Objective Function==<br />
<br />
Combining the above-described terms, we can write the overall objective function. Let <math display="inline">Q_S</math> denote the monolingual dataset for the source language, and let <math display="inline">Q_T</math> denote the monolingual dataset for the target language. Let <math display="inline">D_S:= D(\cdot, S)</math> and<math display="inline">D_T= D(\cdot, T)</math> (i.e. <math display="inline">D_S, D_T</math>) be the decoder restricted to the source or target language, respectively. Let <math display="inline"> M_S </math> and <math display="inline"> M_T </math> denote the target-to-source and source-to-target translation models of the previous epoch. Then our objective function is<br />
<br />
\begin{align}<br />
\mathcal{L}(D,E,R)=\text{T Translation Loss}+\text{T De-noising Loss} +\text{T Adversarial Loss} +\text{S Translation Loss} +\text{S De-noising Loss} +\text{S Adversarial Loss}\\<br />
\end{align}<br />
\begin{align}<br />
=\sum_{q\in Q_T}\left( -\log D_T \circ E \circ C \circ M _S(q) (q) -\log D_T \circ E \circ C (q) (q)+(1-\alpha)\log (R\circ E(q)) +\alpha\log(1-R\circ E(q)) \right)+\sum_{q\in Q_S}\left( -\log D_S \circ E \circ C \circ M_T (q) (q) -\log D_S \circ E \circ C (q) (q)+(1-\beta) \log(1-R \circ E(q)) +\beta\log (R\circ E(q) \right).<br />
\end{align}<br />
<br />
They alternate between iterations minimizing <math display="inline">\mathcal{L} </math> with respect to <math display="inline">E, D</math> and iterations maximizing with respect to <math display="inline">R</math>. ADAM is used for minimization, while RMSprop is used for maximization. After each epoch, M is updated so that <math display="inline">M_S=D_S \circ E</math> and <math display="inline">M_T=D_T \circ E</math>, after which <math display="inline"> M </math> is frozen until the next epoch.<br />
<br />
==Validation==<br />
The authors' aim is for their method to be completely unsupervised, and so do not use parallel corpora even for the selection of hyper-parameters. Instead, they validate by translating sentences to the other language and back, and comparing the resulting sentence with the original according to BLEU, a similarity metric frequently used in translation (Papineni et al. 2002).<br />
<br />
==Experimental Procedure and Results==<br />
<br />
The authors test their method on four data sets. The first is from the English-French translation task of the Workshop on Machine Translation 2014 (WMT14). This data set consists of parallel data. The authors generate a monolingual English corpus by randomly sampling 15 million sentence pairs, and choosing only the English sentences. They then generate a French corpus by selecting the French sentences from those pairs that were not previous chosen. Importantly, this means that the monolingual data sets have no parallel sentences. The second data set is generated from the English-German translation task from WMT14 using he same procedure.<br />
<br />
The third and fourth data sets are generated from Multi30k data set, which consists of multilingual captions of various images. The images are discarded and the English, French, and German captions are used to generate monolingual data sets in the manner described above. These monolingual corpora are much smaller, consisting of 14500 sentence each.<br />
<br />
The unsupervised translation scheme performs well, though not as well as a supervised translation scheme. It converges after a small number of epochs. Besides supervised translation, the authors compare their method with three other baselines: "Word-by-Word" uses only the previously-discussed word-alignment scheme; "Word-Reordering" uses a simple LSTM based language model and a greedy algorithm to select a reordering of the words produced by "Word-by-Word. "Oracle Word Reordering" means the optimal reordering of the words produced by "Word-by-Word".<br />
<br />
==Result Figures==<br />
[[File:MC_Translation Results.png]]<br />
[[File:MC_Translation_Convergence.png]]<br />
<br />
==Commentary==<br />
This paper's results are impressive: that it is even possible to translate between languages without parallel data suggests that languages are more similar than we might initially suspect, and that the method the authors present has, at least in part, discovered some common deep structure. As the authors point out, using no parallel data at all, their method is able to produce results comparable to those produced by neural machine translation methods trained on hundreds of thousands of a parallel sentences on the WMT dataset. On the other hand, the results they offer come with a few significant caveats.<br />
<br />
The first caveat is that the workhorse of the method is the unsupervised word-vector alignment scheme presented in Conneau et al. (2017) (that paper shares 3 authors with this one). As the ablation study reveals, without word-vector alignment, this method preforms extremely poorly. Moreover, word-by-word translation using word-vector alignment alone performs well, albeit not as well as this method. This suggests that the method of this paper mainly learns to perform (sometimes significant) corrections to word-by-word translations by reordering and occasional word substitution. Presumably, it does this by learning something of the natural structure of sentences in each of the two languages, so that it can correct the errors made by word-by-word translation.<br />
<br />
The second caveat is that the best results are attained translating between English and French, two very closely related languages, and the quality of translation between English and German, a slightly-less related pair, is significantly worse ( according to the ''Shorter Oxford English Dictionary'', 28.3 percent of the English vocabulary is French-derived, 28.2 percent is Latin-derived, and 25 percent is derived from Germanic languages. This probably understates the degree of correspondence between the French and English vocabularies, since French likely derives from Latin many of the same words English does. ). The authors do not report results with more distantly-related pairs, but it is reasonable to expect that performance would degrade significantly, especially since Conneau et al. (2017) shows that the word-alignment scheme performs much worse on more distant language pairs. This may be because there are more one-to-one correspondences between the words of closely related languages than there are between more distant languages. The authors suggest that their scheme could be a useful tool for translating between language pairs for which their are few parallel corpora. However, language pairs lacking parallel corpora are often (though not always) distantly related, and it is for such pairs that the performance of the present method suffers.<br />
<br />
The proposed method always beats Oracle Word Reordering on the Multi30k data set, but sometimes does not on the WMT data set. This may be because the WMT sentences are much more syntactically complex than the simple image captions of the Multi30k data set.<br />
<br />
The ablation study also reveals the importance of the corruption process <math display="inline">C</math>: the absence of <math display="inline">C</math> significantly degrades translation quality, though not as much as the absence of word-vector alignment. We can understand this in two related ways. First of all, if we view the model as learning to correct structural errors in word-by-word translations, then the corruption process introduces more errors of this kind, and so provides additional data upon which the model can train. Second, as Vincent et al. (2008) point out, de-noising auto-encoder training encourages a model to learn the structure of the manifold from which the data is drawn. By learning the structure of the source and target languages, the model can better correct the errors of word-by-word translation.<br />
<br />
[[File:MC_Alignment_Results.png|frame|none|alt=Alt text|From Conneau et al. (2017). The final row shows the performance of alignment method used in the present paper. Note the degradation in performance for more distant languages.]]<br />
<br />
[[File:MC_Translation_Ablation.png|frame|none|alt=Alt text|From the present paper. Results of an ablation study. Of note are the first, third, and forth rows, which demonstrate that while the translation component of the loss is relatively unimportant, the word vector alignment scheme and de-noising auto-encoder matter a great deal.]]<br />
<br />
==Future Work==<br />
The principal of performing unsupervised translation by starting with a rough but reasonable guess, and then improving it using knowledge of the structure of target language seem promising. Word by word translation using word-vector alignment works well for closely related languages like English and French, but is unlikely to work as well for more distant languages. For those languages, a better method for getting an initial guess is required.<br />
<br />
==References==<br />
#Bahdanau, Dzmitry, Kyunghyun Cho, and Yoshua Bengio. "Neural machine translation by jointly learning to align and translate." arXiv preprint arXiv:1409.0473 (2014).<br />
#Conneau, Alexis, Guillaume Lample, Marc’Aurelio Ranzato, Ludovic Denoyer, Hervé Jégou. "Word Translation without Parallel Data". arXiv:1710.04087, (2017)<br />
# Dictionary, Shorter Oxford English. "Shorter Oxford english dictionary." (2007).<br />
#Goodfellow, Ian. "NIPS 2016 tutorial: Generative adversarial networks." arXiv preprint arXiv:1701.00160 (2016).<br />
# Hill, Felix, Kyunghyun Cho, and Anna Korhonen. "Learning distributed representations of sentences from unlabelled data." arXiv preprint arXiv:1602.03483 (2016).<br />
# Lample, Guillaume, Ludovic Denoyer, and Marc'Aurelio Ranzato. "Unsupervised Machine Translation Using Monolingual Corpora Only." arXiv preprint arXiv:1711.00043 (2017).<br />
#Papineni, Kishore, et al. "BLEU: a method for automatic evaluation of machine translation." Proceedings of the 40th annual meeting on association for computational linguistics. Association for Computational Linguistics, 2002.<br />
# Mikolov, Tomas, Quoc V Le, and Ilya Sutskever. "Exploiting similarities among languages for machine translation." arXiv preprint arXiv:1309.4168. (2013).<br />
#Sennrich, Rico, Barry Haddow, and Alexandra Birch. "Improving neural machine translation models with monolingual data." arXiv preprint arXiv:1511.06709 (2015).<br />
# Sutskever, Ilya, Oriol Vinyals, and Quoc V. Le. "Sequence to sequence learning with neural networks." Advances in neural information processing systems. 2014.<br />
# Vincent, Pascal, et al. "Extracting and composing robust features with denoising autoencoders." Proceedings of the 25th international conference on Machine learning. ACM, 2008.</div>Pa2forsyhttp://wiki.math.uwaterloo.ca/statwiki/index.php?title=stat946w18/Unsupervised_Machine_Translation_Using_Monolingual_Corpora_Only&diff=32062stat946w18/Unsupervised Machine Translation Using Monolingual Corpora Only2018-02-22T18:41:02Z<p>Pa2forsy: /* Objective Function */</p>
<hr />
<div><br />
[[File:MC_Translation_Example.png]]<br />
== Introduction ==<br />
Neural machine translation systems must be trained on large corpora consisting of pairs of pre-translated sentences. This paper (Lample et al. 2017) proposes an unsupervised neural machine translation system, which can be trained without such parallel data.<br />
<br />
== Overview of unsupervised translation system ==<br />
The unsupervised translation scheme has the following outline:<br />
* The word-vector embedding of the source and target languages are aligned in an unsupervised manner.<br />
* Sentences from the source and target language are mapped to a common latent vector space by an encoder, and then mapped to probability distributions over sentences in the target or source language by a decoder.<br />
* A de-noising auto-encoder loss encourages the latent-space representations to be insensitive to noise.<br />
* An adversarial loss encourages the latent-space representations of source and target sentences to be indistinguishable from each other. It is intended that the latent-space representation of a sentence should reflect its meaning, and not the particular language in which it is expressed.<br />
* A reconstruction loss encourages the model to improve on the translation model of the previous epoch. <br />
I will next discuss this scheme in more detail.<br />
<br />
==Notation==<br />
Let <math display="inline">S</math> denote the set of words in the source language, and let <math display="inline">T</math> denote the set of words in the target language. Let <math display="inline">H \subset \mathbb{R}^{n_H}</math> denote the latent vector space. Moreover, let <math display="inline">S'</math> and <math display="inline">T'</math> denote the sets of finite sequences of words in the source and target language, and let <math display="inline">H'</math> denote the set of finite sequences of vectors in the latent space. For any set X, elide measure-theoretic details and let <math display="inline">\mathcal{P}(X)</math> denote the set of probability distributions over X.<br />
<br />
==Word vector alignment ==<br />
<br />
Conneau et al. (2017) describe an unsupervised method for aligning word vectors across languages. By "alignment", I mean that their method maps words with related meanings to nearby vectors, regardless of the language of the words. Moreover, if two words are one another's literal translations, their word vectors tend to be mutual nearest neighbors. <br />
<br />
The underlying idea of the alignment scheme can be summarized as follows: methods like word2vec or GLoVe generate vectors for which there is a correspondence between semantics and geometry. If <math display="inline">f</math> maps English words to their corresponding vectors, we have the approximate equation<br />
\begin{align}<br />
f(\text{king}) -f(\text{man}) +f(\text{woman})\approx f(\text{queen}).<br />
\end{align}<br />
Furthermore, if <math display="inline">g</math> maps French words to their corresponding vectors, then <br />
\begin{align}<br />
g(\text{roi}) -f(\text{homme}) +f(\text{femme})\approx f(\text{reine}).<br />
\end{align}<br />
<br />
Thus if <math display="inline">W</math> maps the word vectors of English words to the word vectors of their French translations, we should expect <math display="inline">W</math> to be linear. As was observed by Mikolov et al. (2013), the problem of word-vector alignment then becomes a problem of learning the linear transformation that best aligns two point clouds, one from the source language and one from the target language. For more on the history of the word-vector alignment problem, see my CS698 project ([https://uwaterloo.ca/scholar/sites/ca.scholar/files/pa2forsy/files/project_dec_3_0.pdf link]).<br />
<br />
Conneau et al. (2017)'s word vector alignment scheme is unique in that it requires no parallel data, and uses only the shapes of the two word-vector point clouds to be aligned. I will not go into detail, but the heart of the method is a special GAN, in which only the discriminator is a neural network, and the generator is the map corresponding to an orthogonal matrix.<br />
<br />
This unsupervised alignment method is crucial to the translation scheme of the current paper. From now on we denote by <br />
<math display="inline">A: S' \cup T' \to \mathcal{Z}'</math> the function that maps a source- or target- language word sequence to the corresponding aligned word vector sequence.<br />
<br />
==Encoder ==<br />
The encoder <math display="inline">E </math> reads a sequence of word vectors <math display="inline">(z_1,\ldots, z_m) \in \mathcal{Z}'</math> and outputs a sequence of hidden states <math display="inline">(h_1,\ldots, h_m) \in H'</math> in the latent space. Crucially, because the word vectors of the two languages have been aligned, the same encoder can be applied to both. That is, to map a source sentence <math display="inline">x=(x_1,\ldots, x_M)\in S'</math> to the latent space, we compute <math display="inline">E(A(x))</math>, and to map a target sentence <math display="inline">y=(y_1,\ldots, y_K)\in T'</math> to the latent space, we compute <math display="inline">E(A(y))</math>.<br />
<br />
The encoder consists of two LSTMs, one of which reads the word-vector sequence in the forward direction, and one of which reads it in the backward direction. The hidden state sequence is generated by concatenating the hidden states produced by the forward and backward LSTMs at each word vector.<br />
<br />
==Decoder==<br />
<br />
The decoder is a mono-directional LSTM that accepts a sequence of hidden states <math display="inline">h=(h_1,\ldots, h_m) \in H'</math> from the latent space and a language <math display="inline">L \in \{S,T \}</math> and outputs a probability distribution over sentences in that language. We have<br />
<br />
\begin{align}<br />
D: H' \times \{S,T \} \to \mathcal{P}(S') \cup \mathcal{P}(T').<br />
\end{align}<br />
<br />
The decoder makes use of the attention mechanism of Bahdanau et al. (2014). To compute the probability of a given sentence <math display="inline">y=(y_1,\ldots,y_K)</math> , the LSTM processes the sentence one word at a time, accepting at step <math display="inline">k</math> the aligned word vector of the previous word in the sentence <math display="inline">A(y_{k-1})</math> and a context vector <math display="inline">c_k\in H</math> computed from the hidden sequence <math display="inline">h\in H'</math>, and outputting a probability distribution over possible next words. The LSTM is initiated with a special, language-specific start-of-sequence token. Otherwise, the decoder is does not depend on the language of the sentence it is producing. The context vector is computed as described by Bahdanau et al. (2014), where we let <math display="inline">l_{k}</math> denote the hidden state of the LSTM at step <math display="inline">k</math>, and where <math display="inline">U,W</math> are learnable weight matrices, and <math display="inline">v</math> is a learnable weight vector:<br />
\begin{align}<br />
c_k&= \sum_{m=1}^M \alpha_{k,m} h_m\\<br />
\alpha_{k,m}&= \frac{\exp(e_{k,m})}{\sum_{m'=1}^M\exp(e_{k,m'}) },\\<br />
e_{k,m} &= v^T \tanh (Wl_{k-1} + U h_m ).<br />
\end{align}<br />
<br />
<br />
By learning <math display="inline">U,W</math> and <math display="inline">v</math>, the decoder can learn to decide which vectors in the sequence <math display="inline">h</math> are relevant to computing which words in the output sentence.<br />
<br />
At step <math display="inline">k</math>, after receiving the context vector <math display="inline">c_k\in H</math> and the aligned word vector of the previous word in the sequence,<math display="inline">A(y_{k-1})</math>, the LSTM outputs a probability distribution over words, which should be interpreted as the distribution of the next word according to the decoder. The probability the decoder assigns to a sentence is then the product of the probabilities computed for each word in this manner.<br />
<br />
==Overview of objective ==<br />
The objective function is the sum of three terms:<br />
# The de-noising auto-encoder loss,<br />
# The translation loss,<br />
# The adversarial loss.<br />
I shall describe these in the following sections.<br />
<br />
==De-noising Auto-encoder Loss == <br />
A de-noising auto-encoder is a function optimized to map a corrupted sample from some dataset to the original un-corrupted sample. De-noising auto-encoders were introduced by Vincent et al. (2008), who provided numerous justifications, one of which is particularly illuminating. If we think of the dataset of interest as a thin manifold in a high-dimensional space, the corruption process is likely perturb a datapoint off the manifold. To learn to restore the corrupted datapoint, the de-noising auto-encoder must learn the shape of the manifold.<br />
<br />
Hill et al. (2016), used a de-noising auto-encoder to learn vectors representing sentences. They corrupted input sentences by randomly dropping and swapping words, and then trained a neural network to map the corrupted sentence to a vector, and then map the vector to the un-corrupted sentence. Interestingly, they found that sentence vectors learned this way were particularly effective when applied to tasks that involved generating paraphrases. This makes some sense: for a vector to be useful in restoring a corrupted sentence, it must capture something of the sentence's underlying meaning.<br />
<br />
The present paper uses the principal of de-noising auto-encoders to compute one of the terms in its loss function. In each iteration, a sentence is sampled from the source or target language, and a corruption process <math display="inline"> C</math> is applied to it. <math display="inline"> C</math> works by deleting each word in the sentence with probability <math display="inline">p_C</math> and applying to the sentence a permutation randomly selected from those that do not move words more than <math display="inline">k_C</math> spots from their original positions. The authors select <math display="inline">p_C=0.1</math> and <math display="inline">k_C=3</math>. The corrupted sentence is then mapped to the latent space using <math display="inline">E\circ A</math>. The loss is then the negative log probability of the original un-corrupted sentence according to the decoder <math display="inline">D</math> applied to the latent-space sequence.<br />
<br />
The explanation of Vincent et al. (2008) can help us understand this loss-function term: the de-noising auto-encoder loss forces the translation system to learn the shapes of the manifolds of the source and target languages.<br />
<br />
==Translation Loss==<br />
To compute the translation loss, we sample a sentence from one of the languages, translate it with the encoder and decoder of the previous epoch, and then corrupt its output with <math display="inline">C</math>. We then use the current encoder <math display="inline">E</math> to map the corrupted translation to a sequence <math display="inline">h \in H'</math> and the decoder <math display="inline">D</math> to map <math display="inline">h</math> to a probability distribution over sentences. The translation loss is the negative log probability the decoder assigns to the original uncorrupted sentence. <br />
<br />
It is interesting and useful to consider why this translation loss, which depends on the the translation model of the previous iteration, should promote an improved translation model in the current iteration. One loose way to understand this is to think of the translator as a de-noising translator. We are given a sentence perturbed from the manifold of possible sentences from a given language both by the corruption process and by the poor quality of the translation. The model must learn to both project and translate. The technique employed here resembles that used by Sennrich et al. (2014), who trained a neural machine translation system using both parallel and monolingual data. To make use of the monolingual target-language data, they used an auxiliary model to translate it to the source language, then trained their model to reconstruct the original target-language data from the source-language translation. Sennrich et al. argued that training the model to reconstruct true data from synthetic data was more robust than the opposite approach. The authors of the present paper use similar reasoning.<br />
<br />
==Adversarial Loss ==<br />
The intuition underlying the latent space is that it should encode the meaning of a sentence in a language-independent way. Accordingly, the authors introduce an adversarial loss, to encourage latent-space vectors mapped from the source and target languages to be indistinguishable. Central to this adversarial loss is the discriminator <math display="inline">R:H' \to [0,1]</math>, which makes use of <math display="inline">r: H\to [0,1]</math> a three-layer fully-connected neural network with 1024 hidden units per layer. Given a sequence of latent-space vectors <math display="inline">h=(h_1,\ldots,h_m)\in H'</math> the discriminator assigns probability <math display="inline">R(h)=\prod_{i=1}^m r(h_i)</math> that they originated in the target space. Each iteration, the discriminator is trained to maximize the objective function<br />
<br />
\begin{align}<br />
I_T(q) \log (R(E(q))) +(1-I_T(q) )\log(1-R(E(q)))<br />
\end{align}<br />
<br />
where <math display="inline">q</math> is a randomly selected sentence, and <math display="inline">I_T(q)</math> is 1 when <math display="inline">q\in I_T</math> is from the source language and 0 if <math display="inline">q\in I_S</math><br />
<br />
The same term is added to the primary objective function, which the encoder and decoder are trained to minimize. The result is that the encoder and decoder learn to fool the discriminator by mapping sentences from the source and target language to similar sequences of latent-space vectors.<br />
<br />
<br />
The authors note that they make use of label smoothing, a technique recommended by Goodfellow (2016) for regularizing GANs, in which the objective described above is replaced by <br />
<br />
\begin{align}<br />
I_T(q)( (1-\alpha)\log (R(E(q))) +\alpha\log(1-R(E(q))) )+(1-I_T(q) ) ( (1-\beta) \log(1-R(E(q))) +\beta\log (R(E(q)) ))<br />
\end{align}<br />
for some small nonnegative values of <math display="inline">\alpha, \beta</math>, the idea being to prevent the discriminator from making extreme predictions. While one-sided label smoothing (<math display="inline">\beta = 0</math>) is generally recommended, the present model differs from a standard GAN in that it is symmetric, and hence two-sided label smoothing would appear more reasonable.<br />
<br />
==Objective Function==<br />
<br />
Combining the above-described terms, we can write the overall objective function. Let <math display="inline">Q_S</math> denote the monolingual dataset for the source language, and let <math display="inline">Q_T</math> denote the monolingual dataset for the target language. Let <math display="inline">D_S:= D(\cdot, S)</math> and<math display="inline">D_T= D(\cdot, T)</math> (i.e. <math display="inline">D_S, D_T</math>) be the decoder restricted to the source or target language, respectively. Let <math display="inline"> M_S </math> and <math display="inline"> M_T </math> denote the target-to-source and source-to-target translation models of the previous epoch. Then our objective function is<br />
<br />
\begin{align}<br />
\mathcal{L}(D,E,R)=\text{T Translation Loss}+\text{T De-noising Loss} +\text{T Adversarial Loss} +\text{S Translation Loss} +\text{S De-noising Loss} +\text{S Adversarial Loss}\\<br />
\end{align}<br />
\begin{align}<br />
=\sum_{q\in Q_T}\left( -\log D_T \circ E \circ C \circ M _S(q) (q) -\log D_T \circ E \circ C (q) (q)+(1-\alpha)\log (R\circ E(q)) +\alpha\log(1-R\circ E(q)) \right)+\sum_{q\in Q_S}\left( -\log D_S \circ E \circ C \circ M_T (q) (q) -\log D_S \circ E \circ C (q) (q)(1-\beta) \log(1-R \circ E(q)) +\beta\log (R\circ E(q) \right).<br />
\end{align}<br />
<br />
They alternate between iterations minimizing <math display="inline">\mathcal{L} </math> with respect to <math display="inline">E, D</math> and iterations maximizing with respect to <math display="inline">R</math>. ADAM is used for minimization, while RMSprop is used for maximization. After each epoch, M is updated so that <math display="inline">M_S=D_S \circ E</math> and <math display="inline">M_T=D_T \circ E</math>, after which <math display="inline"> M </math> is frozen until the next epoch.<br />
<br />
==Validation==<br />
The authors' aim is for their method to be completely unsupervised, and so do not use parallel corpora even for the selection of hyper-parameters. Instead, they validate by translating sentences to the other language and back, and comparing the resulting sentence with the original according to BLEU, a similarity metric frequently used in translation (Papineni et al. 2002).<br />
<br />
==Experimental Procedure and Results==<br />
<br />
The authors test their method on four data sets. The first is from the English-French translation task of the Workshop on Machine Translation 2014 (WMT14). This data set consists of parallel data. The authors generate a monolingual English corpus by randomly sampling 15 million sentence pairs, and choosing only the English sentences. They then generate a French corpus by selecting the French sentences from those pairs that were not previous chosen. Importantly, this means that the monolingual data sets have no parallel sentences. The second data set is generated from the English-German translation task from WMT14 using he same procedure.<br />
<br />
The third and fourth data sets are generated from Multi30k data set, which consists of multilingual captions of various images. The images are discarded and the English, French, and German captions are used to generate monolingual data sets in the manner described above. These monolingual corpora are much smaller, consisting of 14500 sentence each.<br />
<br />
The unsupervised translation scheme performs well, though not as well as a supervised translation scheme. It converges after a small number of epochs. Besides supervised translation, the authors compare their method with three other baselines: "Word-by-Word" uses only the previously-discussed word-alignment scheme; "Word-Reordering" uses a simple LSTM based language model and a greedy algorithm to select a reordering of the words produced by "Word-by-Word. "Oracle Word Reordering" means the optimal reordering of the words produced by "Word-by-Word".<br />
<br />
==Result Figures==<br />
[[File:MC_Translation Results.png]]<br />
[[File:MC_Translation_Convergence.png]]<br />
<br />
==Commentary==<br />
This paper's results are impressive: that it is even possible to translate between languages without parallel data suggests that languages are more similar than we might initially suspect, and that the method the authors present has, at least in part, discovered some common deep structure. As the authors point out, using no parallel data at all, their method is able to produce results comparable to those produced by neural machine translation methods trained on hundreds of thousands of a parallel sentences on the WMT dataset. On the other hand, the results they offer come with a few significant caveats.<br />
<br />
The first caveat is that the workhorse of the method is the unsupervised word-vector alignment scheme presented in Conneau et al. (2017) (that paper shares 3 authors with this one). As the ablation study reveals, without word-vector alignment, this method preforms extremely poorly. Moreover, word-by-word translation using word-vector alignment alone performs well, albeit not as well as this method. This suggests that the method of this paper mainly learns to perform (sometimes significant) corrections to word-by-word translations by reordering and occasional word substitution. Presumably, it does this by learning something of the natural structure of sentences in each of the two languages, so that it can correct the errors made by word-by-word translation.<br />
<br />
The second caveat is that the best results are attained translating between English and French, two very closely related languages, and the quality of translation between English and German, a slightly-less related pair, is significantly worse ( according to the ''Shorter Oxford English Dictionary'', 28.3 percent of the English vocabulary is French-derived, 28.2 percent is Latin-derived, and 25 percent is derived from Germanic languages. This probably understates the degree of correspondence between the French and English vocabularies, since French likely derives from Latin many of the same words English does. ). The authors do not report results with more distantly-related pairs, but it is reasonable to expect that performance would degrade significantly, especially since Conneau et al. (2017) shows that the word-alignment scheme performs much worse on more distant language pairs. This may be because there are more one-to-one correspondences between the words of closely related languages than there are between more dis