13 May 2019

Generative Autoencoders Beyond VAEs: (Sliced) Wasserstein Autoencoders

Variational Autoencoders¹ or VAEs have been a popular choice of neural generative models since their introduction in 2014. The goal of this post is to compare VAEs to more recent alternatives based on Autoencoders like Wasserstein² and Sliced-Wasserstein³ Autoencoders. Specifically, we will evaluate these models on their ability to model 2 dimensional Gaussian Mixture Models (GMMs) of varying degrees of complexity and discuss some of the advantages of (S)WAEs over VAEs.

Background

Autoencoders⁴ or AEs constitute an important class of models in the deep learning toolkit for self-supervised learning, and more generally, representation learning. The key idea is that a useful representation of samples in a dataset may be obtained by training a model to reconstruct the input at its output while appropriately constraining (by using a bottleneck, sparsity constraints, penalizing derivatives etc.) the intermediate representation or encoding. Typically, both the encoder and the decoder are deterministic, one-to-one functions.

Variational Autoencoders or VAEs are a generalization of autoencoders that allow us to model a distribution in a way that enables efficient sampling, an inherently stochastic operation. Unlike AEs, the encoder in a VAE is a stochastic function that maps an input to a distribution in the latent space. A sample drawn from this distribution is passed through a deterministic decoder to generate an output. During training, the distribution that each training sample maps to is constrained to follow a known prior distribution that is easy to sample from, like a Normal distribution. At test time, to approximate sampling from the underlying distribution of training samples, random samples are drawn from the prior distribution and fed through the decoder (the encoder is discarded).

The problem with VAEs

VAEs involve minimizing a reconstruction loss (like an AE) along with a regularization term – a KL Divergence loss that minimizes the disparity between the prior and latent distribution predicted for each training sample. For Standard Normal distribution ($\mathcal{N}$), a commonly chosen prior, the KL loss looks like

\[E_{x\sim \mathcal{X}} \left[ D_{KL}\left(\mathcal{Q}_{VAE}(z|x) \| \mathcal{N}(z) \right) \right]\]

The following figure beautifully shows the problem that this formulation presents. During training, the KL loss encourages encoded distribution (red circles) for each training sample to become increasingly similar to the prior distribution (white circles). Over time, the distributions of different training samples would start overlapping and leads to the following inconsistency:

Points like the green triangle in the overlapping regions are treated as a valid encodings for all training points; but at the same time the decoder is deterministic and one-to-one!

This inconsistency would place the goal of achieving good reconstruction fundamentally at odds with that of minimizing KL divergence. One could argue that this is usually the case with any regularization used in machine learning today. However, note that in those cases optimizing the regularization objective is not a direct goal. For instance, in case of L2 regularization used in logistic regression, we don’t really care if our model achieves a low L2 but rather that it generalizes on test data. On the other hand, for a VAE, inference directly relies on sampling from the prior distribution. Hence it is crucial that encoded distributions match the prior!

The fix: WAE

Instead of trying to match encoded distribution of each point to a prior distribution, we could match the distribution of the entire encoding space to the prior. In other words, if we took the encodings of all our training data points, distribution of those encoded points should resemble the prior distribution. WAE tries to achieve this by learning a discriminator to tell apart samples drawn from encoded and prior distributions, and training the encoder to fool the discriminator. The reconstruction loss remains intact.

SWAE

The idea behind WAE is neat, but if you have ever trained a GAN, the thought of training a discriminator should make your squirm. Discriminator training is notoriously sensitive to hyper-parameters. However, there is another way to minimize the distribution between encoded points and samples from the prior that does not involve learning a discriminator – minimizing the sliced-wasserstein distance between the two distributions. The trick is to project or slice the distribution along multiple randomly chosen directions, and minimize the wasserstein distance along each of those one-dimensional spaces (which can be done without introducing any additional parameters).

The following figure visualizes the slicing process for a 2D distribution $p_X$ along two directions $\theta_1$ and $\theta_2$. The resulting 1D distributions $\mathcal{R}_{p_X}(\cdot,\theta_i)$ for \(i \in \{1,2\}\) are shown on the right.

I found the following to be a very intuitive interpretation of wasserstein distance in 1D. I would recommend looking at Algorithm 1 box in ² for a quick overview of how to compute the wasserstein distance.

Visualizing samples from trained models

In the figure below, I have visualized samples draw from the trained models. Specifically, we are looking at contour plots of densities estimated from model samples. The density of contour lines tells us how steep the modes are, which in case of Target Distribution depends on the choice of mixing coefficients of the GMM.

From the plots above, we can make a few observations about the ability of VAE, WAE, and SWAE to learn modes. Modes are the home of the most representative samples from any distribution. Hence it is crucial to check if models learn the location of the modes as well as assign correct densities to those modes.

• Mode Location: VAE has the most precise mode locations followed by WAE and then SWAE. For instance, see the mode locations (0.5,-0.5),(0.0,0.5) for the 3 component GMM, and (1,1),(1,-1) for the 5 component GMM. For the 5 component GMM, VAE and WAE missed the low density mode at (0,0). Don’t be fooled by those concentric circles around (0,0) in case of VAE and WAE! Notice how the contour lines get darker as they approach (0,0) indicating a decrease in density as opposed to the increase shown by the target distribution. SWAE doesn’t localize this mode well either but at least the density does not decrease as you approach the mode from most directions.

• Mode density assignment: To gauge density assignment, we need to observe the color (yellow/lighter colors mean higher density), and also the density of contour lines which indicate how steep the peaks are. Note that VAE tends to assign significant density to the space between modes (e.g. around (0,1) in 5 component GMM) and doesn’t assign significant density to lower density modes like (0.0,0.5),(0.5,-0.5) in 3 component GMM, and (0,0) in 5 component GMM. WAE is slightly unpredictable in density assignment. For instance, WAE assigns too much density to (0.5,-0.5) in the 3 component GMM, and too little to all the high density modes in the 5 component GMM. SWAE’s densities are slightly better calibrated but it has difficulty preserving symmetries both local (around the modes), and global.

Quantitative evaluation using Anderson-Darling statistic

While informative, there is only so much you that you can tell by staring at density plots. In this section, we will present a quantitative evaluation to directly measure how similar the distribution of samples from our models are to samples from the target distributions.

Anderson-Darling statistic is used to test if two set of samples come from the same distribution or if a set of samples is drawn from a given distribution. The AD statistic between two empirical CDFs $F$ (the original distribution of data) and $F’$ (the distribution of samples from learned models) is computed as follows

\[n\int_{-\infty}^{\infty}\frac{(F'(x)-F(x))^2}{F(x)(1-F(x))}dx\]

where n is the number of samples used to estimate the empirical CDFs. Basically, the statistic is a weighted sum of squared differences between the two CDFs with higher weights to the tails of the original data CDF. The smaller the value of the statistic, the more confident we can be of the two sets of samples coming from the same underlying distribution.

But our data lives in 2D instead of 1D, so how do we use AD statistic? We slice! We choose a set of random directions and project both sets of samples onto these directions and compute AD statistic in the induced 1D spaces. The 1D statistics are then averaged to get a measure of dissimilarity between the 2D distributions. Note that when comparing different algorithms, we need to be careful to use the same set of random directions for projection.

The following figure compares the AD statistic for the 3 models on the 3 distributions shown above. Smaller the statistic the better. The dotted lines show the lowest AD achieved by each model during training on the test set.

Select target distribution:
2 components     3 components     5 components

The key observations are as follows:

• Best achieved performance (the dotted lines): SWAE outperforms VAE in all 3 cases. WAE outperforms VAE in 2 of the 3 cases and performs similarly in case of 5 components.

• Stability of training: While SWAE and WAE perform similarly in most cases. However, in case of 3 component GMM, we see a sharp rise in AD statistic for WAE around 60000 iterations and the value plateaus to a much larger value than what was already achieved in the first 60000 iterations of training! SWAEs on the other hand are relatively more stable to train.

Encoding Space

AEs and their variants are also commonly used for representation learning. VAEs, WAEs, and SWAEs, all impose constraints on this encoding or representation space. Specifically, they try to match the distribution of samples in the encoding space to that of the prior, which in our case is Standard Normal. The following plots compare the encodings from different models.

Select target distribution:
2 components     3 components     5 components

Interestingly, in all cases the components of the GMM remain well separated in the embedding space. In the 2 component case, while VAEs lead to 2 distinct modes in the encoding space, in WAE and SWAE the two modes are merged together in order to match the distribution to the Standard Normal prior. The difference is less visible in 3 and 5 component case.

The Human Factor

Any clever or sophisticated machine learning algorithm is ultimately implemented by a “human” (at least for now). So, it matters what the experience of the person implementing the algorithm is. Does the algorithm simply “work” with a “textbook” implementation at least for toy problems/data like the ones above? Or does the algorithm require “tuning” hyper-parameters? And if so, how sensitive is the algorithm to hyper-parameters - does one need to be roughly in the ballpark to get decent performance or does one need to get them just right? Is there even a signal to guide you to the right hyper-parameters?

In my experience of training these models and assuming a correct implementation, SWAEs start showing meaningful results with typical hyper-parameters! In fact, the only hyper-parameters SWAEs add to AEs is the number of slices and regularization weight. The losses are easy to interpret and help guide the hyper-parameter search. The experience with VAEs was pretty similar except that I had to adjust the weight corresponding to the KL term (the original VAEs did not have this weight but it was introduced later in $\beta$-VAEs⁵).

WAEs on the other hand were significantly more difficult to find hyper-parameters for. It took me a couple hours and playing with different depths of the discriminator to start seeing results that looked even remotely meaningful. The version that generated results above has a 7 layers deep discriminator which is ridiculous in comparison to my encoder and decoder which are 3 layers deep each. The discriminator is also twice as wide as the encoder and decoder. So SWAEs start appearing quite lucrative when you consider the possibility of loosing the bulky appendage that is the discriminator!

Conclusion

In this post, we discussed the problem with VAEs and looked at WAEs and SWAEs as viable recent alternatives. To compare them we visualized the distributions and encoding spaces learned by these models on samples drawn from three different GMMs. To quantify the similarity of learned distributions to the original distributions, we looked at sliced Anderson-Darling statistic.

Overall, given the conceptual and training simplicity of SWAEs, I personally found them to be a lucrative alternative to VAEs and WAEs. Completely deterministic encoder and decoders, and not requiring a discriminator during learning are two factors going in favor of SWAEs over VAEs and WAEs. SWAEs do have some problems with preserving symmetries, which the VAEs and WAEs are surprisingly good at (especially local symmetry). Hopefully, future research would fix this (one way would be by carefully choosing slicing directions instead of random).

Hope you found the discussion, visualizations and analysis informative! Please feel free to send any feedback, corrections, and comments to my email which you can find here http://tanmaygupta.info/contact/.

References

09 Jul 2017

What does it mean to reason in AI?

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By end of the first VQA challenge, most researchers agreed that the reasoning capabilities of the first generation VQA models were fairly limited or non-existent. Now a new wave of models are flooding the market which attempt to address this issue. Learning to Reason, Inferring and Executing Programs for Visual Reasoning, and Relation Networks are few examples of these next generation models. In many of these papers, authors do not explicitly define reasoning since what they intend to convey is often clear from context and the examples in the papers. However reasoning is a farily complex concept with many flavors and interesting connections to psychology, logic, and pattern recognition. This warrants a more thorough discussion about what it means to reason in the realm of AI and that’s what this post is all about.

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What does it mean to reason in layman’s terms?

A reason is any justification intended to convince fellow human beings of the correctness of our actions or decisions.

In other words, if the sequence of steps or arguments you used to reach a certain conclusion were shown to other people and they reached the same conclusion without questioning any of the arguments, then your reasons have held their ground. Note that what counts as a valid reason is often context-dependent. For instance, consider justifying calling a person crazy. When used in social context, say during an argument with a friend, divergence from commonly expected behaviour or established norms of morality is a valid reason. But these arguments without a clinical diagnosis do not hold any merit in the courthouse of clinical psychology.

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When would we call an AI agent to be capable of reasoning?

Applying the layman’s definition, let us ask the question - What would convince us of a machine’s actions or help us accept the predictions coming from an AI system? Maybe the AI needs to be able to list out the steps of inference which humans can manually inspect and execute to reach the same prediction. But here’s a trick question -

Would we still call the agent capable of reasoning if inference consists of only a single step of pattern matching or look up in a database?

While it may seem counter-intuitive to some, using our definition of a reason above, I would like to argue that such an agent may still be said to reason in a particular domain of discourse if the following hold:

• the approach makes correct predictions for any input in the domain of discourse if the pattern matching is correctly executed,
• the agent defines the pattern matching criterion in a human understandable way that allows the item retrieved to be manually verified to be a correct match given the criterion, and
• given the definition of matching criterion, and a correct match a human makes the same prediction.

What this implies is that a QA system (as shown above) which only expects one of \(N\) questions, and answers them by matching the question string to a database of question strings paired with their correct answer string, could be called a reasoning system. This is because the system tells you exactly how it computed the answer in a way that can be manually verified (does the retrieved question string match the query question string) and executed to get the same answer. However, if we expand the domain of discourse to include a question that is not stored in the database, the system fails to produce an answer. Therefore, it is a brittle system which is pretty much useless for the domains of discourse that humans usually deal with or would like AI to deal with.

It is easy to build a system that reasons, just not one that works for a meaningfully large domain of discourse.

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Reasoning, Pattern Recognition and Composition

Taking the table lookup approach to its extreme, any reasoning problem can be solved by a table lookup provided you could construct a python dictionary with keys as every possible query expected with their values being the corresponding outputs. There is a practical problem though (you think so Sherlock!). This dictionary in most AI applications would be gigantic if not infinitely large. Consider VQA for example. We would need to map every possible image in the world and every possible question that can be asked about it to an answer. People will definitely call you crazy for trying to pull off something like this. But this is where statistical pattern recognition and compositional nature of the queries comes to our rescue by helping us reduce the dictionary size.

Statistical Pattern Recognition

In QA a question may be posed in multiple ways using different syntax. For example,

Mary and John have been married for 5 years. Mary gave birth to a child an year after their marriage. How old is the child now?

and

How old is Mary and John's child now if Mary gave birth to the child the year after they got married? They got married 5 years ago.

are semantically the same question but the strings are quite different. For most questions there are plenty of such semantic duplicates. One way to reduce the size of the dictionary is to store only one representative for each unique semantic sense and use statistical pattern recognition (SPR) for learning a semantic matcher instead of a simple string matcher. A naive way is encoding sentences using an RNN trained as a language model and using nearest neighbor lookups in the embedding space. But generally this won’t work nearly as well as we’d hope. This is where smart ways of encoding the sentence using parsing helps.

Composition

Composition may be though of as another aid to SPR. In the previous example, we can replace the names Mary and John with any other names but the meaning of the question relative to those names and its answer doesn’t change. So the real challenge is to match the following structure which is composed of different types of nodes and edges. These elements combined together represent a unique semantic sense.

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Further improving tractability through math and logic

So far, our lookup based approach would need to match a query to an instantiation of this structure in terms of some or all of its variables. For instance, in this case 5 years and 1 year are instantiations of variables that are crucial for answering the question. This means that in our database we need to have the question and answer stored for every possible instantiation of variables Duration of Marriage and Years after marriage. This observation provides yet another opportunity for a reduction in the dictionary size by recognizing that the answer is a formula or a program that can be executed - Child's Age = Duration of Marriage - Years after marriage. Useful math primitives include algebraic operators (\(+,-,\times,\div\)), comparators (\(>,<,=\)), and logical operators (\(\land,\lor,\neg\)). Now our inference consists of two steps -

1. match question structure in the database to retrieve the math or logical formula, and
2. execute the formula on the instantiation provided by the query to get the answer.

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Let’s talk implementation

Implementation of VQA systems based on some of the above ideas can be found in Learning to Reason and Inferring and Executing Programs for Visual Reasoning. Both works construct agents which given a question about an image list a sequence of steps, or a program if you will, that can be executed on an image to get the answer. Each consists of passing the image or a feature map through a neural module whose output is a feature map, an attention map, or an answer. Each neural module is dedicated to a certain task such as finding a particular object or attribute, counting, comparison etc. Once assembled the modules form a network that can be trained using backpropagation. This also leads to very efficient data usage for learning through module reuse across different questions.

Limitations: While these models seem quite powerful and general enough for a large class of questions, they are not perfect. One issues is that these models need to relearn any math or logical operation from scratch while there is really no need to learn things like greater_than using a neural network when most programming languages provide > operator. Some of the operations that the modules try to learn in a weakly supervised way, such as detection and segmentation, have completely supervised datasets and fairly sophisticated models dedicated to these tasks alone. The VQA Machine: Learning How to Use Existing Vision Algorithms to Answer New Questions is a recent paper that uses object, attribute and relationship predictions from specialized and separately trained models as facts in VQA inference.

Another limitation of these models is that they need the agent that predicts the program to be jump started using some kind of imitation learning using ground truth programs during training which more often than not are hard to come by. Albeit once pretrained, the agents can be finetuned without ground truth programs using REINFORCE. Again these are reasoning models because humans can execute the program in our brains once we see an image and get the answer to the question.

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Dynamic vs static composition

An important thing to remember here is that a model doesn’t necessarily have to be dynamic to be doing compositional reasoning. Sometimes the explanation we seek in a reasoning model is not instance specific or even explicit but rather baked into a static architecture. For example, Relation Networks are supposed to be doing relational reasoning but the model does not really provide any instance specific explanation for the predictions. However, the architecture is designed to do more than just single-shot, black-box whole image feature extraction and prediction. Instead the model looks at all possible pairs of regions, encodes each region-pair and question tuple using a shared module, pools these representations using element-wise sum, and finally makes an answer prediction using this image representation. Here the explanation for each prediction (and the reason why we would trust the prediction) is that for answering questions about relations of two objects, we would expect to get the right answer only by looking at pairs of objects, and that is built into this particular model’s inference but is not enforced in the first generation VQA models which perform abysmally on relational questions.

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Logical rules in Reasoning

Logical operators are not the only thing that we can borrow from formal logic to improve tractability of learning. Logic sometimes also provides a way to encode rules and constrains learning. Since most of machine learning deals with probabilities, we also want to encode these rules as soft constraints. Probabilistic soft logic (PSL) is one way to do it. To get a flavor of PSL, let us try and encode modus ponens, which refers to the following logical rule in propositional logic

If, $A$ is True, and $A\rightarrow B$, then, $B$ is True

Note that this can be compactly written as

$(A \land (A \rightarrow B)) \rightarrow B$

Now in PSL, \(A\) and \(B\) are not booleans but rather floating values in the range \(\left[0,1\right]\), which could be predictions coming from a network with sigmoid activation in the last layer. This allows operations like \(\land\) and \(\rightarrow\) to be computed using simple math operations such as

$A \land B \implies \text{max}\{A+B-1,0\}$
$A \lor B \implies \text{max}\{A+B,1\}$
$\neg A \implies 1-A$
$A \rightarrow B \implies \neg A \lor B \implies \text{max}\{1-A+B,1\}$

Using these modus ponens becomes

$\text{max}\{1-$ $\text{max}\{A+$ $\text{max}\{1-A+B,1\}$ $-1,0\}$ $+B,1\}$

with corresponding colors denoting $(A \land$$(A\rightarrow B)$$)$. This means that if we have probabilistic truth values of $A$ and $B$, we can plug it into the above equation and get a value between $[0,1]$ with $1$ indicating the rule being exactly satisfied and $0$ indicating the rule not being satisfied at all. Note that each branch of logic - propositional logic, predicate logic etc. provides a bunch of these general rules of inference. But depending on the problem, one may be able to come up with specific logical rules. For example, an object detector might find it useful to know that cats don’t fall from the sky (usually).

This is pretty neat, but how does one use these soft logical constraints in training deep networks? Harnessing Deep Neural Networks with Logic Rules proposes simultaneously training a student and a teacher network for any classification task with provided PSL rules. The teacher network projects predictions made by the student in a rule regularized space. The student network tries to predict the correct class (cross entropy loss) while also trying to match the predictions of the rule compliant teacher predictions (L2 loss between teacher and student predictions). Given, the student predictions $p_{\theta}(y|x)$ for input $x$, and a set of rules applicable to $x$, \(\{ r_l(x,y)=1 |\; l=1,\cdots,L \}\), the authors find a closed form solution for the projection into the rule constrained space, which is

$q(y|x) \propto p_{\theta}(y|x) \; e^{-C \sum_l \lambda_l\{1-r_l(x,y)\} }$

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What has reasoning got to do with intelligence in humans?

To answer this question, it is necessary to understand what is it that humans call intelligence. Intelligence is commonly defined as ^[7]

The mental ability to learn from experiences, solve problems, and use knowledge to adapt to new situations.

While there seems to be some concensus about its definition, the debate about whether there is one core general intelligence or multiple intelligences seems to be still on. The following short video provides a comprehensive overview of 4 major theories of intelligence in psychology and the ongoing debate.

For the impatient reader who did not watch the video, below I have summarized, the different factors associated with each theory of intelligence. Notice that reasoning appears in some form or the other in each theory. And just from personal experience, most people would agree that intelligent people do effectively apply the knowledge and skills they have already acquired to solve novel problems. But in order to do so they need to be able to decompose any previously solved problem into a sequence of clearly defined and easily executable inference steps. This allows them to recognize previously seen inference steps (or primitive operations) in novel compositional problem structures. And then it is only a matter of executing those inference steps correctly. Hence in some ways reasoning allows for our finite knowledge to be applicable to much larger set of problems than those we have already seen.

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Conclusion

In this post our thread of thought began at a layman’s definition of reasoning. We used the layman’s definition to understand reasoning in context of artificial intelligence. There we argued that any reasoning problem can be solved by lookup in a table containing all knowledge man had, has, or will ever seek to have. We acknowledged the practical ridiculousness of this idea and described how math, logic, compositionality, and pattern recognition make the problem more tractable (allowing us to answer more questions from smaller databases). We briefly looked at existing systems that implement some of these ideas. Finally just to mess with your mind, we dabbled in psychology trying to figure out what does reasoning have to do with intelligence in humans. It was a long and winding path. For some of you it might have been a dive into a completely nonsensical rabbit hole. For others it might serve to provide some perspective. Irrespective, feel free to send me an email if you feel strongly about any idea we discussed here, inaccuracies, comments, or suggestions.

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References

21 Apr 2017

What 2D data reveals about deep nets?

Neural networks (NN) have seen unparalleled success in a wide range of applications. They have ushered in a revolution of sorts in some industries such as transportation and grocery shopping by facilitating technologies like self-driving cars and just-walk-out stores. This has partly been made possible by a deluge of mostly empirically verified tweaks to NN architectures that have made training neural networks easier and sample efficient. The approach used for empirical verification has unanimously been to take a task and dataset, for instance CIFAR for visual classification in the vision community, apply the proposed architecture to the problem, and report the final accuracy. This final number while validating the merit of the proposed approach, reveals only part of the story. This post attempts to add to that story by asking the following question - what effect do different architectural choices have on the prediction surface of a neural network?

How is this blog different from other efforts to visually understand neural networks?

There are 2 important differences:

• Complexity and Dimensionality of Datasets: Most works try to understand what the network has learned on large, complex, high dimensional datasets like ImageNet, CIFAR or MNIST. In contrast this post only deals with 2D data with mathematically well defined decision boundaries.

• Visualizing the learned decision function vs visualizing indirect measures of it: Due to the above mentioned issues of dimensionality and size of the datasets, researchers are forced to visualize features, activations or filters (see this tutorial for an overview). However, these visualizations are an indirect measure of the functional mapping from input to output that the network represents. This post takes the most direct route - take simple 2D data and well defined decision boundaries, and directly visualize the prediction surface.
  

Source Code

The code used for this post is available on Github. Its written in Python using Tensorflow and NumPy. One of the purposes of this post is to provide the readers with an easy to use code base that is easily extensible and allows further experimentation along this direction. Whenever you come across a new paper on arXiv proposing a new activation function or initialization or whatever, try it out quickly and see results within minutes. The code can easily be run on a laptop without GPU.

Data: A fixed number of samples are uniformly sampled in a 2D domain, \([-2,2] \times [-2,2]\) for all experiments presented here. Each sample \((x,y)\) is assigned a label using \(l=sign(f(x,y))\), where \(f:\mathbb{R}^2 \rightarrow \mathbb{R}\) is a function of you choice. In this post, all results are computed for the following 4 functions of increasing complexity:

1. \(y-x\)
2. \(y-|x|\)
3. \(1-\frac{x^2}{4}-y^2\)
4. \(y-\sin(\pi x)\)

NN Architecture: We will use a simple feedforward architecture with:

1. an input layer of size \(2\),
2. \(n\) hidden layers with potentially different number of units in each layer,
3. a non-linear activation function on each unit,
4. batch normalization and residual connections are used in some experiments
5. an output layer of size \(2\) that produces the logits,
6. a softmax layer to convert logits to probability of each label \(l \in \{1,-1\}\).

Training: The parameters are learned by minimizing a binary cross entropy loss using SGD. Unless specified otherwise, assume \(10000\) training samples, a mini-batch size of \(1000\), and \(100\) epochs. All experiments use a learning rate of \(0.01\). Xavier initialization is used to initialize weights in all layers.

Experiments: We will explore the following territory:

1. Effect of activations
2. Effect of depth
3. Effect of width
4. Effect of dropout
5. Effect of batch normalization
6. Effect of residual connections

Each experiment will try to vary as few factors as possible while keeping others fixed. Unless specified otherwise, the architecture consists of 4 hidden layers with 10 units each and relu activation. No dropout or batch normalization is used in this barebones model.

Effect of activation

This experiment compares 4 activation functions - sigmoid, tanh, relu, elu, and the latest from Google Brain - swish. There are a number of interesting things to observe:

• Sigmoid performs poorly. This is because of its large saturation regions which yield very small gradients. It is possible that results could be improved by using a higher learning rate.
• Given that sigmoid fails miserably, its surprising that tanh works so well since it has a shape similar to sigmoid but shifted to lie between \([-1,1]\) instead of \([0,1]\). In fact tanh works better than relu in this case. It might simply be because it receives larger gradients than sigmoid or relu because of larger slope (~ \(2\)) around \(0\). The elu paper also points to some references that provide theoretical justification for using centered activations.
• Many people don’t realize that NN with relus, no matter how deep, produce a piecewise linear prediction surface. This can be clearly observed in jagged contour plots of ellipse and sine functions.
• It is also interesting to compare elu and relu. While relus turn negative input \(x\) to 0, elus instead use \(e^x-1\). See the original elu paper for theoretical justification of this phenomenon.
• Update Oct 19 2017 - The figure above has been updated with swish, a new activation function which is a weird mix of relu and sigmoid and is defined as \(f(x) = x \cdot \text{sigmoid}(x)\). Qualitatively the results look similar to relu but smoother. Checkout the paper for more details.
  

Effect of depth

As mentioned in the previous section, relus produce piecewise linear functions. From the figure above we observe that the approximation becomes increasingly accurate with higher confidence predictions and crisper decision boundaries as the depth increases.

Effect of width

All networks in this experiment have 4 hidden layers but the number of hidden units vary:

• Narrow Uniform: \([5,5,5,5]\)
• Medium Uniform: \([10,10,10,10]\)
• Wide Uniform: \([20,20,20,20]\)
• Very Wide Uniform: \([40,40,40,40]\)
• Decreasing: \([40,20,10,5]\)
• Increasing: \([5,10,20,40]\)
• Hour Glass: \([20,10,10,20]\)

As with depth, increasing the width improves performance. However, comparing Very Wide Uniform with 8 hidden layers network of the previous section (same width as the Medium Uniform network), increasing depth seems to be a significantly more efficient use of parameters (~5k vs ~1k). This result is theoretically proved in the paper Benefits of depth in neural networks. One might want to reduce the parameters in Very Wide Uniform by reducing width with depth (layers closer to output are narrower). The effect of this can be seen in Decreasing. The effect of reversing the order can be seen in Increasing. I have also included results for an Hour Glass architecture whose width first decreases and then increases. More experiments are needed to comment on the effect of the last three configurations.

Effect of dropout

Dropout was introduced as a means to regularize neural networks and so it does in the above results. The amount of regularization is inversely related to the keep probability. Comparing the last row with 4 hidden layers network in the Effect of Depth section, we see quite significant regularization effect even with a high keep probability. But this comparison is not entirely fair since there is no noise in the data.

Effect of Batch Normalization

Batch normalization is reported to speed up training by a factor of 13 for ImageNet classification tasks. The above figure shows the benefit gained by using batch normalization. You will find that this model with batch normalization beats the 8 hidden layer network without batch normalization in the Effect of Depth section. You can also compare it to elu in Effect of Activations section. elu was proposed as a cheaper alternative to Batch normalization, and indeed they compare well.

Effect of Residual Connection

The above figure compares networks with and without residual connections trained on different number of training examples. Whenever the input and output size of a layer matches. the residual connection adds the input back to the output before applying the activation and passing on the result to the next layer. Recall that the residual block used in the original ResNet paper used 2 convolutional layers with relus in between. Here a residual block consists of a single fully connected layer and no relu activation. Even then, residual connections noticeably improve performance. The other purpose of this experiment was to see how the prediction surface changes with the number of training samples available. Mini-batch size of 100 was used for this experiment since the smallest training set size used is 100.

Conclusion

Neural networks are complicated. But so are high dimensional datasets that are used to understand them. Naturally, trying to understand what deep networks learn on complex datasets is an extremely challenging task. However, as shown in this post a lot of insight can be gained quite easily by using simple 2D data, and we have barely scratched the surface here. For instance, we did not even touch the effects of noise, generalization, training procedures, and more. Hopefully, readers will find the provided code useful for exploring these issues, build intuitions about the numerous architectural changes that are proposed on arXiv every day, and to understand currently successful practices better.

BibTeX Citation

@misc{gupta2016nnpredsurf,
  author = {Gupta, Tanmay},
  title = {What 2D data reveals about deep nets?},
  year = {2016},
  howpublished = {https://bigredt.github.io/2017/04/21/deepvis/}
}

03 Jul 2016

CVPR16 Highlights @ Caesar Palace, Vegas

This year CVPR saw a staggering attendance of over 3600 participants. As I sat in one of those chandelier lit colossal ballrooms of Caesar Palace (checkout some cool photos from my Vegas trip above :smiley: ), I had only one aim in mind - to keep an eye for honest research which is of some interest to me. Needless to say publications featuring CNNs and RNNs were in abundance. However it seems like the vision community as a whole has matured a little in dealing with these obscure learning machines and the earlier fascination with the tool itself has been replaced by growing interest in its creative and meaningful usage. In this post I will highlight some of the papers that seem to be interesting based on a quick glance. This blog post is very similar in spirit to the excellent posts by Tomasz Malisiewicz where he highlights a few papers from every conference he visits. I think it is a good exercise for anyone to keep up with the breadth and depth of a fast moving field like Computer Vision. As always many a grad students (including me) and possibly other researchers are waiting for Tomasz’ highlight of CVPR 2016 (which based on his recent tweet is tentatively named ‘How deep is CVPR 2016?’). But while we are waiting for it let us dive in!

Vision and Language

One of the major attractions for me were the oral and spotlight sessions on Vision and Language and the VQA workshop. There were some good papers from Trevor Darrell’s group in UC Berkeley -

• Neural Module Network: This work proposes composing neural networks for VQA from smaller modules each of which performs a specific task. The architecture of the network comes from a parse of the question which specifies the sequence in the which operations need to be performed on the image to answer the question. A follow up work is also available here where instead of committing to a single network layout, multiple layouts are proposed and a ranking is learned using the REINFORCE rule in a policy gradient framework.

  

• Deep Compositional Captioning: Describing Novel Object Categories without Paired Training Data: Most image captioning models required paired image-caption data to train and cannot generate captions describing novel object categories that aren’t mentioned in any of the captions in the paired data. The main contribution of the paper is to propose a method to transfer knowledge of novel object categories from object classification datasets.

  

• Multimodal Compact Bilinear Pooling for Visual Question Answering and Visual Grounding: Most deep learning approaches to tasks involving image and language require combining the features from two modalities. The paper proposes MCB as an alternative to simple concatenation/addition/elementwise multiplication of the image and language features. MCB is essentially an efficient way of performing an outer product between two vectors and so results in a straightforward feature space expansion.

  

There was also this work born out of a collaboration between Google, UCLA, Oxford and John Hopkins that deals with referring expressions which are expressions used to uniquely identify an object or a region in an image -

• Generation and Comprehension of Unambiguous Object Descriptions: The main idea of the paper is to jointly model generation and interpretation of referring expressions. A smart thing about this is that unlike independent image caption generation which is hard to evaluate, this joint model can be evaluated using simple pixel distance to the object being referred to.

  

Shifting the attention from caption generation to use of captions as training data for zero-shot learning, researchers from University of Michigan and Max-Planck Institute introduced a new task and dataset in -

• Learning Deep Representations of Fine-Grained Visual Descriptions: The task is to learn to classify novel categories of birds given exemplar captions that describe them and paired image caption data for other bird categories. The proposed model learns to score captions and images and classifies any new bird image as belonging to the category of the nearest neighbor caption.

  

During the VQA workshop Jitendra Malik brought something unexpected to the table by pointing out that vision and language while being the most researched pillars of AI aren’t the only ones. There is a third pillar which has to do with embodied cognition, the idea that an agent’s cognitive processing is deeply dependent in a causal way on the characteristics of the agent’s physical, beyond-the-brain body. He argued that in order for a robot to ingrain the intuitive concepts of physics which allow humans to interact with the environment on a daily basis, the robot needs to be able to interact with the world and see the effect of its actions. This theme reflects in some of the recent research efforts lead by his student Pulkit Agrawal at UC Berkeley -

• Learning to Poke by Poking: Experiential Learning of Intuitive Physics: The main idea here is to let a robot run amok in the world and learn to perform specific actions in the process. In this case the robot repeatedly pokes different objects and records the images before and after the interaction collecting 400hrs of training data. Next a network is trained to produce the action parameters given the initial and final image. In this way the robot learns certain rules of physics like the relation between force, mass, acceleration and other factors that affect the change like friction due to a particular surface texture. At test time, this allows the robot to predict an action that will produce a desired effect (in this case specified by initial and final images). Also refer to their previous work Learning to See by Moving where egomotion is used as supervision for feature learning.

  

Some other invited speakers at VQA workshop were - Ali Farhadi, Mario Fritz, Margaret Mitchell, Alex Berg, Kevin Murphy and Yuandong Tian. Do checkout their websites for their latest works on Vision and Language.

Object Detection

Collaborations between FAIR, CMU and AI2 brought us some interesting advances in object detection -

• Training Region-Based Object Detectors with Online Hard Example Mining: This work is a reformulation of the classic hard negative mining framework that was once popular for training object detectors with far more negative/background samples than positive examples. The training involved beginning with an active and possibly balanced set of positives and negatives. A classifier trained on this active set was applied to the complete training set to identify hard false positives which were then added to the active set and the process was repeated. This paper presents an online version of this methodology which rhymes well with current object detectors like Fast-RCNN which are trained using SGD with mini-batches.

• You Only Look Once: Unified, Real-Time Object Detection: A little pony used its magical powers to create a really fast and simple object detector. The key idea is to use a single forward pass through a convolutional network and produce a fixed number of bounding box proposals with corresponding confidence values, and class probabilities for every cell on a grid. The pony also took great risk but delivered a mind blowing live demo of YOLO Object Detector during its oral session at CVPR.

  

Folks from ParisTech also presented their work on improving localization of object detectors -

• LocNet: Improving Localization Accuracy for Object Detection: This work proposes an approach to improve object localization. First the initial bounding box is enlarged. Then a network is used to predict for each row/column the probability of it belonging inside the bounding box (or alternatively the probability of that row/column being an edge of the final bounding box).

Recognition and Parsing in 3D

Object detection in 3D seems to be gradually catching up with its 2D counterpart -

• Deep Sliding Shapes for Amodal 3D Object Detection in RGB-D Images: This paper from Jianxiong Xiao’s group in Princeton shows how to extend the Region Proposal Network (RPN) from Faster-RCNN to do 3D region proposal. This region proposal is amodal meaning that the proposed bounding box includes the entire object in 3D volume irrespective of truncation or occlusion. For recognition they combine deep learned 2D appearance features from image with 3D geometric features from depth extracted using 2D/3D ConvNets respectively.

• 3D Semantic Parsing of Large-Scale Indoor Spaces: This paper due to a collaboration between Stanford, Cornell and University of Cambridge deals with parsing of large scale 3D scenes. For instance consider parsing an entire building into individual rooms and hallways, and then further identifying different components of each room such as walls, floor, ceiling, doors, furnitures etc. The paper also suggests novel applications that can result from this kind of parsing - generating building statistics, analysing workspace efficiency, and manipulation of space such as removing a wall partition between two adjacent rooms. 

• GOGMA: Globally-Optimal Gaussian Mixture Alignment: Gaussian mixture alignment can be used for aligning point clouds where both the source and target point clouds are first parameterized by a GMM. The alignment is then posed as minimizing a discrepancy measure between the 2 GMMs. This paper presents a globally optimal solution when the discrepancy measure used is L2 distance between the densities.

Weakly Supervised and Unsupervised Learning

With availability of good deep learning frameworks like Tensorflow, Caffe etc, supervised learning in many cases is a no-brainer given the data. But collecting and curating data is a huge challenge in itself. There are even startups like Spare5 (they also had a booth at CVPR) which provide data annotation as a service. Nonetheless sometimes collecting data is not even an option. Say for example the task of finding dense correspondence between images. A dataset for this task would have a huge impact on other tasks like optical flow computation or depth estimation. However we are yet to see any dataset for this purpose. The following paper has an interesting take on solving this problem in an unsupervised fashion

• Learning Dense Correspondence via 3D-guided Cycle Consistency: The idea in this paper is to match both source and target real images with images rendered from a synthetic mesh. This establishes a consistency cycle where the correspondence between the matched synthetic images is known from the rendering engine by construction. This is used as a supervisory signal for training a ConvNet that produces a flow field given source and target images.

It wouldn’t be wrong to say - Piotr Dollar is to Edge Detection what Ross Girshick is to Object Detection. In this next paper Piotr and collaborators from Georgia Tech introduce an unsupervised method for edge detection -

• Unsupervised Learning of Edges: The key insight is that motion discontinuities always correspond to edges even though the relation does not hold in the other direction. Their algorithm alternates between computing optical flow and using it to train their edge detector. The output of the improved detector is then further used to improve the optical flow and the process continues. Over time both optical flow and edge detection benefit from each other.

Denoising Autoencoders have been used before for unsupervised feature learning. However depending on the noise characteristics the encoder may not need to capture the semantics and often the clean output can be produced from low level features. The next paper shows how to use an encoder-decoder framework to do unsupervised feature learning in a way that forces features to capture higher level semantics and as a byproduct produces reasonable inpainting results -

• Context Encoders: Feature Learning by Inpainting: Given an image with a region cropped out from it, the encoder produces a feature representation. These features are passed on to a decoder that produces the missing region. One of their results is that pretraining a recognition network in this fashion achieves better performance than random initialization of weights.

The next couple papers are about weakly supervised learning where annotation is available though not in the most direct form.

• Deep Structured Scene Parsing by Learning with Image Descriptions: The paper proposes a method for hierarchically parsing an image using sentence descriptions. For example given a sentence - A man holding a red bottle sits on the chair standing by a monitor on the table, the task is to parse the scene into something like

• WarpNet: Weakly Supervised Matching for Single-view Reconstruction: Consider the problem of reconstructing a bird from a single image. One way to do this is to use a fine grained birds dataset and use key-point matches across instances to do non-rigid structure from motion. However traditional key-point detectors like SIFT would fail to give good correspondence due to appearance variations. This paper proposes WarpNet, a network that produces a Thin Plate Splines warping that is used as a shape prior for achieving robust matches. Furthermore the network is trained in an unsupervised manner without any part annotations.

Semantic Segmentation

FCNs and Hypercolumn have been the dominant approaches for semantic segmentation since CVPR 2015. But it is not trivial to adapt these to instance segmentation which is the goal of the next paper-

• Multi-scale Patch Aggregation for Simultaneous Detection and Segmentation: The proposed approach is to densely sample predefined regions in the image and for each region produce a segmentation mask (basically foreground/background segmentation) and a classification. The segmentation masks are then fused together to get instance level segmentation.

In a previous work from Vladlen Koltun’s group, inference over fully connected CRFs was proposed as an effective and efficient (using filtering in a high dimensional space with permutohedral lattice) way to do semantic segmentation. The following is an extension of that work to semantic segmentation in videos -

• Feature Space Optimization for Semantic Video Segmentation: For semantic segmentation in still images the fully connected CRFs could operate in a very simple feature space - RGB color channels and pixel coordinates. An obvious extension to video using time as an extra feature does not work since moving objects take apart pixels belonging to the same object in consecutive frames. The proposed approach learns an optimal feature space over which fully connected CRFs with gaussian edge potentials can be applied as before.

The following paper from UCLA and Google proposes an alternative to using fully connected CRFs -

• Semantic Image Segmentation with Task-Specific Edge Detection Using CNNs and a Discriminatively Trained Domain Transform: In this work, features produced by a ConvNet are used to produce both a semantic segmentation mask as well as an edge map. The produced edge map is then used to perform domain transform on the segmentation mask for edge preserving filtering. The entire system is trained end-to-end.

Image Captioning

Andrej Karpathy has been a pioneer in image caption generation technology (even though it is far from being perfect yet). In the following paper he and Justin Johnson from Fei Fei Li’s group in Stanford extend their captioning algorithm to generate captions densely on an image -

• DenseCap: Fully Convolutional Localization Networks for Dense Captioning: An image is fed into a network that produces convolutional feature maps. A novel localization layer then takes this feature maps and produces region proposals. Bilinear interpolation is used to get region features from the full size feature maps which are fed into another network with RNN at the end that generates a description for that region.

Stereo and Monocular Vision

The following paper from Noah Snavely and team from Google shows us how to generate high quality images from deep nets -

• Deep Stereo: Learning to predict new views from world’s imagery: The task is that of novel view synthesis from a given pair of images from different views. A plane sweep volume is feed into a network with a carefully designed architecture that consists of a Color Tower which learns to warp pixels and a Selection Tower which learns to select pixels from the warped image produced by the _Color Tower.

Current stereo depth estimation algorithms work very well for images of static scenes with sufficient baseline. However, depth estimation in low baseline conditions where the foreground and background move relative to each other is still a hard problem. The next paper from Vladlen Koltun’s group talks about dense depth estimation of scenes with moving objects from consecutive frames captured using a monocular camera.

• Dense Monocular Depth Estimation in Complex Dynamic Scenes: Due to moving objects present in the scene, the optical flow field is first segmented. Depth is estimated using separate epipolar geometry of each segment while enforcing the condition that each segment is connected to the scene.