Generative Adversarial Networks (GANs) are an exciting technique, a kernel of an effective concept that has been shown to be able to overcome many of the problems of previous generative models: particularly the fuzziness of VAEs. But, as I’ve mentioned before, and as you’ve doubtless read if you’re read any material about the topic, they’re finicky things, difficult to train in a stable way, and particularly difficult to not devolve into mode collapse. Mode collapse is a phenomenon where, at each iteration, the generator places all of its mass on one single output or dense cluster of outputs, instead of representing the full distribution of output space, they way we’d like it to. One proposed solution to this is the one I discussed yesterday, of explicitly optimizing the generator according to not only what the discriminator thinks about its current allocation of probability, but what the discriminator’s next move will be (thus incentivizing the generator not to take indefensible strategies like “put all your mass in one location the discriminator can push down next round”. 

An orthogonal approach to that one is the one described in LSGANs: to change the objective function of the network, away from sigmoid cross-entropy, and instead to a least squares loss. While I don’t have the latex capabilities to walk through the exact mathematics in this format, what this means on a conceptual level is that instead of incentivizing the generator to put all of its mass on places that the discriminator is sure is a “true data” region, we’re instead incentivizing the generator to put mass right on the true/fake data decision boundary. Likely this doesn’t make very much sense yet (it didn’t for me, at this point in reading). Occasionally, delving deeper into math and theory behind an idea provides you rigor, but without much intuition. I found the opposite to be true in this case, where learning more (for the first time!) about f divergences actually made this method make more sense. So, bear with me, and hopefully trust me not to take you to deep into the weeds without a good reason. 

On a theoretical level, this paper’s loss function means that you end up minimizing a chi squared divergence between the distributions, instead of a KL divergence. "F divergences" are a quantity that calculates a measure of how different two distributions are from one another, and does that by taking an average of the density q, weighted at each point by f, which is some function of the ratio of densities, p/q. (You could also think of this as being an average of the function f, weighted by the density q; they’re equivalent statements). For the KL divergence, this function is x*logx. For chi squared it’s (x-1)^2. 

All of this starts to coalesce into meaning with the information that, typically the behavior of a typical GAN looks like the divergence FROM the generator’s probability mass, TO the discriminator’s probability mass. That means that we take the ratio of how much mass a generator puts somewhere to how much mass the data has there, and we plug it into the x*logx function seen below.

 https://i.imgur.com/BYRfi0u.png

Now, look how much the function value spikes when that ratio goes over 1. Intuitively, what this means is that we heavily punish the generator when it puts mass in a place that’s unrealistic, i.e. where there isn’t representation from the data distribution. But - and this is the important thing - we don’t symmetrically punish it when it its mass at a point is far higher than the mass put their in the real data; or when the ratio is much smaller than one. This means that we don’t have a way of punishing mode collapse, the scenario where the generator puts all of its mass on one of the modes of the data; we don’t do a good job of pushing the generator to have mass everywhere that the data has mass. By contrast, the Chi Squared divergence pushes the ratio of (generator/data) to be equal to 1 *from both directions*. So, if there’s more generator mass than data mass somewhere, that’s bad, but it’s also bad for there to be more data mass than generator mass. This gives the network a stronger incentive to not learn mode collapsed solutions.

If you’ve ever read a paper on Generative Adversarial Networks (from now on: GANs), you’ve almost certainly heard the author refer to the scourge upon the land of GANs that is mode collapse. When a generator succumbs to mode collapse, that means that, instead of modeling the full distribution, of input data, it will choose one region where there is a high density of data, and put all of its generated probability weight there. Then, on the next round, the discriminator pushes strongly away from that region (since it now is majority-occupied by fake data), and the generator finds a new mode. 

In the view of the authors of the Unrolled GANs paper,  one reason why this happens is that, in the typical GAN, at each round the generator implicitly assumes that it’s optimizing itself against the final and optimal discriminator. And, so, it makes its best move given that assumption, which is to put all its mass on a region the discriminator assigns high probability. Unfortunately for our short-sighted robot friend, this isn’t a one-round game, and this mass-concentrating strategy gives the discriminator a really good way to find fake data during the next round: just dramatically downweight how likely you think data is in the generator’s prior-round sweet spot, which it’s heavy concentration allows you to do without impacting your assessment of other data. Unrolled GANs operate on this key question: what if we could give the generator an ability to be less short-sighted, and make moves that aren’t just optimizing for the present, but are also defensive against the future, in ways that will hopefully tamp down on this running-around-in-circles dynamic illustrated above. If the generator was incentivized not only to make moves that fool the current discriminator, but also make moves that make the next-step discriminator less likely to tell it apart, the hope is that it will spread out its mass more, and be less likely to fall into the hole of a mode collapse. 

This intuition was realized in UnrolledGANs, through a mathematical approach that is admittedly a little complex for this discussion format. Essentially, in addition to the typical GAN loss (which is based on the current values of the generator and discriminator), this model also takes one “step forward” of the discriminator (calculates what the new parameters of the discriminator would be, if it took one update step), and backpropogates backward through that step. The loss under the next-step discriminator parameters is a function of both the current generator, and the next-step parameters, which come from the way the discriminator reacts to the current generator. When you take the gradient with respect to the generator of both of these things, you get something very like the ideal we described earlier: a generator that is trying to put its mass into areas the current discriminator sees as high-probability, but also change its parameters such that it gives the discriminator a less effective response strategy. 

https://i.imgur.com/0eEjm0g.png

Empirically: UnrolledGANs do a quite good job at their stated aim of reducing mode collapse, and the unrolled training procedure is now a common building-block technique used in other papers. 

Over the last five years, artificial creative generation powered by ML has blossomed. We can now imagine buildings based off of a sketch, peer into the dog-tiled “dreams” of a convolutional net, and, as of 2017, turn images of horses into ones of zebras. This last problem - typically termed image-to-image translation- is the one that CycleGAN focuses on. The kinds of transformations that can full under this category is pretty conceptually broad: zebras to horses, summer scenes to winter ones, images to Monet paintings. (Note: I switch between using horse/zebra as my explanatory example, and using summer/winter. Both have advantages for explaining different conceptual poinfts) However, the idea is the same: you start with image a, which belongs to set A, and you want to generate a mapping of that image into set B, where the only salient change is that it’s now in set B. As a clarifying example: if you started out with a horse, and your goal was to translate it into a zebra, you would hope that the animal is in the same size, relative position, and pose, and that the only element that changed was changing the quality of “horseness” for the quality of “zebraness”. 

https://i.imgur.com/NCExS7A.png

The real trick of CycleGAN is the fact that, unlike prior attempts to solve this problem, they didn’t use paired data. This is understandable, given the prior example: while it’s possible to take a picture of a scene in both summer and winter, you obviously can’t convert a horse into a zebra so that you can take a “paired” picture of it in both forms. When you have paired data, this is a reasonably well-defined problem: you want to learn some mathematical transformation to turn a specific summer image into a specific winter one, and you can use the ground truth winter image as explicit supervision. Since they lack this per-image cross-domain ground truth, the authors of this paper take what would be one question (“is the winter version of this image that the network generated close to the actual known winter version of this image”) and decompose it into two: 
Does the winter version of this original summer image looks like it belongs to the set of winter images? This is enforced by a GAN-style discriminator, which takes in outputs of the summer->winter generator, and true images of winter, and tries to tell them apart. This loss component pushes generated winter images to have the quality of “winterness”. This is the “Adversarial Loss” 
Does the winter version of this image contain enough information about this specific original summer image to accurately reconstruct it with an inverted (winter -> summer) generator? This constraint pushes the generator to actually translate aspects of this specific image between summer and winter. Without it, as the authors of the paper showed, the model has no incentive to actually do translation, and instead just generates winter images that have nothing to do with the summer image (and, frequently experience mode collapse: only generating a single winter image over and over again). This is termed the “Cycle Consistency Loss” 

It’s actually the case that there are two versions of both of the above networks; that’s what puts the “cycle” in CycleGAN. In addition to a loss ensuring you can map summer -> winter -> summer, there’s another one ensuring the other direction, winter -> summer -> winter holds as well. And, for both of those directions, we use the adversarial loss on the middle “translated” image, and a cycle consistency loss on the last “reconstructed” image. A key point here is that, because of the inherent structure of this loss function requires mapping networks going in both directions, training a winter->summer generator gets you a summer-> winter one for free. 

(Note: this is a totally different model architecture than most of the “style transfer” applications you likely previously seen, though when applied to photograph -> painting translation, it can have similar results) 

Concern about the issue of fairness (or the lack of it) in machine learning models is gaining widespread visibility among general public, the governments as well as the researchers. This is especially alarming as AI enabled systems are becoming more and more pervasive in our society as decisions are being taken by AI agents in healthcare to autonomous driving to criminal justice and so on. Bias in any dataset is, in some way or other, a reflection of the general attitude of humankind towards different activities which are typified by certain gender, race or ethnicity. As these datasets are the sources of knowledge  for these AI models (especially the multimodal end-to-end models which depend only on the human annotated training datasets for literally everything), their decision making ability also gets shadowed by the bias in the dataset. This paper makes an important observation about the image captioning models that these models not only explore the bias in the dataset but tend to exaggerate them during inference. This is definitely a shortcoming of the current supervised models which are marked by their over-reliance on image context. The related works section of the paper (Section 2 first part: “Unwanted Dataset Bias”) gives an extensive review of the types of bias in the dataset and of the few recent works trying to address them. Gender bias (Presence of woman in kitchen makes most of us to guess a woman in a kitchen scene in case the person is not clearly apprehensible in the scene or a male is supposed to snowboard more often than a woman) and reporting biases (over reporting less common co-occurrences, such as “male nurse” or “green banana”) are two of the many present in machine learning datasets.

The paper addresses the problem of fair caption generation that would not presume a specific gender without appropriate evidence for that gender. This is done by introducing an ‘Equalizer Model’. This includes two complementary losses in addition to the normal cross entropy loss for the image captioning systems. The Appearance Confusion Loss (ACL) encourages the model to generate gender neutral words (for example ‘person’) when an image does not contain enough evidence of gender. During training, images of persons are masked out and the loss term encourages the gender words (“man” and “woman”) to have equal probability i.e., the model is encouraged to get confused when it should get confused instead of hallucinating from the context. The loss expression is pretty much intuitive (eqn (2) and (3)). However, it is not a good idea to make a model confused only. Thus the other loss (the Confident Loss (Conf)) is introduced. This loss encourages the model to predict gender words and predict them correctly when there is enough evidence of gender in the image. The loss function (eqns. (4) and (5)) has an intelligent use of the quotient between predicted probabilities of male and female gender words. If I have to give a single take away line from the paper then it will be the following which summarizes the working principle behind the two losses very succinctly.
> > “These complementary losses allow the Equalizer model to encourage models to be cautious in the absence of gender information and discriminative in its presence.”

The experiments are also well thought out. For experimentations, 3 different versions of the MSCOCO dataset is created - MSCOCO-Bias, MSCOCO-Confident and MSCOCO-Balanced. The bias in the gender gradually decreases in these 3 datasets. Three different metrics are also used to evaluate the model - Error rate (fraction of man/woman misclassifications), gender ratio (how close the gender ratio in the predicted captions of the test set is to the ground truth gender ratio), right for right reasons (whether the visual evidence used by the model for the prediction of the gender words coincide with the person images). There are a few baseline models and ablation studies. The baselines considered a naive image captioning model (‘Show and Tell’ approach), an approach where images corresponding to less common gender are sampled more while training and another baseline where the gender words are given higher weights in the cross-entropy loss. The ablation models considered the two losses (ACL and Conf) separately. For all the datasets, the proposed equalizer model consistently performed well according to all the 3 metrics. The experiments also show that, as the evaluation datasets become more and more balanced (i.e., the gender distribution departs more and more from the biased gender distribution in the training dataset), the performance of all the models falls away. However, the proposed model performs the best with the least inconsistency of performance among the the datasets. The qualitative examples with grad-cam and sliding window saliency maps for the gender words are also a positive point of the paper.

Things I would have liked the paper to contain:
* There are a few confusions in the expression of the conf loss in eqn. (4). Specifically, I am not sure what is the difference between $w_t$ and $\tilde{w}_t$. It seems the first one is the ground truth word and the later is the predicted word. It would have been good to have a clarification.

Overall, the paper is very new in defining the problem and in solving it. The solution strategy is very intuitive and easy to grasp. The paper is well written too. We can, sincerely, hope that this type of works addressing problems at the intersection of machine learning and societal issues would come more frequently and the discussed paper is a very significant first step towards it.

They created a really nice trick to optimize the $ {L}_{0} $ Pseudo Norm - Regularization on the sorted (By magnitude) values of the optimization variable.

Their code is available at - [The Trimmed Lasso: Sparsity and Robustness](https://github.com/copenhaver/trimmedlasso).