Last edited February 26, 2017.
I really, really like the Wasserstein GAN paper. I know it’s already gotten a lot of hype, but I feel like it could use more.
I also think the theory in the paper scared off a lot of people, which is a bit of a shame. This is my contribution to make the paper more accessible, while hopefully retaining the thrust of the argument.
Why Is This Paper Important?
There’s a giant firehose of machine learning papers - how do you decide which ones are worth reading closely?
For Wasserstein GAN, it was mostly compelling word of mouth.
- The paper proposes a new GAN training algorithm that works well on the common GAN datasets.
- Said training algorithm is backed up by theory. In deep learning, not all theory-justified papers have good empirical results, but theory-justified papers with good empirical results have really good empirical results. For those papers, it’s very important to understand their theory, because the theory usually explains why they perform so much better.
- I heard that in Wasserstein GAN, you can (and should) train the discriminator to convergence. If true, it would remove needing to balance generator updates with discriminator updates, which feels like one of the big sources of black magic for making GANs train.
- The paper shows a correlation between discriminator loss and perceptual quality. This is actually huge if it holds up well. In my limited GAN experience, one of the big problems is that the loss doesn’t really mean anything, thanks to adversarial training, which makes it hard to judge if models are training or not. Reinforcement learning has a similar problem with its loss functions, but there we at least get mean episode reward. Even a rough quantitative measure of training progress could be good enough to use automated hyperparam optimization tricks, like Bayesian optimization. (See this post and this post for nice introductions to automatic hyperparam tuning.)
Additionally, I buy the argument that GANs have close connections to actor-critic reinforcement learning. (See Pfau & Vinyals 2017.) RL is definitely one of my research interests. Also, GANs are taking over the world; I should probably keep an eye on GAN papers anyways.
At this point, you may want to download the paper yourself, especially if you want more of the theoretical details. To aid anyone who takes me up on this, the section names in this post will match the ones in the paper.
The paper begins with background on generative models.
When learning generative models, we assume the data we have comes from some unknown distribution . (The r stands for real.) We want to learn a distribution that approximates , where are the parameters of the distribution.
You can imagine two approaches for doing this.
- Directly learn the probability density function . Meaning, is some differentiable function such that and . We optimize through maximum likelihood estimation.
- Learn a function that transforms an existing distribution into . Here, is some differentiable function, is a common distribution (usually uniform or Gaussian), and .
The paper starts by explaining why the first approach runs into problems.
Given function , the MLE objective is
In the limit, this is equivalent to minimizing the KL-divergence .
Aside: Why Is This True?
Recall that for continuous distributions and , the KL divergence is
In the limit (as ), samples will appear based on the data distribution , so
(Derivations in order: limit of summation turns into integral, flip max to min by negating, add a constant that doesn’t depends on , and apply definition of KL divergence.)
Note that if at an where , the KL divergence goes to . This is bad for the MLE if has low dimensional support, because it’ll be very unlikely that all of lies within that support. If even a single data point lies outside ’s support, the KL divergence will explode.
To deal with this, we can add random noise to when training the MLE. This ensures the distribution is defined everywhere. But now we introduce some error, and empirically people have needed to add a lot of random noise to make models train. That kind of sucks. Additionally, even if we learn a good density , it may be computationally expensive to sample from .
This motivates the latter approach, of learning a (a generator) to transform a known distribution . The other motivation is that it’s very easy to generate samples. Given a trained , simply sample random noise , and evaluate . (The downside of this approach is that we don’t explicitly know what , but in practice this isn’t that important.)
To train (and by extension ), we need a measure of distance between distributions.
(Note: I will use metric, distance function, and divergence interchangeably. I know this isn’t technically accurate. In particular metric and divergence mean different things. I apologize in advance, the three are all heavily conflated in my head.)
Different metrics (different definitions of distance) induce different sets of convergent sequences. We say distance is weaker than distance if every sequence that converges under converges under .
Looping back to generative models, given a distance , we can treat as a loss function. Minimizing with respect to will bring close to . This is principled as long as the mapping is continuous (which will be true if is a neural net).
We know we want to minimize , but how do we define ? This section compares various distances and their properties.
Now, I’ll be honest, my measure theory is pretty awful. And the paper immediately starts talking about compact metric sets, Borel subsets, and so forth. This is admirable from a theory standpoint. However, in machine learning, we’re usually working with functions that are “nice enough” (differentiable almost everywhere), and can therefore ignore many of the precise definitions without destroying the argument too much. As long as we aren’t doing any bullshit like the Cantor set, we’re good.
On to the distances at play.
- The Total Variation (TV) distance is
The Kullback-Leibler (KL) divergence is
This isn’t symmetric. The reverse KL divergence is defined as .
The Jenson-Shannon (JS) divergence: Let be the mixture distribution . Then
Finally, the Earth Mover (EM) or Wasserstein distance: Let be the set of all joint distributions whose marginal distributions are and . Then.
Aside: What’s Up With The Earth Mover Definition?
The EM distance definition is a bit opaque. It took me a while to understand it, but I was very pleased once I did.
First, the intuitive goal of the EM distance. Probability distributions are defined by how much mass they put on each point. Imagine we started with distribution , and wanted to move mass around to change the distribution into . Moving mass by distance costs effort. The earth mover distance is the minimal effort we need to spend.
Why does the infimum over give the minimal effort? You can think of each as a transport plan. To execute the plan, for all move mass from to .
Every strategy for moving weight can be represented this way. But what properties does the plan need to satisfy to transform into ?
- The amount of mass that leaves is . This must equal , the amount of mass originally at .
- The amount of mass that enters is . This must equal , the amount of mass that ends up at .
This shows why the marginals of must be and . For scoring, the effort spent is Computing the infinum of this over all valid gives the earth mover distance.
Now, the paper introduces a simple example to argue why we should care about the Earth-Mover distance.
Consider probability distributions defined over . Let the true data distribution be , with sampled uniformly from . Consider the family of distributions , where , with also sampled from .
Real and fake distribution when
We’d like our optimization algorithm to learn to move to , As , the distance should decrease. But for many common distance functions, this doesn’t happen.
- Total variation: For any , let . This gives
- KL divergence and reverse KL divergence: Recall that the KL divergence is if there is any point where and . For , this is true at . For , this is true at .
- Jenson-Shannon divergence: Consider the mixture , and now look at just one of the KL terms.
For any where , , so this integral works out to . The same is true of , so the JS divergence is
- Earth Mover distance: Because the two distributions are just translations of one another, the best way transport plan moves mass in a straight line from to . This gives
This example shows that there exist sequences of distributions that don’t converge under the JS, KL, reverse KL, or TV divergence, but which do converge under the EM distance.
This example also shows that for the JS, KL, reverse KL, and TV divergence, there are cases where the gradient is always . This is especially damning from an optimization perspective - any approach that works by taking the gradient will fail in these cases.
Admittedly, this is a contrived example because the supports are disjoint, but the paper points out that when the supports are low dimensional manifolds in high dimensional space, it’s very easy for the intersection to be measure zero, which is enough to give similarly bad results.
This argument is then strengthened by the following theorem.
Let be a fixed distribution. Let be a random variable. Let be a deterministic function parametrized by , and let . Then,
- If is continuous in , so is .
- If is sufficiently nice, then is continuous everywhere, and differentiable almost everywhere.
- Statements 1-2 are false for the Jensen-Shannon divergence and all the KLs.
You’ll need to refer to the paper to see what “sufficiently nice” means, but for our purposes it’s enough to know that it’s satisfied for feedfoward networks that use standard nonlinearites. Thus, out of JS, KL, and Wassertstein distance, only the Wasserstein distance has guarantees of continuity and differentiability, which are both things you really want in a loss function.
The second theorem shows that not only does the Wasserstein distance give better guarantees, it’s also the weakest of the group.
Let be a distribution, and be a sequence of distributions. Then, the following are true about the limit.
- The following statements are equivalent.
- with the total variation distance.
- with the Jensen-Shannon divergence.
- The following statements are equivalent.
- , where represents convergence in distribution for random variables.
- or imply the statements in (1).
- The statements in (1) imply the statements in (2).
Together, this proves that every distribution that converges under the KL, reverse-KL, TV, and JS divergences also converges under the Wasserstein divergence. It also proves that a small earth mover distance corresponds to a small difference in distributions.
Combined, this shows the Wasserstein distance is a compelling loss function for generative models.
Unfortunately, computing the Wasserstein distance exactly is intractable. Let’s repeat the definition.
The paper now shows how we can compute an approximation of this.
A result from Kantorovich-Rubinstein duality shows is equivalent to
where the supremum is taken over all -Lipschitz functions.
Aside: What Does Lipschitz Mean?
Let and be distance functions on spaces and . A function is -Lipschitz if for all ,
Intuitively, the slope of a -Lipschitz function never exceeds , for a more general definition of slope.
If we replace the supremum over -Lipschitz functions with the supremum over -Lipschitz functions, then the supremum is instead. (This is true because every -Lipschitz function is a -Lipschitz function if you divide it by , and the Wasserstein objective is linear.)
The supremum over -Lipschitz functions is still intractable, but now it’s easier to approximate. Suppose we have a parametrized function family , where are the weights and is the set of all possible weights. Further suppose these functions are all -Lipschitz for some . Then we have
For optimization purposes, we don’t even need to know what is! It’s enough to know that it exists, and that it’s fixed throughout training process. Sure, gradients of will be scaled by an unknown , but they’ll also be scaled by the learning rate , so will get absorbed into the hyperparam tuning.
If contains the true supremum among -Lipschitz functions, this gives the distance exactly. This probably won’t be true. In that case, the approximation’s quality depends on what -Lipschitz functions are missing from .
Now, let’s loop all this back to generative models. We’d like to train to match . Intuitively, given a fixed , we can compute the optimal for the Wasserstein distance. We can then backprop through to get the gradient for .
The training process has now broken into three steps.
- For a fixed , compute an approximation of by training to convergence.
- Once we find the optimal , compute the gradient by sampling several .
- Update , and repeat the process.
There’s one final detail. This entire derivation only works when the function family is -Lipschitz. To guarantee this is true, we use weight clamping. The weights are constrained to lie within , by clipping after every update to .
The full algorithm is below.
Aside: Compare & Contrast: Standard GANs
Let’s compare the WGAN algorithm with the standard GAN algorithm.
In GANS, the discriminator maximizes
where we constraint to always be a probabiity .
In WGANs, nothing requires to output a probability. This explains why the authors tend to call the critic instead of the discriminator - it’s not explicitly trying to classify inputs as real or fake.
- The original GAN paper showed that in the limit, the maximum of the discriminator objective above is the Jenson-Shannon divergence, up to scaling and constant factors.
In WGANs, it’s the Wasserstein distance instead.
- Although GANs are formulated as a min max problem, in practice we we never train to convergence. In fact, usually the discriminator is too strong, and we need to alternate gradient updates between and to get reasonable generator updates.
We aren’t updating against the Jenson-Shannon divergence, or even an approximation of the Jenson-Shannon divergence, we’re updating against an objective that kind of aims towards the JS divergence, but doesn’t go all the way. It certainly works, but in light of the points this paper makes about gradients of the JS divergence, it’s a bit surprising it does work.
In contrast, because the Wasserstein distance is differentiable nearly everywhere, we can (and should) train to convergence before each generator update, to get as accurate an estimate of as possible. (The more accurate is, the more accurate the gradient .)
First, the authors set up a small experiment to showcase the difference between GAN and WGAN. There are two 1D Gaussian distributions, blue for real and green for fake. Train a GAN discriminator and WGAN critic to optimality, then plot their values over the space. The red curve is the GAN discriminator output, and the cyan curve is the WGAN critic output.
Both identify which distribution is real and which is fake, but the GAN discriminator does so in a way that makes gradients vanish over most of the space. In contrast, the weight clamping in WGAN gives a reasonably nice gradient over everything.
Next, the Wasserstein loss seems to correlate well with image quality. Here, the authors plot the loss curve over time, along with the generated samples.
After reading through the paper, this isn’t too surprising. Since we’re training the critic to convergence, these critic’s value should be good approximations of , where is whatever the Lipschitz constant is. As argued before, a low means and are “close” to one another. It would be more surprising if the critic value didn’t correspond to visual similarity.
The image results also look quite good. Compared to the DCGAN baseline on the bedroom dataset, it performs about as well.
Top: WGAN with the same DCGAN architecture. Bottom: DCGAN
If we remove batch norm from the generator, WGAN still generates okay samples, but DCGAN fails completely.
Top: WGAN with DCGAN architecture, no batch norm. Bottom: DCGAN, no batch norm.
Finally, we make the generator a feedforward net instead of a convolutional one. This keeps the number of parameters the same, while removing the inductive bias from convolutional models. The WGAN samples are more detailed, and don’t mode collapse as much as standard GAN. In fact, they report never running into mode collapse at all for WGANs!
Top: WGAN with MLP architecture. Bottom: Standard GAN, same architecture.
The read-through of the paper ends here. If you’re interested in the Related Work, or the theorem proofs in the Appendix, you’ll need to read the paper.
This is a rich enough paper to have several natural follow-up questions.
The weights in are clamped to . How important is for performance? Based on lurking /r/MachineLearning, the tentative results say that low trains more reliably, but high trains faster when it does work. I imagine it’s because there’s a discrepancy between and , and that discrepancy changes with . There could be interesting work in describing that discrepancy, or in finding ways to bring closer to -Lipschitz functions while still be optimizable.
Given a fixed critic architecture and fixed for clamping, can we quantitatively compare different generators by computing the Wasserstein estimate of both? Again, remember there’s an approximation error from optimizing over instead of , so we may not be able to do much. However, because we fix both the critic architecture and , the hope is that most of the error is some universal error that appears in all distributions. If the approximation error doesn’t change too much between distributions, this would give a way to judge generation quality without relying on Mechanical Turk. (And if the error does change a lot, it would probably be interesting to investigate when that happens.)
The constant depends on both and the model architecture, and therefore we can’t directly compare the critics between models with different architectures. Is there a way to estimate ? Recall the critic objective converges to , so dividing by would normalize the difference between architectures.
(This actually seems pretty straightforward. Take either a random generator or pretrained generator, then train critics from varying architectures and compare their final values. Again, the approximation error could complicate this, but this could be a way to analyze the approximation error itself. Given a few different generators, the change in estimated between different distributions would show how important the distribution is to the approximation error.)
How important is it to train the critic to convergence? A converged critic gives the most accurate gradient, but it takes more time. In settings where that’s impractical, can a simple alternating gradient scheme work?
What ideas from this work are applicable to actor-critic RL? At a first glance, I’m now very interested in investigating the magnitude of the actor gradients. If they tend to be very large or very small, we may have a similar saturation problem, and adding a Lipschitz bound through weight clamping could help.
These ideas apply not just to generative models, but to general distribution matching problems. Are there any low-hanging distribution matching problems that use the Jenson-Shannon or KL divergence instead of the Wasserstein distance? One example of this is the Generative Adversarial Imitation Learning paper. After a decent amount of theory, it derives a GAN-like algorithm for imitation learning. Switching the discriminator to a WGAN approach may give some straightforward wins.
The ideas here are all old news, but some anvils need to be dropped.
Many arguments, especially political ones, fail because both parties argue from a default position. The implicit logic goes roughly like this.
- I believe X.
- The other person believes Y.
- X is the right belief.
- If I can show that the other person’s reasons for believing Y are wrong, they’ll believe X instead.
The first two are prerequisites to an argument. The third is a matter of debate. The fourth is where everything goes to shit.
Stop me if you’ve heard this story before. Two people on the Internet are arguing. One believes X, the other believes Y. The X believer comes up with a reason someone would believe Y, then writes a scathing comment attacking it. The Y believer comes up with a reason someone would believe X, and attacks that too. Meanwhile, neither has addressed the actual reason one believes X and the other believes Y.
Y supporters read all the stupid rebuttals made against X supporters. “Wow, Y supporters think we believe X because of that?” X supporters read all the stupid rebuttals made against Y supporters. “Wow, X supporters think we believe Y because of that?”
Both, at least, agree on one thing. “Why should I listen to this people? Their reasoning is garbage, and whenever you try to point it out they hurl garbage back.”
But the real problem is that you think their reasoning is garbage, and they think your reasoning is garbage, when neither may be true.
Your position is not the default position. No one’s is. Pointing out problems in other people’s arguments only shows there are problems in those arguments. It doesn’t make your argument any more valid. Besides, who says your argument would stand up to the same level of scrutiny? All you’ve really shown is that the people you’re talking to don’t have strong counterarguments. How do you know no strong counterarguments exist? And if they do exist, why should people believe you’re any more right?
Okay, look. I’ll be blunt. Some people do believe in things for stupid reasons. The harm comes when you generalize the stupidity of some arguments to justify ignoring all arguments. If you’re an X thinker, and want to understand how anyone could believe Y, you should assume that at some point, somebody came up with an intellectually consistent argument for Y. (If no good argument for Y exists, how did anyone start believing Y in the first place?) As belief in Y spread from person to person, the original argument got muddled, like a game of ideological telephone. By the time it got to your filter bubble, it’s been distorted into something weaker than it was. So don’t generalize the distorted argument that reaches your eyes and ears. Look harder.
Or, if you don’t have time to look? Take things slowly. Recognize that problems are complicated, and there’s more than meets the eye. Recognize the diversity of belief.
(And yes, I notice the irony of me dictating all these ideas, of arguing against the default position problem from my own default position. Far be it from me to expect people to believe my ideas about epistemology without question. Just please, keep these ideas in mind. Remember it’s hard to argue well, and that it’s worth thinking about problems like this. Something tells me we’re going to need good epistemology a lot in the upcoming years.)
I’ve been working on my current research project for around four to six months. The results haven’t been that good. I have a meeting about it tomorrow, and I’d say there’s about a 50-50 chance we decide I should stop working on the project and switch to another one. If it ends that way, well, that sucks. But after reflecting on it for a while, I think I’m mentally prepared for that outcome.
In preparation for that meeting, my manager recommended I write a document describing what the project is, how it got to where it is now, and all the things I tried along the way. I agreed this was a good idea, although I was starting to get sick of explaining my project. In the past, I wrote a doc when I proposed the project, then made a presentation after it got refined, then wrote another doc when I pivoted. Ironically, all that documentation made it hard to figure out where the project was, because each doc built on the previous ones.
From a naive viewpoint, writing a summary shouldn’t have taken that long. I’ve been living and breathing this project for months, and I had already written several documents on the project. How long could writing one more take?
Well, turns out it took me about a day. That’s actually not too long, all things considered, but why did it have to take that long at all? It was only summarizing things I’d already written.
The tricky part of research is that the author is trying to take these grand ideas, that have never been arranged the way the author’s arranged them, and then he or she needs to convert them into text that gets the point across.
Yasha Berchenko-Kogan (who I actually kinda know in real life) has a great quote about communicating research.
Communicating these ideas is a bit like trying to explain a vacuum cleaner to someone who has never seen one, except you’re only allowed to use words that are four letters long or shorter.
A lot of the metaphorical four letter words I used weren’t useful anymore, and filtering it to just the key points was most of the work.
When I got comments back, I had a shocking revelation: the only one who fully understood my project was me.
On the surface, that’s not so surprising, so let me clarify. I’ve been talking to most of these people about my project for months. They’ve received and read every doc I wrote about my project, from start to finish. I’ve had weekly meetings with several of these people, to explain issues I ran into and get tips on how to get past them.
And the whole time, I was the only one who understood everything that was going on?
What the hell?
I used to think good writing was important because nobody wants to extend research that’s hard to understand.
But now I’ve realized there’s another big benefit. If you want people to give useful feedback, it’s very important they actually understand what the problem is.
I’m about to describe one of those ideas that’s obviously true. Are you ready? Okay, good.
When people say “That sounds good to me”, they’re implicitly saying “Based on my understanding of the problem, that sounds good to me.” Worryingly, if they missed a few details you tried to convey, their brains will fill in the details by themselves. And the details they fill in will be ones that make sense to them, which likely differs from the details you were trying to convey in the first place.
The relevant point: if you ask for help from your advisor, and your problem is in one of those details you failed to convey, you’re not going to get good advice.
If you yourself are a researcher, and this was a new idea for you, read over it a few times. I think it’s worth knowing.
In hindsight, the biggest shifts in my research didn’t happen when I had a new idea. They happened when experienced researchers finally grasped a detail I thought they’d already understood. They would then immediately make an obvious argument for why doing it that way was dumb, and then I’d feel like an idiot.
If everyone’s busy climbing your chain of reasoning, they won’t notice the weak links, because they’ll assume all the links make sense.
That suggests two things:
- Prefer simple solutions over complicated ones. Complicated solutions are hard to explain, meaning it’s harder to get good feedback, and complicated solutions are the ones that need feedback the most.
- If you do need to use a complicated solution, be very, very careful that you understand why you’re doing everything you’re doing, and that the reasons you made up four months ago still make sense in the present.
If I had explained my research project better, I could have saved a few weeks of work, because I could have avoided a lot of silly ideas that weren’t worth doing in retrospect.
Research miscommunication is inevitable. It takes a long time to explain things properly, and usually people don’t have the time for that. I suspect that’s the main reason research takes so long - it’s the blind leading the blind. Sometimes the blind one is you. Sometimes it’s someone else. It’s hard to tell the difference.
If the pattern holds, people aren’t going to understand everything I wanted them to understand from this post. Oh well. I tried.