Wasserstein Generative Adversarial Networks

1 Wasserstein Generative Adversarial NetworksMartin Arjovsky1 Soumith Chintala2L eon Bottou1 2 AbstractWe introduce a new algorithm named WGAN,an alternative to traditional GAN training. Inthis new model, we show that we can improvethe stability of learning, get rid of problems likemode collapse, and provide meaningful learningcurves useful for debugging and hyperparametersearches. Furthermore, we show that the cor-responding optimization problem is sound, andprovide extensive theoretical work highlightingthe deep connections to different distances be-tween IntroductionThe problem this paper is concerned with is that of unsu-pervised learning.

2 Mainly, what does it mean to learn aprobability distribution? The classical answer to this is tolearn a probability density. This is often done by defininga parametric family of densities(P ) Rdand finding theone that maximized the likelihood on our data: if we havereal data examples{x(i)}mi=1, we would solve the problemmax Rd1mm i=1logP (x(i))If the real data distributionPradmits a density andP is thedistribution of the parametrized densityP , then, asymp-totically, this amounts to minimizing the Kullback-LeiblerdivergenceKL(Pr P ).For this to make sense, we need the model densityP toexist.

3 This is not the case in the rather common situationwhere we are dealing with distributions supported by lowdimensional manifolds. It is then unlikely that the modelmanifold and the true distribution s support have a non-negligible intersection (see (Arjovsky & Bottou, 2017)),and this means that the KL distance is not defined (or sim-ply infinite).1 Courant Institute of Mathematical Sciences, NY2 FacebookAI Research, NY. Correspondence to: Martin of the34thInternational Conference on MachineLearning, Sydney, Australia, PMLR 70, 2017. Copyright 2017by the author(s).The typical remedy is to add a noise term to the model dis-tribution.

4 This is why virtually all Generative models de-scribed in the classical machine learning literature includea noise component. In the simplest case, one assumes aGaussian noise with relatively high bandwidth in order tocover all the examples. It is well known, for instance, thatin the case of image generation models, this noise degradesthe quality of the samples and makes them blurry. For ex-ample, we can see in the recent paper (Wu et al., 2016)that the optimal standard deviation of the noise added tothe model when maximizing likelihood is around toeach pixel in a generated image, when the pixels were al-ready normalized to be in the range[0,1].

5 This is a veryhigh amount of noise, so much that when papers report thesamples of their models, they don t add the noise term onwhich they report likelihood numbers. In other words, theadded noise term is clearly incorrect for the problem, but isneeded to make the maximum likelihood approach than estimating the density ofPrwhich may not ex-ist, we can define a random variableZwith a fixed dis-tributionp(z)and pass it through a parametric functiong :Z X(typically a neural network of some kind)that directly generates samples following a certain distribu-tionP . By varying , we can change this distribution andmake it close to the real data distributionPr.

6 This is use-ful in two ways. First of all, unlike densities, this approachcan represent distributions confined to a low dimensionalmanifold. Second, the ability to easily generate samples isoften more useful than knowing the numerical value of thedensity (for example in image superresolution or semanticsegmentation when considering the conditional distributionof the output image given the input image). In general, itis computationally difficult to generate samples given anarbitrary high dimensional density (Neal, 2001).Variational Auto-Encoders (VAEs) (Kingma & Welling,2013) and Generative Adversarial Networks (GANs)(Goodfellow et al.)

7 , 2014) are well known examples of thisapproach. Because VAEs focus on the approximate likeli-hood of the examples, they share the limitation of the stan-dard models and need to fiddle with additional noise offer much more flexibility in the definition of theobjective function, including Jensen-Shannon (Goodfellowet al., 2014), and allf-divergences (Nowozin et al., 2016)as well as some exotic combinations (Huszar, 2015). OnWasserstein Generative Adversarial Networksthe other hand, training GANs is well known for being del-icate and unstable, for reasons theoretically investigated in(Arjovsky & Bottou, 2017).

8 In this paper, we direct our attention on the various ways tomeasure how close the model distribution and the real dis-tribution are, or equivalently, on the various ways to definea distance or divergence (P ,Pr). The most fundamen-tal difference between such distances is their impact on theconvergence of sequences of probability distributions. Asequence of distributions(Pt)t Nconverges if and only ifthere is a distributionP such that (Pt,P )tends to zero,something that depends on how exactly the distance isdefined. Informally, a distance induces a weaker topol-ogy when it makes it easier for a sequence of distributionto 2 clarifies how popular probabilitydistances differ in that order to optimize the parameter , it is of course desir-able to define our model distributionP in a manner thatmakes the mapping 7 P continuous.

9 Continuity meansthat when a sequence of parameters tconverges to , thedistributionsP talso converge toP . However, it is essen-tial to remember that the notion of the convergence of thedistributionsP tdepends on the way we compute the dis-tance between distributions. The weaker this distance, theeasier it is to define a continuous mapping from -space toP -space, since it s easier for the distributions to main reason we care about the mapping 7 P to becontinuous is as follows. If is our notion of distance be-tween two distributions, we would like to have a loss func-tion 7 (P ,Pr)that is continuous, and this is equivalentto having the mapping 7 P be continuous when usingthe distance between distributions.

10 The contributions of this paper are: In Section 2, we provide a comprehensive theoreticalanalysis of how the Earth Mover (EM) distance be-haves in comparison to popular probability distancesand divergences used in the context of learning distri-butions. In Section 3, we define a form of GAN calledWasserstein-GAN that minimizes a reasonable and ef-ficient approximation of the EM distance, and we the-oretically show that the corresponding optimizationproblem is sound. In Section 4, we empirically show that WGANs curethe main training problems of GANs. In particular,training WGANs does not require maintaining a care-ful balance in training of the discriminator and the1 More exactly, the topology induced by is weaker than thatinduced by when the set of convergent sequences under is asuperset of that under.