Evaluating GANs via Duality

1 INTRODUCTION . In the past few years , generative models have become extremely popular in the machine learning community . This is largely due to the recent advances in the field of deep learning , which allowed deep neural generators to produce remarkable results for various tasks , including for example image generation ( Radford et al. , 2015 ) . Two notable approaches in this area are variational auto-encoders ( VAEs ) ( Kingma & Welling , 2013 ; Rezende et al. , 2014 ) , and generative adversarial networks ( GAN ) ( Goodfellow et al. , 2014 ) . In this paper , we focus on GANs , which are especially attractive as they circumvent the notoriously hard optimization of the data likelihood and instead use an adversarial game approach for training a generator . Let us denote the data distribution by pdata ( x ) and the model distribution by pu ( x ) . A probabilistic discriminator is denoted by Dv : x 7→ [ 0 ; 1 ] and a generator by Gu : z 7→ x . The GAN objective is : min u max v M ( u , v ) = 1 2 Ex∼pdata logDv ( x ) + 1 2 Ez∼pz log ( 1−Dv ( Gu ( z ) ) ) . ( 1 ) Each of the two players ( generator/discriminator ) tries to optimize their own objective , which is exactly balanced by the loss of the other player , thus yielding a two-player zero-sum minimax game . Standard GAN approaches aim at finding a pure Nash Equilibrium by using traditional gradientbased techniques to minimize each players cost in an alternating fashion . However , the minimax objective of GANs makes the optimization process challenging . One of the central open issues is the non-convergence problem , which in practice leads to oscillations between different kinds of generated samples ( Metz et al. , 2016 ) . Many different techniques have been proposed to address the shortcomings of the original technique introduced by Goodfellow et al . ( 2014 ) . Given the plethora of GAN-like approaches as well as different training methods , the community is now facing the problem of determining which approaches produce ” better ” generative models . This is in itself a challenging question as there is no clear evaluation criterion . In a recent large-scale empirical study , Lucic et al . ( 2018 ) showed that using various evaluation measures , many state-ofthe-art models are able to reach similar scores with enough tuning . In a recent survey , Borji ( 2018 ) discussed the merits of various evaluation metrics , pointing out that there is no clear consensus regarding which metric is the most appropriate to evaluate the quality of a generative model . While many metrics achieve reasonable discriminability ( i.e . ability to distinguish generated samples from real ones ) , they also tend to have a high computational complexity . Many existing metrics are also specific to datasets of natural images , but we are also starting to notice the success of GANs in other fields such as in cosmology ( Rodriguez et al. , 2018 ) or in the medical domain ( Schlegl et al. , 2017 ) . Another problem related to the evaluation of samples produced by a GAN is the lack of evaluation procedures to detect convergence . While various approaches have analyzed the convergence properties of alternating gradient-based techniques , most of them require the objective function to be convex-concave ( Goodfellow et al. , 2014 ; Nowozin et al. , 2016 ) with the exception of Grnarova et al . ( 2018 ) who prove convergence for semi-concave zero-sum games . In general , alternating gradient descent can fail to converge ( Salimans et al. , 2016 ) . While this non-convergence behavior can in practice be visually recognized for some low-dimensional examples ( such as a 2D mixture of Gaussian ) , this is in general more difficult in high-dimensional spaces due to the lack of a convergence metric . This has been pointed out before ( see e.g . Mescheder et al . ( 2018 ) and ( Fedus et al. , 2018 ) who examine convergence on 2D problems and problems for which the true data distribution is known . One of the common challenges that practitioners are facing is when to stop training . In particular , it is well known that the curves of the discriminator and generator losses oscillate ( see Figures 12 and 13 ) and are non-informative as to whether the model is improving or not ( Arjovsky et al. , 2017 ) . This is especially troublesome when a GAN is trained on non-image data in which case one might not be able to use visual inspection or FID/Inception score as a proxy . In this paper , our main contribution is to leverage existing ideas from game theory to propose a novel and computationally efficient convergence metric for GANs . The principle we follow is based on the theory of duality that was developed after Von Neumann ( 1928 ) derived the minimax theorem for zero-sum games . From duality , it follows that the minimax and maximin 1 are dual to each other and strong duality holds for feasible problems . This duality also implicitly gives us a way to evaluate the deviation between minimax and maximin . As an example , let ’ s consider a two-player game between the discriminator D and the generator G. The duality gap measures the difference between the values of the game under two scenarios : when D first commits to a strategy , and then G gets to respond ; and when G commits first , and then D gets to respond . In this paper , we advocate the use of the duality gap as a measure of convergence for GANs and demonstrate its effectiveness for various settings commonly considered in the literature . As a second contribution , we also demonstrate how to use the minimax loss value in order to detect mode collapse and measure sample quality . Unlike FID or the Inception score that require labelled data or a domain dependent classifier , our metric is domain independent and does not require labels . Our experiments demonstrate that the duality gap and minimax loss are valuable tools to measure convergence and correlate well with existing evaluation metrics . 2 RELATED WORK . Despite the impressive empirical performance achieved by the latest GAN models ( Karras et al. , 2017 ) , they are still subject to many unanswered questions . Among them is the issue of a fair evaluation procedure . Since the log-likelihood of the data is a common objective function to train a generative model , it would appear to be a sensible metric for GANs . However , its computation is often intractable and Theis et al . ( 2015 ) also demonstrate that it has severe limitations as it might yield low visual quality samples despite of a high likelihood . Perhaps the most popular evaluation metric for GANs is the inception score introduced by Salimans et al . ( 2016 ) that measures both diversity of the generated samples and discriminability . While diversity is measured as the entropy of the output distribution , the discriminability aspect requires a pretrained neural network to assign high scores to images close to training images . Various modifications of the inception score have been suggested . Gurumurthy et al . ( 2017 ) adds a term in the cost function to take into account diversity within samples in a particular class . Heusel et al . ( 2017 ) uses features from a hidden layer of the Inception Net , which are modelled as two multivariate Gaussians for the generated and true data . They then suggest using the Frechet Inception Distance ( FID ) between these two Gaussians to assess the quality of the samples . The authors demonstrate improvements over the inception score , especially more robustness to noise , as well as being able to detect intra-class mode dropping . Furthermore , they also demonstrate high degrees of consistency with human judgments . Although this measure appears to have nice properties from an empirical point of view , the Gaussian assumption might not hold in practice . Furthermore , FID requires labelled data in order to train a classifier . Without labels , transfer learning is possible to datasets under limited conditions ( i.e . the source and target distributions should not be too dissimilar ) . 1In game theory , the maximin value of a player is the highest value that the player is ensured to receive without knowing the actions of the other player ( s ) while the minimax value of a player is the smallest value that the other player ( s ) can force the player to receive , without knowing the player ’ s actions . Another popular metric introduced by Gretton et al . ( 2012 ) is the Maximum Mean Discrepancy ( MMD ) which measures the dissimilarity between pdata and pu using independently drawn samples . It is a specific instance of an integral probability metric . A potential hurdle for MMD is that its computational complexity is quadratic in the sample size , although some linear approximation do exist ( Gretton et al. , 2012 ) . Finally , it also depends on the choice of the kernel . There are many other metrics that have been suggested over the past few years . We refer the reader to Borji ( 2018 ) for a detailed survey . Perhaps the two approaches that are the most relevant to our approach are inspired by skill rating systems in games and were both introduced in Olsson et al . ( 2018 ) . The first method named tournament win rate considers a single model playing against past and future versions of itself . This measure is designed to monitor the progress of a single model as it learns during training . The second method named skill rating measures the aptitude of two different fully trained models . In some way , skill rating and tournament win rate can be seen as an approximation of the worst minimax value we advocate in this paper , where instead of doing a full-on optimization in order to find the best adversary for the fixed generator , the search space is limited to discriminators that are snapshots from training , or discriminators trained with different seeds . We would like to conclude our discussion of related work by pointing out to the vast literature on duality used in the optimization community as a convergence criterion for min-max saddle point problem , see e.g . Nemirovski et al . ( 2009 ) ; Komodakis & Pesquet ( 2015 ) . Some recent work by Chen et al . ( 2018 ) also used duality in order to derive a Lagrangian objective to train GANs . Although we also make use of duality , there are some major differences that are worth pointing out . Unlike prior work , our contribution does not relate to optimising GANs , but instead in showing that the duality gap can be empirically used as a proxy to track convergence . We also demonstrate it can be computed at low cost . 3 DUALITY GAP AS A NATURAL PERFORMANCE MEASURE . Standard learning tasks are often described as ( stochastic ) optimization problems ; this applies to common Deep Learning scenarios as well as to classical tasks such as logistic and linear regression . This formulation gives rise to a natural performance measure , namely the test loss2 . In contrast , GANs are formulated as ( stochastic ) zero-sum games . Unfortunately , this fundamentally different formulation does not allow us to use the same performance metric . In this section , we describe a performance measure for GANs , which naturally arises from a game theoretic perspective . We start this section with a brief overview of zero-sum games , including a description of the Duality gap metric and some of its properties . Then we illustrate the usefulness of this metric by analyzing the idealized setting where both G and D have unbounded capacity ( this setting was previously discussed in Goodfellow et al . ( 2014 ) ) . A zero-sum game is defined by two players P1 and P2 who choose a decision from their respective decision sets K1 and K2 . A game objective M : K1 × K2 7→ R , sets the utilities of the players . Concretely , upon choosing a pure strategy ( u , v ) ∈ K1 × K2 the utility of P1 is −M ( u , v ) , while the utility of P2 is M ( u , v ) . The goal of either P1/P2 is to maximize their worst case utilities ; thus , min u∈K1 max v∈K2 M ( u , v ) ( Goal of P1 ) , & max v∈K2 min u∈K1 M ( u , v ) ( Goal of P2 ) ( 2 ) The above formulation raises the question of whether there exists a solution ( u∗ , v∗ ) to which both players may jointly converge . The latter only occurs if there exists ( u∗ , v∗ ) such that neither P1 nor P2 may increase their utility by unilateral deviation . Such a solution is called a pure equilibrium , and is formally defined as follows , max v∈K2 M ( u∗ , v ) = min u∈K1 M ( u , v∗ ) ( Pure Equilibrium ) . While a pure equilibrium does not always exist , the seminal work of Nash et al . ( 1950 ) shows that an extended notion of equilibrium always does . Specifically , there always exists a distribution D1 over elements of K1 , and a distribution D2 over elements of K2 , such that the following holds , max v∈K2 Eu∼D1M ( u , v ) = min u∈K1 Ev∼D2M ( u , v ) ( Mixed Nash Equilibrium ) . 2For classification tasks using the zero-one test error is also very natural . Nevertheless , in regression tasks the test loss is often the only reasonable performance measure . Such a solution is called a Mixed Nash Equilibrium ( MNE ) . This notion of equilibrium gives rise to the following natural performance measure of a given pure/mixed strategy . Definition 1 ( Duality Gap ) . Let D1 and D2 be fixed distributions over elements from K1 and K2 respectively . Then the duality gap of ( D1 , D2 ) is defined as follows , DualGap : = max v∈K2 Eu∼D1M ( u , v ) − min u∈K1 Ev∼D2M ( u , v ) . ( 3 ) Particularly , for a given pure strategy ( u , v ) ∈ K1 ×K2 we define , DualGap : = max v∈K2 M ( u , v ) − min u∈K1 M ( u , v ) . ( 4 ) A well-known , straightforward property of the duality gap is that it is always non-negative . Moreover , by definition , the gap is exactly zero in ( mixed ) Nash Equilibrium solutions . This property is very appealing from a practical point of view , since it means that the duality gap gives us an immediate handle for measuring convergence . Next we illustrate the usefulness of the duality gap metric by analyzing the ideal case where both the generator and discriminator have unbounded capacity . The latter basically means that the generator can represent any distribution , and the discriminator can represent any decision rule . The next proposition shows that in this case , as long as G is not equal to the true distribution then the duality gap is always positive . In particular we show that duality gap is at least as large as the JensenShannon divergence between true and fake distributions ( which is always non-negative ) . We also show that if G outputs the true distribution , then there exists a discriminator such that the duality gap is zero . Proposition 1 . Consider the GAN game objective appearing in Equation ( 1 ) , and assume that the generator and discriminator networks have unbounded capacity . Also , let ( G , D ) be a fixed solution . Then the duality gap of ( G , D ) is larger than the Jensen-Shannon divergence between the true distribution and the fake distribution generated by G. Moreover , if G outputs the true distribution , then there exists a discriminator D such that the Duality gap of ( G , D ) is zero .

Summary: This work proposes to use duality gap and minimax loss as measures for monitoring the progress of training GANs. The authors first showed a relationship between duality gap(DG) and Jensen-Shannon divergence and non-negativeness on DG. Then, a comprehensive discussion was presented on how to estimate and efficiently compute DG. A series of experiments were designed on synthetic data and real-world image data to show 1) how duality gap is sensitive to capture non-convergence during training and 2) how minimax loss efficiently reflects the sample quality from generator.