Title: Finding Mixed Nash Equilibria of Generative Adversarial Networks

Abstract: We reconsider the training objective of Generative Adversarial Networks (GANs) from the mixed Nash Equilibria (NE) perspective. Inspired by the classical prox methods, we develop a novel algorithmic framework for GANs via an infinite-dimensional two-player game and prove rigorous convergence rates to the mixed NE. We then propose a principled procedure to reduce our novel prox methods to simple sampling routines, leading to practically efficient algorithms. Finally, we provide experimental evidence that our approach outperforms methods that seek pure strategy equilibria, such as SGD, Adam, and RMSProp, both in speed and quality.
Assume that the matrix A is too expensive to evaluate whereas the (stochastic) gradients of (1) are easy to obtain. Under such settings, a celebrated algorithm, the so-called entropic Mirror Descent (entropic MD), learns an O T -1 /2 -NE: Let φ(z) := d i=1 z i log z i be the entropy function and φ (y) := log d i=1 e yi = sup z∈∆ d { z, y -φ(z)} be its Fenchel dual.
1 Strictly speaking, our derivation requires mild regularity (see Appendix A.1) assumptions on the probability measure and function classes, which are met by most practical applications.

1. Introduction:

The Generative Adversarial Network (GAN) (Goodfellow et al., 2014) has become one of the most powerful paradigms in learning real-world distributions, especially for image-related data. It has been successfully applied to a host of applications such as image translation (Isola et al., 2017; Kim et al., 2017; Zhu et al., 2017) , super-resolution imaging (Wang et al., 2015) , pose editing (Pumarola et al., 2018b) , and facial animation (Pumarola et al., 2018a) .

Despite of the many accomplishments, the major hurdle blocking the full impact of GAN is its notoriously difficult training phase. In the language of game theory, GAN seeks for a pure strategy equilibrium, which is well-known to be ill-posed in many scenarios (Dasgupta & Maskin, 1986) . Indeed, it is known that a pure strategy equilibrium might not exist (Arora et al., 2017) , might be degenerate (Sønderby et al., 2017) , or cannot be reliably reached by existing algorithms (Mescheder et al., 2017) .

Empirically, it has also been observed that common algorithms, such as SGD or Adam (Kingma & Ba, 2015) , lead to unstable training. While much efforts have been devoted into understanding the training dynamics of GANs (Balduzzi et al., 2018; Gemp & Mahadevan, 2018; Gidel et al., 2018a; b; Liang & Stokes, 2018) , a provably convergent algorithm for general GANs, even under reasonably strong assumptions, is still lacking.

In this paper, we address the above problems with the following contributions:

1. We propose to study the mixed Nash Equilibrium (NE) of GANs: Instead of searching for an optimal pure strategy which might not even exist, we optimize over the set of probability distributions over pure strategies of the networks. The existence of a solution to such problems was long established amongst the earliest game theory work (Glicksberg, 1952) , leading to well-posed optimization problems.

2. We demonstrate that the prox methods of (Nemirovsky & Yudin, 1983; Nemirovski, 2004) , which are fundamental building blocks for solving two-player games with finitely many strategies, can be extended to continuously many strategies, and hence applicable to training GANs. We provide an elementary proof for their convergence rates to learning the mixed NE.

3. We construct a principled procedure to reduce our novel prox methods to certain sampling tasks that were empirically proven easy by recent work (Chaudhari et al., 2017; 2018; Dziugaite & Roy, 2018) . We further establish heuristic guidelines to greatly scale down the memory and computational costs, resulting in simple algorithms whose per-iteration complexity is almost as cheap as SGD.

4. We experimentally show that our algorithms consistently achieve better or comparable performance than popular baselines such as SGD, Adam, and RMSProp (Tieleman & Hinton, 2012) .

Related Work: While the literature on training GANs is vast, to our knowledge, there exist only few papers on the mixed NE perspective. The notion of mixed NE is already present in (Goodfellow et al., 2014) , but is stated only as an existential result. The authors of (Arora et al., 2017) advocate the mixed strategies, but do not provide a provably convergent algorithm. (Oliehoek et al., 2018 ) also considers mixed NE, but only with finitely many parameters. The work (Grnarova et al., 2018) proposes a provably convergent algorithm for finding the mixed NE of GANs under the unrealistic assumption that the discriminator is a single-layered neural network. In contrast, our results are applicable to arbitrary architectures, including popular ones (Arjovsky et al., 2017; Gulrajani et al., 2017) .

Due to its fundamental role in game theory, many prox methods have been applied to study the training of GANs (Daskalakis et al., 2018; Gidel et al., 2018a; Mertikopoulos et al., 2018) . However, these works focus on the classical pure strategy equilibria and are hence distinct from our problem formulation. In particular, they give rise to drastically different algorithms from ours and do not provide convergence rates for GANs.

In terms of analysis techniques, our framework is closely related to (Balandat et al., 2016) , but with several important distinctions. First, the analysis of (Balandat et al., 2016) is based on dual averaging (Nesterov, 2009) , while we consider Mirror Descent and also the more sophisticated Mirror-Prox (see Section 3). Second, unlike our work, (Balandat et al., 2016) do not provide any convergence rate for learning mixed NE of two-player games. Finally, (Balandat et al., 2016) is only of theoretical interest with no practical algorithm.

Notation: Throughout the paper, we use z to denote a generic variable and Z ⊆ R d its domain. We denote the set of all Borel probability measures on Z by M(Z), and the set of all functions on Z by F(Z). 1 We write dµ = ρdz to mean that the density function of µ ∈ M(Z) with respect to the Lebesgue measure is ρ. All integrals without specifying the measure are understood to be with respect to Lebesgue. For any objective of the form min x max y F (x, y), we say that 

(x T , y T ) is an O T -1 /2 -NE if max x,y {F (x T , y) - F (x, y T )} = O T -1 /2

2. Problem Formulation:

We review standard results in game theory in Section 2.1, whose proof can be found in (Bubeck, 2013a; b; c) . Section 2.2 relates training of GANs to the two-player game in Section 2.1, thereby suggesting to generalize the prox methods to infinite dimension.

2.1. Preliminary: Prox Methods for Finite Games:

Consider the classical formulation of a two-player game with finitely many strategies:

EQUATION

where A is a payoff matrix, a is a vector, and

∆ d := z ∈ R d | d i=1 z i = 1

is the probability simplex, representing the mixed strategies (i.e., probability distributions) over d pure strategies. A pair (p NE , q NE ) achieving the min-max value in (1) is called a mixed NE.

For a learning rate η and an arbitrary vector b ∈ R d , define the MD iterates as

z = MD η (z, b) ≡ z = ∇φ (∇φ(z) -ηb) ≡ z i = z i e -ηbi d i=1 z i e -ηbi , ∀1 ≤ i ≤ d. (2)

The equivalence of the last two formulas in (2) can be readily checked.

Denote by pT := 1 T T t=1 p t and qT := 1 T T t=1 q t the ergodic average of two sequences {p t } T t=1 and {q t } T t=1 . Then, with a properly chosen step-size η, we have

p t+1 = MD η p t , -A q t q t+1 = MD η (q t , -a + Ap t ) ⇒ ( pT , qT ) is an O T -1 /2 -NE.

Moreover, a slightly more complicated algorithm, called the entropic Mirror-Prox (entropy MP) (Nemirovski, 2004) , achieves faster rate than the entropic MD:

p t = MD η pt , -A qt , pt+1 = MD η pt , -A q t q t = MD η ( qt , -a + A pt ) , qt+1 = MD η ( qt , -a + Ap t ) ⇒ ( pT , qT ) is an O T -1 -NE.

If, instead of deterministic gradients, one uses unbiased stochastic gradients for entropic MD and MP, then both algorithms achieve O T -1 /2 -NE in expectation.

2.2. Mixed Strategy Formulation for Generative Adversarial Networks:

For illustration, let us focus on the Wasserstein GAN (Arjovsky et al., 2017) , and we perform a common bilinearization trick that dates back at least to the early game theory literature (Glicksberg, 1952) , and is also well-known in optimal transport theory (Villani, 2003) .

The training objective of Wasserstein GAN is

EQUATION

where Θ is the set of parameters for the generator and W the set of parameters for the discriminator f , typically both taken to be neural nets. As mentioned in the introduction, such an optimization problem can be ill-posed, which is also supported by empirical evidence.

The high-level idea of our approach is, instead of solving (3) directly, we focus on the mixed strategy formulation of (3). In other words, we consider the set of all probability distributions over Θ and W, and we search for the optimal distribution that solves the following program:

EQUATION

Define the function g : W → R by g(w) := E X∼P real [f w (X)] and the operator G : M(Θ) → F(W) as (Gν)(w) := E θ∼ν,X∼P θ [f w (X)]. Denoting µ, h := E µ h 2 for any probability measure µ and function h, we may rewrite (4) as

EQUATION

Furthermore, the Fréchet derivative (the analogue of gradient in infinite dimension) of ( 5) with respect to µ is simply g -Gν, and the derivative of (5) with respect to ν is -G † µ, where G † : M(W) → F(Θ) is the adjoint operator of G defined via the relation

EQUATION

One can easily check that (G † µ)(θ) := E X∼P θ ,w∼µ [f w (X)] achieves the equality in (6).

To summarize, the mixed strategy formulation of Wasserstein GAN is (5), whose derivatives can be expressed in terms of g and G. We now make the crucial observation that (5) is the infinite-dimensional analogue of (1): The distributions over finite strategies are replaced with probability measures over a continuous parameter set, the vector a is replaced with a function g, the matrix A is replaced with a linear operator3 G, and the gradients are replaced with Fréchet derivatives. Based on Section 2.1, it is then natural to ask:

Can the entropic Mirror Descent and Mirror-Prox be extended to infinite dimension to solve (5)? Can we retain the convergence rates?

We provide an affirmative answer to both questions in the next section.

Remark. The derivation in Section 2.2 can be applied to any GAN objective.

3 Infinite-Dimensional Prox Methods

This section builds a rigorous infinite-dimensional formalism in parallel to the finitedimensional prox methods and proves their convergence rates. While simple in retrospect, to our knowledge, these results are new.

3.1. Preparation: The Mirror Descent Iterates:

We first recall the notion of (Fréchet) derivative in infinite-dimensional spaces. A (nonlinear) functional Φ : M(Z) → R is said to possess a derivative at µ ∈ M(Z) if there exists a function dΦ(µ) ∈ F(Z) such that, for all µ ∈ M(Z), we have

Φ(µ + µ ) = Φ(µ) + µ , dΦ(µ) + o( ).

Similarly, a (nonlinear) functional Φ : F(Z) → R is said to possess a derivative at h ∈ F(Z) if there exists a measure dΦ (h) ∈ M(Z) such that, for all h ∈ F(Z), we have

Φ (h + h ) = Φ (h) + dΦ (h), h + o( ).

The most important functionals in this paper are the (negative) Shannon entropy µ ∈ M(Z), Φ(µ) := dµ log dµ dz and its Fenchel dual h ∈ F(Z), Φ (h) := log e h dz.

The first result of our paper is to show that, in direct analogy to (2), the infinite-dimensional MD iterates can be expressed as: Theorem 1 (Infinite-Dimensional Mirror Descent, informal). For a learning rate η and an arbitrary function h, we can equivalently define

EQUATION

)

Moreover, most the essential ingredients in the analysis of finite-dimensional prox methods can be generalized to infinite dimension.

See Theorem 4 of Appendix A for precise statements and a long list of "essential ingredients of prox methods" generalizable to infinite dimension.

3.2. Infinite-Dimensional Prox Methods and Convergence Rates:

Armed with results in Section 3.1, we now introduce two "conceptual" algorithms for solving the mixed NE of Wasserstein GANs: The infinite-dimensional entropic MD in Algorithm 1 and MP in Algorithm 2. These algorithms iterate over probability measures and cannot be directly used in practice, but they possess rigorous convergence rates, and hence motivate the reduction procedure in Section 4 to come.

Algorithm 1: Infinite-Dimensional Entropic MD Input:

Initial distributions µ 1 , ν 1 , learning rate η for t = 1, 2, . . . , T -1 do ν t+1 = MD η ν t , -G † µ t , µ t+1 = MD η (µ t , -g + Gν t ); return νT = 1 T T t=1 ν t and μT = 1 T T t=1 µ t .

Algorithm 2: Infinite-Dimensional Entropic MP Input: Initial distributions μ1 , ν1 , learning rate η for t = 1, 2, . . . , T do 2. Assume that we have access to stochastic derivatives

ν t = MD η νt , -G † μt , µ t = MD η (μ t , -g + Gν t ); νt+1 = MD η νt , -G † µ t , μt+1 = MD η (μ t , -g + Gν t ); return νT = 1 T T t=1 ν t and μT = 1 T T t=1 µ t . Theorem 2 (Convergence Rates). Let Φ(µ) = dµ log dµ dz . Let M be a constant such that max -g + Gν L ∞ , G † µ L ∞ ≤ M , and L be such that G(ν -ν ) L ∞ ≤ L ν -ν TV and G † (µ -µ ) L ∞ ≤ L µ -µ TV . Let D(•, •)

-Ĝ † µ t T t=1 and ĝ -Ĝν T t=1 such that max E -ĝ + Ĝν L ∞ , E Ĝ † µ L ∞

≤ M , and the variance is upper bounded by σ 2 . Assume also that the bias of stochastic derivatives sat- 4) , and Algorithm 2 with stochastic derivatives achieves

isfies max E[-ĝ + Ĝν] + g -Gν L ∞ , E[ Ĝ † µ] -G † µ L ∞ ≤ τ . Then Algo- rithm 1 with stochastic derivatives achieves O T -1 /2 -NE in expectation with η = D0 T (4τ+ M /

O T -1 /2 + O(τ ) -NE in expectation with η = min 4 √ 3L , 2D0 3T σ 2 .

The proof can be found in Appendix B and C.

Remark. If, as in previous work (Arora et al., 2017) , we assume the output of the discriminator to be bounded by U , then we have M, M ≤ 2U and L ≤ U in Theorem 2. The constant error term for stochastic MP is standard; see, e.g., (Juditsky et al., 2011) .

4 From Theory to Practice Section 4.1 reduces Algorithm 1 and Algorithm 2 to a sampling routine (Welling & Teh, 2011) that has widely been used in machine learning. Section 4.2 proposes to further simplify the algorithms by summarizing a batch of samples by their mean.

For simplicity, we will only derive the algorithm for entropic MD; the case for entropic MP is similar but requires more computation. To ease the notation, we assume η = 1 throughout this section as η does not play an important role in the derivation below.

4.1. Implementable Entropic MD: From Probability Measure to Samples:

Consider Algorithm 1. The reduction consists of three steps.

Step 1: Reformulating Entropic Mirror Descent Iterates

The definition of the MD iterate (7) relates the updated probability measure µ t+1 to the current probability measure µ t , but it tells us nothing about the density function of µ t+1 , from which we want to sample. Our first step is to express (7) in a more tractable form. By recursively applying (7) and using Theorem 4.10 in Appendix A, we have, for some

constants C 1 , ..., C T -1 , dΦ(µ T ) = dΦ(µ T -1 ) -(-g + Gν T -1 ) + C T -1 = dΦ(µ T -2 ) -(-g + Gν T -2 ) -(-g + Gν T -1 ) + C T -1 + C t-2 = • • • = dΦ(µ 1 ) --(T -1)g + G T -1 s=1 ν s + T -1 s=1 C s .

For simplicity, assume that µ 1 is uniform so that dΦ(µ 1 ) is a constant function. Then, by ( 13) and that dΦ (dΦ(µ T )) = dµ T , we see that the density function of µ T is simply

dµ T = exp{(T -1)g-G T -1 s=1 νs}dw exp{(T -1)g-G T -1 s=1 νs}dw . Similarly, we have dν T = exp{G † T -1 s=1 µs}dθ exp{G † T -1 s=1 µs}dθ .

Step 2: Empirical Approximation for Stochastic Derivatives

The derivatives of (5) involve the function g and operator G. Recall that g requires taking expectation over the real data distribution, which we do not have access to. A common approach is to replace the true expectation with its empirical average:

g(w) = E X∼P real [f w (X)] 1 n n i=1 f w (X real i ) ĝ(w)

where X i 's are real data and n is the batch size. Clearly, ĝ is an unbiased estimator of g.

On the other hand, Gν t and G † µ t involve expectation over ν t and µ t , respectively, and also over the fake data distribution P θ . Therefore, if we are able to draw samples from µ t and ν t , then we can again approximate the expectation via the empirical average:

θ (1) , θ (2) , ..., θ (n ) ∼ ν t , X (j) i n i=1 ∼ P θ (j) , Ĝν t (w) 1 nn n i=1 n j=1 f w X (j) i w (1) , w (2) , ..., w (n ) ∼ µ t , {X i } n i=1 ∼ P θ , Ĝ † µ t (θ) 1 nn n i=1 n j=1 f w (j) (X i ) .

Now, assuming that we have obtained unbiased stochastic derivatives -

t s=1

Ĝ † µ s and t s=1 -ĝ + Ĝν s , how do we actually draw samples from µ t+1 and ν t+1 ? Provided we can answer this question, then we can start with two easy-to-sample distributions (µ 1 , ν 1 ), and then we will be able to draw samples from (µ 2 , ν 2 ). These samples in turn will allow us to draw samples from (µ 3 , ν 3 ), and so on. Therefore, it only remains to answer the above question. This leads us to:

Step 3: Sampling by Stochastic Gradient Langevin Dynamics For any probability distribution with density function e -h dz, the Stochastic Gradient Langevin Dynamics (SGLD) (Welling & Teh, 2011) iterates as

EQUATION

)

where γ is the step-size, ∇h is an unbiased estimator of ∇h, is the thermal noise, and ξ k ∼ N (0, I) is a standard normal vector, independently drawn across different iterations.

Suppose we start at (µ 1 , ν 1 ). Plugging h ← -Ĝ † µ 1 and h ← -ĝ + Ĝν 1 into (8), we obtain, for

{X i } n i=1 ∼ P θ k , {w (j) } n j=1 ∼ µ 1 , standard normal ξ k , ξ k , and X real i ∼ P real , {θ (j) } n j=1 ∼ ν 1 , {X (j)

i } ∼ P θ (j) , the following update rules:

θ k+1 = θ k + γ∇ θ   1 nn n i=1 n j=1 f w (j) (X i )   + 2γ ξ k w k+1 = w k + γ∇ w   1 n n i=1 f w k (X real i ) - 1 nn n i=1 n j=1 f w k X (j) i   + 2γ ξ k .

The theory of (Welling & Teh, 2011) states that, for large enough k, the iterates of SGLD above (approximately) generate samples according to the probability measures (µ 2 , ν 2 ). We can then apply this process recursively to obtain samples from (µ 3 , ν 3 ), (µ 4 , ν 4 ), ...(µ T , ν T ). Finally, since the entropic MD and MP output the averaged measure (μ T , νT ), it suffices to pick a random index t ∈ {1, 2, ..., T } and then output samples from (µ t, ν t).

Putting

Step 1-3 together, we obtain Algorithm 4 and 5 in Appendix D.

Remark. In principle, any first-order sampling method is valid above. In the experimental section, we also use a RMSProp-preconditioned version of the SGLD (Li et al., 2016) .

4.2. Summarizing Samples by Averaging::

A Simple yet Effective Heuristic Although Algorithm 4 and 5 are implementable, they are quite complicated and resourceintensive, as the total computational complexity is O(T 2 ). This high complexity comes from the fact that, when computing the stochastic derivatives, we need to store all the historical samples and evaluate new gradients at these samples.

An intuitive approach to alleviate the above issue is to try to summarize each distribution by only one parameter. To this end, the mean of the distribution is the most natural candidate, as it not only stablizes the algorithm, but also is often easier to acquire than the actual samples. For instance, computing the mean of distributions of the form e -h dz, where h is a loss function defined by deep neural networks, has been empirically proven successful in (Chaudhari et al., 2017; 2018; Dziugaite & Roy, 2018) via SGLD. In this paper, we adopt the same approach as in (Chaudhari et al., 2017) where we use exponential damping (the β term in Algorithm 3) to increase stability. Algorithm 3, dubbed the Mirror-GAN, shows how to encompass this idea into entropic MD; the pseudocode for the similar Mirror-Prox-GAN can be found in Algorithm 6 of Appendix D.

Algorithm 3: Mirror-GAN: Approximate Mirror Decent for GANs

Input: w1 , θ1 ← random initialization, {γ t } T t=1 , { t } T t=1 , {K t } T -1 t=1 , β (see Appendix D for meaning of the hyperparameters), standard normal noise ξ k , ξ k . for t = 1, 2, . . . , T -1 do wt , w (1) t ← w t ; θt , θ (1) t ← θ t ; for k = 1, 2, . . . , K t do Generate A = {X 1 , . . . , X n } ∼ P θ (k) t ; θ (k+1) t = θ (k) t + γt n ∇ θ Xi∈A f wt (X i ) + √ 2γ t t ξ k ; Generate B = {X real 1 , . . . , X real n } ∼ P real ; Generate B = {X 1 , . . . , X n } ∼ P θt ; w (k+1) t = w (k) t + γ t n ∇ w X real i ∈B f w (k) t (X real i ) - γ t n ∇ w X i ∈B f w (k) t (X i ) + 2γ t t ξ k ; wt ← (1 -β) wt + βw (k+1) t ; θt ← (1 -β) θt + βθ (k+1) t ; w t+1 ← (1 -β)w t + β wt ; θ t+1 ← (1 -β)θ t + β θt ; return w T , θ T .

5. Experimental Evidence:

The purpose of our experiments is twofold. First, we use established baselines to demonstrate that Mirror-and Mirror-Prox-GAN consistently achieve better or comparable performance than common algorithms. Second, we report that our algorithms are stable and always improve as the training process goes on. This is in contrast to unstable training algorithms, such as Adam, which often collapse to noise as the iteration count grows. (Cha, 2017) .

We use visual quality of the generated images to evaluate different algorithms. We avoid reporting numerical metrics, as recent studies (Barratt & Sharma, 2018; Borji, 2018; Lucic et al., 2018) suggest that these metrics might be flawed. Setting of the hyperparameters and more auxiliary results can be found in Appendix E.

5.1. Synthetic Data:

We repeat the synthetic setup as in (Gulrajani et al., 2017) . The tasks include learning the distribution of 8 Gaussian mixtures, 25 Gaussian mixtures, and the Swiss Roll. For both the generator and discriminator, we use two MLPs with three hidden layers of 512 neurons. We choose SGD and Adam as baselines, and we compare them to Mirror-and Mirror-Prox-GAN. All algorithms are run up to 10 5 iterations4 . The results of 25 Gaussian mixtures are shown in Figure 1 ; An enlarged figure of 25 Gaussian Mixtures and other cases can be found in Appendix E.1. As Figure 1 shows, SGD performs poorly in this task, while the other algorithms yield reasonable results. However, compared to Adam, Mirror-and Mirror-Prox-GAN fit the true distribution better in two aspects. First, the modes found by Mirror-and Mirror-Prox-GAN are more accurate than the ones by Adam, which are perceivably biased. Second, Mirror-and Mirror-Prox-GAN perform much better in capturing the variance (how spread the blue dots are), while Adam tends to collapse to modes. These observations are consistent throughout the synthetic experiments; see Appendix E.1.

5.2. Real Data:

For real images, we use the LSUN bedroom dataset (Yu et al., 2015) . We have also conducted a similar study with MNIST; more results can be found in Appendix E.2.

We use the same architecture (DCGAN) as in (Radford et al., 2015) with batch normalization. As the networks become deeper in this case, the gradient magnitudes differ significantly across different layers. As a result, non-adaptive methods such as SGD or SGLD do not perform well in this scenario. To alleviate such issues, we replace SGLD by the RMSProppreconditioned SGLD (Li et al., 2016) for our sampling routines. For baselines, we consider two adaptive gradient methods: RMSprop and Adam. We also include two contemporary algorithms, the Simultaneous and Alternated Extra-Adam, from the concurrent ICLR submission (Gidel et al., 2018a) .

Figure 2 shows the results at the 10 5 th iteration. 

6. Conclusions:

Our goal of systematically understanding and expanding on the game theoretic perspective of mixed NE along with stochastic Langevin dynamics for training GANs is a promising research vein. While simple in retrospect, we provide guidelines in developing approximate infinite-dimensional prox methods that mimic closely the provable optimization framework to learn the mixed NE of GANs. Our proposed Mirror-and Mirror-Prox-GAN algorithm feature cheap per-iteration complexity while rapidly converging to solutions of good quality.

A A Framework for Infinite-Dimensional Mirror Descent:

A.1 A note on the regularity It is known that the (negative) Shannon entropy is not Fréchet differentiable in general. However, below we show that the Fréchet derive can be well-defined if we restrict the probability measures to within the set M(Z) :={all probability measures on Z that admit densities w.r.t. the Lebesgue measure, and the density is continuous and positive almost everywhere on Z}.

We will also restrict the set of functions to be bounded and integrable:

F(Z) := all bounded continuous functions f on Z such that e -f < ∞ .

It is important to notice that µ ∈ M(Z) and h ∈ F(Z) implies µ = MD η (µ, h) ∈ M(Z); this readily follows from the formula (7).

A.2 Properties of Entropic Mirror Map:

The total variation of a (possibly non-probability) measure µ ∈ M(Z) is defined as (Halmos, 2013)

µ TV = sup h L ∞ ≤1 hdµ = sup h L ∞ ≤1 µ, h .

Recall the standard topology induced by • TV and • L ∞ for measures and functions (Halmos, 2013), respectively. Whenever we speak about continuity or differentiability below, it is understood to be w.r.t. to the standard topology. Notice also that the G operator defined in ( 5) is bounded if the discriminator f w is bounded, and hence continous (Halmos, 2013).

We depart from the fundamental Gibbs Variational Principle, which dates back to the earliest work of statistical mechanics (Gibbs, 1902) . For two probability measures µ, µ , denote their relative entropy by (the reason for this notation will become clear in ( 14))

D Φ (µ, µ ) := Z dµ log dµ dµ .

By the definition of M(Z), it is clear that the relative entropy is well-defined for any µ, µ ∈ M(Z).

Theorem 3 (Gibbs Variation Principle). Let h ∈ F(Z) and µ ∈ M(Z) be a reference measure. Then

EQUATION

)

and equality is achieved by dµ = e h dµ Z e h dµ .

Part of the following theorem is folklore in the mathematics and learning community. However, to the best of our knowledge, the relation to the entropic MD has not been systematically studied before, as we now do.

Theorem 4. For a probability measure dµ = ρdz, let Φ(µ) = ρ log ρdz be the negative Shannon entropy, and let Φ (h) = log Z e h dz. Then 1. Φ is the Fenchel conjugate of Φ:

EQUATION

)

2. The derivatives admit the expression

EQUATION

)

3. The Bregman divergence of Φ is the relative entropy:

EQUATION

)

4. Φ is 4-strongly convex with respect to the total variation norm: For all λ ∈ (0, 1),

EQUATION

)

5. The following duality relation holds: For any constant C, we have

EQUATION

)

7. Alternative to (17), we have the equivalent characterization of Φ :

EQUATION

)

8. Similar to (16), we have

EQUATION

)

9. The following three-point identity holds for all µ, µ , µ ∈ M(Z):

EQUATION

)

10. Let the Mirror Descent iterate be defined as in (7). Then the following statements are equivalent:

(a) µ + = MD η (µ, h). (b) There exists a constant C such that dΦ(µ + ) = dΦ(µ) -ηh + C.

In particular, for any µ , µ ∈ M(Z) we have

EQUATION

Proof.

1. Equation ( 10) is simply the Gibbs variational principle (9) with dµ ← dz.

By (10), we know that

EQUATION

)

But for dµ = ρdz, the function h := 1 + log ρ saturates the equality in ( 22).

2. We prove a more general result on the Bregman divergence D Φ in ( 23) below.

Let dµ = ρdz, dµ = ρ dz, and dµ = ρ dz ∈ M(Z). Let > 0 be small enough such that (ρ + ρ )dz is absolutely continuous with respect to dµ ; note that this is possible because µ, µ , and µ ∈ M(Z). We compute

D Φ (ρ + ρ , ρ ) = Z (ρ + ρ ) log ρ + ρ ρ = Z ρ log ρ ρ + Z ρ log 1 + ρ ρ + Z ρ log ρ ρ + Z ρ log 1 + ρ ρ (i) = Z ρ log ρ ρ + Z ρ + Z ρ log ρ ρ + 2 Z ρ 2 ρ + o( ) = D Φ (ρ, ρ ) + Z ρ 1 + log ρ ρ + o( ),

where (i) uses log(1 + t) = t + o(t) as t → 0. In short, for all µ , µ ∈ M(Z),

d µ D Φ (µ, µ )(µ ) = µ , 1 + log ρ ρ (23) which means d µ D Φ (µ, µ ) = 1 + log ρ ρ .

The formula ( 12) is the special case when dµ ← dz.

We now turn to (13). For every h ∈ F(Z), we need to show that the following holds for every h ∈ F(Z): 3. Let dµ = ρdz and dµ = ρ dz. We compute

Φ (h + h ) -Φ (h) = log Z e h+ h dz -log Z e h dz =

D Φ (µ, µ ) = Φ(µ) -Φ(µ ) -µ -µ , dΦ(µ ) = Z ρ log ρdz - Z ρ log ρ dz -µ -µ , 1 + log ρ by (12) = Z ρ log ρ ρ dz = Z dµ log dµ dµ .

4. Define µ λ = λµ + (1 -λ)µ . By ( 14) and the classical Pinsker's inequality (Gray, 2011), we have

EQUATION

Equation ( 15) follows by multiplying with λ and 1 -λ respectively and summing the two inequalities up.

5. Let µ = ρdz and µ = ρ dz. Then, by the definition of Bregman divergence and ( 12), ( 13),

D Φ (dΦ(µ ), dΦ(µ)) = Φ (dΦ(µ )) -Φ (dΦ(µ)) - e 1+log ρ dz Z e 1+log ρ , 1 + log ρ -1 -log ρ = log Z e 1+log ρ -log Z e 1+log ρ + Z ρ log ρ ρ = Z ρ log ρ ρ = D Φ (µ, µ )

since Z ρdz = Z ρ dz = 1. This proves the first equality.

For the second equality, we write

D Φ (dΦ(µ ) + C, dΦ(µ)) = Φ (dΦ(µ ) + C) -Φ (dΦ(µ)) - e 1+log ρ dz Z e 1+log ρ , 1 + log ρ + C -1 -log ρ = log Z e 1+log ρ +C -log Z e 1+log ρ + Z ρ log ρ ρ -C = Z ρ log ρ ρ = D Φ (µ, µ ) = D Φ (dΦ(µ ), dΦ(µ))

where we have used the first equality in the last step.

6. Let µ h = dΦ (h), µ h = dΦ (h ), and µ λ = λµ h + (1 -λ)µ h for some λ ∈ (0, 1). By Pinsker's inequality and ( 14), we have

EQUATION

EQUATION

Now, notice that

µ λ -µ h , dΦ(µ h ) = µ λ -µ h , dΦ(dΦ (h)) = µ λ -µ h , dΦ e h dz Z e h by (13) = µ λ -µ h , 1 + h -log Z e h by (12) = µ λ -µ h , h

and, similarly, we have µ λ -µ h , dΦ(µ h ) = µ λ -µ h , h . Multiplying ( 27) by λ and ( 28) by 1 -λ, summing the two up, and using the above equalities, we get

Φ(µ λ ) -λΦ(µ h ) + (1 -λ)Φ(µ h ) + λ(1 -λ) µ h -µ h , h -h ≥ 2λ(1 -λ) µ h -µ h 2 TV . By (15), we know that Φ(µ λ ) -λΦ(µ h ) + (1 -λ)F (µ h ) ≤ -2λ(1 -λ) µ h -µ h 2 TV .

Moreover, by definition of the total variation norm, it is clear that

EQUATION

Combing the last three inequalities gives (17).

7. Let K be a positive integer and k ∈ {0, 1, 2, . . . , K}. Set

EQUATION

By convexity of Φ , we have

EQUATION

By ( 29) and ( 17), we may further upper bound (31) as

EQUATION

Summing up (32) over k, we get, in view of (30),

EQUATION

Since K is arbitrary, we may take K → ∞ in (33), which is (18).

8.. Straightforward calculation shows:

D Φ (h, h ) = log Z e h -log Z e h - Z e h e h (h -h ) .

On the other hand, by definition of the Bregman divergence and ( 12), (13), we have

D Φ (dΦ (h ), dΦ (h)) = Z e h Z e h h -log Z e h - Z e h Z e h h + log Z e h - Z 1 + h -log Z e h e h Z e h - e h Z e h = Z e h e h (h -h) -log Z e h + log Z e h = Φ (h) -Φ (h ) -dΦ (h ), h -h = D Φ (h, h ).

9. By definition of the Bregman divergence, we have

D Φ (µ, µ ) = Φ(µ) -Φ(µ ) -µ -µ , dΦ(µ ) , D Φ (µ , µ) = Φ(µ ) -Φ(µ) -µ -µ, dΦ(µ) , D Φ (µ , µ ) = Φ(µ ) -Φ(µ ) -µ -µ , dΦ(µ ) .

Equation ( 20) then follows by straightforward calculations.

10. First, let µ + = MD η (µ, h). Then if µ + = ρ + dz and µ = ρdz, then (7) implies

ρ + = ρe -ηh Z ρe -ηh .

By (12), we therefore have

dΦ(µ + ) = 1 + log ρ + = 1 + log ρ -ηh -log Z ρe -ηh whence (21) holds with C = -log Z ρe -ηh .

Conversely, assume that dΦ(µ + ) = dΦ(µ) -ηh + C for some constant C, and apply dΦ to both sides. The left-hand side becomes

dΦ dΦ(µ + ) = dΦ (1 + log ρ + ) = ρ + dz ρ + dz = ρ + dz = dµ + ,

where as the formula (13) implies that

dΦ (dΦ(µ) -ηh + C) = e 1+log ρ-ηh+C Z e 1+log ρ-ηh+C dz = ρe -ηh dz Z ρe -ηh = e -ηh dµ Z e -ηh dµ

.

Combining the two equalities gives dµ + = e -ηh dµ Z e -ηh dµ which exactly means µ + = MD η (µ, h).

B Proof of Convergence Rates for Infinite-Dimensional:

Mirror Descent

B.1 Mirror Descent, Deterministic Derivatives:

By the definition of the algorithm, (21), and the three-point identity (20), we have, for any µ ∈ M(W),

EQUATION

By item 10 of Theorem 4, there exists a constant C t such that

EQUATION

Using ( 16), we see that

D Φ (µ t , µ t+1 ) = D Φ (dΦ(µ t+1 ), dΦ(µ t )) = D Φ dΦ(µ t+1 ) -C t , dΦ(µ t ) ≤ 1 8 dΦ(µ t+1 ) -C t -dΦ(µ t ) 2 L ∞ by (18) = η 2 8 -g + Gν t 2 L ∞ by (35) ≤ η 2 M 2 8 .

Consequently, we have

EQUATION

)

Exactly the same argument applied to ν t 's yields, for any ν ∈ M(Θ),

EQUATION

)

Summing up ( 36) and ( 37), substituting µ ← µ NE , ν ← ν NE and dividing by T , we get

EQUATION

)

The left-hand side of ( 38) can be simplified to

EQUATION

By definition of the Nash Equilibrium, we have μT , g -

EQUATION

Combining ( 51)-( 54), we conclude that

η = 2 M D 0 T ⇒ | μT , g -Gν T -µ NE , g -Gν NE | ≤ M D 0 T .

B.2 Mirror Descent, Stochastic Derivatives:

We first write

µ t -µ, η(-ĝ + Ĝν t ) = µ t -µ, η(-g + Gν t ) + µ t -µ, η -ĝ + Ĝν t + g -Gν t .

Taking conditional expectation and using the bias estimate of stochastic derivatives, we conclude that

E µ t -µ, η(-ĝ + Ĝν t ) ≤ µ t -µ, η(-g + Gν t ) + µ t -µ TV • ητ ≤ µ t -µ, η(-g + Gν t ) + 2ητ .

Therefore, using exactly the same argument leading to (36), we may obtain

E T t=1 µ t -µ, -ĝ + Ĝν t ≤ ED Φ (µ, µ 1 ) η + ηM 2 T 8 + 2ηT τ.

The rest is the same as with deterministic derivatives.

C Proof of Convergence Rates for Infinite-Dimensional Mirror-Prox:

We first need a technical lemma, which is Lemma 6.2 of (Juditsky & Nemirovski, 2011) tailored to our infinite-dimensional setting. We give a slightly different proof.

Lemma 5. Given any µ ∈ M(Z) and h, h ∈ F(Z), let µ = MD η (μ, h) and μ+ = MD η (μ, h ). Let Φ be α-strongly convex (recall that α = 4 when Φ is the entropy). Then, for any µ ∈ M(Z), we have

EQUATION

Proof. Recall from (15) that entropy is α-strongly convex with respect to • TV . We first write

EQUATION

For the first term, ( 20) and ( 21) implies

EQUATION

)

Similarly, the second term of the right-hand side of ( 43) can be written as

EQUATION

Hölder's inequality for the third term gives

EQUATION

Finally, recall that Φ is α-strongly convex, and hence we have

EQUATION

The lemma follows by combining inequalities ( 44)-( 47) in (43).

C.1 Mirror-Prox, Deterministic Derivatives:

Let α = 4, μT := 1 T T t=1 µ t , and νT := 1 T T t=1 ν t . In Lemma 5, substituting µ ← µ NE , μ ← μt , h ← -g + Gν t (so that µ = µ t ) and h ← -g + Gν t (so that μ+ = μt+1 ), we get 48) and (49) yields

EQUATION

) Since G(ν t -νt ) L ∞ ≤ L • ν t -νt TV and G † (µ t -μt ) L ∞ ≤ L • µ t -μt TV , summing up (

µ t -µ NE , η(-g + Gν t ) + ν t -ν NE , -ηG † µ t ) ≤ D Φ (µ NE , μt ) -D Φ (µ NE , μt+1 ) + D Φ (ν NE , νt ) -D Φ (ν NE , νt+1 ) + η 2 L 2 2α - α 2 μt -µ t 2 TV + νt -ν t 2 TV ≤ D Φ (µ NE , μt ) -D Φ (µ NE , μt+1 ) + D Φ (ν NE , νt ) -D Φ (ν NE , νt+1 ) if η ≤ α L = 4 L .

Summing up the last inequality over t and using D Φ (•, •) ≥ 0, we obtain

1 T T t=1 µ t -µ NE , η(-g + Gν t ) + ν t -ν NE , -ηG † µ t ) ≤ D Φ (µ NE , μ1 ) + D Φ (ν NE , ν1 ) T = D 0 T .

(50) The left-hand side of (50) can be simplified to

EQUATION

By definition of the (µ NE , ν NE ), we have

EQUATION

Combining ( 50)-( 53), we conclude In Lemma 5, substituting µ ← µ NE , μ ← μt , h ← -ĝ + Ĝν t (so that µ = µ t ), and h ← -ĝ + Ĝν t (so that μ+ = μt+1 ), we get

η ≤ 4 L ⇒ | μT , g -Gν T -µ NE , g -Gν NE | ≤ D 0 T η .

µ t -µ NE , η(-ĝ + Ĝν t ) ≤ D Φ (µ NE , μt )-D Φ (µ NE , μt+1 )+ η 2 2α Ĝν t -Ĝν t 2 L ∞ - α 2 μt -µ t 2 TV . (54) Note that E Ĝν t -Ĝν t 2 L ∞ ≤ 3 E Ĝν t -Gν t 2 L ∞ + E Gν t -Gν t 2 L ∞ + E Gν t -Ĝν t 2 L ∞ ≤ 6σ 2 + 3L 2 E ν t -νt 2 TV .

Therefore, taking expectation conditioned on the history for both sides of (54) and using the bias estimates of the stochastic derivatives, we get

µ t -µ NE , η(-g + Gν t ) ≤ ED Φ (µ NE , μt ) -ED Φ (µ NE , μt+1 ) + 3η 2 σ 2 α + 3η 2 L 2 2α E ν t -νt 2 TV - α 2 E μt -µ t 2 TV + 2ητ.

Similarly, we have

ν t -ν NE , -ηG † µ t ≤ ED Φ (ν NE , νt ) -ED Φ (ν NE , νt+1 ) + 3η 2 σ 2 α + 3η 2 L 2 2α E µ t -μt 2 TV - α 2 E νt -ν t 2 TV + 2ητ. Algorithm SGD RMSProp Adam Entropic MD/MP Dataset S M L S M L S M L

Step-size γ 10 -2 10 -4 10 -4 10 -2 10 -4

Gradient penalty λ 0. K t , γ t and t , we use the simple exponential scheduling:

K t = (1 + 10 -5 ) t . γ t = γ × (1 -10 -5 ) t , γ in Table 1. t = × (1 -5 × 10 -5 ) t , in Table 1.

The idea is that the initial iterations are very noisy, and hence it makes sense to take less SGLD steps. As the iteration counts grow, the algorithms learn more meaningful parameters, and we should increase the number of SGLD steps as well as decreasing the step-size γ t and thermal noise t to make the sampling more accurate. This is akin to the warmup steps in the sampling literature. Prox-GAN can better capture the variance of the true distribution, as well as finding the unbiased modes.

E.1 Synthetic Data:

In Figure 6 , we plot the data generated after 10 4 , 2 × 10 4 , 5 × 10 4 , 8 × 10 4 , and 10 5 iterations by different algorithms fro 25 Gaussian mixtures. It is clear that Mirror-and Mirror-Prox-GAN find the modes of the distribution faster. In practice, it was observed that the noise introduced by SGLD quickly drives the iterates to non-trivial parameter regions, whereas SGD tends to get stuck at very bad local minima. Adam, as an adaptive algorithm, is capable of escaping bad local minima, however at a rate slower than Mirror-and Mirror-Prox-GAN. The quality of Adam's final solution is also not as good as Mirror-and Mirror-Prox-GAN; see the discussions in Section 5.1.

E.2 Real Data:



E.2.1 MNSIT:

Results on MNIST dataset are shown in Figure 7 . The models are trained by each algorithm for 10 5 iterations. We can see that all algorithms achieve comparable performance. Therefore, the dataset seems too weak to be a discriminator for different algorithms. More results on the LSUN bedroom dataset are shown in Figure 8 . We show images generated after 4 × 10 4 , 8 × 10 4 , and 10 5 iterations by each algorithm. We can see that the Mirror-GAN and Alternated Extra-Adam outperform vanilla RMSProp. Adam was able to obtain meaningful images in early stages of training. However, further iterations do not improve the image quality of Adam. In contrast, they lead to severe mode collapse at the 8 × 10 4 th iteration, and converge to noise later on. Simultaneous Extra-Adam completely fails in this task.

E.2.2 LSUN Bedroom:

Finally, for reference, we report the Inception Score in Table 2 . 2 × 10 4 iterations 5 × 10 4 iterations 8 × 10 4 iterations 10 5 iterations 

:

It should be noted that µ, h is NOT an inner product, and rather is the dual pairing in Banach spaces(Halmos, 2013).

The linearity of G trivially follows from the linearity of expectation.

One iteration here means using one mini-batch of data. It does not correspond to the T in our algorithms, as there might be multiple SGLD iterations within each time step t.

Their code is available on https://github.com/igul222/improved_wgan_training.

Figures and Tables:

be the relative entropy, and denote by D 0 := D(µ NE , µ 1 ) + D(ν NE , ν 1 ) the initial distance to the mixed NE. Then 1. Assume that we have access to the deterministic derivatives -G † µ t T t=1 and {g -Gν} T t=1 . Then Algorithm 1 achieves O T -1 /2 -NE with η = 2 M D0 T , and Algorithm 2 achieves O T -1 -NE with η = 4 L .

Figure 1: Fitting 25 Gaussian mixtures up to 10 5 iterations. Blue dots represent the true distribution and red ones are from the trained generator.

Figure 2: Dataset LSUN bedroom, 10 5 iterations.

e h e h dz. Notice that T (0) = 0 and T is smooth as a function of . Thus, by the Intermediate Value Theorem, Φ (h + h ) -Φ (h) = T ( ) -T (e h+ h dz. Hence it suffices to prove e h+ h Z e h+ h = e h Z e h + o(1) in . To this end, let C = sup |h | < ∞. Then e h Z e h e -2 C ≤ e h+ h Z e h+ h ≤ e h Z e h e 2 C . It remains to use e t = 1 + t + o(t) and ≤ .

Figure 3: Fitting 8 Gaussian mixtures up to 10 5 iterations.

Figure 3, 4, and 5  show results on learning 8 Gaussian mixtures, 25 Gaussian mixtures, and the Swiss Roll. As in the case for 25 Gaussian mixtures, we find that Mirror-and Mirror-

Figure 5: Fitting 25 Gaussian mixtures up to 10 5 iterations.

Figure 6: Learning 25 Gaussian mixtures accross different iterations.

Figure 7: True MNIST images and samples generated by different algorithms.

Hyperparameter setting. "S", "M", "L" stands for synthetic data, MNIST and LSUN bedroom, respectively. MD for LSUN bedroom uses a RMSProp preconditioner, so the stepsize is the same as one in RMSProp.

Table content: <table><tr><td/><td>1</td><td>10</td><td/><td>0.1</td><td>10</td><td/><td>0.1</td><td>10</td><td/></tr><tr><td>Noise</td><td/><td/><td/><td/><td/><td/><td colspan="3">10 -2 10 -3 10 -6</td></tr><tr><td>Batch Size n</td><td>1024</td><td>50</td><td>64</td><td>1024</td><td>50</td><td>64</td><td>1024</td><td>50</td><td>64</td></tr></table>

Inception Score of generator trained on LSUN dataset. The reported scores are based on the average of 6400 images from each generator.

Table content: <table/>

Peer review (iclr2019):
 This paper proposes to consider the mixed equilibrium objective function for GANS. The authors generalize the mirror descent/mirror prox to handle continuous games. The technical challenge is to write those algorithms in infinite dimensional spaces. This reviewer finds this however to be a mere technicality, and there seems to be no conceptual obstruction. In fact other paper have already written this, see for example ``Mirror Descent Learning in Continuous Games" by Zhou et al. at CDC 2017 (I*m sure there are other references too). While the theory part is not particularly exciting, the paper could be saved by the experiments. However as far I can tell the authors are only able to reproduce the results obtained with more classical approaches.