Title: STACKELBERG GAN: TOWARDS PROVABLE MINI-MAX EQUILIBRIUM VIA MULTI-GENERATOR ARCHI-TECTURES

Abstract: We study the problem of alleviating the instability issue in the GAN training procedure via new architecture design. The discrepancy between the minimax and maximin objective values could serve as a proxy for the difficulties that the alternating gradient descent encounters in the optimization of GANs. In this work, we give new results on the benefits of multi-generator architecture of GANs. We show that the minimax gap shrinks to as the number of generators increases with rate O(1/ ). This improves over the best-known result of O(1/ 2 ). At the core of our techniques is a novel application of Shapley-Folkman lemma to the generic minimax problem, where in the literature the technique was only known to work when the objective function is restricted to the Lagrangian function of a constraint optimization problem. Our proposed Stackelberg GAN performs well experimentally in both synthetic and real-world datasets, improving Fréchet Inception Distance by 14.61% over the previous multi-generator GANs on the benchmark datasets.

1. INTRODUCTION:

Generative Adversarial Nets (GANs) are emerging objects of study in machine learning, computer vision, natural language processing, and many other domains. In machine learning, study of such a framework has led to significant advances in adversarial defenses (Xiao et al., 2018; Samangouei et al., 2018) and machine security (Athalye et al., 2018; Samangouei et al., 2018) . In computer vision and natural language processing, GANs have resulted in improved performance over standard generative models for images and texts (Goodfellow et al., 2014) , such as variational autoencoder (Kingma & Welling, 2013) and deep Boltzmann machine (Salakhutdinov & Larochelle, 2010) . A main technique to achieve this goal is to play a minimax two-player game between generator and discriminator under the design that the generator tries to confuse the discriminator with its generated contents and the discriminator tries to distinguish real images/texts from what the generator creates. Despite a large amount of variants of GANs, many fundamental questions remain unresolved. One of the long-standing challenges is designing universal, easy-to-implement architectures that alleviate the instability issue of GANs training. Ideally, GANs are supposed to solve the minimax optimization problem (Goodfellow et al., 2014) , but in practice alternating gradient descent methods do not clearly privilege minimax over maximin or vice versa (page 35, Goodfellow (2016) ), which may lead to instability in training if there exists a large discrepancy between the minimax and maximin objective values. The focus of this work is on improving the stability of such minimax game in the training process of GANs. To alleviate the issues caused by the large minimax gap, our study is motivated by the so-called Stackelberg competition in the domain of game theory. In the Stackelberg leadership model, the players of this game are one leader and multiple followers, where the leader firm moves first and then the follower firms move sequentially. It is known that the Stackelberg model can be solved to find a subgame perfect Nash equilibrium. We apply this idea of Stackelberg leadership model to the architecture design of GANs. That is, we design an improved GAN architecture with multiple generators (followers) which team up to play against the discriminator (leader). We therefore name our model Stackelberg GAN. Our theoretical and experimental results establish that: GANs with multi-generator architecture have smaller minimax gap, and enjoy more stable training performances. Our Contributions. This paper tackles the problem of instability during the GAN training procedure with both theoretical and experimental results. We study this problem by new architecture design. • We propose Stackelberg GAN framework of having multiple generators in the GAN architecture. Our framework is general that can be applied to all variants of GANs, e.g., vanilla GAN, Wasserstein GAN, etc. It is built upon the idea of jointly optimizing an ensemble of GAN losses w.r.t. all pairs of discriminator and generator.

Differences with prior work. Although the idea of having multiple generators in the GAN architecture is not totally new, e.g., MIX+GAN (Arora et al., 2017) and MGAN (Hoang et al., 2018) , there are key differences between Stackelberg GAN and prior work. a) In MGAN (Hoang et al., 2018) , various generators are combined as a mixture of probabilistic models with assumption that the generators and discriminator have enough capacity. In contrast, in the Stackelberg GAN model we uniformly ensemble the losses of various standard GAN without any assumption on the model capacity. b) In MIX+GAN (Arora et al., 2017) , the losses are ensembled with learned weights and an extra regularization term, which discourages the weights being too far away from uniform. We find it slightly unnecessary because the expressive power of each generator already allows implicit scaling of each generator. To the contrary, in the Stackelberg GAN we apply equal weights for all generators.

• We prove that the minimax duality gap shrinks as the number of generators increases (see • We propose Stackelberg GAN framework of having multiple generators in the GAN architecture. Our framework is general that can be applied to all variants of GANs, e.g., vanilla GAN, Wasserstein GAN, etc. It is built upon the idea of jointly optimizing an ensemble of GAN losses w.r.t. all pairs of discriminator and generator.

Differences with prior work. Although the idea of having multiple generators in the GAN architecture is not totally new, e.g., MIX+GAN (Arora et al., 2017) and MGAN (Hoang et al., 2018) , there are key differences between Stackelberg GAN and prior work. a) In MGAN (Hoang et al., 2018) , various generators are combined as a mixture of probabilistic models with assumption that the generators and discriminator have enough capacity. In contrast, in the Stackelberg GAN model we uniformly ensemble the losses of various standard GAN without any assumption on the model capacity. b) In MIX+GAN (Arora et al., 2017) , the losses are ensembled with learned weights and an extra regularization term, which discourages the weights being too far away from uniform. We find it slightly unnecessary because the expressive power of each generator already allows implicit scaling of each generator. To the contrary, in the Stackelberg GAN we apply equal weights for all generators.

• We prove that the minimax duality gap shrinks as the number of generators increases (see Theorem 1 and Corollary 2). Unlike the previous work, our result has no assumption on the 2 • We propose the Stackelberg GAN framework of multiple generators in the GAN architecture.

Our framework is general since it can be applied to all variants of GANs, e.g., vanilla GAN, Wasserstein GAN, etc. It is built upon the idea of jointly optimizing an ensemble of GAN losses w.r.t. all pairs of discriminator and generator.

Differences from prior work. Although the idea of having multiple generators in the GAN architecture is not totally new, e.g., MIX+GAN (Arora et al., 2017) , MGAN (Hoang et al., 2018) , MAD-GAN (Ghosh et al., 2017) and GMAN (Durugkar et al., 2016) , there are key differences between Stackelberg GAN and prior work. a) In MGAN (Hoang et al., 2018) and MAD-GAN (Ghosh et al., 2017) , various generators are combined as a mixture of probabilistic models with assumption that the generators and discriminator have infinite capacity. Also, they require that the generators share common network parameters. In contrast, in the Stackelberg GAN model we allow various sampling schemes beyond the mixture model, e.g., each generator samples a fixed but unequal number of data points independently. Furthermore, each generator has free parameters. We also make no assumption on the model capacity in our analysis. This is an important research question as raised by Arora et al. (2018) . b) In MIX+GAN (Arora et al., 2017) , the losses are ensembled with learned weights and an extra regularization term, which discourages the weights being too far away from uniform. We find it slightly unnecessary because the expressive power of each generator already allows implicit scaling of each generator. In the Stackelberg GAN, we apply equal weights for all generators and obtain improved guarantees. c) In GMAN (Durugkar et al., 2016) , there are multiple discriminators while it is unclear in theory why multi-discriminator architecture works well. In this paper, we provide formal guarantees for our model. • We prove that the minimax duality gap shrinks as the number of generators increases (see Theorem 1 and Corollary 2). Unlike the previous work, our result has no assumption on the expressive power of generators and discriminator, but instead depends on their non-convexity.

With extra condition on the expressive power of generators, we show that Stackelberg GAN is able to achieve -approximate equilibrium with O(1/ ) generators (see Theorem 3). This improves over the best-known result in (Arora et al., 2017) which requires generators as many as O(1/ 2 ). At the core of our techniques is a novel application of the Shapley-Folkman lemma to the generic minimax problem, where in the literature the technique was only known to work when the objective function is restricted to the Lagrangian function of a constrained optimization problem (Zhang et al., 2018) . This results in tighter bounds than that of the covering number argument as in (Arora et al., 2017) . We also note that MIX+GAN is a heuristic model which does not exactly match the theoretical analysis in (Arora et al., 2017) , while this paper provides formal guarantees for the exact model of Stackelberg GAN. • We empirically study the performance of Stackelberg GAN for various synthetic and real datasets. We observe that without any human assignment, surprisingly, each generator automatically learns balanced number of modes without any mode being dropped (see Figure 1 ). Compared with other multi-generator GANs with the same network capacity, our experiments show that Stackelberg GAN enjoys 26.76 Fréchet Inception Distance on CIFAR-10 dataset while prior results achieve 31.34 (smaller is better), achieving an improvement of 14.61%. 

2. STACKELBERG GAN:

Before proceeding, we define some notations and formalize our model setup in this section.

Notations. We will use bold lower-case letter to represent vector and lower-case letter to represent scalar. Specifically, we denote by θ ∈ R t the parameter vector of discriminator and γ ∈ R g the parameter vector of generator. Let D θ (x) be the output probability of discriminator given input x, and let G γ (z) represent the generated vector given random input z. For any function f (u), we denote by f * (v) := sup u {u T vf (u)} the conjugate function of f . Let clf be the convex closure of f , which is defined as the function whose epigraph is the convex closed hull of that of function f . We define clf := -cl(-f ). We will use I to represent the number of generators.

2.1. MODEL SETUP:

Preliminaries. The key ingredient in the standard GAN is to play a zero-sum two-player game between a discriminator and a generator -which are often parametrized by deep neural networks in practice -such that the goal of the generator is to map random noise z to some plausible images/texts G γ (z) and the discriminator D θ (•) aims at distinguishing the real images/texts from what the generator creates.

For every parameter implementations γ and θ of generator and discriminator, respectively, denote by the payoff value

φ(γ; θ) := E x∼P d f (D θ (x)) + E z∼Pz f (1 -D θ (G γ (z)))

, where f (•) is some concave, increasing function. Hereby, P d is the distribution of true images/texts and P z is a noise distribution such as Gaussian or uniform distribution. The standard GAN thus solves the following saddle point problems:

EQUATION

For different choices of function f , problem (1) leads to various variants of GAN. For example, when f (t) = log t, problem (1) is the classic GAN; when f (t) = t, it reduces to the Wasserstein GAN. We refer interested readers to the paper of Nowozin et al. (2016) for more variants of GANs.

Stackelberg GAN. Our model of Stackelberg GAN is inspired from the Stackelberg competition in the domain of game theory. Instead of playing a two-player game as in the standard GAN, in Stackelberg GAN there are I + 1 players with two firms -one discriminator and I generators. One can make an analogy between the discriminator (generators) in the Stackelberg GAN and the leader (followers) in the Stackelberg competition.

Stackelberg GAN is a general framework which can be built on top of all variants of standard GANs. The objective function is simply an ensemble of losses w.r.t. all possible pairs of generators and discriminator: Φ(γ 1 , ..., γ I ; θ) := I i=1 φ(γ i ; θ). Thus it is very easy to implement. The Stackelberg GAN therefore solves the following saddle point problems:

w * := inf γ1,...,γ I ∈R g sup θ∈R t 1 I Φ(γ 1 , ..., γ I ; θ), or q * := sup θ∈R t inf γ1,...,γ I ∈R g 1 I Φ(γ 1 , ..., γ I ; θ).

We term w *q * the minimax (duality) gap. We note that there are key differences between the naïve ensembling model and ours. In the naïve ensembling model, one trains multiple GAN models independently and averages their outputs. In contrast, our Stackelberg GAN shares a unique discriminator for various generators, thus requires jointly training. Figure 2 shows the architecture of our Stackelberg GAN.

How to generate samples from Stackelberg GAN? In the Stackelberg GAN, we expect that each generator learns only a few modes. In order to generate a sample that may come from all modes, we use a mixed model. In particular, we generate a uniformly random value i from 1 to I and use the i-th generator to obtain a new sample. Note that this procedure in independent of the training procedure.

3. ANALYSIS OF STACKELBERG GAN:

In this section, we develop our theoretical contributions and compare our results with the prior work.

3.1. MINIMAX DUALITY GAP:

We begin with studying the minimax gap of Stackelberg GAN. Our main results show that the minimax gap shrinks as the number of generators increases.

To proceed, denote by We have the following guarantees on the minimax gap of Stackelberg GAN.

h i (u i ) := inf γi∈R g (-φ(γ i ; •)) * (u i ),

clh i ( u) := inf {a j },{u j i }    t+2 j=1 a j h i (u j i ) : u = t+2 j=1 a j u j i , t+2 j=1 a j = 1, a j ≥ 0    . clh i represents

Theorem 1. Let ∆ i γ := sup u∈R t {h i (u) -clh i (u)} ≥ 0 and ∆ worst γ := max i∈[I] ∆ i γ .

Denote by t the number of parameters of discriminator, i.e., θ ∈ R t . Suppose that h i (•) is continuous and domh i is compact and convex. Then the duality gap can be bounded by

0 ≤ w * -q * ≤ ∆ minimax θ + ∆ maximin θ + ,

provided that the number of generators I > t+1 ∆ worst γ . Remark 1. Theorem 1 makes mild assumption on the continuity of loss and no assumption on the model capacity of discriminator and generators. The analysis instead depends on their nonconvexity as being parametrized by deep neural networks. In particular, ∆ i γ measures the divergence between the function value of h i and its convex relaxation

clh i ; When φ(γ i ; θ) is convex w.r.t. argument γ i , ∆ i γ is exactly 0. The constant ∆ worst γ

is the maximal divergence among all generators, which does not grow with the increase of I. This is because ∆ worst , we have 0 ≤ w *q * ≤ .

3.2. EXISTENCE OF APPROXIMATE EQUILIBRIUM:

The results of Theorem 1 and Corollary 2 are independent of model capacity of generators and discriminator. When we make assumptions on the expressive power of generator as in (Arora et al., 2017) , we have the following guarantee (2) on the existence of -approximate equilibrium. Theorem 3. Under the settings of Theorem 1, suppose that for any ξ > 0, there exists a generator G such that E x∼P d ,z∼Pz G(z)x 2 ≤ ξ. Let the discriminator and the generators be L-Lipschitz w.r.t. inputs and parameters, respectively. Then for any > 0, there exist

I = t+1 ∆ worst γ generators G γ * 1 , ..., G γ * I and a discriminator D θ * such that for some value V ∈ R, ∀γ 1 , ..., γ I ∈ R g , Φ(γ 1 , ..., γ I ; θ * ) ≤ V + , ∀θ ∈ R t , Φ(γ * 1 , ..., γ * I ; θ) ≥ V -.

(2)

Related Work. While many efforts have been devoted to empirically investigating the performance of multi-generator GAN, little is known about how many generators are needed so as to achieve certain equilibrium guarantees. Probably the most relevant prior work to Theorem 3 is that of (Arora et al., 2017) . In particular, Arora et al. (2017) showed that there exist I = 100t 2 ∆ 2 generators and one discriminator such that -approximate equilibrium can be achieved, provided that for all x and any ξ > 0, there exists a generator G such that

E z∼Pz G(z) -x 2 ≤ ξ. Hereby, ∆ is a global upper bound of function |f |, i.e., f ∈ [-∆, ∆].

In comparison, Theorem 3 improves over this result in two aspects: a) the assumption on the expressive power of generators in (Arora et al., 2017) implies our condition E x∼P d ,z∼Pz G(z)x 2 ≤ ξ. Thus our assumption is weaker. b) The required number of generators in Theorem 3 is as small as t+1 ∆ worst γ . We note that ∆ worst γ 2∆ by the definition of ∆ worst γ . Therefore, Theorem 3 requires much fewer generators than that of (Arora et al., 2017) .

4. ARCHITECTURE, CAPACITY AND MODE COLLAPSE/DROPPING:

In this section, we empirically investigate the effect of network architecture and capacity on the mode collapse/dropping issues for various multi-generator architecture designs. Hereby, the mode dropping refers to the phenomenon that generative models simply ignore some hard-to-represent modes of real distributions, and the mode collapse means that some modes of real distributions are "averaged" by generative models. For GAN, it is widely believed that the two issues are caused by the large gap between the minimax and maximin objective function values (see page 35, Goodfellow (2016)). Our experiments verify that network capacity (change of width and depth) is not very crucial for resolving the mode collapse issue, though it can alleviate the mode dropping in certain senses. Instead, the choice of architecture of generators plays a key role. To visualize this discovery, we test the performance of varying architectures of GANs on a synthetic mixture of Gaussians dataset with 8 modes and 0.01 standard deviation. We observe the following phenomena: Naïvely increasing capacity of one-generator architecture does not alleviate mode collapse. It shows that the multi-generator architecture in the Stackelberg GAN effectively alleviates the mode collapse issue. Though naïvely increasing capacity of one-generator architecture alleviates mode dropping issue, for more challenging mode collapse issue, the effect is not obvious (see Figure 3 ). Generators tend to learn balanced number of modes when they have same capacity. We observe that for varying number of generators, each generator in the Stackelberg GAN tends to learn equal number of modes when the modes are symmetric and every generator has same capacity (see Figure 5 ). 

5. EXPERIMENTS:

In this section, we verify our theoretical contributions by the experimental validation.

5.1. MNIST DATASET:

We first show that Stackelberg GAN generates more diverse images on the MNIST dataset (LeCun et al., 1998) than classic GAN. We follow the standard preprocessing step that each pixel is normalized via subtracting it by 0.5 and dividing it by 0.5. The detailed network setups of discriminator and generators are in Table 4 .

Figure 6 shows the diversity of generated digits by Stackelberg GAN with varying number of generators. When there is only one generator, the digits are not very diverse with many repeated "1"'s and much fewer "2"'s. As the number of generators increases, the generated images tend to be more diverse. In particular, for 10-generator Stackelberg GAN, each generator is associated with one or two digits without any digit being missed.

5.2. FASHION-MNIST DATASET:

We also observe better performance by the Stackelberg GAN on the Fashion-MNIST dataset. Fashion-MNIST is a dataset which consists of 60,000 examples. Each example is a 28 × 28 grayscale image associating with a label from 10 classes. We follow the standard preprocessing step that each pixel is normalized via subtracting it by 0.5 and dividing it by 0.5. We specify the detailed network setups of discriminator and generators in Table 4 .

Figure 7 shows the diversity of generated fashions by Stackelberg GAN with varying number of generators. When there is only one generator, the generated images are not very diverse without any bags being found. As the number of generators increases, the generated images tend to be more diverse. In particular, for 10-generator Stackelberg GAN, each generator is associated with one class without any class being missed.

5.3. CIFAR-10 DATASET:

We then implement Stackelberg GAN on the CIFAR-10 dataset. CIFAR-10 includes 60,000 32×32 training images, which fall into 10 classes (Krizhevsky & Hinton, 2009) ). The architecture of generators and discriminator follows the design of DCGAN in (Radford et al., 2015) . We train models with 5, 10, and 20 fixed-size generators. The results show that the model with 10 generators performs the best. We also train 10-generator models where each generator has 2, 3 and 4 convolution layers.

We find that the generator with 2 convolution layers, which is the most shallow one, performs the best. So we report the results obtained from the model with 10 generators containing 2 convolution layers. Figure 8a shows the samples produced by different generators. The samples are randomly drawn instead of being cherry-picked to demonstrate the quality of images generated by our model.

For quantitative evaluation, we use Inception score and Fréchet Inception Distance (FID) to measure the difference between images generated by models and real images.

Results of Inception Score. The Inception score measures the quality of a generated image and is correlated well with human's judgment (Salimans et al., 2016) . We report the Inception score obtained by our Stackelberg GAN and other baseline methods in Table 1 . For fair comparison, we only consider the baseline models which are completely unsupervised model and do not need any label information. Instead of directly using the reported Inception scores by original papers, we replicate the experiment of MGAN using the code, architectures and parameters reported by their original papers, and evaluate the scores based on the new experimental results. Table 1 shows that our model achieves a score of 7.62 in CIFAR-10 dataset, which outperforms the state-of-the-art models. For fairness, we configure our Stackelberg GAN with the same capacity as MGAN, that is, the two models have comparative number of total parameters. When the capacity of our Stackelberg GAN is as small as DCGAN, our model improves over DCGAN significantly.

Results of Fréchet Inception Distance. We then evaluate the performance of models on CIFAR-10 dataset using the Fréchet Inception Distance (FID), which better captures the similarity between generated images and real ones (Heusel et al., 2017) . As Table 1 shows, under the same capacity as DCGAN, our model reduces the FID by 20.74%. Meanwhile, under the same capacity as MGAN, our model reduces the FID by 14.61%. This improvement further indicates that our Stackelberg GAN with multiple light-weight generators help improve the quality of the generated images. (Arjovsky et al., 2017) 3.82 ± 0.06 -MIX+WGAN (Arora et al., 2017) 4.04 ± 0.07 -Improved-GAN (Salimans et al., 2016) 4.36 ± 0.04 -ALI (Dumoulin et al., 2017) 5.34 ± 0.05 -BEGAN (Berthelot et al., 2017) 5.62 -MAGAN (Wang et al., 2017) 5.67 -GMAN (Durugkar et al., 2016) 6.00 ± 0.19 -DCGAN (Radford et al., 2015) 6.40 ± 0.05 37.7 Ours (capacity as DCGAN)

7.02 ± 0.07 29.88 D2GAN (Nguyen et al., 2017) 7.15 ± 0.07 -MAD-GAN (our run) (Ghosh et al., 2017) 6.67 ± 0.07 34.10 MGAN (our run) (Hoang et al., 2018) 7.52 ± 0.1 31.34 Ours (capacity 1×MGAN≈ 1.8×DCGAN) 7.62 ± 0.07 26.76

5.4. TINY IMAGENET DATASET:

We also evaluate the performance of Stackelberg GAN on the Tiny ImageNet dataset. The Tiny ImageNet is a large image dataset, where each image is labelled to indicate the class of the object inside the image. We resize the figures down to 32 × 32 following the procedure described in (Chrabaszcz et al., 2017) . Figure 8b shows the randomly picked samples generated by 10-generator Stackelberg GAN. Each row has samples generated from one generator. Since the types of some images in the Tiny ImageNet are also included in the CIFAR-10, we order the rows in the similar way as Figure 8a .

6. CONCLUSIONS:

In this work, we tackle the problem of instability during GAN training procedure, which is caused by the huge gap between minimax and maximin objective values. The core of our techniques is a multi-generator architecture. We show that the minimax gap shrinks to as the number of generators increases with rate O(1/ ), when the maximization problem w.r.t. the discriminator is concave. This improves over the best-known results of O(1/ 2 ). Experiments verify the effectiveness of our proposed methods.

A SUPPLEMENTARY EXPERIMENTS Figure 9 shows how the architecture of generators affects the distributions of samples by each generators. The enlarged versions of samples generated by Stackelberg GAN with architectures shown in Table 5 and 

B PROOFS OF MAIN RESULTS:

B.1 PROOFS OF THEOREM 1 AND COROLLARY 2: MINIMAX DUALITY GAP

The statement 0 ≤ w *q * is by the weak duality. Thus it suffices to prove the other side of the inequality. All notations in this section are defined in Section 3.1. We first show that inf γ1,...,γ I ∈R g sup

θ∈R t 1 I Φ(γ 1 , ..., γ I ; θ) -sup θ∈R t inf γ1,...,γ I ∈R g 1 I Φ(γ 1 , ..., γ I ; θ) ≤ .

Denote by p(u) := inf γ1,...,γ I ∈R g sup θ∈R t Φ(γ 1 , ..., γ I ; θ)u T θ .

We have the following lemma.

Lemma 4. We have sup θ∈R t inf γ1,...,γ I ∈R g Φ(γ 1 , ..., γ I ; θ) = ( clp)(0) ≤ p(0) = inf γ1,...,γ I ∈R g sup θ∈R t Φ(γ 1 , ..., γ I ; θ).

Proof. By the definition of p(0), we have p(0) = inf γ1,...,γ I ∈R g sup θ∈R t Φ(γ 1 , ..., γ I ; θ). Since ( clp)(•) is the convex closure of function p(•) (a.k.a. weak duality theorem), we have ( clp)(0) ≤ p(0). We now show that sup θ∈R t inf γ1,...,γ I ∈R g Φ(γ 1 , ..., γ I ; θ) = ( clp)(0).

Note that p(u) = inf γ1,...,γ I ∈R g p γ1,...,γ I (u), where p γ1,...,γ I (u) = sup θ∈R t { Φ(γ 1 , ..., γ I ; θ)u T θ} = (-Φ(γ 1 , ..., γ I ; •)) * (-u), and that inf u∈R t {p γ1,...,γ I (u) + u T µ} =sup where u 1 , ..., u I , u ∈ R t . Therefore,

p(0) = inf u1,...,u I ∈R t I i=1 h i (u i ), s.t. I i=1 u i = 0.

Consider the subset of R t+1 :

Y i := y i ∈ R t+1 : y i = [u i , h i (u i )] , u i ∈ domh i , i ∈ [I].

Define the vector summation 

Y := Y 1 + Y 2 + ... + Y I . Since h i (•) is continuous and domh i is compact, the set {(u i , h i (u i )) : u i ∈ domh i } is compact. So Y, conv(Y), Y i ,

Y i + i∈I(y) conv(Y i )   .

We apply Lemma 6 to prove Lemma 5 with m = t + 1. Let (r, w) ∈ conv(Y) be such that r = 0, and w = clp(0).

Applying the above Shapley-Folkman lemma to the set Y = I i=1 Y i , we have that there are a subset I ⊆ [I] of size t + 1 and vectors

EQUATION

EQUATION

Representing elements of the convex hull of Y i ⊆ R t+1 by Carathéodory theorem, we have that for each i ∈ I, there are vectors {u j i } t+2 j=1 and scalars {a j i } t+2 j=1 ∈ R such that t+2 j=1

EQUATION

Recall that we define

EQUATION

∆ i γ := sup u∈R t {h i (u) -clh i (u)} ≥ 0.

We have for i ∈ I,

w i ≥ clh i   t+2 j=1 a j i u j i   (by the definition of clh i (•)) ≥ h i   t+2 j=1 a j i u j i   -∆ i γ (by the definition of ∆ i γ ) = h i (u i ) -∆ i γ .

(by Eqn. ( 6))

Thus, by Eqns. ( 4) and ( 6), we have

EQUATION

Under review as a conference paper at ICLR 2019 Therefore, we have By Lemmas 4 and 5, we have proved that inf γ1,...,γ I ∈R g sup

EQUATION

θ∈R t 1 I Φ(γ 1 , ..., γ I ; θ) -sup θ∈R t inf γ1,...,γ I ∈R g 1 I Φ(γ 1 , ..., γ I ; θ) ≤ .

To prove Theorem 1, we note that w *q * := inf γ1,...,γ I ∈R g sup

θ∈R t 1 I Φ(γ 1 , ..., γ I ; θ) -sup θ∈R t inf γ1,...,γ I ∈R g 1 I Φ(γ 1 , ..., γ I ; θ) = inf γ1,...,γ I ∈R g sup θ∈R t 1 I Φ(γ 1 , ..., γ I ; θ) - inf γ1,...,γ I ∈R g sup θ∈R t 1 I Φ(γ 1 , ..., γ I ; θ) + inf γ1,...,γ I ∈R g sup θ∈R t 1 I Φ(γ 1 , ..., γ I ; θ) -sup θ∈R t inf γ1,...,γ I ∈R g 1 I Φ(γ 1 , ..., γ I ; θ) + sup θ∈R t inf γ1,...,γ I ∈R g 1 I Φ(γ 1 , ..., γ I ; θ) -sup θ∈R t inf γ1,...,γ I ∈R g 1 I Φ(γ 1 , ..., γ I ; θ) ≤ ∆ minimax θ + ∆ maximin θ + , as desired.

When φ(γ i ; θ) is concave and closed w.r.t. discriminator parameter θ, we have clφ = φ. Thus, ∆ minimax θ = ∆ maximin θ = 0 and 0 ≤ w *q * ≤ .

B.2 PROOFS OF THEOREM 3: EXISTENCE OF APPROXIMATE EQUILIBRIUM:

We first show that the equilibrium value V is 2f (1/2). For the discriminator D θ which only outputs 1/2, it has payoff 2f (1/2) for all possible implementations of generators G γ1 , ..., G γ I . Therefore, we have V ≥ 2f (1/2). We now show that V ≤ 2f (1/2). We note that by assumption, for any ξ > 0, there exists a closed neighbour of implementation of generator G ξ such that E x∼P d ,z∼Pz G ξ (z)x 2 ≤ ξ for all G ξ in the neighbour. Such a neighbour exists because the generator is Lipschitz w.r.t. its parameters. Let the parameter implementation of such neighbour of G ξ be Γ. The Wasserstein distance between G ξ and P d is ξ. Since the function f and the discriminator are L f -Lipschitz and L-Lipschitz, respectively, we have

E z∼G ξ f (1 -D θ (z)) -E x∼P d f (1 -D θ (x)) ≤ O(L f Lξ).

Thus, for any fixed γ, we have

sup θ∈R t E x∼P d f (D θ (x)) + E z∼G ξ f (1 -D θ (z)) ≤ O(L f Lξ) + sup θ∈R t E x∼P d f (D θ (x)) + E x∼P d f (1 -D θ (x)) ≤ O(L f Lξ) + 2f (1/2) → 2f (1/2), (ξ → +0)

which implies that sup θ∈R t Φ(γ 1 , ..., γ I ; θ) = 2f (1/2) for all γ 1 , ..., γ I ∈ Γ. So we have V = 2f (1/2). This means that the discriminator cannot do much better than a random guess. The above analysis implies that the equilibrium is achieved when D θ * only outputs 1/2. Denote by Θ the small closed neighbour of such θ * such that Φ(γ 1 , ..., γ I ; θ) is concave w.r.t. θ ∈ Θ for any fixed γ 1 , ..., γ I ∈ Γ. We thus focus on the loss on Θ ⊆ R t and Γ ⊆ R g : Lemma 8 (Proposition 3.4 (b), Bertsekas (2009) ). For any function p(u), denote by q(µ) := inf u∈R t {p(u) + µ T u}. We have sup µ∈R t q(µ) = clp(0). Clearly, the vanilla GAN optimization can be understood as projecting under L: inf γ∈R g L(P d , P Gγ (z) ).

Φ(γ 1 , ..., γ I ; θ) := I i=1 [E x∼P d f (D θ (x)) + E z∼Pz f (1 -D θ (G γi (z)))] , θ ∈ Θ, γ 1 , ..., γ I ∈ Γ.

D DISTRIBUTIONAL APPROXIMATION PROPERTIES OF STACKELBERG GAN:

In the Stackelberg GAN setting, we are projecting under a different distance L which is defined as L = sup 

We note that f is strictly concave and the discriminator has capacity large enough implies the followings: L(P 1 , P 2 ), as a function of P 2 , achieves the global minimum if and only if P 2 = P 1 . The theorem then follows from this fact and ( 11). 

E NETWORK SETUP:



:



Figures and Tables:

Figure 1: Stackelberg GAN stabilizes the training procedure on a toy 2D mixture of 8 Gaussians. Top Row: Standard GAN training. It shows that several modes are dropped. Bottom Row: Stackelberg GAN training with 8 generator ensembles, each of which is denoted by one color. We can see that each generator exactly learns one mode of the distribution without any mode being dropped.

Figure 1: Stackelberg GAN stabilizes the training procedure on a toy 2D mixture of 8 Gaussians. Top Row: Standard GAN training. It shows that several modes are dropped. Bottom Row: Stackelberg GAN training with 8 generator ensembles, each of which is denoted by one color. We can see that each generator exactly learns one mode of the distribution without any mode being dropped.

Figure 2: Stackelberg GAN stabilizes the training procedure on a toy 2D mixture of 8 Gaussians. Top Row: Standard GAN training. It shows that several modes are dropped. Bottom Row: Stackelberg GAN training with 8 generator ensembles, each of which is denoted by one color. We can see that each generator exactly learns one mode of the distribution without any mode being dropped.

Figure 2: Stackelberg GAN stabilizes the training procedure on a toy 2D mixture of 8 Gaussians. Left Figure, Top Row: Standard GAN training. It shows that several modes are dropped. Left Figure, Bottom Row: Stackelberg GAN training with 8 generator ensembles, each of which is denoted by one color. Right Figure: Stackelberg GAN training with 10 generator ensembles on real images without cherry pick, where each row corresponds to one generator. We can see that each generator exactly learns one mode of the distribution without any mode being dropped.[Pengtao: It is kind of abrupt that you say "Stackelberg GAN stabilizes the training procedure" in the beginning sentence, then the rest talks about losing mode. In the introduction, a convincing tie between instability and mode collapse is still missing.]

Figure 1: Left Figure, Top Row: Standard GAN training on a toy 2D mixture of 8 Gaussians. Left Figure, Bottom Row: Stackelberg GAN training with 8 generator ensembles, each of which is denoted by one color. Right Figure: Stackelberg GAN training with 10 generator ensembles on fashion-MNIST dataset without cherry pick, where each row corresponds to one generator.

Figure 2: Architecture of Stackelberg GAN. We ensemble the losses of various generator and discriminator pairs with equal weights.

measures the divergence of only one generator and when each generator for example has the same architecture, we have ∆ worst γ = ∆ 1 γ = ... = ∆ I γ . Similarly, the terms ∆ minimax θ and ∆ maximin θ characterize the non-convexity of discriminator. When the discriminator is concave such as logistic regression and support vector machine, ∆ minimax θ = ∆ maximin θ = 0 and we have the following straightforward corollary about the minimax duality gap of Stackelberg GAN. Corollary 2. Under the settings of Theorem 1, when φ(γ i ; θ) is concave and closed w.r.t. discriminator parameter θ and the number of generators I > t+1 ∆ worst γ

Figure 3: Comparison of mode collapse/dropping issue of one-generator and multi-generator architectures with varying model capacities. (a) and (b) show that increasing the model capacity can alleviate the mode dropping issue, though it does not alleviate the mode collapse issue. (c) Multi-generator architecture with even small capacity resolves the mode collapse issue. Stackelberg GAN outperforms multi-branch models. We compare performance of multi-branch GAN and Stackelberg GAN with objective functions: (Multi-Branch GAN) φ 1 I I

Figure 4: Comparison of mode collapse issue of multi-branch and multi-generator architectures with varying model capacities. (a) and (b) show that increasing the model capacity can alleviate the mode dropping issue, though it does not alleviate the mode collapse issue. (c) Multi-generator architecture with much smaller capacity resolves the mode collapse issue.Generators tend to learn balanced number of modes when they have same capacity. We observe that for varying number of generators, each generator in the Stackelberg GAN tends to learn equal number of modes when the modes are symmetric and every generator has same capacity (see Figure5).

Figure 5: Stackelberg GAN with varying number of generators of architecture 2-128-256-512-1024-2.

Figure 6: Standard GAN vs. Stackelberg GAN on the MNIST dataset without cherry pick. Left Figure: Digits generated by the standard GAN. It shows that the standard GAN generates many "1"'s which are not very diverse. Middle Figure: Digits generated by the Stackelberg GAN with 5 generators, where every two rows correspond to one generator. Right Figure: Digits generated by the Stackelberg GAN with 10 generators, where each row corresponds to one generator.

Figure 7: Generated samples by Stackelberg GAN on CIFAR-10 dataset without cherry pick. Left Figure: Examples generated by the standard GAN. It shows that the standard GAN fails to generate bags. Middle Figure: Examples generated by the Stackelberg GAN with 5 generators, where every two rows correspond to one generator. Right Figure: Examples generated by the Stackelberg GAN with 10 generators, where each row corresponds to one generator.

(a) Samples on CIFAR-10. (b) Samples on Tiny ImageNet.

Figure 8: Examples generated by Stackelberg GAN on CIFAR-10 (left) and Tiny ImageNet (right) without cherry pick, where each row corresponds to samples from one generator.For fairness, we configure our Stackelberg GAN with the same capacity as MGAN, that is, the two models have comparative number of total parameters. When the capacity of our Stackelberg GAN is as small as DCGAN, our model improves over DCGAN significantly. Results of Fréchet Inception Distance. We then evaluate the performance of models on CIFAR-10 dataset using the Fréchet Inception Distance (FID), which better captures the similarity between generated images and real ones(Heusel et al., 2017). As Table1shows, under the same capacity as DCGAN, our model reduces the FID by 20.74%. Meanwhile, under the same capacity as MGAN, our model reduces the FID by 14.61%. This improvement further indicates that our Stackelberg GAN with multiple light-weight generators help improve the quality of the generated images.

Figure 9: Effects of generator architecture of Stackelberg GAN on a toy 2D mixture of Gaussians, where the number of generators is set to be 8. Top Row: The generators have one hidden layer. Middle Row: The generators have two hidden layers. Bottom Row: The generators have three hidden layer. It shows that with the number of hidden layers increasing, each generator tends to learn more modes. However, mode collapse never happens for all three architectures.

u∈R t {u T (-µ)p γ1,...,γ I (u)} = -(p γ1,...,γ I ) * (-µ) (by the definition of conjugate function) = -(-Φ(γ 1 , ..., γ I ; •)) * * (µ)= Φ(γ 1 , ..., γ I ; µ). (by conjugate theorem) u∈R t {p(u) + u T µ} (by Lemma 8) ..,γ I ∈R g {p γ1,...,γ I (u) + u T µ} (by the definition of p(u)) = supµ∈R t inf γ1,...,γ I ∈R g inf u∈R t {p γ1,...,γ I (u) + u T µ} = sup µ∈R t inf γ1,...,γ I ∈R g Φ(γ 1 , ..., γ I ; µ).(by Eqn. (3)) By Lemma 4, it suffices to show p(0) -( clp)(0) ≤ (t + 1)∆ worst γ . We have the following lemma. Lemma 5. Under the assumption in Theorem 1, p(0) -( clp)(0) ≤ (t + 1)∆ worst γ . Proof. We note that p(u) := inf γ1,...,γ I ∈R g sup θ∈R t Φ(γ 1 , ..., γ I ; θ)u T θ = inf γ1,...,γ I ∈R g sup θ∈R t I i=1 clφ(γ i ; θ)u T θ (by the definition of Φ) = inf γ1,...,γ I ∈R g I i=1 -clφ(γ i ; •) * (-u) (by the definition of conjugate function) = inf γ1,...,γ I ∈R g inf u1+...+u I =-u I i=1 (-clφ(γ i ; •)) * (u i ) (γ i ; •)) * (u i ) (by conjugate theorem) = inf u1+...+u I =-u inf γ1,...,γ I ∈R g {(-φ(γ 1 ; •)) * (u 1 ) + ... + (-φ(γ I ; •)) * (u I )} =: inf u1+...+u I =-u {h 1 (u 1 ) + ... + h I (u I )}, (by the definition of h i (•))

and conv(Y i ), i ∈ [I] are all compact sets. According to the definition of Y and the standard duality argument(Bertsekas, 2009), we have p(0) = inf {w : there exists (r, w) ∈ Y such that r = 0} , and clp(0) = inf {w : there exists (r, w) ∈ conv (Y) such that r = 0} .We are going to apply the following Shapley-Folkman lemma. Lemma 6 (Shapley-Folkman, Starr (1969)). Let Y i , i ∈ [I] be a collection of subsets of R m . Then for every y ∈ conv( I i=1 Y i ), there is a subset I(y) ⊆ [I] of size at most m such that

Since Φ(γ 1 , ..., γ I ; θ) is concave w.r.t. θ ∈ Θ for all γ 1 , ..., γ I ∈ Γ, by Corollary 2, we haveinf γ1,...,γ I ∈Γ sup θ∈Θ 1 I Φ(γ 1 , ..., γ I ; θ)sup θ∈Θ inf γ1,...,γ I ∈Γ 1 I Φ(γ 1 , ..., γ I ; θ) ≤ .The optimal implementations of γ 1 , ..., γ I is achieved by argmin γ1,...,γ I ∈Γ sup θ∈Θ 1 I Φ(γ 1 , ..., γ I ; θ).C USEFUL LEMMASLemma 7. Given the function(f 1 + ... + f I )(θ) := f 1 (θ) + ... + f I (θ),wheref i : R t → R, i ∈ [I] are closed proper convex functions. Denote by f * 1 ⊕ ... ⊕ f * I the infimal convolution (f * 1 ⊕ ... ⊕ f * I )(u) := inf u1+...+u I =u {f * 1 (u 1 ) + ... + f * I (u I )}, u ∈ R t .Provided that f 1 + ... + f I is proper, then we have(f 1 + ... + f I ) * (u) = cl(f * 1 ⊕ ... ⊕ f * I )(u), ∀u ∈ R t . Proof. For all θ ∈ R t , we have f 1 (θ) + ... + f I (θ) = sup u1 {θ T u 1f * 1 (u 1 )} + ... + sup u I {θ T u If * I (u I )} = sup u1,...,u I {θ T (u 1 + ... + u I )f * 1 (u 1 ) -...f * I (u I )} = sup u sup u1+...+u I =u θ T uf * 1 (u 1 ) -...f * I (u I ) = sup u θ T uinfu1+...+u I =u f * 1 (u 1 ) -...f * I (u I ) = sup u θ T u -(f * 1 ⊕ ... ⊕ f * I )(u) = (f * 1 ⊕ ... ⊕ f * I ) * (θ).

1 ⊕ ... ⊕ f * I )(u) = cl(f * 1 ⊕ ... ⊕ f * I )(u) = (f * 1 ⊕ ... ⊕ f * I ) * * (u) = (f 1 + ... + f I ) * (u), where the first equality holds because (f * 1 ⊕ ... ⊕ f * I ) is convex, the second quality is by standard conjugate theorem, and the last equality holds by conjugating the both sides of Eqn. (9).

Suppose that f is strictly concave and the discriminator has infinite capacity. Then, the global optimum of Stackelberg GAN is achieved if and only if1 I I i=1 Gγ i (z) = P d .Proof. We defineL(P d , P Gγ (z) ) = sup θ∈R t E x∼P d f (D θ (x)) + E z∼Pz f (1 -D θ (G γ (z))).

z∼Pz f (1 -D θ (G γi (z))) Gγ i (z) .

Figure 11: Examples generated by Stackelberg GAN with 10 generators on CIFAR-10 dataset, where each row corresponds to samples from one generator.

Figure 12: Examples generated by Stackelberg GAN with 10 generators on Tiny ImageNet dataset, where each row corresponds to samples from one generator.

of objective function w.r.t. argument θ. For example, it is equal to 0 if and only if φ(γ i ; θ) is concave and closed w.r.t. discriminator parameter θ.

Table content: <table><tr><td>non-convexity</td><td>θ</td><td>+ ∆ minimax θ</td><td>measures the</td></tr></table>

Quantitative evaluation of various GANs on CIFAR-10 dataset. All results are either reported by the authors themselves or run by us with codes provided by the authors. Every model is trained without label. Methods with higher inception score and lower Fréchet Inception Distance are better.

Table content: <table><tr><td>Model</td><td colspan="2">Inception Score Fréchet Inception Distance</td></tr><tr><td>Real data WGAN</td><td>11.24 ± 0.16</td><td>-</td></tr></table>

Architecture and hyper-parameters for the mixture of Gaussians dataset.

Table content: <table><tr><td colspan="3">Operation Input Dim Output Dim BN?</td><td>Activation</td></tr><tr><td>Generator G(z) : z ∼ N (0, 1) Linear Linear Discriminator Linear Linear Linear Number of generators Batch size for real data Number of iterations Slope of Leaky ReLU Learning rate Optimizer</td><td>2 16 2 512 256 8 64 200 0.2 0.0002 Adam</td><td>2 16 2 512 256 1</td><td>Tanh Leaky ReLU Leaky ReLU Sigmoid</td></tr></table>

Architecture and hyper-parameters for the MNIST datasets.

Table content: <table><tr><td colspan="3">Operation Input Dim Output Dim BN?</td><td>Activation</td></tr><tr><td>Generator G(z) : z ∼ N (0, 1) Linear Linear Discriminator Linear Linear Linear Number of generators Batch size for real data Slope of Leaky ReLU Learning rate Optimizer</td><td>100 512 2 512 256 10 100 0.2 0.0002 Adam</td><td>2 512 784 512 256 1</td><td>Tanh Leaky ReLU Leaky ReLU Sigmoid</td></tr></table>

Peer review (iclr2019):
 This paper proposes the Stackelberg GAN framework of multiple generators in the GAN architecture. The architecture is similar with previous multiple-generator GANs (MAD-GAN and MGAN). In fact, it*s even simpler in the sense that Stackelberg GAN has simpler loss function for the discriminator compared with the previous two. The authors prove that the minimax duality gap shrinks as the number of generators increases. And this proof has no assumption on the expressive power of generators and discriminator. With this proof, the authors argues that because the duality gap shrinks as the number of generators increases, the training of GANs gets more stable. From the algorithm part, I think the algorithm is very similar (and even simpler) than MAD-GAN and MGAN. The MAD-GAN and MGAN even proposed some specific loss for the discriminator so that it will encourage different generator to generate different modes in the target distribution. The Stackelberg GAN does not do this, but "partially" achieved the same goal. However, from Figure 9, we see that the simpler the generator is, the easier different generator will capture different modes. I think that this is due to the simplicity of discriminator loss. Therefore, on the algorithm part, the author may want to address the difference between Stackelberg GAN and MAD-GAN and MGAN. On the experiment part, we need to see more comparison between these three methods. In the current experiment, MGAN result is very similar to the proposed method, and MAD-GAN result is missing. Personally, I think that on cifar dataset (or larger datasets), these three methods should have very similar behavior. From the theoretical part, the authors derived a bound of the minimax duality gap for the Stackelberg GAN, without the assumption on the expressive power of generators and discriminator. Although the bound may not be practical, these are nice efforts. There are many typos in the paper (and appendix), which make me difficult to follow the proofs. For example, "Let clf (bclf) be the convex(concave) closure of f, which is defined as the function whose epigraph (subgraph) is the convex (concave) closed hull of that of function f." Do we have concave closed hull of subgraph of function f? What is the concave closed hull of a set? The usage of sub(sup)-script is also very confusing, like in the definition of h_i(u_i). The authors may want to correct typos and improve the presentation. In the conclusion, the authors conclude "We show that the minimax gap shrinks to .eps as the number of generators increases with rate e O(1/.eps)." This is an over-claim, because the authors only proved this under the assumption of concavity of the maximization w.r.t. discriminators. Finally, the authors may want to provide some simple results of the Stackelberg GAN from the perspective of density approximation, even assuming infinite capacity of the discriminator set, as other GANs does. Whether the distance defined by the maximization problem a distance or divergence. If we exactly minimizing that objective function, do we get the target distribution?