STACKELBERG GAN: TOWARDS PROVABLE MINI-
Year: 2018

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 Õ(1/ ). This improves over the best-known result of Õ(1/ ). 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.

heading: 1 INTRODUCTION
text: 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.
Under review as a conference paper at ICLR 2019 Under review as a conference paper at ICLR 2019
(a) Step 0. (b) Step 6k. (c) Step 19k. (d) Step 25k.
(e) Step 0. (f) Step 6k. (g) Step 19k. (h) Step 25k.
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.
Under review as a conference paper at ICLR 2019
(a) Step 0. (b) Step 6k. (c) Step 13k. (d) Step 19k. (e) Step 25k.
(f) Step 0. (g) Step 6k. (h) Step 13k. (i) Step 19k. (j) Step 25k.
Figure 1: Stackelberg GAN stabiliz s the training procedure on a toy 2D mixture of 8 G ussians. Top Row: Standard GAN trai ing. It shows that several modes are dropp d. Bottom Row: Stack lberg 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.
• We propose Stackelberg GAN framework of having multiple generators in the GAN archi-
tecture. Our framework is general that can be applied to all variants of GANs, e.g., vanilla
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.
probabil stic models with assumption that the g nerator and discriminator have infinite ca-
pacity. Also, they require that h generators sha e common network parameters. In contrast,
in e 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.
Fur ermore, each gene ator has free par meters. We also make no assumption on the model

heading: 2 STACKELBERG GAN
text: 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 ∈ Rt the parameter vector of discriminator and γ ∈ Rg the parameter vector of generator. Let Dθ(x) be the output probability of discriminator given input x, and let Gγ(z) represent the generated vector given random input z. For any function f(u), we denote by f∗(v) := supu{uTv − f(u)} the conjugate function of f . Let c̆lf 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 ĉlf := −c̆l(−f). We will use I to represent the number of generators.

heading: 2.1 MODEL SETUP
text: 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 Gγ(z) and the discriminator Dθ(·) 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) := Ex∼Pdf(Dθ(x)) + Ez∼Pzf(1−Dθ(Gγ(z))), where f(·) is some concave, increasing function. Hereby, Pd is the distribution of true images/texts and Pz is a noise distribution such as Gaussian or uniform distribution. The standard GAN thus solves the following saddle point problems:
inf γ∈Rg sup θ∈Rt φ(γ; θ), or sup θ∈Rt inf γ∈Rg φ(γ; θ). (1)
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(γ1, ..., γI ; θ) := ∑I i=1 φ(γi; θ). Thus it is very easy to implement. The Stackelberg GAN therefore solves the following saddle point problems:
w∗ := inf γ1,...,γI∈Rg sup θ∈Rt
1 I Φ(γ1, ..., γI ; θ), or q∗ := sup θ∈Rt inf γ1,...,γI∈Rg 1 I Φ(γ1, ..., γI ; 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.

heading: 3 ANALYSIS OF STACKELBERG GAN
text: In this section, we develop our theoretical contributions and compare our results with the prior work.

heading: 3.1 MINIMAX DUALITY GAP
text: 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 hi(ui) := infγi∈Rg (−φ(γi; ·))∗(ui), where the conjugate operation is w.r.t. the second argument of φ(γi; ·). We clarify here that the subscript i in hi indicates that the function hi is derived from the i-th generator. The argument of hi should depend on i, so we denote it by ui. Intuitively, hi serves as an approximate convexification of −φ(γi, ·) w.r.t the second argument due to the conjugate operation. Denote by c̆lhi the convex closure of hi:
c̆lhi(ũ) := inf {aj},{uji}
   t+2∑
j=1
ajhi(u j i ) : ũ =
t+2∑
j=1
ajuji ,
t+2∑
j=1
aj = 1, aj ≥ 0    .
c̆lhi represents the convex relaxation of hi because the epigraph of c̆lhi is exactly the convex hull of epigraph of hi by the definition of c̆lhi. Let ∆minimaxθ = infγ1,...,γI∈Rg supθ∈Rt 1 IΦ(γ1, ..., γI ; θ)− infγ1,...,γI∈Rg supθ∈Rt 1 I Φ̃(γ1, ..., γI ; θ), and ∆ maximin θ = supθ∈Rt infγ1,...,γI∈Rg 1 I Φ̃(γ1, ..., γI ; θ)− supθ∈Rt infγ1,...,γI∈Rg 1 IΦ(γ1, ..., γI ; θ), where Φ̃(γ1, ..., γI ; θ) := ∑I i=1 ĉlφ(γi; θ) and −ĉlφ(γi; θ) is the convex closure of −φ(γi; θ) w.r.t. argument θ. Therefore, ∆maximinθ + ∆minimaxθ measures the non-convexity of objective function w.r.t. argument θ. For example, it is equal to 0 if and only if φ(γi; θ) is concave and closed w.r.t. discriminator parameter θ. We have the following guarantees on the minimax gap of Stackelberg GAN.
Theorem 1. Let ∆iγ := supu∈Rt{hi(u) − c̆lhi(u)} ≥ 0 and ∆worstγ := maxi∈[I] ∆iγ . Denote by t the number of parameters of discriminator, i.e., θ ∈ Rt. Suppose that hi(·) is continuous and domhi is compact and convex. Then the duality gap can be bounded by
0 ≤ w∗ − q∗ ≤ ∆minimaxθ + ∆maximinθ + , 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, ∆iγ measures the divergence between the function value of hi and its convex relaxation c̆lhi; When φ(γi; θ) is convex w.r.t. argument γi, ∆iγ 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 ∆worstγ measures the divergence of only one generator and when each generator for example has the same architecture, we have ∆worstγ = ∆ 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, ∆minimaxθ = ∆ 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(γi; θ) is concave and closed w.r.t. discriminator parameter θ and the number of generators I > t+1 ∆
worst γ , we have 0 ≤ w∗ − q∗ ≤ .

heading: 3.2 EXISTENCE OF APPROXIMATE EQUILIBRIUM
text: 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 Ex∼Pd,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
Gγ∗1 , ..., Gγ∗I and a discriminator Dθ∗ such that for some value V ∈ R, ∀γ1, ..., γI ∈ Rg, Φ(γ1, ..., γI ; θ∗) ≤ V + , ∀θ ∈ Rt, 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 Ez∼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 Ex∼Pd,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
∆worstγ . Therefore, Theorem 3 requires much fewer generators than that of (Arora et al., 2017).

heading: 4 ARCHITECTURE, CAPACITY AND MODE COLLAPSE/DROPPING
text: 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).
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∑
i=1
γi; θ
) vs. (Stackelberg GAN) 1
I
I∑
i=1
φ(γi; θ).
Hereby, the multi-branch GAN has made use of extra information that the real distribution is Gaussian mixture model with probability distribution function 1I ∑I i=1 pNi(x), so that each γi tries to fit one component. However, even this we observe that with same model capacity, Stackelberg GAN significantly outperforms multi-branch GAN (see Figure 4 (a)(c)) even without access to the extra information. The performance of Stackelberg GAN is also better than multi-branch GAN of much larger capacity (see Figure 4 (b)(c)).
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).

heading: 5 EXPERIMENTS
text: In this section, we verify our theoretical contributions by the experimental validation.

heading: 5.1 MNIST DATASET
text: 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.

heading: 5.2 FASHION-MNIST DATASET
text: We also observe better performance by the Stackelberg GAN on the Fashion-MNIST dataset. FashionMNIST 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.

heading: 5.3 CIFAR-10 DATASET
text: 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.

heading: 5.4 TINY IMAGENET DATASET
text: 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 .

heading: 6 CONCLUSIONS
text: 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 Õ(1/ ), when the maximization problem w.r.t. the discriminator is concave. This improves over the best-known results of Õ(1/ 2). Experiments verify the effectiveness of our proposed methods.

heading: A SUPPLEMENTARY EXPERIMENTS
text: 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 Table 6 are deferred to Figures 10, 11, 12 and 13.

heading: B PROOFS OF MAIN RESULTS
text: 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,...,γI∈Rg sup θ∈Rt
1 I Φ̃(γ1, ..., γI ; θ)− sup θ∈Rt inf γ1,...,γI∈Rg 1 I Φ̃(γ1, ..., γI ; θ) ≤ .
Denote by
p(u) := inf γ1,...,γI∈Rg sup θ∈Rt
{ Φ̃(γ1, ..., γI ; θ)− uT θ } .
We have the following lemma.
Lemma 4. We have
sup θ∈Rt inf γ1,...,γI∈Rg Φ̃(γ1, ..., γI ; θ) = (c̆lp)(0) ≤ p(0) = inf γ1,...,γI∈Rg sup θ∈Rt Φ̃(γ1, ..., γI ; θ).
Proof. By the definition of p(0), we have p(0) = infγ1,...,γI∈Rg supθ∈Rt Φ̃(γ1, ..., γI ; θ). Since (c̆lp)(·) is the convex closure of function p(·) (a.k.a. weak duality theorem), we have (c̆lp)(0) ≤ p(0). We now show that
sup θ∈Rt inf γ1,...,γI∈Rg Φ̃(γ1, ..., γI ; θ) = (c̆lp)(0).
Note that p(u) = infγ1,...,γI∈Rg pγ1,...,γI (u), where pγ1,...,γI (u) = supθ∈Rt{Φ̃(γ1, ..., γI ; θ) − uT θ} = (−Φ̃(γ1, ..., γI ; ·))∗(−u), and that
inf u∈Rt {pγ1,...,γI (u) + uTµ} = − sup u∈Rt {uT (−µ)− pγ1,...,γI (u)}
= −(pγ1,...,γI )∗(−µ) (by the definition of conjugate function) = −(−Φ̃(γ1, ..., γI ; ·))∗∗(µ) = Φ̃(γ1, ..., γI ;µ). (by conjugate theorem) (3)
So we have
(c̆lp)(0) = sup µ∈Rt inf u∈Rt {p(u) + uTµ} (by Lemma 8)
= sup µ∈Rt inf u∈Rt inf γ1,...,γI∈Rg
{pγ1,...,γI (u) + uTµ} (by the definition of p(u))
= sup µ∈Rt inf γ1,...,γI∈Rg inf u∈Rt {pγ1,...,γI (u) + uTµ}
= sup µ∈Rt inf γ1,...,γI∈Rg Φ̃(γ1, ..., γI ;µ). (by Eqn. (3))
By Lemma 4, it suffices to show p(0)− (c̆lp)(0) ≤ (t+ 1)∆worstγ . We have the following lemma. Lemma 5. Under the assumption in Theorem 1, p(0)− (c̆lp)(0) ≤ (t+ 1)∆worstγ .
Proof. We note that
p(u) := inf γ1,...,γI∈Rg sup θ∈Rt
{ Φ̃(γ1, ..., γI ; θ)− uT θ }
= inf γ1,...,γI∈Rg sup θ∈Rt
{ I∑
i=1
ĉlφ(γi; θ)− uT θ } (by the definition of Φ̃)
= inf γ1,...,γI∈Rg
( I∑
i=1
−ĉlφ(γi; ·) )∗ (−u) (by the definition of conjugate function)
= inf γ1,...,γI∈Rg inf u1+...+uI=−u
{ I∑
i=1
(−ĉlφ(γi; ·))∗(ui) } (by Lemma 7)
= inf γ1,...,γI∈Rg inf u1+...+uI=−u
{ I∑
i=1
(−φ(γi; ·))∗(ui) } (by conjugate theorem)
= inf u1+...+uI=−u inf γ1,...,γI∈Rg
{(−φ(γ1; ·))∗(u1) + ...+ (−φ(γI ; ·))∗(uI)}
=: inf u1+...+uI=−u
{h1(u1) + ...+ hI(uI)}, (by the definition of hi(·))
where u1, ...,uI ,u ∈ Rt. Therefore,
p(0) = inf u1,...,uI∈Rt
I∑
i=1
hi(ui), s.t. I∑
i=1
ui = 0.
Consider the subset of Rt+1:
Yi := { yi ∈ Rt+1 : yi = [ui, hi(ui)] ,ui ∈ domhi } , i ∈ [I].
Define the vector summation Y := Y1 + Y2 + ...+ YI . Since hi(·) is continuous and domhi is compact, the set {(ui, hi(ui)) : ui ∈ domhi}
is compact. So Y , conv(Y), Yi, and conv(Yi), 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 c̆lp(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 Yi, i ∈ [I] be a collection of subsets of Rm. Then for every y ∈ conv(∑Ii=1 Yi), there is a subset I(y) ⊆ [I] of size at most m such that
y ∈
  ∑
i 6∈I(y)
Yi + ∑
i∈I(y)
conv(Yi)   .
We apply Lemma 6 to prove Lemma 5 with m = t+ 1. Let (r, w) ∈ conv(Y) be such that r = 0, and w = c̆lp(0). Applying the above Shapley-Folkman lemma to the set Y = ∑Ii=1 Yi, we have that there are a subset I ⊆ [I] of size t+ 1 and vectors
(ri, wi) ∈ conv(Yi), i ∈ I and ui ∈ domhi, i 6∈ I, such that ∑
i6∈I
ui + ∑
i∈I
ri = r = 0, (4)
∑
i6∈I
hi(ui) + ∑
i∈I
wi = c̆lp(0). (5)
Representing elements of the convex hull of Yi ⊆ Rt+1 by Carathéodory theorem, we have that for each i ∈ I, there are vectors {uji}t+2j=1 and scalars {aji}t+2j=1 ∈ R such that
t+2∑
j=1
aji = 1, a j i ≥ 0, j ∈ [t+ 2],
ri =
t+2∑
j=1
ajiu j i =: ui ∈ domhi, wi =
t+2∑
j=1
ajihi(u j i ). (6)
Recall that we define
c̆lhi(ũ) := inf {aj},{uji}
   t+2∑
j=1
ajhi(u j i ) : ũ =
t+2∑
j=1
ajuji ,
t+2∑
j=1
aj = 1, aj ≥ 0    ,
and ∆iγ := supu∈Rt{hi(u)− c̆lhi(u)} ≥ 0. We have for i ∈ I,
wi ≥ c̆lhi
  t+2∑
j=1
ajiu j i   (by the definition of c̆lhi(·))
≥ hi
  t+2∑
j=1
ajiu j i  −∆iγ (by the definition of ∆iγ)
= hi (ui)−∆iγ . (by Eqn. (6))
(7)
Thus, by Eqns. (4) and (6), we have I∑
i=1
ui = 0, ui ∈ domhi, i ∈ [I]. (8)
Therefore, we have
p(0) =
I∑
i=1
hi(ui) (by Eqn. (8))
≤ c̆lp(0) + ∑
i∈I
∆iγ (by Eqns. (5) and (7))
≤ c̆lp(0) + |I|∆worstγ = c̆lp(0) + (t+ 1)∆worstγ , (by Lemma 6)
as desired.
By Lemmas 4 and 5, we have proved that
inf γ1,...,γI∈Rg sup θ∈Rt
1 I Φ̃(γ1, ..., γI ; θ)− sup θ∈Rt inf γ1,...,γI∈Rg 1 I Φ̃(γ1, ..., γI ; θ) ≤ .
To prove Theorem 1, we note that
w∗ − q∗ := inf γ1,...,γI∈Rg sup θ∈Rt
1 I Φ(γ1, ..., γI ; θ)− sup θ∈Rt inf γ1,...,γI∈Rg 1 I Φ(γ1, ..., γI ; θ)
= inf γ1,...,γI∈Rg sup θ∈Rt
1 I Φ(γ1, ..., γI ; θ)− inf γ1,...,γI∈Rg sup θ∈Rt 1 I Φ̃(γ1, ..., γI ; θ)
+ inf γ1,...,γI∈Rg sup θ∈Rt
1 I Φ̃(γ1, ..., γI ; θ)− sup θ∈Rt inf γ1,...,γI∈Rg 1 I Φ̃(γ1, ..., γI ; θ)
+ sup θ∈Rt inf γ1,...,γI∈Rg
1 I Φ̃(γ1, ..., γI ; θ)− sup θ∈Rt inf γ1,...,γI∈Rg 1 I Φ(γ1, ..., γI ; θ)
≤ ∆minimaxθ + ∆maximinθ + , as desired. When φ(γi; θ) is concave and closed w.r.t. discriminator parameter θ, we have ĉlφ = φ. Thus, ∆minimaxθ = ∆ 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 Dθ which only outputs 1/2, it has payoff 2f(1/2) for all possible implementations of generators Gγ1 , ..., GγI . 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 Gξ such that Ex∼Pd,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 Gξ be Γ. The Wasserstein distance between Gξ and Pd is ξ. Since the function f and the discriminator are Lf -Lipschitz and L-Lipschitz, respectively, we have∣∣Ez∼Gξf(1−Dθ(z))− Ex∼Pdf(1−Dθ(x))
∣∣ ≤ O(LfLξ). Thus, for any fixed γ, we have
sup θ∈Rt
Ex∼Pdf(Dθ(x)) + Ez∼Gξf(1−Dθ(z))
≤ O(LfLξ) + sup θ∈Rt Ex∼Pdf(Dθ(x)) + Ex∼Pdf(1−Dθ(x))
≤ O(LfLξ) + 2f(1/2)→ 2f(1/2), (ξ → +0) which implies that supθ∈Rt Φ(γ1, ..., γI ; θ) = 2f(1/2) for all γ1, ..., γI ∈ 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 Dθ∗ only outputs 1/2. Denote by Θ the small closed neighbour of such θ∗ such that Φ(γ1, ..., γI ; θ) is concave w.r.t. θ ∈ Θ for any fixed γ1, ..., γI ∈ Γ. We thus focus on the loss on Θ ⊆ Rt and Γ ⊆ Rg:
Φ(γ1, ..., γI ; θ) :=
I∑
i=1
[Ex∼Pdf(Dθ(x)) + Ez∼Pzf(1−Dθ(Gγi(z)))] , θ ∈ Θ, γ1, ..., γI ∈ Γ.
Since Φ(γ1, ..., γI ; θ) is concave w.r.t. θ ∈ Θ for all γ1, ..., γI ∈ Γ, by Corollary 2, we have
inf γ1,...,γI∈Γ sup θ∈Θ
1 I Φ(γ1, ..., γI ; θ)− sup θ∈Θ inf γ1,...,γI∈Γ 1 I Φ(γ1, ..., γI ; θ) ≤ .
The optimal implementations of γ1, ..., γI is achieved by argminγ1,...,γI∈Γ supθ∈Θ 1 IΦ(γ1, ..., γI ; θ).

heading: C USEFUL LEMMAS
text: Lemma 7. Given the function
(f1 + ...+ fI)(θ) := f1(θ) + ...+ fI(θ),
where fi : Rt → R, i ∈ [I] are closed proper convex functions. Denote by f∗1 ⊕ ...⊕ f∗I the infimal convolution
(f∗1 ⊕ ...⊕ f∗I )(u) := inf u1+...+uI=u {f∗1 (u1) + ...+ f∗I (uI)}, u ∈ Rt.
Provided that f1 + ...+ fI is proper, then we have
(f1 + ...+ fI) ∗(u) = cl(f∗1 ⊕ ...⊕ f∗I )(u), ∀u ∈ Rt.
Proof. For all θ ∈ Rt, we have f1(θ) + ...+ fI(θ) = sup
u1 {θTu1 − f∗1 (u1)}+ ...+ sup uI {θTuI − f∗I (uI)}
= sup u1,...,uI
{θT (u1 + ...+ uI)− f∗1 (u1)− ...− f∗I (uI)}
= sup u sup u1+...+uI=u
{ θTu− f∗1 (u1)− ...− f∗I (uI) }
= sup u
{ θTu− inf
u1+...+uI=u f∗1 (u1)− ...− f∗I (uI)
}
= sup u
{ θTu− (f∗1 ⊕ ...⊕ f∗I )(u) }
= (f∗1 ⊕ ...⊕ f∗I )∗(θ).
(9)
Therefore,
cl(f∗1 ⊕ ...⊕ f∗I )(u) = c̆l(f∗1 ⊕ ...⊕ f∗I )(u) = (f∗1 ⊕ ...⊕ f∗I )∗∗(u) = (f1 + ...+ fI)∗(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).
Lemma 8 (Proposition 3.4 (b), Bertsekas (2009)). For any function p(u), denote by q(µ) := infu∈Rt{p(u) + µTu}. We have supµ∈Rt q(µ) = c̆lp(0).

heading: D DISTRIBUTIONAL APPROXIMATION PROPERTIES OF STACKELBERG GAN
text: Theorem 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 if
1
I
I∑
i=1
PGγi (z) = Pd.
Proof. We define
L(Pd,PGγ(z)) = sup θ∈Rt Ex∼Pdf(Dθ(x)) + Ez∼Pzf(1−Dθ(Gγ(z))).
Clearly, the vanilla GAN optimization can be understood as projecting under L:
inf γ∈Rg
L(Pd,PGγ(z)).
In the Stackelberg GAN setting, we are projecting under a different distance L̃ which is defined as
L̃ = sup θ∈Rt
Ex∼Pdf(Dθ(x)) + 1
I
I∑
i=1
Ez∼Pzf(1−Dθ(Gγi(z))) (10)
= L ( Pd, 1
I
I∑
i=1
PGγi (z) ) . (11)
We note that f is strictly concave and the discriminator has capacity large enough implies the followings: L(P1,P2), as a function of P2, achieves the global minimum if and only if P2 = P1. The theorem then follows from this fact and (11).

heading: E NETWORK SETUP

Review summaries:
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.