Title: SMOOTHNESS AND STABILITY IN GANS

Abstract: Generative adversarial networks, or GANs, commonly display unstable behavior during training. In this work, we develop a principled theoretical framework for understanding the stability of various types of GANs. In particular, we derive conditions that guarantee eventual stationarity of the generator when it is trained with gradient descent, conditions that must be satisfied by the divergence that is minimized by the GAN and the generator's architecture. We find that existing GAN variants satisfy some, but not all, of these conditions. Using tools from convex analysis, optimal transport, and reproducing kernels, we construct a GAN that fulfills these conditions simultaneously. In the process, we explain and clarify the need for various existing GAN stabilization techniques, including Lipschitz constraints, gradient penalties, and smooth activation functions.
Generative adversarial networks (Goodfellow et al., 2014) , or GANs, are a powerful class of generative models defined through minimax game. GANs and their variants have shown impressive performance in synthesizing various types of datasets, especially natural images. Despite these successes, the training of GANs remains quite unstable in nature, and this instability remains difficult to understand theoretically.

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Since the introduction of GANs, there have been many techniques proposed to stabilize GANs training, including studies of new generator/discriminator architectures, loss functions, and regularization techniques. Notably, Arjovsky et al. (2017) proposed Wasserstein GAN (WGAN), which in principle avoids instability caused by mismatched generator and data distribution supports. In practice, this is enforced by Lipschitz constraints, which in turn motivated developments like gradient penalties (Gulrajani et al., 2017) and spectral normalization (Miyato et al., 2018) . Indeed, these stabilization techniques have proven essential to achieving the latest state-of-the-art results (Karras et al., 2018; Brock et al., 2019) .

On the other hand, a solid theoretical understanding of training stability has not been established. Several empirical observations point to an incomplete understanding. For example, why does applying a gradient penalty together spectral norm seem to improve performance (Miyato et al., 2018) , even though in principle they serve the same purpose? Why does applying only spectral normalization with the Wasserstein loss fail (Miyato, 2018) , even though the analysis of Arjovsky et al. (2017) suggests it should be sufficient? Why is applying gradient penalties effective, even outside their original context of the Wasserstein GAN (Fedus et al., 2018) ?

In this work, we develop a framework to analyze the stability of GAN training that resolves these apparent contradictions and clarifies the roles of these regularization techniques. Our approach considers the smoothness of the loss function used. In optimization, smoothness is a well-known condition that ensures that gradient descent and its variants become stable (see e.g., Bertsekas (1999) ). For example, the following well-known proposition is the starting point of our stability analysis: Proposition 1 (Bertsekas (1999) , Proposition 1.2.3). Suppose f : R p → R is L-smooth and bounded below. Let

x k+1 := x k -1 L ∇f (x k ). Then ||∇f (x k )|| → 0 as k → ∞.

This proposition says that under a smoothness condition on the function, gradient descent with a constant step size 1 L approaches stationarity (i.e., the gradient norm approaches zero). This is a rather weak notion of convergence, as it does not guarantee that the iterates converge to a point, and even if the iterates do converge, the limit is a stationary point and not necessarily an minimizer. Nevertheless, empirically, not even this stationarity is satisfied by GANs, which are known to frequently destabilize and diverge during training. To diagnose this instability, we consider the smoothness of the GAN's loss function. GANs are typically framed as minimax problems of the form inf

EQUATION

where J is a loss function that takes a generator distribution µ θ and discriminator ϕ, and θ ∈ R p denotes the parameters of the generator. Unfortunately, the minimax nature of this problem makes stability and convergence difficult to analyze. To make the analysis more tractable, we define J(µ) = sup ϕ J (µ, ϕ), so that (1) becomes simply inf θ J(µ θ ).

(2)

This choice corresponds to the common assumption that the discriminator is allowed to reach optimality at every training step. Now, the GAN algorithm can be regarded as simply gradient descent on the R p → R function θ → J(µ θ ), which may be analyzed using Proposition 1. In particular, if this function θ → J(µ θ ) satisfies the smoothness assumption, then the GAN training should be stable in that it should approach stationarity under the assumption of an optimal discriminator.

In the remainder of this paper, we investigate whether the smoothness assumption is satisfied for various GAN losses. Our analysis answers two questions:

Q1. Which existing GAN losses, if any, satisfy the smoothness condition in Proposition 1? Q2. Are there choices of loss, regularization, or architecture that enforce smoothness in GANs?

As results of our analysis, our contributions are as follows:

1. We derive sufficient conditions for the GAN algorithm to be stationary under certain assumptions (Theorem 1). Our conditions relate to the smoothness of GAN loss used as well as the parameterization of the generator. 2. We show that most common GAN losses do not satisfy the all of the smoothness conditions, thereby corroborating their empirical instability. 3. We develop regularization techniques that enforce the smoothness conditions. These regularizers recover common GAN stabilization techniques such as gradient penalties and spectral normalization, thereby placing their use on a firmer theoretical foundation. 4. Our analysis provides several practical insights, suggesting for example the use of smooth activation functions, simultaneous spectral normalization and gradient penalties, and a particular learning rate for the generator.

1.1. RELATED WORK:

Divergence minimization Our analysis regards the GAN algorithm as minimizing a divergence between the current generator distribution and the desired data distribution, under the assumption of an optimal discriminator at every training step. This perspective originates from the earliest GAN paper, in which Goodfellow et al. (2014) show that the original minimax GAN implicitly minimizes the Jensen-Shannon divergence. Since then, the community has introduced a large number of GAN or GAN-like variants that learn generative models by implicitly minimizing various divergences, including f -divergences (Nowozin et al., 2016) , Wasserstein distance (Arjovsky et al., 2017) , and maximum-mean discrepancy (Li et al., 2015; Unterthiner et al., 2018) . Meanwhile, the non-saturating GAN (Goodfellow et al., 2014) has been shown to minimize a certain Kullback-Leibler divergence (Arjovsky & Bottou, 2017) . Several more theoretical works consider the topological, geometric, and convexity properties of divergence minimization (Arjovsky & Bottou, 2017; Liu et al., 2017; Bottou et al., 2018; Farnia & Tse, 2018; Chu et al., 2019) , perspectives that we draw heavily upon. Sanjabi et al. (2018) also prove smoothness of GAN losses in the specific case of the regularized optimal transport loss. Their assumption for smoothness is entangled in that it involves a composite condition on generators and discriminators, while our analysis addresses them separately.

Table 1 : Common GANs, their corresponding loss functions, and their optimal discriminators.

Loss function J(µ):

Optimal discriminator Φ µ (x)

Minimax GAN D JS (µ || µ 0 ) 1 2 log µ(x) µ(x)+µ0(x) Non-saturating GAN D KL ( 1 2 µ + 1 2 µ 0 || µ 0 ) -1 2 log µ0(x) µ(x)+µ0(x) Wasserstein GAN W 1 (µ, µ 0 ) arg max f ∈Lip 1 E y∼µ [f (y)] -E y∼µ0 [f (y)] GMMN, Coulomb GAN 1 2 MMD 2 (µ, µ 0 ) E y∼µ [K(x, y)] -E y∼µ0 [K(x, y)] IPM-GAN IPM F (µ, µ 0 ) arg max f ∈F E y∼µ [f (y)] -E y∼µ0 [f (y)]

Other approaches Even though many analyses, including ours, operate under the assumption of an optimal discriminator, this assumption is unrealistic in practice. Li et al. (2017b) contrast this optimal discriminator dynamics with first-order dynamics, which assumes that the generator and discriminator use alternating gradient updates and is what is used computationally. As this is a differing approach from ours, we only briefly mention some results in this area, which typically rely on game-theoretic notions (Kodali et al., 2017; Grnarova et al., 2018; Oliehoek et al., 2018) or local analysis (Nagarajan & Kolter, 2017; Mescheder et al., 2018) . Some of these results rely on continuous dynamics approximations of gradient updates; in contrast, our work focuses on discrete dynamics.

1.2. NOTATION:

Let R := R ∪ {∞, -∞}. We let P(X) denote the set of all probability measures on a compact set X ⊆ R d . We let M(X) and C(X) denote the dual pair consisting of the set of all finite signed measures on X and the set of all continuous functions X → R. For any statement A, we let χ{A} be 0 if A is true and ∞ if A is false. For a Euclidean vector x, its Euclidean norm is denoted by x 2 , and the operator norm of a matrix A is denoted by A 2 , i.e.,

A 2 = sup x 2≤1 Ax 2 / x 2 . A function f : X → Y between two metric spaces is α-Lipschitz if d Y (f (x 1 ), f (x 2 )) ≤ αd X (x 1 , x 2 ). A function f : R d → R is β-smooth if its gradients are β-Lipschitz, that is, for all x, y ∈ R d , ∇f (x) -∇f (y) 2 ≤ β x -y 2 .

2. SMOOTHNESS OF GAN LOSSES:

This section presents Theorem 1, which provides concise criteria for the smoothness of GAN losses.

In order to keep our analysis agnostic to the particular GAN used, let J : P(X) → R be an arbitrary convex loss function, which takes a distribution over X ⊂ R d and outputs a real number. Note that the typical minimax formulation of GANs can be recovered from just the loss function J using convex duality. In particular, recall that the convex conjugate J : C(X) → R of J satisfies the following remarkable duality, known as the Fenchel-Moreau theorem:

EQUATION

Based on this duality, minimizing J can be framed as the minimax problem inf µ∈P(X)

EQUATION

recovering the well-known adversarial formulation of GANs. We now define the notion of an optimal discriminator for an arbitrary loss function J, based on this convex duality: Definition 1 (Optimal discriminator). Let J : M(X) → R be a convex, l.s.c., proper function. An optimal discriminator for a probability distribution µ ∈ P(X) is a continuous function Φ µ : X → R that attains the maximum of the second equation in (3), i.e., J(µ

) = Φ µ (x) dµ -J (Φ µ ).

This definition recovers the optimal discriminators of many existing GAN and GAN-like algorithms (Farnia & Tse, 2018; Chu et al., 2019) , most notably those in Table 1 . Our analysis will apply to any algorithm in this family of algorithms. See Appendix B for more details on this perspective.

We also formalize the notion of a family of generators: Definition 2 (Family of generators). A family of generators is a set of pushforward probability measures {µ θ = f θ# ω : θ ∈ R p }, where ω is a fixed probability distribution on Z (the latent variable) and f θ : Z → X is a measurable function (the generator).

Now, in light of Proposition 1, we are interested in the smoothness of the mapping θ → J(µ θ ), which would guarantee the stationarity of gradient descent on this objective, which in turn implies stationarity of the GAN algorithm under the assumption of an optimal discriminator. The following theorem is our central result, which decomposes the smoothness of θ → J(µ θ ) into conditions on optimal discriminators and the family of generators.

Theorem 1 (Smoothness decomposition for GANs). Let J : M(X) → R be a convex function whose optimal discriminators Φ µ : X → R satisfy the following regularity conditions:

(D1) x → Φ µ (x) is α-Lipschitz, (D2) x → ∇ x Φ µ (x) is β 1 -Lipschitz, (D3) µ → ∇ x Φ µ (x) is β 2 -Lipschitz w.r.t. the 1-Wasserstein distance.

Also, let µ θ = f θ# ω be a family of generators that satisfies:

EQUATION

G2) θ → D θ f θ (z) is B-Lipschitz in expectation for z ∼ ω, i.e., E z∼ω [ D θ1 f θ1 (z) - D θ2 f θ2 (z) 2 ] ≤ B θ 1 -θ 2 2 . Then θ → J(µ θ ) is L-smooth, with L = αB + A 2 (β 1 + β 2 ).

Theorem 1 connects the smoothness properties of the loss function J with the smoothness properties of the optimal discriminator Φ µ , and once paired with Proposition 1, it suggests a quantitative value 1 L for a stable generator learning rate. In order to obtain claims of stability for practically sized learning rates, it is important to tightly bound the relevant constants.

In Sections 4 to 6, we carefully analyze which GAN losses satisfy (D1), (D2), and (D3), and with what constants. We summarize our results in Table 2 : it turns out that none of the listed losses, except for one, satisfy (D1), (D2), and (D3) simultaneously with a finite constant. The MMD-based loss satisfies the three conditions, but its constant for (D1) grows as α = O( √ d), which is an unfavorable dependence on the data dimension d that forces an unacceptably small learning rate. See for complete details of each condition. This failure of existing GANs to satisfy the stationarity conditions corroborates the observed instability of GANs. Theorem 1 decomposes smoothness into conditions on the generator and conditions on the discriminator, allowing a clean separation of concerns. In this paper, we focus on the discriminator conditions (D1), (D2), and (D3) and only provide an extremely simple example of a generator that satisfies (G1) and (G2), in Section 7. Because analysis of the generator conditions may become quite complicated and will vary with the choice of architecture considered (feedforward, convolutional, ResNet, etc.), we leave a detailed analysis of the generator conditions (G1) and (G2) as a promising avenue for future work. Indeed, such analyses may lead to new generator architectures or generator regularization techniques that stabilize GAN training.

3. ENFORCING SMOOTHNESS WITH INF-CONVOLUTIONS:

In this section, we present a generic regularization technique that imposes the three conditions sufficient for stable learning on an arbitrary loss function J, thereby stabilizing training. In Section 2, we observe that the Wasserstein, IPM, and MMD losses respectively satisfy (D1), (D2), and (D3) individually, but not all of of them at the same time. Using techniques from convex analysis, we convert these three GAN losses into three regularizers that, when applied simultaneously, causes the resulting loss to satisfy all the three conditions. Here, we only outline the technique; the specifics of each case are deferred to Sections 4 to 6.

We start with an arbitrary base loss function J to be regularized. Next, we take an existing GAN loss that satisfies the desired regularity condition and convert it into a regularizer function R : M(X) → R. Then, we consider J ⊕ R, which denotes the inf-convolution defined as

EQUATION

This new function J ⊕ R inherits the regularity of R, making it a stable candidate as a GAN loss. Moreover, because the inf-convolution is a commutative operation, we can sequentially apply multiple regularizers R 1 , R 2 , and R 3 without destroying the added regularity. In particular, if we carefully choose functions R 1 , R 2 , and R 3 , then J = J ⊕ R 1 ⊕ R 2 ⊕ R 3 will satisfy (D1), (D2), and (D3) simultaneously. Moreover, under some technical assumptions, this composite function J inherits the original minimizers of J, making it a sensible GAN loss:

Proposition 2 (Invariance of minimizers). Let R 1 (ξ) := ξ KR , R 2 (ξ) := ξ S * , and R 3 (ξ) := 1 2

ξ 2 H be the three regularizers defined by (8), ( 12), and (19) respectively. Assume that J : M(X) → R has a unique minimizer at µ 0 with J(µ 0 ) = 0, and J(µ) ≥ c μ -μ0 H for some c > 0. Then the inf-convolution J = J ⊕ R 1 ⊕ R 2 ⊕ R 3 has a unique minimizer at µ 0 with J(µ 0 ) = 0.

The duality formulation (4) provides a practical method for minimizing this composite function. We leverage the duality relation

EQUATION

This minimax problem can be seen as a GAN whose discriminator objective has three added regularization terms.

The concrete form of these regularizers are summarized in Table 3 . Notably, we observe that we recover standard techniques for stabilizing GANs:

• (D1) is enforced by Lipschitz constraints (i.e., spectral normalization) on the discriminator. • (D2) is enforced by spectral normalization and a choice of Lipschitz, smooth activation functions for the discriminator.

Table 3 : Smoothness-inducing regularizers and their convex conjugates. To enforce a regularity condition on a loss function J, we take a source loss that satisfies it and view it as a regularizer R.

We then consider the inf-convolution J ⊕ R, which corresponds to an added regularization term R on the discriminator ϕ. These regularization terms correspond to existing GAN techniques.

Purpose Source loss

R(ξ) R (ϕ) GAN techniques (D1) W 1 ξ KR χ{ ϕ Lip ≤ 1} spectral norm (D2) IPM S ξ S * χ{ϕ ∈ S} smooth activations, spectral norm (D3) MMD 2 1 2 ξ 2 H 1 2 ϕ 2 H gradient penalties

• (D3) is enforced by gradient penalties on the discriminator.

Our analysis therefore puts these regularization techniques on a firm theoretical foundation (Proposition 1 and Theorem 1) and provides insight into their function.

4. ENFORCING (D1) WITH LIPSCHITZ CONSTRAINTS:

In this section, we show that enforcing (D1) leads to techniques and notions commonly used to stabilize GANs, including the Wasserstein distance, Lipschitz constraints and spectral normalization.

Recall that (D1) demands that the optimal discriminator Φ µ is Lipschitz:

(D1) x → Φ µ (x) is α-Lipschitz for all µ ∈ P(X), i.e., |Φ µ (x) -Φ µ (y)| ≤ α||x -y|| 2 .

If Φ µ is differentiable, this is equivalent to that the optimal discriminator has a gradient with bounded norm. This is a sensible criterion, since a discriminator whose gradient norm is too large may push the generator too hard and destabilize its training.

To check (D1), the following proposition shows that it suffices to check whether |J(µ) -J(ν)| ≤ αW 1 (µ, ν) for all distributions µ, ν: Arjovsky et al. (2017) show that this property does not hold for common divergences based on the Kullback-Leibler or Jensen-Shannon divergence, while it does hold for the Wasserstein-1 distance. Indeed, it is this desirable property that motivates their introduction of the Wasserstein GAN. Framed in our context, their result is summarized as follows: Proposition 4. The minimax and non-saturating GAN losses do not satisfy (D1) for some µ 0 .

Proposition 3. (D1) holds if and only if J is α-Lipschitz w.r.t. the Wasserstein-1 distance.

Proposition 5. The Wasserstein GAN loss satisfies (D1) with α = 1 for any µ 0 .

Our stability analysis therefore deepens the analysis of Arjovsky et al. (2017) and provides an alternative reason that the Wasserstein distance is desirable as a metric: it is part of a sufficient condition that ensures stationarity of gradient descent.

4.1. FROM WASSERSTEIN DISTANCE TO LIPSCHITZ CONSTRAINTS:

Having identified the Wasserstein GAN loss as one that satisfies (D1), we next follow the strategy outlined in Section 3 to convert it into a regularizer for an arbitrary loss function. Towards this, we define the regularizer R 1 : M(X) → R and compute its convex conjugate R 1 : C(X) → R:

EQUATION

)

This norm is the Kantorovich-Rubinstein norm (KR norm), which extends the Wasserstein-1 distance to M(X); it holds that µν KR = W 1 (µ, ν) for µ, ν ∈ P(X). Then, its inf-convolution with an arbitrary function inherits the Lipschitz property held by R 1 :

Proposition 6 (Pasch-Hausdorff). Let J : M(X) → R be a function, and define J := J ⊕R 1 . Then J is α-Lipschitz w.r.t. the distance induced by the KR norm, and hence the Wasserstein-1 distance when restricted to P(X).

Due to Proposition 3, we now obtain a transformed loss function J that automatically satisfies (D1). This function is a generalization of the Pasch-Hausdorff envelope (see Chapter 9 in Rockafeller & Wets (1998)), also known as Lipschitz regularization or the McShane-Whitney extension (McShane, 1934; Whitney, 1934; Kirszbraun, 1934; Hiriart-Urruty, 1980) .

The convex conjugate computation in (8) shows that J can be minimized in practice by imposing Lipschitz constraints on discriminators. Indeed, by (4),

EQUATION

Farnia & Tse (2018) consider this loss J in the special case of an f -GAN with J(µ) = D f (µ, µ 0 ); they showed that minimizing J corresponds to training a f -GAN normally but constraining the discriminator to be α-Lipschitz. We show that this technique is in fact generic for any J: minimizing the transformed loss can be achieved by training the GAN as normal, but imposing a Lipschitz constraint on the discriminator.

Our analysis therefore justifies the use of Lipschitz constraints, such as spectral normalization (Miyato et al., 2018) and weight clipping (Arjovsky & Bottou, 2017) , for general GAN losses. However, Theorem 1 also suggests that applying only Lipschitz constraints may not be enough to stabilize GANs, as (D1) alone does not ensure that the GAN objective is smooth.

5. ENFORCING (D2) WITH DISCRIMINATOR SMOOTHNESS:

(D2) demands that the optimal discriminator Φ µ is smooth:

(D2) x → ∇Φ µ (x) is β 1 -Lipschitz for all µ ∈ P(X), i.e., ∇Φ µ (x) -∇Φ µ (y) 2 ≤ β 1 x -y 2 .

Intuitively, this says that for a fixed generator µ, the optimal discriminator Φ µ should not provide gradients that change too much spatially.

Although the Wasserstein GAN loss (D1), we see that it, along with the minimax GAN and the non-saturating GAN, do not satisfy (D2): Proposition 7. The Wasserstein, minimax, and non-saturating GAN losses do not satisfy (D2) for some µ 0 .

We now construct a loss that by definition satisfies (D2). Let S be the class of 1-smooth functions, that is, for which ∇f (x) -∇f (y) 2 ≤ xy 2 , and consider the integral probability metric (IPM) (Müller, 1997) w.r.t. S, defined by

EQUATION

)

The optimal discriminator for the loss IPM S (µ, µ 0 ) is the function that maximizes the supremum in the definition. This function by definition belongs to S and therefore is 1-smooth. Hence, this IPM loss satisfies (D2) with β 1 = 1 by construction.

5.1. FROM INTEGRAL PROBABILITY METRIC TO SMOOTH DISCRIMINATORS:

Having identified the IPM-based loss as one that satisfies (D2), we next follow the strategy outlined in Section 3 to convert it into a regularizer for an arbitrary loss function. To do so, we define a regularizer R 2 : M(X) → R and compute its convex conjugate R 2 : C(X) → R:

EQUATION

)

The norm is the dual norm to S, which extends the IPM to signed measures; it holds that IPM S (µ, ν) = µν S * for µ, ν ∈ P(X). Similar to the situation in the previous section, inf-convolution preserves the smoothness property of R 2 : Proposition 8. Let J : M(X) → R be a convex, proper, lower semicontinuous function, and define J := J ⊕ R 2 . Then the optimal discriminator for J is β 1 -smooth.

Applying ( 4) and ( 12), we see that we can minimize this transformed loss function by restricting the family of discriminators to only β 1 -smooth discriminators:

EQUATION

In practice, we can enforce this by applying spectral normalization (Miyato et al., 2018) and using a Lipschitz, smooth activation function such as ELU (Clevert et al., 2016) or sigmoid.

Proposition 9. Let f : R d → R be a neural network consisting of k layers whose linear transformations have spectral norm 1 and whose activation functions are 1-Lipschitz and 1-smooth. Then f is k-smooth.

6 ENFORCING (D3) WITH GRADIENT PENALTIES (D3) is the following smoothness condition:

(D3) µ → ∇Φ µ (x) is β 2 -Lipschitz for any x ∈ X, i.e., ∇Φ µ (x) -∇Φ ν (x) 2 ≤ β 2 W 1 (µ, ν).

(D3) requires that the gradients of the optimal discriminator do not change too rapidly in response to changes in µ. Indeed, if the discriminator's gradients are too sensitive to changes in the generator, the generator may not be able to accurately follow those gradients as it updates itself using a finite step size. In finite-dimensional optimization of a function f : R d → R, this condition is analogous to f having a Lipschitz gradient.

We now present an equivalent characterization of (D3) that is easier to check in practice. We define the Bregman divergence of a convex function J : M(X) → R by

EQUATION

where Φ µ is the optimal discriminator for J at µ. Then, (D3) is characterized in terms of the Bregman divergence and the KR norm as follows: Proposition 10. Let J : M(X) → R be a convex function. Then J satisfies (D3) if and only if

D J (ν, µ) ≤ β2 2 µ -ν 2 KR for all µ, ν ∈ M(X).

It is straightforward to compute the Bregman divergence corresponding to several popular GANs:

EQUATION

EQUATION

EQUATION

)

The first two Bregman divergences are not bounded above by ||µ-ν|| 2 KR for reasons similar to those discussed in Section 4, and hence: Proposition 11. The minimax and non-saturating GAN losses do not satisfy (D3) for some µ 0 .

Even so, the Bregman divergence for the non-saturating loss is always less than that of the minimax GAN, suggesting that the non-saturating loss should be stable in more situations than the minimax GAN. On the other hand, the MMD-based loss (Li et al., 2015) does satisfy (D3) when its kernel is the Gaussian kernel K(x, y) = e -π||x-y|| 2 : Proposition 12. The MMD loss with Gaussian kernel satisfies (D3) with β 2 = 2π for all µ 0 .

6.1. FROM MAXIMUM MEAN DISCREPANCY TO GRADIENT PENALTIES:

Having identified the MMD-based loss as one that satisfies (D3), we next follow the strategy outlined in Section 3 to convert it into a regularizer for an arbitrary loss function. To do so, we define the regularizer R 3 : M(X) → R and compute its convex conjugate R 3 : C(X) → R:

EQUATION

)

The norm is the norm of a reproducing kernel Hilbert space norm (RKHS) H with Gaussian kernel; this norm extends the MMD to signed measures, as it holds that MMD(µ, ν) = μν H for µ, ν ∈ P(X). Here, ξ = K(x, •) ξ(dx) ∈ H denotes the mean embedding of a signed measure ξ ∈ M(X); we also adopt the convention that ||ϕ|| H = ∞ if ϕ ∈ H. Similar to the situation in the previous sections, inf-convolution preserves the smoothness property of R 3 : Proposition 13 (Moreau-Yosida regularization). Suppose J : M(X) → R is convex, and define J := J ⊕ R 3 . Then J is convex, and

D J (ν, µ) ≤ β2 2 ||µ -ν|| 2 KR .

By Proposition 10, this transformed loss function satisfies (D3), having inherited the regularity properties of the squared MMD. This transformed function is a generalization of Moreau-Yosida regularization or the Moreau envelope (see Chapter 1 in Rockafeller & Wets (1998) ). It is well-known that in the case of a function f : R n → R, this regularization results in a function with Lipschitz gradients, so it is unsurprising that this property carries over to the infinite-dimensional case.

Applying ( 4) and ( 19), we see that the transformed loss function can be minimized as a GAN by implementing an RKHS squared norm penalty on the discriminator: 

EQUATION

EQUATION

In an ideal world, we would use this expression as a penalty on the discriminator to enforce (D3). Of course, as an infinite series, this formulation is computationally impractical. However, the first two terms are very close to common GAN techniques like gradient penalties (Gulrajani et al., 2017) and penalizing the output of the discriminator (Karras et al., 2018) . We therefore interpret these common practices as partially applying the penalty given by the RKHS norm squared, approximately enforcing (D3). We view the choice of only using the leading terms as a disadvantageous but practical necessity.

Interestingly, according to our analysis, gradient penalties and spectral normalization are not interchangeable, even though both techniques were designed to constrain the Lipschitz constant of the discriminator. Instead, our analysis suggests that they serve different purposes: gradient penalties enforce the variational smoothness (D3), while spectral normalization enforces Lipschitz continuity (D1). This demystifies the puzzling observation of Miyato (2018) that GANs using only spectral normalization with a WGAN loss do not seem to train well; it also explains why using both spectral normalization and a gradient penalty is a reasonable strategy. It also motivates the use of gradient penalties applied to losses other than the Wasserstein loss (Fedus et al., 2018) .

7. VERIFYING THE THEORETICAL LEARNING RATE:

In this section, we empirically test the theoretical learning rate given by Theorem 1 and Proposition 1 as well as our regularization scheme (7) based on inf-convolutions. We approximately implement our composite regularization scheme (7) on a trivial base loss of J(µ) = χ{µ = µ 0 } by alternating stochastic gradient steps on

EQUATION

)

where μ is a random interpolate between samples from µ and µ 0 , as used in Gulrajani et al. (2017) .

The regularization term is a truncation of the series for the squared RKHS norm ( 22) and approximately enforces (D3). The discriminator is a 7-layer convolutional neural network with spectral normalization1 and ELU activations, an architecture that enforces (D1) and (D2). We include a final scalar multiplication by α so that by Proposition 9, β 1 = 7α. We take two discriminator steps for every generator step, to better approximate our assumption of an optimal discriminator. For the generator, we use an extremely simple particle-based generator which satisfies (G1) and (G2), in order to minimize the number of confounding factors in our experiment. Let ω be the discrete uniform distribution on Z = {1, . . . , N }. For an N × d matrix θ and z ∈ Z, define

EQUATION

and it satisfies (G2) with B = 0, since D θ f θ (z) is constant w.r.t. θ. With this setup, Theorem 1 suggests a theoretical learning rate of

EQUATION

)

We randomly generated hyperparameter settings for the Lipschitz constant α, the smoothness constant β 2 , the number of particles N , and the learning rate γ. We trained each model for 100,000 steps on CIFAR-10 and evaluate each model using the Fréchet Inception Distance (FID) of Heusel et al. (2017) . We hypothesize that stability is correlated with image quality; Figure 1 plots the FID for each hyperparameter setting in terms of the ratio of the true learning rate γ and the theoretically motivated learning rate γ 0 . We find that the best FID scores are obtained in the region where γ/γ 0 is between 1 and 1000. For small learning rates γ/γ 0 1, we observe that the convergence is too slow to make a reasonable progress on the objective, whereas as the learning rate gets larger γ/γ 0 1, we observe a steady increase in FID, signalling unstable behavior. It also makes sense that learning rates slightly above the optimal rate produce good results, since our theoretical learning rate is a conservative lower bound. Note that our intention is to test our theory, not to generate good images, which is difficult due to our weak choice of generator. Overall, this experiment shows that our theory and regularization scheme are sensible.

8. FUTURE WORK:

Inexact gradient descent In this paper, we employed several assumptions in order to regard the GAN algorithm as gradient descent. However, real-world GAN algorithms must be treated as "inexact" descent algorithms. As such, future work includes: (i) relaxing the optimal discriminator assumption (cf. Sanjabi et al. (2018) ) or providing a stability result for discrete simultaneous gradient descent (cf. continuous time analysis in Nagarajan & Kolter (2017); Mescheder et al. (2018) ), (ii) addressing stochastic approximations of gradients (i.e., SGD), and (iii) providing error bounds for the truncated gradient penalty used in (23).

Generator architectures Another important direction of research is to seek more powerful generator architectures that satisfy our smoothness assumptions (G1) and (G2). In practice, generators are often implemented as deep neural networks, and involve some specific architectures such as deconvolution layers (Radford et al., 2015) and residual blocks (e.g., Gulrajani et al. (2017) ; Miyato et al. (2018) ). In this paper, we did not provide results on the smoothness of general classes of generators, since our focus is to analyze stability properties influenced by the choice of loss function J (and therefore optimal discriminators). However, our conditions (G1) and (G2) shed light on how to obtain smoothly parameterized neural networks, which is left for future work.

A INF-CONVOLUTION IN R d:

To gain intuition on the inf-convolution, we present a finite-dimensional analogue of the techniques in Section 3. For simplicity of presentation, we will omit any regularity conditions (e.g., lower semicontinuity). We refer readers to Chapter 12 of Bauschke & Combettes (2011) for a detailed introduction.

Let J and R be convex functions on R d . The inf-convolution of J and R is a function J R defined as

(J ⊕ R)(x) := inf z∈R d J(z) + R(x -z).

The inf-convolution is often called the epigraphic sum since the epigraph of J R coincides with the Minkowski sum of epigraphs of J and R, as Figure 2 illustrates. The inf-convolution is associative and commutative operation; that is, it is always true that

(J 1 ⊕ J 2 ) ⊕ J 3 = J 1 ⊕ (J 2 ⊕ J 3 ) =: J 1 ⊕ J 2 ⊕ J 3 and J 1 ⊕ J 2 = J 2 ⊕ J 1 .

There are two important special cases of inf-convolutions: The first one is the Pasch-Hausdorff envelope J α , which is the inf-convolution between J and α • 2 (α > 0). It is known that J α becomes α-Lipschitz. The second important example is the Moreau envelope J β = J ⊕ 1 2β • 2 2 , i.e., the inf-convolution with the quadratic regularizer 1 2β • 2 2 . The Moreau envelope J β is always differentiable, and the gradient of J β is β-Lipschitz (thus J β is β-smooth).

It is worth noting that the set of minimizers does not change after these two operations. More generally, we have the following result: Proposition 15. Let J, R : R d → R be proper and lower semicontinuous functions with min J > -∞ and min R = 0. Suppose R(0) = 0 and R(x) ≥ ψ( x 2 ) for some increasing function ψ : R ≥0 → R. Then, min J ⊕ R = min J and arg min J ⊕ R = arg min J. Top row: Generally, inf-convolutions inherit the regularity of their parent functions. In this example, J 1 (x) = |x| is 1-Lipschitz but not smooth, while J 2 (x) = x 2 /2 is 1-smooth but not Lipschitz. The inf-convolution J 1 ⊕ J 2 is the well-known Huber function, which is both 1-Lipschitz and 1-smooth. Bottom row: Convex conjugates of the functions in the top row. The conjugate of J 1 ⊕ J 2 is given as the sum of conjugates J 1 (z) = χ{|z| ≤ 1} and J 2 (z) = z 2 /2.

To sum up, given a function J, we can always construct a regularized alternative J β α that is α-Lipschitz and β-smooth and has the same minimizers as J.

The next question is how to implement the inf-convolution in GAN-like optimization problems. For this, it is convenient to consider the convex conjugate. Recall that the Fenchel-Moreau theorem says that there is a duality between a convex function J and its convex conjugate J as J(x) = sup z∈R d x, z -J (z) and J (z) = sup x∈R d x, z -J(x). The important property is that the convex conjugate of the inf-convolution is the sum of convex conjugates, that is, we always have

(J ⊕ R) (z) = J (z) + R (z).

This property can be useful for implementing the regularized objective J β α as follows. First, we can check that the convex conjugates of the norm and the squared norm are given as

( • 2 ) = χ{ • ≤ 1} and ( 1 2 • 2 2 ) = 1 2 • 2 2

. Hence, we have

J β α (x) := J ⊕ α • 2 ⊕ 1 2β • 2 2 (x) = sup z: z 2≤α x, z -J (z) - β 2 z 2 2 ,

which means that minimizing J β α can be recast in min-max problem with the norm clipping and 2 -regularization on the dual variable z.

B COMMON GAN LOSSES:

For completeness and clarity, we explicitly write out the expressions for the losses listed in Table 1 . For more detailed computations of optimal discriminators, see Chu et al. (2019) ; for more details on the convex duality interpretation, see Farnia & Tse (2018) .

Minimax GAN Goodfellow et al. (2014) originally proposed the minimax GAN and showed that the corresponding loss function for the minimax GAN is the Jensen-Shannon divergence, defined as

J(µ) := D JS (µ || µ 0 ) := 1 2 D KL (µ || 1 2 µ + 1 2 µ 0 ) + 1 2 D KL (µ 0 || 1 2 µ + 1 2 µ 0 ),

where µ 0 ∈ P(X) is a fixed probability measure (usually the empirical measure of the data), and D KL (µ || ν) is the Kullback-Leibler divergence between µ and ν. The optimal discriminator in the sense of Definition 1 is given as

Φ µ (x) = 1 2 log dµ d(µ + µ 0 ) (x),

where dµ d(µ+µ0) is the Radon-Nikodym derivative. If µ and µ 0 have densities µ(x) and µ 0 (x), then dµ d(µ

+ µ 0 ) (x) = µ(x) µ(x) + µ 0 (x)

, so our optimal discriminator matches that of Goodfellow et al. (2014) up to a constant factor and logarithm. To recover the minimax formulation, the convex duality (4) yields:

inf µ D JS (µ, µ 0 ) = inf µ sup ϕ E x∼µ [ϕ(x)] -(-1 2 E x∼µ0 [log(1 -e 2ϕ(x)+log 2 )] -1 2 log 2 (DJS(•,µ0)) (ϕ) ) = inf µ sup D 1 2 E x∼µ [log(1 -D(x))] + 1 2 E x∼µ0 [log D(x)],

using the substitution ϕ = 1 2 log(1 -D) -1 2 log 2.

Non-saturating GAN Goodfellow et al. (2014) also proposed the heuristic non-saturating GAN. Theorem 2.5 of Arjovsky & Bottou (2017) shows that the loss function minimized is

J(µ) := D KL ( 1 2 µ + 1 2 µ 0 || µ 0 ) = 1 2 D KL (µ || µ 0 ) -D JS (µ || µ 0 ).

The optimal discriminator is

Φ µ (x) = - 1 2 log dµ 0 d(µ + µ 0 ) (x).

Wasserstein GAN Arjovsky et al. (2017) proposed the Wasserstein GAN, which minimizes the Wasserstein-1 distance between the input µ and a fixed measure µ 0 :

J(µ) := W 1 (µ, µ 0 ) := inf π E (x,y)∼π [||x -y||],

where the infimum is taken over all couplings π, probability distributions over X × X whose marginals are µ and µ 0 respectively. The optimal discriminator Φ µ is called the Kantorovich potential in the optimal transport literature (Villani, 2009) . The convex duality (4) recover the Wasserstein GAN:

inf µ W 1 (µ, µ 0 ) = inf µ sup ϕ E x∼µ [ϕ(x)] -(E x∼µ0 [ϕ(x)] + χ{||ϕ|| Lip ≤ 1} (W1(•,µ0)) (ϕ) ) = inf µ sup ||ϕ||Lip≤1 E x∼µ [ϕ(x)] -E x∼µ0 [ϕ(x)],

an expression of Kantorovich-Rubinstein duality. The Lipschitz constraint on the discriminator is typically enforced by spectral normalization (Miyato et al., 2018) , less frequently by weight clipping (Arjovsky et al., 2017) , or heuristically by gradient penalties (Gulrajani et al., 2017) (although this work shows that gradient penalties may serve a different purpose altogether).

Maximum mean discrepancy Given a positive definite kernel K : X × X → R, the maximum mean discrepancy (MMD, Gretton et al. (2012) ) between µ and ν is defined by

J(µ) := 1 2 MMD 2 K (µ, ν) := 1 2 K(x, y) (µ -ν)(dx) (µ -ν)(dy).

where (H, • H ) is the reproducing kernel Hilbert space (RKHS) for K. The generative momentmatching network (GMMN, Li et al. (2015) ) and the Coulomb GAN (Unterthiner et al., 2018) use the squared MMD as the loss function. The optimal discriminator in this case is

Φ µ (x) = E y∼µ [K(x, y)] -E y∼µ0 [K(x, y)],

which in constrast to other GANs, may be approximated by simple Monte Carlo, rather than an auxiliary optimization problem.

Note that MMD-GANs (Li et al., 2017a; Arbel et al., 2018) minimize a modified version of the MMD, the Optimized MMD (Sriperumbudur et al., 2009; Arbel et al., 2018) . These MMD-GANs are adversarial in a way that does not arise from convex duality, so our theory currently does not apply to these GANs.

Integral probability metrics An integral probability metric (Müller, 1997) is defined by

J(µ) := IPM F (µ, µ 0 ) := sup f ∈F f dµ -f dµ 0 ,

where F is a class of functions. The optimal discriminator is the function that maximizes the supremum in the definition. The Wasserstein distance may be thought of as an IPM with F containing all 1-Lipschitz functions. The MMD may be thought of as an IPM with F all functions with RKHS norm at most 1, but no GANs based on MMD are actually trained this way, as it is difficult to constrain the discriminator to such functions.

C OPTIMAL DISCRIMINATORS ARE FUNCTIONAL DERIVATIVES:

Let J : P(X) → R be a convex function. Recall the definition of the optimal discriminator (Definition 1):

Φ µ ∈ arg max ϕ∈C(X) ϕ dµ -J (ϕ).

This definition can be understood as an infinite dimensional analogue of subgradients. In fact, in finite-dimensional convex analysis, z is a subgradient of f : R d → R if and only if it can be written as z ∈ arg max z z , xf (z ). The calculus of subgradients shares many properties with the standard calculus of derivatives, such as chain rules (Rockafeller & Wets, 1998) . This motivate us to investigate derivative-like features of optimal discriminators.

We introduce the functional derivative, also known as the von Mises influence function: Definition 3 (Functional derivative). Let J : P(X) → R be a function of probability measures. We say that a continuous function Φ µ : X → R is a functional derivative of J at µ if

J(µ + ξ) = J(µ) + Φ µ dξ + O( 2 )

holds for any ξ = νµ with ν ∈ P(X).

Under this definition, optimal discriminators are actually functional derivatives.

Proposition 16 (Chu et al. (2019) , Theorem 2). Let J : M(X) → R be a proper, lower semicontinuous, and convex function. If there exists a maximizer Φ µ of ϕ → ϕ dµ -J (ϕ), then Φ µ is a functional derivative of J at µ in the sense of Definition 3.

The following result relates the derivative of the loss function with the derivative of the optimal discriminator: Proposition 17 (Chu et al. (2019) , Theorem 1). Let J : P(X) → R be continuously differentiable, in the sense that the functional derivative Φ µ exists and (µ, ν) → E x∼ν [Φ µ (x)] is continuous. Let θ → µ θ be continuous in the sense that 1 h (µ θ+hµ θ ) converges to a weak limit as h → 0. Then, we have

∇ θ J(µ θ ) = ∇ θ E x∼µ θ [ Φ(x)],

where Φ = Φ µ θ is treated as a function X → R that is not dependent on θ.

We use this important computational tool in many of our proofs. For the case of the generator model µ θ = f θ# ω, an important consequence of Proposition 17 is that

∇ θ J(µ θ ) = ∇ θ E z∼ω [ Φ(f θ (z))] = E z∼ω [∇ x=f θ (z) Φ µ (f θ (z)) • D θ f θ (z)].

We use this fact in the proof of Theorem 1.

D PROOFS FOR SECTIONS 1 AND 2:

The following result is well known in the dynamical systems and the optimization literature. For the sake of completeness, we provide its proof. Proposition 1 (Bertsekas (1999) , Proposition 1.2.3). Suppose f : R p → R is L-smooth and bounded below. Let

x k+1 := x k -1 L ∇f (x k ). Then ||∇f (x k )|| → 0 as k → ∞. Proof. It is known that if f is L-smooth, then f (y) ≤ f (x) + ∇f (x), y -x + L 2 ||y -x|| 2

holds for any x, y ∈ R d (see e.g. Lemma 3.4 in Bubeck (2015) ). Then, we have

f (x k+1 ) ≤ f (x k ) + ∇f (x k ), x k+1 -x k + L 2 ||x k+1 -x k || 2 ≤ f (x k ) - 1 L ||∇f (x k )|| 2 + 1 2L ||∇f (x k )|| 2 = f (x k ) - 1 2L ||∇f (x k )|| 2 .

Summing this inequality over k, we have

1 2L n-1 k=0 ||∇f (x k )|| 2 ≤ f (x 0 ) -f (x n ),

from which we conclude that

min 0≤k≤n-1 ||∇f (x k )|| 2 ≤ 2L n (f (x 0 ) -inf x f (x)).

Next, we move on to prove Theorem 1, which we restate here for readability.

Theorem 1 (Smoothness decomposition for GANs). Let J : M(X) → R be a convex function whose optimal discriminators Φ µ : X → R satisfy the following regularity conditions:

(D1) x → Φ µ (x) is α-Lipschitz, (D2) x → ∇ x Φ µ (x) is β 1 -Lipschitz, (D3) µ → ∇ x Φ µ (x) is β 2 -Lipschitz w.r.t. the 1-Wasserstein distance.

Also, let µ θ = f θ# ω be a family of generators that satisfies:

EQUATION

G2) θ → D θ f θ (z) is B-Lipschitz in expectation for z ∼ ω, i.e., E z∼ω [ D θ1 f θ1 (z) - D θ2 f θ2 (z) 2 ] ≤ B θ 1 -θ 2 2 . Then θ → J(µ θ ) is L-smooth, with L = αB + A 2 (β 1 + β 2 ).

Intuitively, Theorem 1 can be understood as the chain rule. For simplicity, let us consider the smoothness of a composite function D • G, where D and G are functions on R. A sufficient condition for D•G to be smooth is that its second derivative is bounded. For this, suppose that (i) D is α-Lipschitz and β-smooth and (ii) G is A-Lipschitz and B-smooth. By the chain rule, the second derivative is bounded as

(D • G) = (D • G)G + (D • G)(G ) 2 ≤ αB + A 2 β

, which has the same form as the consequence of Theorem 1. For GANs, we need somewhat more involved conditions; We need two types of smoothness (D2) and (D3) for the optimal discriminators. To this end, we utilize the functional gradient point of view that we explained in Appendix C.

Proof of Theorem 1. First, using the functional gradient interpretation of the optimal discriminator, we have

EQUATION

By assumption, there are bounded non-negative numbers α, β 1 , β 2 , A, and B such that

EQUATION

E z∼ω ||D θ1 f θ1 (z) -D θ2 f θ2 (z)|| op ≤ B||θ 1 -θ 2 ||.

Here, we wrote sup x∼µ f (x) for the essential supremum of f w.r.t. µ. From (D1 ) and (G2 ), the first term (a) is bounded as

E z∼ω [∇Φ µ θ 1 (f θ1 (z)) • (D θ1 f θ1 -D θ2 f θ2 )] ≤ αB θ 1 -θ 2 .

From (D3 ), (G1 ), and the Cauchy-Schwarz inequality, the second term (b) is bounded as

E z∼ω [ ∇Φ µ θ 1 (f θ1 (z)) -∇Φ µ θ 2 (f θ1 (z)) • D θ2 f θ2 ] ≤Aβ 2 W 1 (µ θ1 , µ θ2 ) ≤Aβ 2 E z∼ω f θ1 (z) -f θ2 (z) ≤ A 2 β 2 θ 1 -θ 2 ,

where the second inequality holds from the following optimal transport interpretation of W 1 :

W 1 (µ θ1 , µ θ2 ) = inf γ∈P(X×X): coupling of µ θ 1 and µ θ 2 x -y dγ ≤ E z∼ω f θ1 (z) -f θ2 (z) .

Lastly, from (D2 ), (G1 ) and the Cauchy-Schwarz inequality, the term (c) is bounded as

E z∼ω [ ∇Φ µ θ 2 (f θ1 (z)) -∇Φ µ θ 2 (f θ2 (z)) • D θ2 f θ2 ] ≤ Aβ 1 E z∼ω f θ1 (z)-f θ2 (z) ≤ A 2 β 1 θ 1 -θ 2 .

Combining the above upper bounds for (a)-(c), we conclude that

||∇ θ1 J(µ θ1 ) -∇ θ2 J(µ θ2 )|| ≤ (αB + A 2 (β 1 + β 2 )) θ 1 -θ 2 .

E PROOFS FOR SECTION 3:

In the finite-dimensional case, Proposition 15 says that taking inf-convolution with a "coercive" regularizer does not change the set of minimizers of the original objective. A similar invariance holds for GAN objectives, which are defined on infinite-dimensional space of signed measures, for the regularizers R 1 , R 2 and R 3 introduced in Sections 4 to 6:

Proposition 2 (Invariance of minimizers). Let R 1 (ξ) := ξ KR , R 2 (ξ) := ξ S * , and R 3 (ξ) := 1 2 ξ 2

H be the three regularizers defined by (8), ( 12), and (19) respectively. Assume that J : M(X) → R has a unique minimizer at µ 0 with J(µ 0 ) = 0, and J(µ) ≥ c μ -μ0 H for some c > 0. Then the inf-convolution J = J ⊕ R 1 ⊕ R 2 ⊕ R 3 has a unique minimizer at µ 0 with J(µ 0 ) = 0.

Proof. In the following, we endow M(X) with the Gaussian RKHS norm • H , and for convenience • will refer to this norm if not otherwise specified. To show the result, we apply Lemma 1 three times, first on R 3 and R 1 , and then again on R 3 ⊕R 1 and R 2 , and then finally on R 3 ⊕R 1 ⊕R 2 and J(• + µ 0 ). The result follows from noting that

J(µ) = [R 3 ⊕ R 1 ⊕ R 2 ⊕ J(• + µ 0 )](µ -µ 0 ).

In order to apply the lemma, it suffices to show that R 3 is uniformly continuous on bounded sets and coercive, and that there exist constants c 1 and

c 2 such that R 1 (ξ) ≥ c 1 ξ and R 2 (ξ) ≥ c 2 ξ . R 3 is uniformly continuous on bounded sets: suppose ξ < C and ξ < C; then |R 3 (ξ) -R 3 (ξ )| = | 1 2 ξ 2 -1 2 ξ 2 | = | 1 2 ξ -ξ , ξ + ξ | ≤ C ξ -ξ by the Cauchy-Schwarz and triangle inequality. R 3 is also coercive, as R 3 (ξ) = 1 2 ξ 2 → ∞ when ξ → ∞. R 1 satisfies R 1 (ξ) ≥ c 1 ξ by Lemma 7. R 2 satisfies R 2 (ξ) ≥ c 2 ξ by Lemma 5. Recall that a function F is said to be coercive if F (ξ) → ∞ when ξ → ∞.

Lemma 1. Suppose F : M(X) → R has a unique minimizer at 0 with F (0) = 0, and is uniformly continuous on bounded sets and coercive. Suppose G : M(X) → R has a unique minimizer at 0 with G(0) = 0, and G(•) ≥ c • for some c > 0. Then the inf-convolution F ⊕ G has a unique minimizer at 0 with (F ⊕ G)(0) = 0, and is uniformly continuous on bounded sets, coercive, and real-valued.

Proof. From Theorem 2.3 of Strömberg (1996) , inf F ⊕ G = 0, and 0 ∈ arg min F ⊕ G. To show that 0 is the unique minimizer, let ξ ∈ arg min F ⊕ G. From the definition of inf-convolution, for every n there exists an ξ n ∈ M(X) that satisfies

lim n→∞ F (ξ n ) + G(ξ -ξ n ) = (F ⊕ G)(ξ) = 0.

Since F and G are non-negative, F (ξ n ) → 0 and G(ξξ n ) → 0. By our assumption that G(•) ≥ c • , the latter limit implies that ξξ n → 0, which implies that F (ξ n ) → F (ξ) by continuity of F . Comparing our two expressions for the limit of F (ξ n ), we find that F (ξ) = 0, which implies that ξ = 0, since F has a unique minimizer at 0. Hence arg min F ⊕ G = {0}.

From Theorem 2.10 of Strömberg (1996) , F ⊕ G is uniformly continuous on bounded sets and real-valued. To show that F ⊕ G is coercive, we show the equivalent condition its sublevel sets are bounded (see Proposition 11.11 of Bauschke & Combettes (2011)). That is, we show that for all a > 0, there exists a constant b such that if (F ⊕ G)(ξ) ≤ a, then ξ ≤ b. Let a > 0, and let ξ ∈ M(X) be such that (F ⊕ G)(ξ) ≤ a. Let > 0; from the definition of inf-convolution, there exists a ξ ∈ M(X) such that

F ( ξ) + G(ξ -ξ) ≤ (F ⊕ G)(ξ) + ≤ a + .

Because F and G are non-negative, we have that

F ( ξ) ≤ a + , G(ξ -ξ) ≤ a + .

Using the fact that F is coercive and hence its sublevel sets are bounded, let b be such that if F (ν) ≤ a + , then ν ≤ b . By the triangle inequality and our assumption that

G(•) ≥ c • , ξ ≤ ξ + ξ -ξ ≤ b + G(ξ -ξ) c ≤ b + a + c .

Because this constant is independent of ξ, this shows that F ⊕ G is coercive.

F PROOFS FOR SECTION 4:

Proposition 3. (D1) holds if and only if J is α-Lipschitz w.r.t. the Wasserstein-1 distance.

Proof. First, we check the forward implication:

|J(µ) -J(ν)| ≤ αW 1 (µ, ν) ⇒ sup x∼µ ∇ x Φ µ (x) 2 ≤ α.

From the duality between L ∞ -and L 1 -norms, we have

EQUATION

Below, we will abbreviate

• L 1 (µ; R d ) as • L 1 .

We utilize the fact that the optimal discriminator Φ µ is the functional gradient of J in the sense of Definition 3. For any t ≥ 0, let µ t = (id + tv) # µ be the law of x + tv(x) with x ∼ µ. Then, µ t converges weakly to µ = µ 0 as t → 0 since W 1 (µ t , µ) ≤ t v L 1 (µ) → 0. Applying Proposition 17 to J and µ t , we have

d dt J(µ t ) t=0 = d dt Φ µ (x + tv(x)) dµ(x) t=0 = ∇Φ µ (x) • v(x) dµ.

In particular, the right-hand side of ( 26) is bounded from above as

sup v L 1 (µ) =1 ∇Φ µ (x) • v(x) dµ(x) = sup v L 1 (µ) =1 d dt J(µ t ) t=0 = sup v L 1 (µ) =1 lim t→0 J(µ t ) -J(µ) t ≤ sup v L 1 (µ) =1 lim t→0 αW 1 (µ t , µ) t ≤ sup v L 1 (µ) =1 lim t→0 αt v L 1 (µ) t = α,

which proves the first half of the statement.

For the converse, we borrow some techniques from the optimal transport theory. Let p > 1, and let W p (µ, ν) denote the Wasserstein-p distance between µ and ν. Suppose that µ t : [0, 1] → P(X) is an P p (X)-absolutely continuous curve, that is, every µ t has a finite p-moment and there exists v ∈ L 1 ([0, 1]) such that W p (µ s , µ t ) ≤ t s v(r) dr holds for any 0 ≤ s ≤ t ≤ 1. Then, the limit |µ | Wp (t) := lim h→0 |h| -1 W p (µ t+h , µ t ) exists for almost all t ∈ [0, 1] (see Ambrosio et al. (2008) , Theorem 1.1.2). Such |µ | Wp is called the metric derivative of µ t .

Lemma 2 (Ambrosio et al. (2008), Theorem 8.3.1) . Let µ t : [0, 1] → P(X) be an P p (X)-absolutely continuous curve, and let |µ | Wp ∈ L 1 ([0, 1]) be its metric derivative. Then, there exists a vector field

v : (x, t) → v t (x) ∈ R d such that v t ∈ L p (µ t ; X), v t ≤ |µ | Wp (t)

for almost all t ∈ [0, 1], and

1 0 d dt ϕ(x, t)µ t (dx)dt = - 1 0 v t (x), ∇ x ϕ(x, t) µ t (dx)dt

holds for any cylindrical function ϕ(x, t).

Given p > 1, let µ t be a P p (X)-absolutely continuous curve such that µ 0 = µ and µ 1 = ν. Then

|J(µ) -J(ν)| = 1 0 d dt J(µ t ) dt = 1 0 v t • ∇Φ µt dµ t dt ≤ 1 0 ||v t || L p (µt) ∇Φ µt L q (µt) dt ≤ 1 0 ||v t || L p (µt) dt • sup μ ∇Φ μ L q (µ) ≤ 1 0 |µ t | Wp dt • sup μ ∇Φ μ L q (µ) ,

where |µ t | is the metric derivative. The existence of v t and the bound ||v t || L p (µt) ≤ |µ t | Wp is due to Lemma 2. Taking the infimum over all possible such curves, we find that

|J(µ) -J(ν)| ≤ W p (µ, ν) • sup μ ∇Φ μ L q (µ) .

To conclude, we consider the limit p → 1. Using the fact that q → ||f || L q (µ) is increasing in q, we have

|J(µ) -J(ν)| ≤ W p (µ, ν) • sup μ ∇Φ μ L ∞ (µ) ≤ αW p (µ, ν). Then |J(µ) -J(ν)| ≤ inf p>1 αW p (µ, ν) = α inf π∈Π(µ,ν) inf p>1 ||x -y|| L p (π) = α inf π∈Π(µ,ν) ||x -y|| L 1 (π)

= αW 1 (µ, ν).

Proposition 4. The minimax and non-saturating GAN losses do not satisfy (D1) for some µ 0 .

Proposition 5. The Wasserstein GAN loss satisfies (D1) with α = 1 for any µ 0 .

Proof for Proposition 4 and Proposition 5. The counterexample for Proposition 4 is a slight simplification of Example 1 of Arjovsky et al. (2017) . Let δ x denote the distribution that outputs x ∈ R with probability 1. Then, we have

D JS (δ x || δ 0 ) = log 2 x = 0 0 x = 0, D KL ( 1 2 δ x + 1 2 δ 0 || δ 0 ) = ∞ x = 0 0 x = 0, and W 1 (δ x , δ 0 ) = |x|.

Therefore, for J(µ) = D JS (µ || δ 0 ), the inequality |J(δ x ) -J(δ 0 )| ≤ αW 1 (δ x , δ 0 ) cannot hold for sufficiently small x = 0, because the left-hand side equals log 2 while the right-hand side equals α|x|. For J(µ) = D KL ( 1 2 µ + 1 2 δ 0 || δ 0 ), the inequality will not hold for any x = 0. Next, we verify Proposition 5. For J(µ) = W 1 (µ, µ 0 ), we have J(µ) -J(ν) = W 1 (µ, µ 0 ) -W 1 (ν, µ 0 ) ≤ W 1 (µ, ν) by the triangle inequality (see e.g., Section 6 in Villani (2009) ). Reversing the roles of µ and ν, we can conclude that |J(µ) -J(ν)| ≤ W 1 (µ, ν). The result follows from Proposition 3.

Proposition 6 (Pasch-Hausdorff). Let J : M(X) → R be a function, and define J := J ⊕R 1 . Then J is α-Lipschitz w.r.t. the distance induced by the KR norm, and hence the Wasserstein-1 distance when restricted to P(X).

Lemma 4. The convex conjugate of µ → β 1 µ S * is ϕ → χ{ϕ ∈ β 1 S}.

Proof. The proof of Lemma 4 is nearly identical to the proof of Lemma 3. Proposition 9. Let f : R d → R be a neural network consisting of k layers whose linear transformations have spectral norm 1 and whose activation functions are 1-Lipschitz and 1-smooth. Then f is k-smooth.

Proof. In this section, let (f ) and s(f ) denote the Lipschitz constant of f and ∇f respectively. This inequality holds s(f

• g) ≤ s(f ) (g) 2 + (f )s(g), (27) since ||d(f • g)(x) -d(f • g)(y)|| = ||df (g(x))dg(x) -df (g(y))dg(y)|| = ||df (g(x))dg(x) -df (g(y))dg(x) + df (g(y))dg(x) -df (g(y))dg(y)|| ≤ ||df (g(x)) -df (g(y))|| ||dg(x)|| + ||df (g(y))|| ||dg(x) -dg(y)|| ≤ s(f )||g(x) -g(y)|| (g) + (f )s(g)||x -y|| ≤ s(f ) (g) 2 + (f )s(g) ||x -y||.

Let σ be an elementwise activation function with (σ) = s(σ) = 1, and let A be a linear layer with spectral norm 1. Then (A) = 1 and s(A) = 0, so

(σ • A) ≤ (σ) (A) = 1 s(σ • A) ≤ s(σ) (A) 2 + (σ)s(A) = 1.

We note that we can use the same value of s(σ) whether we consider σ : R → R or elementwise as σ : R d → R d , since for an elementwise σ, we have

||dσ(x) -dσ(y)|| 2 = || diag σ (x) -diag σ (y)|| 2 = || diag(σ (x) -σ (y))|| 2 = max i |σ (x i ) -σ (y i )| ≤ max i s(σ)|x i -y i | = s(σ)||x -y|| ∞ ≤ s(σ)||x -y|| 2 .

We apply the inequality ( 27) recursively for all k layers to obtain that the entire network is ksmooth.

Lemma 5. Let H be an RKHS with the Gaussian kernel. Then

||μ|| H ≤ √ 2 σ 2 ||µ|| S * . Proof. It follows from ∂f (x) ∂xi = f, ∂K(x,•) ∂xi that, for f ∈ H, ∇f (x) -∇f (y) 2 = d i=1 ∂f (x) ∂x i - ∂f (y) ∂y i 2 ≤ d i=1 f 2 H ∂K(x, •) ∂x i - ∂K(y, •) ∂y i 2 H = f 2 H d i=1 ∂ 2 K(x, x) ∂x i ∂ xi -2 ∂ 2 K(x, y) ∂x i ∂y i + ∂ 2 K(y, ỹ) ∂y i ∂ ỹi ≤ f 2 H • (Γ(x, y) + Γ(y, x))

, where x and ỹ denote copies of x and y, and

Γ(x, y) = d i=1 ∂ 2 K(x, x) ∂x i ∂ xi - ∂ 2 K(x, y) ∂x i ∂ xi .

From (30), we find that for a Gaussian kernel,

Γ(x, y) = 1 σ 4 exp - x -y 2 2σ 2 x -y 2 ,

and thus,

∇f (x) -∇f (y) ≤ √ 2 σ 2 f H x -y . This implies that μ H = sup f ∈H, f H ≤1 f dµ ≤ sup f S ≤ √ 2 σ 2 f dµ = √ 2 σ 2 µ S * . H PROOFS FOR SECTION 6 Lemma 6. The convex conjugate of µ → λ 2 ||µ|| 2 KR is f → 1 2λ ||f || 2 Lip .

Proof. This proof is generalized from the finite-dimensional case (Boyd & Vandenberghe, 2004) . We can bound the convex conjugate from above by

sup µ f dµ - λ 2 ||µ|| 2 KR ≤ sup µ ||f || Lip ||µ|| KR - λ 2 ||µ|| 2 KR ≤ sup z∈R ||f || Lip z - λ 2 z 2 = ||f || Lip • ||f || Lip λ - λ 2 • ||f || 2 Lip λ 2 = 1 2λ ||f || 2 Lip .

If f is constant, then we may choose µ = 0 to see that

sup µ f dµ - λ 2 ||µ|| 2 KR ≥ 0 = 1 2λ ||f || 2 Lip ,

and we are done.

Otherwise, for f (x) = f (y), let µ x,y be the signed measure given by

µ x,y = ||f || 2 Lip λ(f (x) -f (y)) (δ x -δ y ). Note that ||µ x,y || KR = sup ||g||Lip≤1 g dµ x,y = ||f || 2 Lip λ(f (x) -f (y)) sup ||g||Lip≤1 g(x) -g(y) = ||f || 2 Lip λ(f (x) -f (y)) • d(x, y) and f dµ x,y = ||f || 2 Lip λ(f (x) -f (y)) f (x) -f (y) = 1 λ ||f || 2 Lip . Then sup µ f dµ - λ 2 ||µ|| 2 KR ≥ sup f (x) =f (y) f dµ x,y - λ 2 ||µ x,y || 2 KR = sup f (x) =f (y) 1 λ ||f || 2 Lip - λ 2 1 λ ||f || 2 Lip d(x, y) f (x) -f (y) 2 = 1 λ ||f || 2 Lip - λ 2 1 λ ||f || 2 Lip 1 ||f || Lip 2 = 1 2λ ||f || 2 Lip .

Proposition 10. Let J : M(X) → R be a convex function. Then J satisfies (D3) if and only if

D J (ν, µ) ≤ β2 2 µ -ν 2 KR for all µ, ν ∈ M(X).

Proof. Recall the definition of the Bregman divergence:

D J (ν, µ) := J(ν) -J(µ) -Φ µ (x) d(ν -µ).

Proposition 10 claims that the optimal discriminator Φ µ is β 2 -smooth as a function of µ if and only

EQUATION

This proof is generalized from the finite-dimensional case (Zhou, 2018; Sidford, 2017) . Suppose that ∇Φ µ (x) -∇Φ ν (x) 2 ≤ β 2 µν KR holds for all x. Let µ t (0 ≤ t ≤ 1) be defined as a mixture between µ t = (1t)µ + tν so that

µ 0 = µ, µ 1 = ν, and µ t -µ s KR = |t -s| • µ -ν KR .

Then, we have

|D J (ν, µ)| = 1 0 d dt J(µ t ) dt -Φ µ d(ν -µ) = 1 0 Φ µt d(ν -µ) dt -Φ µ d(ν -µ) = 1 0 (Φ µt -Φ µ ) d(ν -µ) dt ≤ 1 0 Φ µt -Φ µ Lip µ -ν KR dt ≤ 1 0 β 2 µ t -µ KR µ -ν KR dt ≤ 1 0 tβ 2 µ -ν 2 KR dt = β 2 2 µ -ν 2 KR .

Since the choice of µ and ν is arbitrary, we have proved (28) by assuming (D2).

Next, we move on to prove the converse. Before proceeding, note that the Bregman divergence D J (ν, µ) is always non-negative. In fact, for any µ, ν ∈ M(X), we have

Φ µ d(ν -µ) = Φ µ dν -J (Φ µ ) ≤J(ν) - Φ µ dµ -J (Φ µ ) =J(µ) ≤ J(ν) -J(µ), which implies D J (ν, µ) ≥ 0.

Choose ξ ∈ M(X) arbitrarily. If ξ = µ, the inequality ∇Φ µ -∇Φ ξ 2 ≤ β 2 ξµ KR is trivial, so we assume ξ = µ below. By the non-negativity of the Bregman divergence, we have

EQUATION

Since the above inequality holds for all ν ∈ M(X), the last expression is still non-negative if we take the infimum over all ν. In particular,

inf ν∈M(X) [Φ ξ -Φ µ ] d(ν -ξ) + β 2 2 ||ξ -ν|| 2 KR = -sup ν∈M(X) [Φ µ -Φ ξ ] d(ν -ξ) - β 2 2 ||ξ -ν|| 2 KR = -sup ζ∈M(X) [Φ µ -Φ ξ ] dζ - β 2 2 ||ζ|| 2 KR = - 1 2β 2 Φ µ -Φ ξ 2 Lip ,

using Lemma 6 for the last equality.

Continuing from (29), we have

0 ≤ J(ξ) -J(µ) -Φ µ d(ξ -µ) - 1 2β 2 Φ µ -Φ ξ 2 Lip .

Swapping the roles of µ and ξ, we obtain a similar inequality. Adding both sides of thus obtained two inequalities, we obtain

0 ≤ [Φ ξ -Φ µ ] d(ξ -µ) - 1 β 2 Φ µ -Φ ξ 2 Lip .

Finally, we have

Φ µ -Φ ξ 2 Lip ≤ β 2 [Φ ξ -Φ µ ] d(ξ -µ) ≤ β 2 Φ µ -Φ ξ Lip ξ -µ KR ,

and hence we have the desired result:

Φ µ -Φ ξ Lip ≤ β 2 ξ -µ KR . Lemma 7. Let K(x, y) = exp(-||x-y|| 2 2σ 2 ). Then MMD 2 K (µ, ν) ≤ 1 σ 2 ||µ -ν|| 2 KR .

Proof. First, we note that

MMD 2 K (µ, ν) = K(x, y) (µ -ν)(dx) (µ -ν)(dy) ≤ sup x =x (K(x, y) -K(x , y)) (µ -ν)(dy) d(x, x ) ||µ -ν|| KR ≤ sup x =x ,y =y K(x, y) -K(x, y ) -K(x , y) + K(x , y ) d(x, x ) d(y, y ) ||µ -ν|| 2 KR .

Next, we show that

c = sup x =x ,y =y K(x, y) -K(x, y ) -K(x , y) + K(x , y ) d(x, x ) d(y, y ) = 1 σ 2 .

Because K is differentiable, it suffices to check that the operator norm of ∂ xi ∂ yj K(x, y) is bounded.

To see this, note that c = sup

x =x Lip K(x, •) -K(x , •) ||x -x || = sup x =x ,y ||∇ y K(x, y) -∇ y K(x , y)|| 2 ||x -x || ≤ sup x,y ||∇ x ∇ y K(x, y)|| op ,

where we obtained by the last equality by applying the vector-valued mean value theorem (Rudin, 1964) to t → ∇ y K((1t)x + tx , y), thereby obtaining

EQUATION

This is the symmetric matrix 1 σ 2 exp -||x -y|| 2 2σ 2 1 σ 2 (xy)(xy) T -I .

By inspection, we see that xy is an eigenvector with eigenvalue 1 σ 2 exp(-||x-y|| 2 2σ 2 )( 1 σ 2 ||x -y|| 2 -1), and the other eigenvectors are orthogonal to xy with eigenvalues -1 σ 2 exp(-||x-y|| 2 2σ 2 ).

Setting z = ||x-y|| 2 2σ 2 and taking the absolute value of the eigenvalues, we see the maximum operator norm over all x, y is equal to

max sup z≥0 1 σ 2 e -z |2z -1|, sup z≥0 1 σ 2 e -z = 1 σ 2 . Therefore c ≤ 1 σ 2 .

It is not strictly necessary for the proof to show equality. However, to show equality, let u be a unit vector, and set x = y = 0, and x = y = tu. Then c ≥ sup t =0

K(tu, tu) -K(tu, 0) -K(0, tu) + K(0, 0) d(tu, 0) d(tu, 0)

= sup t =0 2 -2 exp(-t 2 2σ 2 ) t 2 ≥ lim t→0 2 -2 exp(-t 2 2σ 2 ) t 2 = lim t→0 -2 exp(-t 2 2σ 2 ) • -2t 2σ 2 2t = 1 σ 2 ,

Proposition 13 (Moreau-Yosida regularization). Suppose J : M(X) → R is convex, and define J := J ⊕ R 3 . Then J is convex, and D J (ν, µ) ≤ β2 2 ||µ -ν|| 2 KR .

Proof. We work on the more general case of K(x, y) = (2πσ 2 ) -d/2 exp(-x-y 2 2σ 2 ), and define

J(µ) = inf μ J(μ) + cL 2 MMD 2 K (µ, μ) for c = σ 2 (2πσ 2 ) d/2 .

Let µ * be the unique minimizer of the infimum, which exists because the function is a strongly convex. By the envelope theorem, we compute that

d d J(µ + χ) =0 = d d cL 2 ||µ + χ -µ * || 2 H =0 = d d cL 2 ||µ -µ * || 2 H + 2 µ -µ * , χ H + 2 ||χ|| 2 H =0 = cL µ -µ * , χ H = cL (µ -µ * ) dχ, so δ J δµ = cL(µ -µ * ). Then D J (ν, µ) = J(ν) -J(µ) - δ J δµ d(ν -µ) = inf ν J(ν) + cL 2 ||ν -ν|| 2 H -J(µ * ) + cL 2 ||µ -µ * || 2 H -cL µ -µ * , ν -µ H ≤ J(µ * ) + cL 2 ||ν -µ * || 2 H -J(µ * ) + cL 2 ||µ -µ * || 2 H -cL µ -µ * , ν -µ H = cL 2 ||µ -ν|| 2 H ≤ cL 2 • (2πσ 2 ) -d/2 • 1 σ 2 ||µ -ν|| 2 KR = L 2 ||µ -ν|| 2 KR ,

where we used Lemma 7 for the second-to-last line. The proposition follows for σ = (2π) -1/2 .

Regarding this choice of σ, it will turn out that in the general case, the dual penalty includes a numerically unfavorable factor of (2πσ 2 ) -d/2 , dependent on the dimension of the problem. In practical applications, such as image generation, d can be quite large, making the accurate computation of (2πσ 2 ) -d/2 completely infeasible. For numerical stability, we propose choosing the critical parameter σ = (2π) -1/2 , which corresponds to the dimension-free kernel K(x, y) = e -π||x-y|| 2 .

Lemma 8. Let H be an RKHS with a continuous kernel K on a compact domain X. To derive a lower bound for general f ∈ C(X), we use the Mercer decomposition of the positive definite kernel K,

K(x, y) = ∞ i=1 γ i φ i (x)φ i (y)

where γ i ≥ 0 are eigenvalues and {φ i } ∞ i=1 is a complete orthonormal sequence of L 2 (X; dx). It is well-known that the corresponding RKHS is given by

H = g ∈ L 2 (X, dx) | ∞ i=1 (g, φ i ) 2 L 2 (X) γ i < ∞ ,

and the norm of g ∈ H by

g 2 H = ∞ j=1 (g, φ i ) 2 L 2 (X) γ i .

Let f ∈ C(X) ⊂ L 2 (X, dx) be an arbitrary function. We replace the supremum of the conjugate function sup This is maximized when b j = a j /(λγ j ), and the maximum value is

EQUATION

)

If f ∈ H, this value is finite, and the lower bound of the conjugate given by is 1 2λ f 2 H , which is the same as the upper bound. If f / ∈ H, the value (32) is infinite, and the conjugate function takes +∞.

Proposition 14. Let H be an RKHS with the Gaussian kernel K(x, y) = e -π||x-y|| 2 . Then for f ∈ H,

EQUATION

)

Proof. We consider the more general case of K(x, y) = (2πσ 2 ) -d/2 e -||x-y|| 2 /2σ 2 , where we have Novak et al. (2018) prove this result for σ = 1. We sketch the proof for the general case here. We use the Fourier transform convention that

||f || 2 H = ∞ k=0 ( 1 2 σ 2 ) k k1+•••+k d =k 1 d i=1 k i ! ||∂ k1 x1 • • • ∂ k d x d f || 2 L 2 (R d ) = ||f || 2 L 2 (R d ) + 1 2 σ 2 ||∇f || 2 L 2 (R d ) + 1 4 σ 4 ||∇ 2 f || 2 L 2 (R d ) + other terms.

f (k) = (2π) -d/2 R d f (x) e -ik•x dx, f (x) = (2π) -d/2 R d f (k) e ik•x dk.

Consider the inner product

f, g = R d

e σ 2 ||k|| 2 /2 f (k) ĝ(k) dk, defined for functions ||f || < ∞. Expanding the exponential in Taylor series, this inner product gives the equation for the norm in the proposition. By use of the Fourier inversion formula, it can be shown that f, K x = f (x), where K x (y) = (2πσ 2 ) -d/2 e -||x-y|| 2 /2σ 2 , so this is an RKHS with the Gaussian kernel.

Whereas Miyato et al. (2018) divide each layer by the spectral norm of the convolutional kernel, we divide by the spectral norm of the convolutional operator itself, computed using the same power iteration algorithm applied to the operator. This is so that the layers are truly 1-Lipschitz, which is critical for our theory.

Figures and Tables:

Figure1: FID of simple particle generators with various learning rates. The horizontal axis shows the ratio γ/γ 0 between the actual learning rate γ and the theoretical learning rate γ 0 suggested by Theorem 1 and Proposition 1. The vertical axis shows the FID after 100,000 SGD iterations.

Figure2: Illustration of inf-convolutions and their convex conjugates in R. Top row: Generally, inf-convolutions inherit the regularity of their parent functions. In this example, J 1 (x) = |x| is 1-Lipschitz but not smooth, while J 2 (x) = x 2 /2 is 1-smooth but not Lipschitz. The inf-convolution J 1 ⊕ J 2 is the well-known Huber function, which is both 1-Lipschitz and 1-smooth. Bottom row: Convex conjugates of the functions in the top row. The conjugate of J 1 ⊕ J 2 is given as the sum of conjugates J 1 (z) = χ{|z| ≤ 1} and J 2 (z) = z 2 /2.

µ ∈ M(X) that has a square-integrable Radon-Nykodim derivative with respect to the Lebesgue measure dx, and then we have a lower bound. We use µ(x) for the Radon-Nykodim derivative with slight abuse of notation. Suppose f (x) = ∞ j=1 a j φ j (x) and µ(x) = ∞ j=1 b j φ j (x) are the expansion. Note that, since the kernel embedding is given by μ(x) = K(x, y)µ(y) dy = ∞ j=1 γ j b j φ j (x),

Regularity of common GAN losses.

Table content: <table><tr><td/><td>(D1) (D2) (D3)</td></tr><tr><td>Minimax GAN</td><td/></tr><tr><td>Non-saturating GAN</td><td/></tr><tr><td>Wasserstein GAN</td><td>?</td></tr><tr><td>IPM S</td><td>?</td></tr><tr><td>MMD 2</td><td>*</td></tr></table>

Computationally, the RKHS norm is difficult to evaluate. We propose taking advantage of the following infinite series representation of ||f || 2 H in terms of the derivatives of f (Fasshauer & Ye, 2011; Novak et al., 2018): Proposition 14. Let H be an RKHS with the Gaussian kernel K

Table content: <table/>

The convex conjugate of µ → λ 2 ||µ|| 2 H is f → 1 2λ ||f || 2 H + χ{f ∈ H}.Proof. First we show an upper bound for f ∈ H:

Table content: <table><tr><td>sup µ</td><td>f dµ -</td><td>λ 2</td><td colspan="3">||µ|| 2 H = sup µ</td><td colspan="2">f, µ H -</td><td>λ 2</td><td>||µ|| 2 H</td></tr><tr><td/><td/><td/><td colspan="3">≤ sup µ</td><td colspan="2">||f || H ||µ|| H -</td><td>λ 2</td><td>||µ|| 2 H</td></tr><tr><td/><td/><td/><td colspan="3">≤ sup z∈R</td><td colspan="2">||f || H z -</td><td>λ 2</td><td>z 2</td></tr><tr><td/><td/><td/><td colspan="4">= ||f || H •</td><td>||f || H λ</td><td>-</td><td>λ 2</td><td>•</td><td>H ||f || 2 λ 2</td></tr><tr><td/><td/><td/><td>=</td><td>1 2λ</td><td colspan="3">||f || 2 H .</td></tr></table>

Peer review (iclr2020):
This paper provides a unified theoretical framework for regularizing GAN losses. It accounts for most regularization technics especially spectral normalization and gradient penalty and explains how those two methods are in fact complementary. So far this was only observed experimentally but without any theoretical insight. The result goes beyond that as the criterion could be applied to general convex cost functional. The main general theorem is Theorem 1 which states 3 conditions on the optimal critic and 2 others on the generator. The paper is mainly concerned by the conditions on the optimal critic and show that the first 2 conditions can be achieved by the Spectral normalization, while the last one can be achieved by some gradient penalty. The paper is clearly written, well structured and pleasant to read. I have the following two remarks: - Proposition 8 provides a way to ensure condition 2 holds (beta-smoothness). It requires spectral normalization and smooth activation functions. In practice, while the spectral normalization is important, the choice of the activation is not in general 1-smooth (Leaky-relu for instance). Does it really matter in practice? Some illustrative experiments could be beneficial to better understand what*s happening. - Is it that hard to obtain generators that satisfy condition G1 and G2, it seems to be a natural consequence on the regularity of the mapping f? If that is the case, it might be worth better explaining how this is challenging. Limitations: The paper considers only the setting where the optimal critic is reached and therefore it is still unclear if the analysis carries on to the training procedures used in practice (non-optimal critic). The authors recognize this limitation and leave it for future work. Overall, I feel that the paper provides good insights on what regularization is important for training gans and why. For that reason, I think this paper should be accepted. ------------------------------------------------------------------------------------------------------------ Revision: I think the paper provides a good theoretical contribution in terms of interpreting many of the tricks used for improving GAN training. In fact the paper also suggests some new regularization methods (prop 13 for conditions D3) which would constrain the RKHS norm of the critic. The authors show how it is related to gradient penalty, in a particular case, but the result also suggests something more general. For instance [1], consider an abstract RKHS space containing deep networks and provide an upper-bound on the rkhs norm of such networks in terms of the spectral norm of their weights and a lower-bound in terms of its Lipschitz constant. I do agree with reviewer 1 that a better discussion of the connection to [2] should be included since that paper was interested in ensuring weak continuity of the loss, which can be thought of as a first requirement to get more regularity of the cost functional. I still think the paper is worth being accepted and raised my score to 8 as I think the authors addressed the major concerns that were raised. [1] A. Bietti, G. Mialon, D. Chen, and J. Mairal. A Kernel Perspective for Regularizing Deep Neural Networks. [2] Michael Arbel, Dougal Sutherland, Miko?aj Binkowski, and Arthur Gretton. On gradient regularizers for MMD GANs. In Advances in Neural Information Processing Systems, pp. 6700–6710, 2018.