# Last-iterate convergence rates for min-max optimization

We study the problem of finding min-max solutions for smooth two-input objective functions. While classic results show average-iterate convergence rates for various algorithms, nonconvex applications such as training Generative Adversarial Networks require last-iterate convergence guarantees, which are more difficult to prove. It has been an open problem as to whether any algorithm achieves non-asymptotic last-iterate convergence in settings beyond the bilinear and convex-strongly concave settings. In this paper, we study the Hamiltonian Gradient Descent (HGD) algorithm, and we show that HGD exhibits a linear convergence rate in a variety of more general settings, including convex-concave settings that are "sufficiently bilinear." We also prove similar convergence rates for the Consensus Optimization (CO) algorithm of [MNG17] for some parameter settings of CO.

## Authors

• 16 publications
• 6 publications
• 11 publications
• ### Convergence Behaviour of Some Gradient-Based Methods on Bilinear Games

Min-max optimization has attracted much attention in the machine learnin...
08/15/2019 ∙ by Guojun Zhang, et al. ∙ 0

We present a new family of min-max optimization algorithms that automati...
10/22/2020 ∙ by Kimon Antonakopoulos, et al. ∙ 0

• ### Stochastic Hamiltonian Gradient Methods for Smooth Games

The success of adversarial formulations in machine learning has brought ...
07/08/2020 ∙ by Nicolas Loizou, et al. ∙ 45

• ### A Single-Loop Smoothed Gradient Descent-Ascent Algorithm for Nonconvex-Concave Min-Max Problems

Nonconvex-concave min-max problem arises in many machine learning applic...
10/29/2020 ∙ by Jiawei Zhang, et al. ∙ 0

• ### Linear Last-iterate Convergence for Matrix Games and Stochastic Games

06/16/2020 ∙ by Chung-Wei Lee, et al. ∙ 0

• ### Super-efficiency of automatic differentiation for functions defined as a minimum

In min-min optimization or max-min optimization, one has to compute the ...
02/10/2020 ∙ by Pierre Ablin, et al. ∙ 13

• ### Higher-order methods for convex-concave min-max optimization and monotone variational inequalities

We provide improved convergence rates for constrained convex-concave min...
07/09/2020 ∙ by Brian Bullins, et al. ∙ 0

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## 1 Introduction

In this paper we consider methods to solve smooth unconstrained min-max optimization problems. In the most classical setting, a min-max objective has the form

 minx1maxx2g(x1,x2)

where is a smooth objective function with two inputs. The usual goal in such problems is to find a saddle point, also known as a min-max solution, which is a pair that satisfies

 g(x∗1,x2)≤g(x∗1,x∗2)≤g(x1,x∗2) (1)

for every and . Min-max problems have a long history, going back at least as far as neumann1928theorie

, which formed the basis of much of modern game theory, and including a great deal of work in the 1950s when algorithms such as

fictitious play were explored brown1951iterative ; robinson1951iterative .

The convex-concave setting, where we assume is convex in and concave in

, is a classic min-max problem that has a number of different applications, such as solving constrained convex optimization problems. While a variety of tools have been developed for this setting, a very popular approach within the machine learning community has been the use of so-called

no-regret algorithms cesa2006prediction ; hazan2016introduction . This trick, which was originally developed by hannan1957approximation and later emerged in the development of boosting freund1999adaptive , provides a simple computational method via repeated play: each of the inputs and are updated iteratively according to no-regret learning protocols, and one can prove that the average-iterates converge to a min-max solution.

Recently, interest in min-max optimization has surged due to the enormous popularity of Generative Adversarial Networks (GANs), whose training involves solving a nonconvex min-max problem where and correspond to the parameters of two different neural nets goodfellow2014generative . Due to the fundamentally nonconvex nature of this problem, it is infeasible to find a “global” solution of the min-max objective. Instead, the typical goal in GAN training is to find a local min-max, namely a pair that satisfies eq. 1 for all in some neighborhood of . Moreover, iterate averaging is no longer desirable in the nonconvex setting, as it lacks the theoretical guarantees present in the convex-concave setting, and in practice, averaging neural net parameters tends to hurt performance. Thus, even though GANs are often trained with no-regret algorithms such as gradient descent, the final neural net parameters used after training are the last iterates for and , rather than the time-averaged iterates. As such, provable guarantees for GAN training must include last-iterate convergence properties.

In this paper, we focus on proving last-iterate convergence rates for min-max problems. Provable convergence rates are useful because they allow for quantitative comparison of different algorithms and can aid in choosing learning rates and architectures to ensure fast convergence in practice. Yet despite the extensive amount of literature on convergence rates for convex optimization, very few last-iterate convergence rates have been proved for min-max problems. Standard analysis of no-regret algorithms says essentially nothing about last-iterate convergence. In fact, widely used no-regret algorithms, such as Simultaneous Gradient Descent/Ascent (SGDA), fail to converge even in the simple bilinear setting where for some arbitrary matrix . SGDA provably cycles in continuous time and diverges in discrete time (see for example daskalakis2018training ; mescheder2018training ). In fact, the full range of Follow-The-Regularized-Leader (FTRL) algorithms provably do not converge in zero-sum games with interior equilibria mertikopoulos2018cycles . This occurs because the iterates of the FTRL algorithms exhibit cyclic behavior, a phenomenon commonly observed when training GANs in practice as well.

Existing work on last-iterate convergence rates has been limited to the bilinear or convex-strongly concave settings tseng1995linear ; liang2018interaction ; du2019linear ; mokhtari2019unified . In particular, the following basic question is still open:

“What last-iterate convergence rates are achievable for convex-concave min-max problems?”

We give a partial answer for this question by proving linear last-iterate convergence rates for an algorithm called Hamiltonian Gradient Descent (HGD) under much weaker assumptions compared to previous results. HGD is gradient descent on the squared norm of the gradient, and it has been mentioned in mescheder2017numerics ; balduzzi2018mechanics . Our results are the first to show non-asymptotic convergence of an efficient algorithm in settings that are not bilinear or strongly convex in either player. We show that HGD has linear convergence in settings where there are no spurious critical points provided that a “sufficiently bilinear” condition on the second-order derivatives holds.111The condition can be fulfilled by adding a large, well-conditioned bilinear term to any objective Our result implies that HGD achieves linear convergence in convex-concave settings that are “sufficiently bilinear,” which is surprising since general smooth convex-concave optimization can have rates no faster than due to lower bounds on smooth convex optimization agarwal2017lower ; arjevani2017oracle . On the practical side, while vanilla HGD has issues training GANs in practice, mescheder2017numerics show that a related algorithm known as Consensus Optimization (CO) can effectively train GANs in a variety of settings, including on CIFAR-10 and celebA. We show that CO can be viewed as a perturbation of HGD, which implies that for some parameter settings, CO converges at the same rate as HGD.

We begin in Section 2 with background material and notation, including some of our key assumptions. In Section 3, we discuss Hamiltonian Gradient Descent (HGD), and we present our linear convergence rates for HGD in various settings. In Section 4, we present some of the key technical components used to prove our results from Section 3. Finally, in Section 5, we present our results for Consensus Optimization. The details of our proofs are in Appendix F.

## 2 Background

### 2.1 Preliminaries

In this section, we discuss some key definitions and notation. We will use

to denote the Euclidean norm for vectors or the operator norm for matrices or tensors. For a symmetric matrix

, we will use and

to denote the smallest and largest eigenvalues of

. For a general real matrix , and

denote the smallest and largest singular values of

.

###### Definition 2.1.

A critical point of is a point such that .

###### Definition 2.2 (Convexity / Strong convexity).

Let . A function is -strongly convex if for any . When is twice-differentiable, is -strongly-convex iff for all . If in either of the above definitions, is called convex.

###### Definition 2.3 (Monotone / Strongly monotone).

Let . A vector field is -strongly monotone if for any . If , is called monotone.

###### Definition 2.4 (Smoothness).

A function is -smooth if is differentiable everywhere and for all satisfies .

##### Notation

Since is a function of and , we will often consider and to be components of one vector . We will use superscripts to denote iterate indices. Following balduzzi2018mechanics , we use to denote the signed vector of partial derivatives. Under this notation, the Simultaneous Gradient Descent/Ascent (SGDA) algorithm can be written as follows:

 x(k+1)=x(k)−ηξ(x(k))

We will write the Jacobian of as:

 J≡∇ξ=⎛⎜ ⎜ ⎜⎝∂2g∂x21∂2g∂x1∂x2−∂2g∂x2∂x1−∂2g∂x22⎞⎟ ⎟ ⎟⎠.

Note that unlike the Hessian in standard optimization, is not symmetric, due to the negative sign in . When clear from the context, we often omit dependence on when writing and other functions. Note that , and are defined for a given objective – we omit this dependence as well for notational clarity. We will always assume is sufficiently differentiable whenever we take derivatives. In particular, we assume second-order differentiability in Section 3.

We will also use the following non-standard definition for notational convenience:

###### Definition 2.5 (Higher-order Lipschitz).

A function is -Lipschitz if for all , and , and for all , .

We will consider a variety of settings for min-max optimization based on properties of the objective function . In the convex-concave setting, is convex as a function of for any fixed and is concave as a function of for any fixed . We can form analogous definitions by replacing the words “convex” and “concave” with words such as “strongly convex/concave”, “linear”, or “nonconvex”. The bilinear setting refers to the case when for some matrix . The strongly monotone setting refers to the case when is a strongly monotone vector field, as in the case when is strongly convex-strongly concave.

### 2.2 Notions of convergence in min-max problems

The convergence rates in this paper will apply to min-max problems where satisfies the following assumption:

###### Assumption 2.6.

All critical points of the objective are global min-maxes (i.e. they satisfy eq. 1).

In other words, we prove convergence rates to min-maxes in settings where convergence to critical points is necessary and sufficient for convergence to min-maxes. This assumption is true for convex-concave settings, but also holds for some nonconvex-nonconcave settings, as we discuss in Appendix D.

We will measure the convergence of our algorithms to -approximate critical points, defined as follows:

###### Definition 2.7.

Let . A point is an -approximate critical point if .

Convergence to approximate critical points is a common goal in standard convex and nonconvex optimization (see for example allen2016variance ; ghadimi2016accelerated ; carmon2017convex ), as it is a necessary condition for convergence to local or global minima. For min-max optimization, it makes even more sense to use to measure how close we are to convergence since the value of at a given point gives no information about how close we are to a min-max.

Our main convergence rate results focus on this first-order notion of convergence, which is sufficient given creftypecap 2.6. We discuss notions of second-order convergence and ways to adapt our results to the general nonconvex setting in Appendix A.

### 2.3 Related work

##### Asymptotic and local convergence

Several recent papers have given asymptotic or local convergence results for min-max problems. mertikopoulos2018optimistic show that the extragradient (EG) algorithm converges asymptotically in a broad class of problems known as coherent saddle point problems, which include quasiconvex-quasiconcave problems. Quasiconvex functions are unimodal and have no local maxima, but they may have critical points that are not local minima. As such, coherent saddle point problems capture some settings for which creftypecap 2.6 does not hold.

For more general smooth nonconvex min-max problems, a number of different papers have given local stability or local asymptotic convergence results for various algorithms mescheder2017numerics ; daskalakis2018limit ; balduzzi2018mechanics ; letcher2018stable ; mazumdar2019finding . We discuss this more in Appendix A.

##### Non-asymptotic convergence rates

Compared to the work on asymptotic convergence, the work on non-asymptotic last-iterate convergence rates has been limited to much more restrictive settings. A classic result by rockafellar1976monotone shows a linear convergence rate for the proximal point method in the bilinear and strongly convex-strongly concave cases. Another classic result, by tseng1995linear , shows a linear convergence rate for the extragradient algorithm in the bilinear case. liang2018interaction show that a number of algorithms achieve a linear convergence rate in the bilinear case, including Optimistic Mirror Descent (OMD) and Consensus Optimization (CO). They also show that SGDA obtains a linear convergence rate in the strongly convex-strongly concave case. mokhtari2019unified show that OMD and EG obtain a linear rate for the strongly convex-strongly concave case, in addition to proving similar results for generalized versions of both algorithms. Finally, du2019linear show that SGDA achieves a linear convergence rate for a convex-strongly concave setting with a full column rank linear interaction term.222Specifically, they assume , where is smooth and convex, is smooth and strongly convex, and has full column rank. We make a brief comparison of our work to that of du2019linear for the convex-strongly concave setting in Appendix C..

Our main algorithm for finding saddle points of is called Hamiltonian Gradient Descent (HGD). HGD consists of performing gradient descent on a particular objective function that we refer to as the Hamiltonian, following the terminology of balduzzi2018mechanics .333We note that the function is not the Hamiltonian as in the sense of classical physics, as we do not use the symplectic structure in our analysis, but rather we only perform gradient descent on . If we let be the vector of (appropriately-signed) partial derivatives, then the Hamiltonian is:

 H(x):=12∥ξ(x)∥2=12(∥∂g∂x1(x)∥2+∥∂g∂x2(x)∥2).

Since a critical point occurs when , we can find a (approximate) critical point by finding a (approximate) minimizer of . Moreover, under creftypecap 2.6, finding a critical point is equivalent to finding a saddle point. This motivates the HGD update procedure on with step-size :

 x(k+1)=x(k)−η∇H(x(k)), (2)

HGD has been mentioned in mescheder2017numerics ; balduzzi2018mechanics , and it strongly resembles the Consensus Optimization (CO) approach of mescheder2017numerics .

The HGD update requires a Hessian-vector product because , making HGD a second-order iterative scheme. However, Hessian-vector products are cheap to compute when the objective is defined by a neural net, taking only two gradient oracle calls pearlmutter1994fast . This makes the Hessian-vector product oracle a theoretically appealing primitive, and it has been used widely in the nonconvex optimization literature. Since Hessian-vector product oracles are feasible to compute for GANs, many recent algorithms for local min-max nonconvex optimization have also utilized Hessian-vector products mescheder2017numerics ; balduzzi2018mechanics ; adolphs2018local ; letcher2018stable ; mazumdar2019finding .

To the best of our knowledge, previous work on last-iterate convergence rates has only focused on how algorithms perform in three particular cases: (a) when the objective is bilinear, (b) when is strongly convex-strongly concave, and (c) when is convex-strongly concave tseng1995linear ; liang2018interaction ; du2019linear ; mokhtari2019unified . The existence of methods with provable finite-time guarantees for settings beyond the aforementioned has remained an open problem. This work is the first to show that an efficient algorithm, namely HGD, can achieve non-asymptotic convergence in settings that are not strongly convex or linear in either player.

### 3.1 Convergence Rates for HGD

We now state our main theorems for this paper, which show convergence to critical points. When creftypecap 2.6 holds, we get convergence to min-maxes. All of our main results will use the following multi-part assumption:

###### Assumption 3.1.

Let .

1. Assume a critical point for exists.

2. Assume is -Lipschitz and let .

Our first theorem shows that HGD converges for the strongly convex-strongly concave case. Although simple, this result will help us demonstrate our analysis techniques.

###### Theorem 3.2.

Let creftypecap 3.1 hold and let be -strongly convex in and -strongly concave in . Then the HGD update procedure described in (2) with step-size starting from some will satisfy

 ∣∣∣∣ξ(x(k))∣∣∣∣≤(1−α2LH)k/2∣∣∣∣ξ(x(0))∣∣∣∣.

Next, we show that HGD converges when is linear in one of its arguments and the cross-derivative is full rank. This setting allows a slightly tighter analysis compared to Theorem 3.4.

###### Theorem 3.3.

Let creftypecap 3.1 hold and let be -smooth in and linear in , and assume the cross derivative is full rank with all singular values at least for all . Then the HGD update procedure described in (2) with step-size starting from some will satisfy

 ∣∣∣∣ξ(x(k))∣∣∣∣≤(1−γ4(2γ2+L2)LH)k/2∣∣∣∣ξ(x(0))∣∣∣∣.

Finally, we show our main result, which requires smoothness in both players and a large, well-conditioned cross-derivative.

###### Theorem 3.4.

Let creftypecap 3.1 hold and let be -smooth in and -smooth in . Let and , and assume the cross derivative is full rank with all singular values lower bounded by and upper bounded by for all . Moreover, let the following “sufficiently bilinear” condition hold:

 (γ2+ρ2)(μ2+γ2)−4L2Γ2>0. (3)

Then the HGD update procedure described in (2) with step-size starting from some will satisfy

 ∣∣∣∣ξ(x(k))∣∣∣∣≤(1−(γ2+ρ2)(γ2+μ2)−4L2Γ2(2γ2+ρ2+μ2)2LH)k/2∣∣∣∣ξ(x(0))∣∣∣∣. (4)

As discussed above, Theorem 3.4 provides the first last-iterate convergence rate that does not require strong convexity or linearity in either player. We even achieve linear convergence in convex-concave settings where eq. 3 holds, which is surprising since linear convergence is impossible in general for convex-concave settings due to lower bounds for convex optimization. Thus, the “sufficiently bilinear” condition eq. 3 is crucial for our linear rate. We give some explanations for this condition in the following section. In simple experiments for HGD on convex-concave and nonconvex-nonconcave objectives, the convergence rate speeds up when there is a larger bilinear component, as expected from our theoretical results. We show these experiments in Appendix H.

We can construct examples of objectives that satisfy the assumptions of Theorem 3.4 but are not strongly monotone or bilinear. One such example is , where and are -smooth convex functions. We discuss a simple example that is not convex-concave in Appendix D.

### 3.2 Explanation of “sufficiently bilinear” condition

In this section, we explain the “sufficiently bilinear” condition eq. 3. Suppose our objective is for a smooth function . Then for sufficiently large values of (i.e. has a large enough bilinear term), we see that satisfies eq. 3. To see this, note that in the worst case (i.e. when the eigenvalues of and are not bounded away from zero), condition eq. 3 requires . Let and be lower and upper bounds on the singular values of . Then eq. 3 becomes , which is true for (i.e. suffices).

We can understand this condition by appealing to an analogous min-max optimization setting. Suppose that we use SGDA on the objective for -smooth convex-concave . According to liang2018interaction , SGDA will converge at a rate of roughly .444The actual rate is , for some parameter that is at least . For , SGDA will diverge in the worst case. For , we get linear convergence, but it will be slow because is large (this can be thought of as a large condition number). Finally, for , we get fast linear convergence, since . Thus, to get fast linear convergence it suffices to make the problem “sufficiently strongly convex-strongly concave” (or “sufficiently strongly monotone”).

Theorem 3.4 and condition eq. 3 show that there exists another class of settings where we can achieve linear rates in the min-max setting. In our case, if we have an objective for a smooth function , we will get linear convergence if and , which ensures that the problem is “sufficiently bilinear.” Intuitively, it makes sense that the “sufficiently bilinear” setting allows a linear rate because the pure bilinear setting allows a linear rate.

Another way to understand condition eq. 3 is that it is a sufficient condition for the existence of a unique critical point in a general class of settings, as we show in the following lemma, which we prove in Appendix E.

###### Lemma 3.5.

Let where and are -smooth. Moreover, assume that and each have a 0 eigenvalue for some and . If eq. 3 holds, then has a unique critical point.

## 4 Proof sketches for HGD convergence rate results

In this section, we go over the key components of the proofs for our convergence rates from Section 3.1. Recall that the intuition behind HGD was that critical points (where ) are global minima of . On the other hand, there is no guarantee that is a convex potential function, and a priori, one would not assume gradient descent on this potential would find a critical point. Nonetheless, we are able to show that in a variety of settings, satisfies the PL condition, which allows HGD to have linear convergence. Proving this requires proving properties about the singular values of .

### 4.1 The Polyak-Łojasiewicz condition for the Hamiltonian

We begin by recalling the definition of the PL condition.

###### Definition 4.1 (Polyak-Łojasiewicz (PL) condition polyak1963 ; lojasiewicz1963 ).

A function satisfies the PL condition with a constant if for all , .

The PL condition is well-known to be the weakest condition necessary to obtain linear convergence rate for gradient methods; see for example karimi2016linear . We will show that satisfies the PL condition, which allows us to use the following classic theorem.

###### Theorem 4.2 (Linear rate under PL polyak1963 ; lojasiewicz1963 ).

Let be -smooth and let . Suppose satisfies the PL condition with parameter . Then if we run gradient descent from with step-size , we have: .

For completeness, we provide the proof of Theorem 4.2 in Appendix B.

All of our results use creftypecap 3.1, so we are guaranteed that has a critical point. This implies that the global minimum of is 0, which allows us to prove the following key lemma:

###### Lemma 4.3.

Assume we have a twice differentiable with associated . Let . If for every , then satisfies the PL condition with parameter .

###### Proof.

Consider the squared norm of the gradient of the Hamiltonian:

 12∥∇H∥2 =12∥J⊤ξ∥2=12⟨ξ,(JJ⊤)ξ⟩≥c2||ξ||2=cH.

The proof is finished by noting that when is a critical point. ∎

To use Theorem 4.2, we will also need to show that is smooth, which holds when is -Lipschitz. The proof of Lemma 4.4 is in Appendix F.

###### Lemma 4.4.

Consider any which is -Lipschitz for constants . Then the Hamiltonian is -smooth.

To use Lemma 4.3, we will need control over the eigenvalues of , which we achieve with the following linear algebra lemmas. We provide their proofs in Appendix F.

###### Lemma 4.5.

Let and let . If and , then for all eigenvalues of , we have .

###### Lemma 4.6.

Let , where C is square and full rank. Then if is an eigenvalue of , then we must have .

### 4.2 Proof sketches for Theorems 3.2, 3.3, and 3.4

We now proceed to sketch the proofs of our main theorems using the techniques we have described. The following lemma shows it suffices to prove the PL condition for for the various settings of our theorems:

###### Lemma 4.7.

Given , suppose satisfies the PL condition with parameter is -smooth. Then if we update some using eq. 2 with step-size , then we have the following:

 ∣∣∣∣ξ(x(k))∣∣∣∣≤(1−αLH)k/2∣∣∣∣ξ(x(0))∣∣∣∣.
###### Proof.

Since satisfies the PL condition with parameter and is -smooth, we know by Theorem 4.2 that gradient descent on with step-size converges at a rate of . Substituting in for gives the lemma. ∎

It remains to show that satisfies the PL condition in the settings of Theorems 3.4, 3.3 and 3.2. First, we show the result for the strongly convex-strongly concave setting of Theorem 3.2.

###### Lemma 4.8 (PL for the strongly convex-strongly concave setting).

Let be -strongly convex in and -strongly concave in . Then satisfies the PL condition with parameter .

###### Proof.

We apply Lemma 4.5 with . Since is -strongly-convex in and -strongly concave in we have and . Then the magnitude of the eigenvalues of is at least . Thus, , so by Lemma 4.3, satisfies the PL condition with parameter . ∎

Next, we show that satisfies the PL condition for the nonconvex-linear setting of Theorem 3.3. We prove this lemma in Section F.4 by using Lemma 4.6.

###### Lemma 4.9 (PL for the smooth nonconvex-linear setting).

Let be -smooth in and linear in . Moreover, for all , let be full rank and square with . Then satisfies the PL condition with parameter .

Finally, we prove that satisfies the PL condition in the nonconvex-nonconvex setting of Theorem 3.4. The proof for Lemma 4.10 is in Section F.5, and it uses Lemma F.2, which is similar to Lemma 4.6.

###### Lemma 4.10 (PL for the smooth nonconvex-nonconvex setting).

Let be -smooth in and -smooth in . Also, let be full rank and let all of its singular values be lower bounded by and upper bounded by for all . Let and . Assume the following condition holds:

 (γ2+ρ2)(μ2+γ2)−4L2Γ2>0.

Then satisfies the PL condition with constant .

Combining Lemmas 4.10, 4.9 and 4.8 with Lemma 4.7 yields Theorems 3.4, 3.3 and 3.2.

## 5 Convergence rates for Consensus Optimization

The Consensus Optimization (CO) algorithm of mescheder2017numerics is as follows:

 x(k+1)=x(k)−η(ξ(x(k))+γ∇H(x(k))) (5)

where . This is essentially a weighted combination of SGDA and HGD. mescheder2017numerics remark that while HGD has poor performance on nonconvex problems in practice, CO can effectively train GANs in a variety of settings, including on CIFAR-10 and celebA. While they frame CO as SGDA with a small modification, they actually set for several of their experiments, which suggests that one can also view CO as a modified form of HGD.

Using this perspective, we prove Theorem 5.1, which implies that we get linear convergence of CO in the same settings as Theorems 3.4, 3.3 and 3.2 provided that is sufficiently large (i.e. the HGD update is large compared to the SGDA update). The key technical component is showing that HGD still performs well even with a certain kind of small arbitrary perturbation. Previously, liang2018interaction proved that CO achieves linear convergence in the bilinear setting, so our result greatly expands the settings where CO has provable non-asymptotic convergence. We prove Theorem 5.1 in Appendix G.

###### Theorem 5.1.

Let creftypecap 3.1 hold. Let be smooth and suppose satisfies the PL condition with parameter . Then if we update some using the CO update eq. 5 with step-size and , we get the following convergence:

 ∣∣∣∣ξ(x(k))∣∣∣∣≤(1−c4)k∣∣∣∣ξ(x(0))∣∣∣∣. (6)

We also show that CO converges in practice on some simple examples in Appendix H.

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## Appendix A General nonconvex min-max optimization

In standard nonconvex optimization, a common goal is to find second-order local minima, which are approximate critical points where is approximately positive definite. Likewise, a common goal in nonconvex min-max optimization is to find approximate critical points where an analogous second-order condition holds, namely that is approximately positive definite and is approximately negative definite. Critical points where this second-order condition holds are called local min-maxes. When creftypecap 2.6 holds, all critical points are global min-maxes, but in more general settings, we may encounter critical points that do not satisfy these conditions. Critical points may be local min-mins or max-mins or indefinite points. A number of recent papers have proposed dynamics for nonconvex min-max optimization, showing local stability or local asymptotic convergence results [MNG17, DP18, BRM18, LFB19, MJS19]. The key guarantee that these papers generally give is that their algorithms will be stable at local min-maxes and unstable at some set of undesirable critical points (such as local max-mins). This essentially amounts to a guarantee that in the convex-concave setting, their algorithms will converge asymptotically and in the strictly concave-strictly convex setting (i.e. where there is only an undesirable max-min), their algorithms will diverge asymptotically. This type of local stability is essentially the best one can ask for in the general nonconvex setting, and we show how to give similar guarantees for our algorithm in Section A.1.

### a.1 Nonconvex extensions for HGD

While the naive version of HGD will try to converge to all critical points, we can modify HGD slightly to achieve second-order stability guarantees as in various related work such as [BRM18, LFB19]. In particular, we consider modifying HGD so that there is some scalar in front of the term as follows:

 x(k+1)=x(k)−ηα∇H(x(k)) (7)

We now present two ways to choose . Our first method is inspired by the Simplectic Gradient Adjustment algorithm of [BRM18], which is as follows:

 x(k+1)=x(k)−η(ξ(x(k))−λA⊤ξ(xk)) (8)

where is the antisymmetric part of and . [BRM18] show that is positive when in a strictly convex-strictly concave region and negative in a strictly concave-strictly convex region. Thus, if we choose , we can ensure that the modified HGD will exhibit local stability around strict min-maxes and local instability around strict max-mins. This follows simply because we will do gradient descent on in the first case and gradient ascent on in the second case.

Another way to choose involves using an approximate eigenvalue computation on and to detect whether is positive semidefinite and is negative semidefinite (which would mean we are in a convex-concave region). We set if we are in a convex-concave region and

otherwise, which will guarantee local stability around min-maxes and local instability around other critical points. This approximate eigenvector computation can be done using a logarithmic number of Hessian-vector products.

## Appendix B Proof of linear convergence rate under PL condition

Here we present a classic proof of Theorem 4.2.

###### Proof of Theorem 4.2.
 f(x(k+1))−f(x∗) ≤f(x(k))−f(x∗)−12L∣∣∣∣∇f(x(k))∣∣∣∣2 ≤f(x(k))−f(x∗)−μL(f(x(k))−f(x∗)) =(1−μL)(f(x(k))−f(x∗))

where the first line comes from smoothness and the update rule for gradient descent, the second inequality comes from the PL condition. Applying the last line recursively gives the result. ∎

## Appendix C Comparison of Theorem 3.4 to [Dh19]

In this section, we compare our results in Theorem 3.4 to those of [DH19]. [DH19] prove a rate for SGDA when is -smooth and convex in and -smooth and -strongly concave in and is some fixed matrix . The specific setting they consider is to find the unconstrained min-max for a function defined as where is convex and smooth, is strongly-convex and smooth, and has rank (i.e. has full column rank).

Their rate uses the potential function , where we have:

 λ =2LΓ(L+Γ2μ)μγ2 (9) ak =∣∣∣∣x(k)1−x∗1∣∣∣∣ (10) bk =∣∣∣∣x(k)2−x∗2∣∣∣∣ (11)

where is the min-max for the objective. Their rate (Theorem 3.1 in [DH19]) is

 Pk+1≤⎛⎜⎝1−cμ2γ4L3Γ2(L+Γ2μ)⎞⎟⎠kPk (12)

for some constant . To translate this rate into bounds on , we can use the smoothness of in both of its arguments to note that and likewise for . So the rate on translates into a rate on with some additional factor in front.

Their rate and our rate are incomparable – neither is strictly better. For instance when is much larger than all other quantities, their rates simplify to , while ours go to . While our convergence rate requires the sufficiently bilinear condition eq. 3 to hold, we do not require convexity in or concavity in . Moreover, we allow to change as long as the bounds on the singular values hold whereas [DH19] require to be a fixed matrix.

## Appendix D Nonconvex-nonconcave setting where creftypecap 2.6 and the conditions for Theorem 3.4 hold

In this section we give a concrete example of a nonconvex-nonconcave setting where creftypecap 2.6 and the conditions for Theorem 3.4 hold. We choose this example for simplicity, but one can easily come up with other more complicated examples.

For our example, we define the following function:

 F(x)=⎧⎪ ⎪⎨⎪ ⎪⎩−3(x+π2)for x≤−π2−3cosxfor −π2π2 (13)

The first and second derivatives of are as follows:

 F′(x)=⎧⎪ ⎪⎨⎪ ⎪⎩−3for x≤−π23sinxfor −π2π2 (14)
 F′′(x)=⎧⎪ ⎪⎨⎪ ⎪⎩0for x≤−π23cosxfor −π2π2 (15)

From Figure 2, we can see that this function is neither convex nor concave.

Our objective will be . Note that because for all . Also, since .

First, we show that satisfies creftypecap 3.1. We see that has a critical point at . Moreover, is -Lipschitz for any finite-sized region of . Thus, if we assume our algorithm stays within a ball of some radius , the -Lipschitz assumption will be satisfied. Since our algorithm does not diverge and indeed converges at a linear rate to the min-max, this assumption is fairly mild.

Next, we show that satisfies condition eq. 3. Condition eq. 3 requires for . We see that this holds because and .

Therefore, the assumptions of Theorem 3.4 are satisfied.

We can also show that this objective satisfies creftypecap 2.6, so we get convergence to the min-max of . We will show that has only one critical point (at ) and that this critical point is a min-max. We first give a “proof by picture” below, showing a plot of in Figure 3, along with plots of and showing that is indeed a min-max.

We can also formally show that is the unique critical point of and that it is a min-max. We prove this for completeness, although the calculations more or less amount to a simple case analysis. Let us look at the derivatives of with respect to and :

 ∂g∂x1(x1,x2)=⎧⎪ ⎪⎨⎪ ⎪⎩−3+4x2for x1≤−π23sinx1+4x2for −π2π2 (16)
 ∂g∂x2(x1,x2)=⎧⎪ ⎪⎨⎪ ⎪⎩3+4x1for x2≤−π2−3sinx2+4x1for −π2π2 (17)

Observe that if then critical points of must satisfy , which implies that . Likewise, if , then critical points of must have . We show that this implies that only has critical points where and are both in the range .

Suppose had a critical point such that . Then this critical point must satisfy . But from our observation above, if a critical point has , then must lie in , which contradicts .

Next, suppose had a critical point such that . Then this critical point must satisfy , which implies that . But then by the observation above, must lie in , which contradicts .

From this we see that any critical point of must have . We can make analogous arguments to show that any critical point of must have .

From this, we can conclude that all critical points of must satisfy the following:

 3sinx1 +4x2=0 (18) −3sinx2 +4x1=0 (19)

These equations imply the following:

 x1 =34sinx2 (20) x2 =−34sinx1 (21) ⇒x1 =34sin(−34sinx2) (22) ⇒x2 =−34