# What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization?

ICML, pp. 4880-4889, 2019.

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Abstract:

Minimax optimization has found extensive applications in modern machine learning, in settings such as generative adversarial networks (GANs), adversarial training and multi-agent reinforcement learning. As most of these applications involve continuous nonconvex-nonconcave formulations, a very basic question arises---``what is a proper def...More

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Introduction

- Minimax optimization refers to problems of two agents—one agent tries to minimize the payoff function f : X × Y → R while the other agent tries to maximize it.
- In the last few years, minimax optimization has found significant applications in machine learning, in settings such as generative adversarial networks (GAN) [Goodfellow et al, 2014], adversarial training [Madry et al, 2017] and multi-agent reinforcement learning [Omidshafiei et al, 2017]
- These minimax problems are often solved using gradient-based algorithms, especially gradient descent ascent (GDA), an algorithm that alternates between a gradient descent step for x and some number of gradient ascent steps for y.
- Most previous work [e.g., Daskalakis and Panageas, 2018, Mazumdar and Ratliff, 2018, Adolphs et al, 2018] studied a notion of local Nash equilibrium which replaces all the global minima or maxima in the definition of Nash equilibrium by their local counterparts

Highlights

- Minimax optimization refers to problems of two agents—one agent tries to minimize the payoff function f : X × Y → R while the other agent tries to maximize it
- Most of the minimax problems arising in modern machine learning applications do not have this simple convex-concave structure
- The main contribution of this paper is to propose the first proper mathematical definition of local optimality for this sequential setting—local minimax, a local surrogate for the global minimax points
- In Section 3.1, we develop a formal notion of local surrogacy for global minimax points which we refer to as local minimax points
- We consider general nonconvex-nonconcave minimax optimization problems. Since most these problems arising in modern machine learning correspond to sequential games, we propose a new notion of local optimality—local minimax—the first proper mathematical definition of local optimality for the two-player sequential setting
- We establish a strong connection to gradient descent ascent—up to some degenerate points, local minimax points are exactly equal to the stable limit points of gradient descent ascent

Results

- The authors pointed out that while many modern applications are sequential games, the problem of finding their optima—global minimax points—is NP-hard in general.
- In Section 3.1, the authors develop a formal notion of local surrogacy for global minimax points which the authors refer to as local minimax points.
- In Section 3.3, the authors establish a close relationship between stable fixed points of GDA and local minimax points.
- To the best of the knowledge, this is the first proper mathematical definition of local optimality for the two-player sequential setting

Conclusion

- The authors consider general nonconvex-nonconcave minimax optimization problems.
- Since most these problems arising in modern machine learning correspond to sequential games, the authors propose a new notion of local optimality—local minimax—the first proper mathematical definition of local optimality for the two-player sequential setting.
- The authors establish a strong connection to GDA—up to some degenerate points, local minimax points are exactly equal to the stable limit points of GDA

Summary

## Introduction:

Minimax optimization refers to problems of two agents—one agent tries to minimize the payoff function f : X × Y → R while the other agent tries to maximize it.- In the last few years, minimax optimization has found significant applications in machine learning, in settings such as generative adversarial networks (GAN) [Goodfellow et al, 2014], adversarial training [Madry et al, 2017] and multi-agent reinforcement learning [Omidshafiei et al, 2017]
- These minimax problems are often solved using gradient-based algorithms, especially gradient descent ascent (GDA), an algorithm that alternates between a gradient descent step for x and some number of gradient ascent steps for y.
- Most previous work [e.g., Daskalakis and Panageas, 2018, Mazumdar and Ratliff, 2018, Adolphs et al, 2018] studied a notion of local Nash equilibrium which replaces all the global minima or maxima in the definition of Nash equilibrium by their local counterparts
## Results:

The authors pointed out that while many modern applications are sequential games, the problem of finding their optima—global minimax points—is NP-hard in general.- In Section 3.1, the authors develop a formal notion of local surrogacy for global minimax points which the authors refer to as local minimax points.
- In Section 3.3, the authors establish a close relationship between stable fixed points of GDA and local minimax points.
- To the best of the knowledge, this is the first proper mathematical definition of local optimality for the two-player sequential setting
## Conclusion:

The authors consider general nonconvex-nonconcave minimax optimization problems.- Since most these problems arising in modern machine learning correspond to sequential games, the authors propose a new notion of local optimality—local minimax—the first proper mathematical definition of local optimality for the two-player sequential setting.
- The authors establish a strong connection to GDA—up to some degenerate points, local minimax points are exactly equal to the stable limit points of GDA

Related work

- Minimax optimization: Since the seminal paper of von Neumann [1928], notions of equilibria in games and their algorithmic computation have received wide attention. In terms of algorithmic computation, the vast majority of results focus on the convex-concave setting [Korpelevich, 1976, Nemirovski and Yudin, 1978, Nemirovski, 2004]. In the context of optimization, these problems have generally been studied in the setting of constrained convex optimization [Bertsekas, 2014]. Results beyond convex-concave setting are much more recent. Rafique et al [2018], Nouiehed et al [2019] consider nonconvex but concave minimax problems where for any x, f (x, ·) is a concave function. In this case, they propose algorithms combining approximate maximization over y and a proximal gradient method for x to show convergence to stationary points. Lin et al [2018] consider a special case of the nonconvex-nonconcave minimax problem, where the function f (·, ·) satisfies a variational inequality. In this setting, they consider a proximal algorithm that requires the solving of certain strong variational inequality problems in each step and show its convergence to stationary points. Hsieh et al [2018] propose proximal methods that asymptotically converge to a mixed Nash equilibrium; i.e., a distribution rather than a point.

Reference

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- Proposition 29 ([Glicksberg, 1952]). Assume that the function f: X × Y → R is continuous and that X ⊂ Rd1, Y ⊂ Rd2 are compact. Then min max E(μ,ν)f (x, y) = max min E(μ,ν)f (x, y).
- Lemma 34 ([Rockafellar, 2015]). Assume the function φ is -weakly convex. Let λ < 1/, and denote x = argminx φ(x ) + (1/2λ) x − x 2. Then ∇φλ(x) ≤ implies: x − x = λ, and min g ≤, g∈∂φ(x)
- The proof of Theorem 35 is similar to the convergence analysis for nonsmooth weakly-convex functions [Davis and Drusvyatskiy, 2018], except here the max-oracle has error. Theorem 35 claims, other than an additive error 4 as a√result of the oracle solving the maximum approximately, that the remaining term decreases at a rate of 1/ T.
- 2. Clearly, the gradient is equal to (0.2y, 0.2x + sin(y)). And, for any fixed x, there are only two maxima y (x) satisfying 0.2x + sin(y ) = 0 where y1(x) ∈ (−3π/2, −π/2) and y2(x) ∈ (π/2, 3π/2). On the other hand, f (x, y1(x)) is monotonically decreasing with respect to x, while f (x, y2(x)) is monotonically increasing, with f (0, y1(0)) = f (0, y2(0)) by symmetry. It is not hard to check y1(0) = −π and y2(0) = π. Therefore, (0, −π) and (0, π) are two global solutions of the minimax problem. However, the gradients at both points are not 0, thus they are not stationary points. By Proposition 18 they are also not local minimax points.

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