Greedy Adversarial Equilibrium: An Efficient Alternative to Nonconvex-Nonconcave Min-Max Optimization

• URL: http://arxiv.org/abs/2006.12363v4
• Date: Tue, 4 May 2021 16:41:54 GMT
• Title: Greedy Adversarial Equilibrium: An Efficient Alternative to Nonconvex-Nonconcave Min-Max Optimization
• Authors: Oren Mangoubi and Nisheeth K. Vishnoi
• Abstract summary: We show that Lipschitzitz's $varepsilon$-greedy adversarial model converges from any starting point to a $max_z f(x, z)$. We also show that Lipschitz's $nabla_y f(x,y)$ is in the dimension $d$, $1/varepsilon$, and the bounds on $nabla2_y f(x,y)$ are $nabla2_y.
• Score: 28.431572772564518
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Min-max optimization of an objective function $f: \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ is an important model for robustness in an adversarial setting, with applications to many areas including optimization, economics, and deep learning. In many applications $f$ may be nonconvex-nonconcave, and finding a global min-max point may be computationally intractable. There is a long line of work that seeks computationally tractable algorithms for alternatives to the min-max optimization model. However, many of the alternative models have solution points which are only guaranteed to exist under strong assumptions on $f$, such as convexity, monotonicity, or special properties of the starting point. We propose an optimization model, the $\varepsilon$-greedy adversarial equilibrium, and show that it can serve as a computationally tractable alternative to the min-max optimization model. Roughly, we say that a point $(x^\star, y^\star)$ is an $\varepsilon$-greedy adversarial equilibrium if $y^\star$ is an $\varepsilon$-approximate local maximum for $f(x^\star,\cdot)$, and $x^\star$ is an $\varepsilon$-approximate local minimum for a "greedy approximation" to the function $\max_z f(x, z)$ which can be efficiently estimated using second-order optimization algorithms. We prove the existence of such a point for any smooth function which is bounded and has Lipschitz Hessian. To prove existence, we introduce an algorithm that converges from any starting point to an $\varepsilon$-greedy adversarial equilibrium in a number of evaluations of the function $f$, the max-player's gradient $\nabla_y f(x,y)$, and its Hessian $\nabla^2_y f(x,y)$, that is polynomial in the dimension $d$, $1/\varepsilon$, and the bounds on $f$ and its Lipschitz constant.

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