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- W4287753776 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." @default.
- W4287753776 created "2022-07-26" @default.
- W4287753776 creator A5006627464 @default.
- W4287753776 creator A5063089732 @default.
- W4287753776 date "2020-06-22" @default.
- W4287753776 modified "2023-09-24" @default.
- W4287753776 title "Greedy Adversarial Equilibrium: An Efficient Alternative to Nonconvex-Nonconcave Min-Max Optimization" @default.
- W4287753776 doi "https://doi.org/10.48550/arxiv.2006.12363" @default.
- W4287753776 hasPublicationYear "2020" @default.
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