Black-Box Min--Max Continuous Optimization Using CMA-ES with Worst-case Ranking Approximation

• URL: http://arxiv.org/abs/2204.02646v1
• Date: Wed, 6 Apr 2022 08:03:39 GMT
• Title: Black-Box Min--Max Continuous Optimization Using CMA-ES with Worst-case Ranking Approximation
• Authors: Atsuhiro Miyagi, Kazuto Fukuchi, Jun Sakuma, Youhei Akimoto
• Abstract summary: A popular approach updates $x$ and $y$ simultaneously or alternatingly. Existing approaches fail if $f$ is not Lipschitz smooth and strongly convex-concave around the optimal solution. We propose minimizing the worst-case objective function $F(x)=max_yf(x,y)$ directly using the covariance matrix adaptation evolution strategy.
• Score: 22.576922942465142
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: In this study, we investigate the problem of min-max continuous optimization in a black-box setting $\min_{x} \max_{y}f(x,y)$. A popular approach updates $x$ and $y$ simultaneously or alternatingly. However, two major limitations have been reported in existing approaches. (I) As the influence of the interaction term between $x$ and $y$ (e.g., $x^\mathrm{T} B y$) on the Lipschitz smooth and strongly convex-concave function $f$ increases, the approaches converge to an optimal solution at a slower rate. (II) The approaches fail to converge if $f$ is not Lipschitz smooth and strongly convex-concave around the optimal solution. To address these difficulties, we propose minimizing the worst-case objective function $F(x)=\max_{y}f(x,y)$ directly using the covariance matrix adaptation evolution strategy, in which the rankings of solution candidates are approximated by our proposed worst-case ranking approximation (WRA) mechanism. Compared with existing approaches, numerical experiments show two important findings regarding our proposed method. (1) The proposed approach is efficient in terms of $f$-calls on a Lipschitz smooth and strongly convex-concave function with a large interaction term. (2) The proposed approach can converge on functions that are not Lipschitz smooth and strongly convex-concave around the optimal solution, whereas existing approaches fail.

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