Black-Box Min–Max Continuous Optimization Using CMA-ES with Worst-case Ranking Approximation
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In this study, we investigate the problem of min-max continuous optimization in a black-box setting min_xmax_yf(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^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_yf(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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