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Journal Articles SIAM Journal on Optimization Year : 2022

Global Linear Convergence of Evolution Strategies on More Than Smooth Strongly Convex Functions

Abstract

Evolution strategies (ESs) are zero-order stochastic black-box optimization heuristics invariant to monotonic transformations of the objective function. They evolve a multivariate normal distribution, from which candidate solutions are generated. Among different variants, CMA-ES is nowadays recognized as one of the state-of-the-art zero-order optimizers for difficult problems. Albeit ample empirical evidence that ESs with a step-size control mechanism converge linearly, theoretical guarantees of linear convergence of ESs have been established only on limited classes of functions. In particular, theoretical results on convex functions are missing, where zero-order and also first order optimization methods are often analyzed. In this paper, we establish almost sure linear convergence and a bound on the expected hitting time of an ES, namely the (1 + 1)-ES with (generalized) one-fifth success rule and an abstract covariance matrix adaptation with bounded condition number, on a broad class of functions. The analysis holds for monotonic transformations of positively homogeneous functions and of quadratically bounded functions, the latter of which particularly includes monotonic transformation of strongly convex functions with Lipschitz continuous gradient. As far as the authors know, this is the first work that proves linear convergence of ES on such a broad class of functions.
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Dates and versions

hal-02941429 , version 1 (17-09-2020)
hal-02941429 , version 2 (15-10-2020)
hal-02941429 , version 3 (20-01-2022)

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Youhei Akimoto, Anne Auger, Tobias Glasmachers, Daiki Morinaga. Global Linear Convergence of Evolution Strategies on More Than Smooth Strongly Convex Functions. SIAM Journal on Optimization, 2022. ⟨hal-02941429v3⟩
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