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

TL;DR

This paper proves the first theoretical guarantees of linear convergence for evolution strategies on a broad class of convex and strongly convex functions, extending understanding of their efficiency.

Contribution

It establishes almost sure linear convergence and bounds on hitting time for the $(1+1)_-ES, a family of evolution strategies, on broad classes of convex functions.

Findings

Proves almost sure linear convergence of ES on broad function classes.

Provides bounds on expected hitting time for ES algorithms.

Extends theoretical understanding of ES performance on convex functions.

Abstract

Evolution strategies (ESs) are zeroth-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 zeroth-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 zeroth-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 \new{ES family, namely the $(1+1)_\kappa$-ES, which includes 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.