Convergence rate of the (1+1)-evolution strategy on locally strongly convex functions with lipschitz continuous gradient

TL;DR

This paper provides theoretical bounds on the convergence rate of the (1+1)-ES algorithm when optimizing locally strongly convex functions with Lipschitz continuous gradients, without prior knowledge of function properties.

Contribution

It derives the first upper and lower bounds for the linear convergence rate of (1+1)-ES on a broad class of convex functions, expanding understanding beyond quadratic cases.

Findings

Upper bound of convergence rate: exp(-Omega_d(L/(d*U)))

Lower bound of convergence rate: exp(-1/d)

No prior knowledge of function properties needed for the analysis

Abstract

Evolution strategy (ES) is one of the promising classes of algorithms for black-box continuous optimization. Despite its broad successes in applications, theoretical analysis on the speed of its convergence is limited on convex quadratic functions and their monotonic transformation. In this study, an upper bound and a lower bound of the rate of linear convergence of the (1+1)-ES on locally $L$-strongly convex functions with $U$-Lipschitz continuous gradient are derived as $\exp\left(-\Omega_{d\to\infty}\left(\frac{L}{d\cdot U}\right)\right)$ and $\exp\left(-\frac1d\right)$, respectively. Notably, any prior knowledge on the mathematical properties of the objective function, such as Lipschitz constant, is not given to the algorithm, whereas the existing analyses of derivative-free optimization algorithms require it.