Convergence Rate of the (1+1)-ES on Locally Strongly Convex and Lipschitz Smooth Functions

Daiki Morinaga, Kazuto Fukuchi, Jun Sakuma, Youhei Akimoto

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 $\exp (-\Omega _{d\to \infty }({L}/({d\cdot U})))$ and $\exp (-1/d)$ , respectively. Notably, any prior knowledge on the mathematical properties of the objective function, such as the Lipschitz constant, is not given to the algorithm, whereas the existing analyses of derivative-free optimization algorithms require it.