Linear Convergence Rate Analysis 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) achieves widespread success in applications as a continuous black-box optimization algorithm. However, theoretical guarantee of its convergence rate has been done only inside convex quadratic functions and their monotonic transformations. In this study, we derive an upper bound and a lower bound of the rates of linear convergence of the (1+1)-ES on locally L-strongly convex and U-smooth functions, as exp [EQUATION] and [EQUATION], respectively. The order of the upper bound derived is competitive to that derived for a derivative-free optimization (DFO) algorithm in the previous study. Unlike DFO studies, any prior knowledge on the mathematical properties of the objective function such as Lipschitz constant is not given to the optimization algorithm.