Halfspace sampling in evolution strategies

Chun-Kit Au, Ho-fung Leung

Abstract: This paper presents a novel halfspace sampling method in single parent elitist evolution strategies (ESs) for unimodal functions. In halfspace sampling, the supporting hyperplane going through a parent separates the search space into a positive halfspace and a negative halfspace. If an offspring lies in the negative halfspace, it will be reflected with respect to the parent so that it lies in the positive halfspace. We derive the convergence rates of a scale-invariant step size (1+1)-ES with halfspace sampling on spherical functions in finite and infinite dimensions. We prove that the lower bounds of convergence rates are improved by a factor of 2 when strategies sample their offspring in the optimal positive halfspace. We also implement halfspace sampling into the (1+1) CMA-ES by introducing the concept of evolution halfspaces. Evolution halfspaces accumulate the significant information of the previous successful and unsuccessful steps in order to estimate the optimal positive halfspace. The (1+1)-CMA-ES with halfspace sampling is benchmarked on the BBOB noise-free testbed and experimentally compared with the standard (1+1)-CMA-ES.