Global linear convergence of evolution strategies with recombination on scaling-invariant functions

Abstract

Evolution Strategies (ESs) are stochastic derivative-free optimization algorithms whose most prominent representative, the CMA-ES algorithm, is widely used to solve difficult numerical optimization problems. We provide the first rigorous investigation of the linear convergence of step-size adaptive ESs involving a population and recombination, two ingredients crucially important in practice to be robust to local irregularities or multimodality. We investigate the convergence of step-size adaptive ESs with weighted recombination on composites of strictly increasing functions with continuously differentiable scaling-invariant functions with a global optimum. This function class includes functions with non-convex sublevel sets and discontinuous functions. We prove the existence of a constant r such that the logarithm of the distance to the optimum divided by the number of iterations converges to r. The constant is given as an expectation with respect to the stationary distribution of a Markov chain—its sign allows to infer linear convergence or divergence of the ES and is found numerically. Our main condition for convergence is the increase of the expected log step-size on linear functions. In contrast to previous results, our condition is equivalent to the almost sure geometric divergence of the step-size on linear functions.