On the Ultimate Convergence Rates for Isotropic Algorithms and the Best Choices Among Various Forms of Isotropy

Abstract

In this paper, we show universal lower bounds for isotropic algorithms, that hold for any algorithm such that each new point is the sum of one already visited point plus one random isotropic direction multiplied by any step size (whenever the step size is chosen by an oracle with arbitrarily high computational power). The bound is 1–O(1/d) for the constant in the linear convergence (i.e. the constant C such that the distance to the optimum after n steps is upper bounded by Cⁿ), as already seen for some families of evolution strategies in [19,12], in contrast with 1–O(1) for the reverse case of a random step size and a direction chosen by an oracle with arbitrary high computational power. We then recall that isotropy does not uniquely determine the distribution of a sample on the sphere and show that the convergence rate in isotropic algorithms is improved by using stratified or antithetic isotropy instead of naive isotropy. We show at the end of the paper that beyond the mathematical proof, the result holds on experiments. We conclude that one should use antithetic-isotropy or stratified-isotropy, and never standard-isotropy.