Consensus-based optimization for linearly separable functions

Abstract

Consensus-based optimization (CBO) is an efficient metaheuristic for global optimisation with attractive mathematical properties, allowing global convergence results even in non-convex settings. In practice it suffers greatly from the curse of dimensionality, as do most particle-based optimisers. Different strategies have been proposed to apply CBO even for high-dimensional optimisation problems, the most prominent being the so-called anisotropic noise model. However, a recent work by Bonandin et al. highlighted that this method performs well primarily on separable objective functions, and even small coordinate rotations severely degrade the performance. Motivated by this observation, in this work we study the case where the objective function is separable only in a linearly transformed coordinate system. To leverage the efficiency of anisotropic CBO we propose a reparametrisation scheme, which provably estimates such a coordinate transformation and then applies CBO in the new variables. Furthermore, we show how our scheme can be interpreted as noise adaptation in CBO. Numerical examples highlight the efficacy of the method, demonstrating significant performance improvements on challenging benchmark objectives.