MirrorCBO: A consensus-based optimization method in the spirit of mirror descent

TL;DR

MirrorCBO introduces a novel consensus-based optimization algorithm that combines the advantages of mirror descent and derivative-free methods, effectively handling non-convex and constrained problems with promising empirical results.

Contribution

The paper proposes MirrorCBO, a new optimization method that generalizes CBO using mirror descent principles, with theoretical convergence guarantees and diverse application demonstrations.

Findings

Demonstrates competitive performance in sparsity-inducing optimization.

Effectively handles convex constraints with convergence guarantees.

Shows potential for optimization on non-convex submanifolds.

Abstract

In this work we propose MirrorCBO, a consensus-based optimization (CBO) method which generalizes standard CBO in the same way that mirror descent generalizes gradient descent. For this we apply the CBO methodology to a swarm of dual particles and retain the primal particle positions by applying the inverse of the mirror map, which we parametrize as the subdifferential of a strongly convex function $\phi$. In this way, we combine the advantages of a derivative-free non-convex optimization algorithm with those of mirror descent. As a special case, the method extends CBO to optimization problems with convex constraints. Assuming bounds on the Bregman distance associated to $\phi$, we provide asymptotic convergence results for MirrorCBO with explicit exponential rate. Another key contribution is an exploratory numerical study of this new algorithm across different application settings, focusing on (i) sparsity-inducing optimization, and (ii) constrained optimization, demonstrating the competitive performance of MirrorCBO. We observe empirically that the method can also be used for optimization on (non-convex) submanifolds of Euclidean space, can be adapted to mirrored versions of other recent CBO variants, and that it inherits from mirror descent the capability to select desirable minimizers, like sparse ones. We also include an overview of recent CBO approaches for constrained optimization and compare their performance to MirrorCBO.