MirrorCBO: A consensus-based optimization method in the spirit of mirror descent
This work addresses optimization challenges in machine learning and AI by extending derivative-free methods to constrained and non-convex settings, though it is incremental as it builds on existing CBO and mirror descent frameworks.
The authors tackled the problem of non-convex optimization by proposing MirrorCBO, a consensus-based optimization method that generalizes standard CBO using mirror descent principles, enabling handling of convex constraints and sparsity. They provided asymptotic convergence results with explicit exponential rates and demonstrated competitive performance in applications like sparsity-inducing and constrained optimization.
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 $φ$. 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 $φ$, 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.