Convergence of Consensus-Based Particle Methods for Nonconvex Bi-Level Optimization

TL;DR

This paper introduces a derivative-free consensus-based optimization method for nonconvex bi-level problems, providing convergence guarantees and demonstrating effectiveness through numerical experiments.

Contribution

It proposes a novel consensus-based approach combining smooth quantile selection and Gibbs approximation, with proven convergence for both mean-field and finite-particle dynamics.

Findings

Convergence to the bi-level solution with explicit exponential rate.

Error bounds and stability conditions are established.

Numerical experiments validate theoretical results.

Abstract

In this paper, we study a consensus-based optimization method for nonconvex bi-level optimization, where the objective is to minimize an upper-level function over the set of global minimizers of a lower-level problem. The proposed approach is derivative-free, and constructs its consensus point via smooth quantile selection combined with a Gibbs-type Laplace approximation. We establish convergence guarantees for both the associated \textit{mean-field} dynamics and its \textit{finite-particle} approximation. In particular, under suitable assumptions on smooth quantile localization, error bounds, and stability, we show that the mean-field law reaches any arbitrary prescribed Wasserstein neighborhood of the target bi-level solution with an explicit exponential rate up to the hitting time. Numerical experiments on a two-dimensional constrained problem and neural network training further support the theoretical results.