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

arXiv: https://arxiv.org/abs/2605.19667 · HTML: https://mlsanity.com/paper/2605.19667/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 52 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 8 of 8)

- [Optimal Hessian/Jacobian-Free Nonconvex-PL Bilevel Optimization](https://mlsanity.com/paper/2407.17823/) (2024)
- [Graph Reinforcement Learning for Network Control via Bi-Level Optimization](https://mlsanity.com/paper/2305.09129/) (2023)
- [CB2O: Consensus-Based Bi-Level Optimization](https://mlsanity.com/paper/2411.13394/) (2024)
- [Superquantile-Gibbs Relaxation for Minima-selection in Bilevel Optimization](https://mlsanity.com/paper/2505.05991/) (2025)
- [Constrained Consensus-Based Optimization and Numerical Heuristics for the Few Particle Regime](https://mlsanity.com/paper/2410.10361/) (2024)
- [Consensus-based optimization for closed-box adversarial attacks and a connection to evolution strategies](https://mlsanity.com/paper/2506.24048/) (2025)
- [Consensus-based optimization (CBO): Towards Global Optimality in Robotics](https://mlsanity.com/paper/2602.06868/) (2026)
- [Well-posedness and mean-field limit estimate of a consensus-based algorithm for min-max problems](https://mlsanity.com/paper/2602.12886/) (2026)

## Abstract (arXiv)

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 textitmean-field dynamics and its textitfinite-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.