Gradient-free Riemannian Langevin Sampler

2026-07-08

AI Summary

This work addresses the slow mixing and tendency to become trapped in local modes that plague conventional Markov chain Monte Carlo (MCMC) methods when sampling from multimodal distributions. To overcome these limitations, the authors propose a gradient-free Riemannian Langevin sampler (GRiLS), which, for the first time, incorporates Riemannian geometry into a gradient-free MCMC framework. By leveraging a Riemannian metric to reshape the local geometric structure of the target distribution, GRiLS facilitates more efficient transitions across modes. Furthermore, it employs an interacting particle mechanism to adaptively estimate the mean and covariance of the target distribution without requiring gradient information. Experimental results demonstrate that GRiLS significantly outperforms both gradient-based and gradient-free MCMC methods on multimodal benchmark problems, achieving superior exploration and mixing performance while remaining entirely gradient-free.

Abstract

We address the problem of efficiently sampling multimodal probability distributions, where standard Markov Chain Monte Carlo methods often suffer from poor mixing and mode trapping. To mitigate these issues, we propose Gradient-free Riemannian Langevin Sampler (GRiLS), a novel proposal that improves exploration without requiring gradient evaluations of the target density. Our approach introduces a Riemannian metric which reshapes the local geometry in order to facilitate transitions across modes. The resulting gradient-free MCMC algorithm is particularly suitable for complex, computationally expensive targets where derivatives are unavailable or impractical. The GRiLS proposal requires knowing the mean and covariance of the target density, which we estimate using an ensemble of interacting particles. Empirical results on multimodal benchmarks demonstrate that GRiLS achieves improved mixing compared to existing gradient-based and gradient-free MCMC approaches.

Problem

• multimodal sampling
• Markov Chain Monte Carlo
• mode trapping
• gradient-free sampling
• probability distribution

Innovation

• Gradient-free
• Riemannian Langevin
• Multimodal sampling
• MCMC
• Geometry-aware