The No-Underrun Sampler: A Locally-Adaptive, Gradient-Free MCMC Method

TL;DR

The No-Underrun Sampler (NURS) is a gradient-free, locally-adaptive MCMC method that effectively samples from complex distributions without gradient information, combining theoretical rigor with practical performance.

Contribution

NURS introduces a novel gradient-free, locally-adaptive MCMC algorithm with proven theoretical properties and demonstrated effectiveness on challenging multi-scale distributions.

Findings

NURS is reversible and has Wasserstein contraction similar to Hit-and-Run.

It effectively samples from Neal's funnel distribution.

Theoretical bounds relate NURS to Hit-and-Run.

Abstract

In this work, we introduce the No-Underrun Sampler (NURS), a locally-adaptive, gradient-free Markov chain Monte Carlo method that blends ideas from Hit-and-Run and the No-U-Turn Sampler. NURS dynamically adapts to the local scale of the target distribution without requiring gradient evaluations, making it especially suitable for applications where gradients are unavailable or costly. We establish key theoretical properties, including reversibility, formal connections to Hit-and-Run and Random Walk Metropolis, Wasserstein contraction comparable to Hit-and-Run in Gaussian targets, and bounds on the total variation distance between the transition kernels of Hit-and-Run and NURS. Empirical experiments, supported by theoretical insights, illustrate the ability of NURS to sample from Neal's funnel, a challenging multi-scale distribution from Bayesian hierarchical inference.