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

Many MCMC methods either rely on gradients – such as the No-U-Turn Sampler (NUTS) – or struggle with multi-scale distributions, where different regions require vastly different exploration strategies.  NURS is a new  locally adaptive MCMC method that overcomes these challenges by combining ideas from Hit-and-Run and NUTS while remaining gradient-free, simple and inherently parallelizable. It selects random update directions uniformly from the unit sphere and introduces an adaptive orbit-based exploration that adjusts to the local scale of the target distribution.

Empirical results on Neal’s funnel, a challenging multi-scale benchmark, show that while NURS moves diffusively in narrow regions, it transitions to ballistic movement in broader ones, enabling efficient sampling across the distribution’s different scales. I will also discuss NURS’s formal connections to Hit-and-Run, quantitative tuning guidelines, and new coupling results. By offering a locally adaptive, theoretically grounded approach to sampling multi-scale distributions, NURS expands the scope of gradient-free MCMC and opens new directions for locally adaptive MCMC.