Accelerating Non-Smooth and Heavy-Tailed Sampling

2026-10-07

AI Summary

This study addresses the problem of low sampling efficiency in non-smooth, heavy-tailed distributions over constrained domains by proposing an irreversible anchored Langevin dynamics method. By introducing a cyclic drift to break reversibility constraints and combining a divergence-free skew-symmetric matrix field with flow potential theory, the approach preserves the target distribution while accelerating convergence without requiring derivatives of the target density. Both theoretical analysis and empirical results demonstrate that the proposed method significantly reduces asymptotic variance and achieves finite-time convergence rates superior to those of reversible algorithms, thereby effectively enhancing sampling efficiency for complex distributions.

Abstract

Anchored Langevin dynamics (ALD) is useful for non-smooth sampling where the density of the target distribution is possibly non-differentiable and heavy-tailed; reflected anchored Langevin dynamics (RALD) can sample possibly non-differentiable target density on a constrained domain. In this paper, we propose and study non-reversible anchored Langevin dynamics (NALD) for sampling possibly non-differentiable and heavy-tailed target density in the Euclidean space and the non-reversible reflected anchored Langevin dynamics (NRALD) for sampling possibly non-differentiable target density in the constrained space. Our construction adds a circulation drift generated by a possibly state-dependent divergence-free skew-symmetric matrix field and a stream potential. It preserves the target distribution without requiring derivatives of target density, admits a random-time-change representation, and applies both on the whole Euclidean space and on bounded domains with normal reflection. By breaking reversibility, we show that NALD and NRALD can converge to their target distributions faster than their reversible counterparts via finite-time non-asymptotic convergence analysis, a large deviations analysis and asymptotic variance reduction. Numerical experiments demonstrate the efficiency of the proposed algorithms.

Problem

• Non-smooth sampling
• Heavy-tailed distribution
• Langevin dynamics
• Constrained domain
• Acceleration

Innovation

• Non-reversible Langevin dynamics
• Non-smooth sampling
• Heavy-tailed distribution
• Circulation drift
• Convergence acceleration