Reflected Anchored Langevin Algorithms

2026-10-07

AI Summary

This study addresses the limitation of traditional Langevin algorithms, which rely on differentiable log-densities and struggle with sampling under non-smooth constraints. To overcome this, we propose Reflected Anchored Langevin Dynamics, a method that introduces a smooth reference potential alongside a state-dependent scaling mechanism. By integrating Euler–Maruyama discretization, projection techniques, and reflected diffusion processes, the proposed approach effectively circumvents sampling limitations within non-differentiable constrained domains. Theoretically, we establish a convergence framework based on the Wasserstein distance, deriving explicit convergence bounds and iteration complexity guarantees. Empirical evaluations confirm the consistency between theoretical predictions and practical performance. This work presents an efficient new sampling paradigm for non-smooth target distributions.

Abstract

First order Langevin algorithms for constrained sampling in machine learning, such as projected Langevin Monte Carlo which are based on discretizations of reflected Langevin dynamics, require differentiable log densities that limits their applicability. This paper introduces reflected anchored Langevin dynamics (RALD), a reflected diffusion that converges to non-differentiable targets on constrained domains. The method uses a smooth anchored reference potential and multiplies the drift and noise covariance of its reflected Langevin dynamics by the same state dependent scaling factor. Its Euler-Maruyama discretization with projection gives reflected anchored Langevin Monte Carlo (RALMC) algorithm. We prove explicit convergence bounds and iteration complexity for RALMC in the 2-Wasserstein distance to the target distribution. Numerical experiments are provided to illustrate the theoretical predictions and the empirical performance of the method.

Problem

• constrained sampling
• Langevin algorithms
• non-differentiable targets
• reflected Langevin dynamics

Innovation

• Reflected Anchored Langevin Dynamics
• Constrained Sampling
• Non-differentiable Targets
• Langevin Monte Carlo
• Wasserstein Convergence