Constrained Langevin Algorithms with L-mixing External Random Variables

TL;DR

This paper improves the theoretical bounds on the convergence of constrained Langevin algorithms with L-mixing data variables, achieving faster deviation rates in Wasserstein distance compared to previous results.

Contribution

It provides a tighter deviation bound of O(T^{-1/2} log T) for non-convex losses with L-mixing data, extending analysis to polyhedral, possibly unbounded constraints.

Findings

Achieves O(T^{-1/2} log T) deviation bound in Wasserstein distance.

Extends analysis to non-convex losses with L-mixing data variables.

Handles polyhedral constraints, not necessarily bounded.

Abstract

Langevin algorithms are gradient descent methods augmented with additive noise, and are widely used in Markov Chain Monte Carlo (MCMC) sampling, optimization, and machine learning. In recent years, the non-asymptotic analysis of Langevin algorithms for non-convex learning has been extensively explored. For constrained problems with non-convex losses over a compact convex domain with IID data variables, the projected Langevin algorithm achieves a deviation of $O(T^{-1/4} (\log T)^{1/2})$ from its target distribution [27] in $1$-Wasserstein distance. In this paper, we obtain a deviation of $O(T^{-1/2} \log T)$ in $1$-Wasserstein distance for non-convex losses with $L$-mixing data variables and polyhedral constraints (which are not necessarily bounded). This improves on the previous bound for constrained problems and matches the best-known bound for unconstrained problems.