Improved Langevin Monte Carlo for stochastic optimization via landscape modification

TL;DR

This paper introduces a landscape modification technique for Langevin Monte Carlo algorithms that reduces energy barriers, leading to faster convergence and improved sampling efficiency in low-temperature regimes.

Contribution

The authors propose a novel landscape transformation for LMC that decreases energy barriers, resulting in polynomial dependence on temperature and barrier height, improving convergence guarantees.

Findings

Reduced energy barriers in transformed landscapes.

Polynomial dependence on temperature and energy barrier.

Enhanced convergence rates for Langevin Monte Carlo.

Abstract

Given a target function $H$ to minimize or a target Gibbs distribution $\pi_{\beta}^0 \propto e^{-\beta H}$ to sample from in the low temperature, in this paper we propose and analyze Langevin Monte Carlo (LMC) algorithms that run on an alternative landscape as specified by $H^f_{\beta,c,1}$ and target a modified Gibbs distribution $\pi^f_{\beta,c,1} \propto e^{-\beta H^f_{\beta,c,1}}$, where the landscape of $H^f_{\beta,c,1}$ is a transformed version of that of $H$ which depends on the parameters $f,\beta$ and $c$. While the original Log-Sobolev constant affiliated with $\pi^0_{\beta}$ exhibits exponential dependence on both $\beta$ and the energy barrier $M$ in the low temperature regime, with appropriate tuning of these parameters and subject to assumptions on $H$, we prove that the energy barrier of the transformed landscape is reduced which consequently leads to polynomial dependence on both $\beta$ and $M$ in the modified Log-Sobolev constant associated with $\pi^f_{\beta,c,1}$. This yield improved total variation mixing time bounds and improved convergence toward a global minimum of $H$. We stress that the technique developed in this paper is not only limited to LMC and is broadly applicable to other gradient-based optimization or sampling algorithms.