Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincar\'e Inequality

TL;DR

This paper extends the analysis of Langevin Monte Carlo (LMC) convergence beyond Poincaré inequalities to include weak Poincaré inequalities, covering heavy-tailed distributions and explicitly quantifying the impact of initial conditions.

Contribution

It establishes upper and lower bounds for Langevin diffusions and LMC under weak Poincaré inequalities, broadening understanding to heavy-tailed densities and initial error dependencies.

Findings

Convergence bounds under weak Poincaré inequalities for heavy-tailed distributions.

Explicit quantification of initial error impact across different tail behaviors.

Demonstration of an unavoidable phase transition in initial error dependency.

Abstract

Langevin diffusions are rapidly convergent under appropriate functional inequality assumptions. Hence, it is natural to expect that with additional smoothness conditions to handle the discretization errors, their discretizations like the Langevin Monte Carlo (LMC) converge in a similar fashion. This research program was initiated by Vempala and Wibisono (2019), who established results under log-Sobolev inequalities. Chewi et al. (2022) extended the results to handle the case of Poincar\'e inequalities. In this paper, we go beyond Poincar\'e inequalities, and push this research program to its limit. We do so by establishing upper and lower bounds for Langevin diffusions and LMC under weak Poincar\'e inequalities that are satisfied by a large class of densities including polynomially-decaying heavy-tailed densities (i.e., Cauchy-type). Our results explicitly quantify the effect of the initializer on the performance of the LMC algorithm. In particular, we show that as the tail goes from sub-Gaussian, to sub-exponential, and finally to Cauchy-like, the dependency on the initial error goes from being logarithmic, to polynomial, and then finally to being exponential. This three-step phase transition is in particular unavoidable as demonstrated by our lower bounds, clearly defining the boundaries of LMC.