Stochastic Langevin Monte Carlo for (weakly) log-concave posterior distributions

TL;DR

This paper analyzes a stochastic Langevin Monte Carlo method for sampling from weakly log-concave posteriors, focusing on computational cost and convergence in high-dimensional, less convex settings using the Kurdyka-iewicz inequality.

Contribution

It provides a convergence analysis of a continuous-time stochastic Langevin algorithm under weak convexity, extending previous results to broader curvature conditions with explicit complexity bounds.

Findings

Final simulation horizon scales with dimension and sample size as specified.

Introduces a Poissonian subsampling scheme for efficiency.

Handles weakly convex posteriors using KL inequality.

Abstract

In this paper, we investigate a continuous time version of the Stochastic Langevin Monte Carlo method, introduced in [WT11], that incorporates a stochastic sampling step inside the traditional over-damped Langevin diffusion. This method is popular in machine learning for sampling posterior distribution. We will pay specific attention in our work to the computational cost in terms of $n$ (the number of observations that produces the posterior distribution), and $d$ (the dimension of the ambient space where the parameter of interest is living). We derive our analysis in the weakly convex framework, which is parameterized with the help of the Kurdyka-\L ojasiewicz (KL) inequality, that permits to handle a vanishing curvature settings, which is far less restrictive when compared to the simple strongly convex case. We establish that the final horizon of simulation to obtain an $\varepsilon$ approximation (in terms of entropy) is of the order $( d \log(n)^2 )^{(1+r)^2} [\log^2(\varepsilon^{-1}) + n^2 d^{2(1+r)} \log^{4(1+r)}(n) ]$ with a Poissonian subsampling of parameter $\left(n ( d \log^2(n))^{1+r}\right)^{-1}$, where the parameter $r$ is involved in the KL inequality and varies between $0$ (strongly convex case) and $1$ (limiting Laplace situation).