Analysis of Primal-Dual Langevin Algorithms

TL;DR

This paper analyzes primal-dual Langevin algorithms for sampling from complex distributions, providing theoretical convergence guarantees, bias bounds, and demonstrating effectiveness on Bayesian imaging problems.

Contribution

It introduces a continuous-time limit and bias analysis for primal-dual Langevin algorithms, extending Langevin Monte Carlo methods to non-smooth convex functions.

Findings

The continuous-time limit is a stochastic differential equation with a unique stationary state.

Discretization introduces bias, but bounds are established for convergence to the target distribution.

Numerical experiments confirm theoretical results on small and large-scale Bayesian inverse problems.

Abstract

We analyze a recently proposed class of algorithms for the problem of sampling from probability distributions $\mu^\ast$ in $\mathbb{R}^d$ with a Lebesgue density of the form $\mu^\ast(x) \propto \exp(-f(Kx)-g(x))$, where $K$ is a linear operator and $f,g$ convex and non-smooth. The method is a generalization of the primal-dual hybrid gradient optimization algorithm to a sampling scheme. We give the iteration's continuous time limit, a stochastic differential equation in the joint primal-dual variable, and its mean field limit Fokker-Planck equation. Under mild conditions, the scheme converges to a unique stationary state in continuous and discrete time. Contrary to purely primal overdamped Langevin diffusion, the stationary state in continuous time does not have $\mu^\ast$ as its primal marginal. Thus, further analysis is carried out to bound the bias induced by the partial dualization, and potentially correct for it in the diffusion. Time discretizations of the diffusion lead to implementable algorithms, but, as is typical in Langevin Monte Carlo methods, introduce further bias. We prove bounds for these discretization errors, which allow to give convergence results relating the produced samples to the target. We demonstrate our findings numerically first on small-scale examples in which we can exactly verify the theoretical results, and subsequently on typical examples of larger scale from Bayesian imaging inverse problems.