On perturbed proximal gradient algorithms

TL;DR

This paper analyzes a perturbed proximal gradient algorithm that uses Monte Carlo methods for gradient approximation, providing convergence guarantees and non-asymptotic bounds for both biased and unbiased cases.

Contribution

It introduces convergence conditions and bounds for a Monte Carlo-based proximal gradient method, covering both biased and unbiased gradient approximations.

Findings

Convergence is guaranteed under specific step size and batch size conditions.

Non-asymptotic bounds are derived for the averaged algorithm.

Applications include sparse generalized linear models and graphical model learning.

Abstract

We study a version of the proximal gradient algorithm for which the gradient is intractable and is approximated by Monte Carlo methods (and in particular Markov Chain Monte Carlo). We derive conditions on the step size and the Monte Carlo batch size under which convergence is guaranteed: both increasing batch size and constant batch size are considered. We also derive non-asymptotic bounds for an averaged version. Our results cover both the cases of biased and unbiased Monte Carlo approximation. To support our findings, we discuss the inference of a sparse generalized linear model with random effect and the problem of learning the edge structure and parameters of sparse undirected graphical models.