Convergence rate of random scan Coordinate Ascent Variational Inference under log-concavity

TL;DR

This paper establishes tight convergence rates for the random scan coordinate ascent variational inference algorithm under log-concavity, showing it scales linearly with problem parameters, improving upon previous quadratic bounds for the deterministic version.

Contribution

It provides the first tight convergence rate analysis for the random scan version of coordinate ascent variational inference under log-concavity, extending recent deterministic results.

Findings

Convergence rate scales linearly with condition number and number of blocks.

Random scan version converges faster than deterministic scan in terms of problem parameters.

Results are based on optimal transport geometry for probability distributions.

Abstract

The Coordinate Ascent Variational Inference scheme is a popular algorithm used to compute the mean-field approximation of a probability distribution of interest. We analyze its random scan version, under log-concavity assumptions on the target density. Our approach builds on the recent work of M. Arnese and D. Lacker, \emph{Convergence of coordinate ascent variational inference for log-concave measures via optimal transport} [arXiv:2404.08792] which studies the deterministic scan version of the algorithm, phrasing it as a block-coordinate descent algorithm in the space of probability distributions endowed with the geometry of optimal transport. We obtain tight rates for the random scan version, which imply that the total number of factor updates required to converge scales linearly with the condition number and the number of blocks of the target distribution. By contrast, available bounds for the deterministic scan case scale quadratically in the same quantities, which is analogue to what happens for optimization of convex functions in Euclidean spaces.