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

arXiv: https://arxiv.org/abs/2406.07292 · HTML: https://mlsanity.com/paper/2406.07292/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 20 references that are in the map.
- First posted on arXiv 11 Jun 2024.

## Builds on (most cited first, 9 of 9)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Coordinate Descent Algorithms](https://mlsanity.com/paper/1502.04759/) (2015)
- [On the Complexity Analysis of Randomized Block-Coordinate Descent Methods](https://mlsanity.com/paper/1305.4723/) (2013)
- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [On the Convergence of Coordinate Ascent Variational Inference](https://mlsanity.com/paper/2306.01122/) (2023)
- [Convergence of coordinate ascent variational inference for log-concave measures via optimal transport](https://mlsanity.com/paper/2404.08792/) (2024)
- [Independent projections of diffusions: Gradient flows for variational inference and optimal mean field approximations](https://mlsanity.com/paper/2309.13332/) (2023)
- [Partially factorized variational inference for high-dimensional mixed models](https://mlsanity.com/paper/2312.13148/) (2023)
- [Wasserstein Proximal Coordinate Gradient Algorithms](https://mlsanity.com/paper/2405.04628/) (2024)

## Built on by (most cited first, 9 of 9)

- [Entropy contraction of the Gibbs sampler under log-concavity](https://mlsanity.com/paper/2410.00858/) (2024)
- [Variational Inference with Mixtures of Isotropic Gaussians](https://mlsanity.com/paper/2506.13613/) (2025)
- [Exponential Convergence of CAVI for Bayesian PCA](https://mlsanity.com/paper/2505.16145/) (2025)
- [Nearly Dimension-Independent Convergence of Mean-Field Black-Box Variational Inference](https://mlsanity.com/paper/2505.21721/) (2025)
- [Rotated Mean-Field Variational Inference and Iterative Gaussianization](https://mlsanity.com/paper/2510.07732/) (2025)
- [Radon--Wasserstein Gradient Flows for Interacting-Particle Sampling in High Dimensions](https://mlsanity.com/paper/2602.05227/) (2026)
- [Wasserstein Contraction of Coordinate Ascent Variational Inference](https://mlsanity.com/paper/2605.30253/) (2026)
- [Mode Collapse of Mean-Field Variational Inference](https://mlsanity.com/paper/2510.17063/) (2025)
- [Stability of Finite-Batch Particle Mean-Field Variational Inference Beyond Strong Convexity](https://mlsanity.com/paper/2608.11486/) (2026)

## Abstract (arXiv)

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, 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.