# Convergence of coordinate ascent variational inference for log-concave measures via optimal transport

arXiv: https://arxiv.org/abs/2404.08792 · HTML: https://mlsanity.com/paper/2404.08792/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 30 references that are in the map.
- First posted on arXiv 12 Apr 2024.

## Builds on (most cited first, 8 of 8)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Analysis of Langevin Monte Carlo via convex optimization](https://mlsanity.com/paper/1802.09188/) (2018)
- [Sampling as optimization in the space of measures: The Langevin dynamics as a composite optimization problem](https://mlsanity.com/paper/1802.08089/) (2018)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Algorithms for mean-field variational inference via polyhedral optimization in the Wasserstein space](https://mlsanity.com/paper/2312.02849/) (2023)
- [Theoretical and Computational Guarantees of Mean Field Variational Inference for Community Detection](https://mlsanity.com/paper/1710.11268/) (2017)
- [On the Convergence of Coordinate Ascent Variational Inference](https://mlsanity.com/paper/2306.01122/) (2023)
- [Iteration Complexity Analysis of Block Coordinate Descent Methods](https://mlsanity.com/paper/1310.6957/) (2013)

## Built on by (most cited first, 14 of 14)

- [Entropy contraction of the Gibbs sampler under log-concavity](https://mlsanity.com/paper/2410.00858/) (2024)
- [Convergence rate of random scan Coordinate Ascent Variational Inference under log-concavity](https://mlsanity.com/paper/2406.07292/) (2024)
- [Variational Inference with Mixtures of Isotropic Gaussians](https://mlsanity.com/paper/2506.13613/) (2025)
- [Stability of Mean-Field Variational Inference](https://mlsanity.com/paper/2506.07856/) (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)
- [Stability of Sequential and Parallel Coordinate Ascent Variational Inference](https://mlsanity.com/paper/2603.20929/) (2026)
- [Random Coordinate Descent on the Wasserstein Space of Probability Measures](https://mlsanity.com/paper/2604.01606/) (2026)
- [Wasserstein Contraction of Coordinate Ascent Variational Inference](https://mlsanity.com/paper/2605.30253/) (2026)
- [Provable Accuracy Bounds for Hybrid Dynamical Optimization and Sampling](https://mlsanity.com/paper/2410.06397/) (2024)
- [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)

Mean field variational inference (VI) is the problem of finding the closest product (factorized) measure, in the sense of relative entropy, to a given high-dimensional probability measure ρ. The well known Coordinate Ascent Variational Inference (CAVI) algorithm aims to approximate this product measure by iteratively optimizing over one coordinate (factor) at a time, which can be done explicitly. Despite its popularity, the convergence of CAVI remains poorly understood. In this paper, we prove the convergence of CAVI for log-concave densities ρ. If additionally log ρ has Lipschitz gradient, we find a linear rate of convergence, and if also ρ is strongly log-concave, we find an exponential rate. Our analysis starts from the observation that mean field VI, while notoriously non-convex in the usual sense, is in fact displacement convex in the sense of optimal transport when ρ is log-concave. This allows us to adapt techniques from the optimization literature on coordinate descent algorithms in Euclidean space.