# Wasserstein Contraction of Coordinate Ascent Variational Inference

arXiv: https://arxiv.org/abs/2605.30253 · HTML: https://mlsanity.com/paper/2605.30253/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 75 references that are in the map.
- First posted on arXiv 28 May 2026.

## Builds on (most cited first, 29 of 29)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices](https://mlsanity.com/paper/1903.08568/) (2019)
- [Theoretical guarantees for approximate sampling from smooth and log-concave densities](https://mlsanity.com/paper/1412.7392/) (2014)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [Statistical guarantees for the EM algorithm: From population to sample-based analysis](https://mlsanity.com/paper/1408.2156/) (2014)
- [Ten Steps of EM Suffice for Mixtures of Two Gaussians](https://mlsanity.com/paper/1609.00368/) (2016)
- [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)
- [Penalising model component complexity: A principled, practical approach to constructing priors](https://mlsanity.com/paper/1403.4630/) (2014)
- [Error bounds for particle gradient descent, and extensions of the log-Sobolev and Talagrand inequalities](https://mlsanity.com/paper/2403.02004/) (2024)
- [Entropy contraction of the Gibbs sampler under log-concavity](https://mlsanity.com/paper/2410.00858/) (2024)
- [Practical bounds on the error of Bayesian posterior approximations: A nonasymptotic approach](https://mlsanity.com/paper/1809.09505/) (2018)
- [Independent projections of diffusions: Gradient flows for variational inference and optimal mean field approximations](https://mlsanity.com/paper/2309.13332/) (2023)
- [Conditionally conjugate mean-field variational Bayes for logistic models](https://mlsanity.com/paper/1711.06999/) (2017)
- [Partially factorized variational inference for high-dimensional mixed models](https://mlsanity.com/paper/2312.13148/) (2023)
- [Convergence rate of random scan Coordinate Ascent Variational Inference under log-concavity](https://mlsanity.com/paper/2406.07292/) (2024)
- [On Mixing of Markov Chains: Coupling, Spectral Independence, and Entropy Factorization](https://mlsanity.com/paper/2103.07459/) (2021)
- [The Mean-Field Approximation: Information Inequalities, Algorithms, and Complexity](https://mlsanity.com/paper/1802.06126/) (2018)
- [Wasserstein Proximal Coordinate Gradient Algorithms](https://mlsanity.com/paper/2405.04628/) (2024)
- [The EM Algorithm is Adaptively-Optimal for Unbalanced Symmetric Gaussian Mixtures](https://mlsanity.com/paper/2103.15653/) (2021)
- [On the rate of convergence of alternating minimization for non-smooth non-strongly convex optimization in Banach spaces](https://mlsanity.com/paper/1911.00404/) (2019)
- [Fast Mixing of Data Augmentation Algorithms: Bayesian Probit, Logit, and Lasso Regression](https://mlsanity.com/paper/2412.07999/) (2024)
- [Mixing times of data-augmentation Gibbs samplers for high-dimensional probit regression](https://mlsanity.com/paper/2505.14343/) (2025)
- [Fast convergence of the Expectation Maximization algorithm under a logarithmic Sobolev inequality](https://mlsanity.com/paper/2407.17949/) (2024)
- [Exponential Convergence of CAVI for Bayesian PCA](https://mlsanity.com/paper/2505.16145/) (2025)
- [Theory and computation for structured variational inference](https://mlsanity.com/paper/2511.09897/) (2025)
- [Stability of Sequential and Parallel Coordinate Ascent Variational Inference](https://mlsanity.com/paper/2603.20929/) (2026)

## Built on by (most cited first, 1 of 1)

- [Stability of Finite-Batch Particle Mean-Field Variational Inference Beyond Strong Convexity](https://mlsanity.com/paper/2608.11486/) (2026)

## Abstract (arXiv)

We study the non-asymptotic contraction in Wasserstein distance of the sequential, parallel, and random-scan coordinate ascent variational inference algorithms. This is shown to hold under a functional smoothness condition of the optimality maps and a transportation-information inequality at their fixed points. Our results are sharp and general, and as opposed to those based on global strong log-concavity assumptions, they allow for local convergence on smooth, non-smooth, and discrete manifolds, including within the context of data augmentation. We consider many applications in statistical physics and Bayesian statistics. These include pairwise Markov Random field models such as Ising and Curie-Weiss, unbalanced Bayesian Gaussian Mixture Models, high-dimensional Bayesian Probit Regression, and high-dimensional Logistic Regression with P'olya--Gamma random variables (i.e. Jaakkola-Jordan's algorithm). In many of these models, these represent the first available convergence results of their kind.