# Batch and match: black-box variational inference with a score-based divergence

arXiv: https://arxiv.org/abs/2402.14758 · HTML: https://mlsanity.com/paper/2402.14758/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 54 references that are in the map.
- First posted on arXiv 22 Feb 2024.

## Builds on (most cited first, 30 of 30)

- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [A Kernelized Stein Discrepancy for Goodness-of-fit Tests and Model Evaluation](https://mlsanity.com/paper/1602.03253/) (2016)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [Automatic Differentiation Variational Inference](https://mlsanity.com/paper/1603.00788/) (2016)
- [Stochastic model-based minimization of weakly convex functions](https://mlsanity.com/paper/1803.06523/) (2018)
- [Measuring Sample Quality with Stein's Method](https://mlsanity.com/paper/1506.03039/) (2015)
- [Variational inference via Wasserstein gradient flows](https://mlsanity.com/paper/2205.15902/) (2022)
- [Stochastic (Approximate) Proximal Point Methods: Convergence, Optimality, and Adaptivity](https://mlsanity.com/paper/1810.05633/) (2018)
- [Minimum Stein Discrepancy Estimators](https://mlsanity.com/paper/1906.08283/) (2019)
- [Forward-backward Gaussian variational inference via JKO in the Bures-Wasserstein Space](https://mlsanity.com/paper/2304.05398/) (2023)
- [Challenges and Opportunities in High-dimensional Variational Inference](https://mlsanity.com/paper/2103.01085/) (2021)
- [Black box variational inference for state space models](https://mlsanity.com/paper/1511.07367/) (2015)
- [Provable convergence guarantees for black-box variational inference](https://mlsanity.com/paper/2306.03638/) (2023)
- [Black-box Variational Inference for Stochastic Differential Equations](https://mlsanity.com/paper/1802.03335/) (2018)
- [Provable Bayesian Inference via Particle Mirror Descent](https://mlsanity.com/paper/1506.03101/) (2015)
- [On the Convergence of Black-Box Variational Inference](https://mlsanity.com/paper/2305.15349/) (2023)
- [Provable Gradient Variance Guarantees for Black-Box Variational Inference](https://mlsanity.com/paper/1906.08241/) (2019)
- [Robust, Accurate Stochastic Optimization for Variational Inference](https://mlsanity.com/paper/2009.00666/) (2020)
- [Faster Stochastic Variational Inference using Proximal-Gradient Methods with General Divergence Functions](https://mlsanity.com/paper/1511.00146/) (2015)
- [Semi-Implicit Variational Inference via Score Matching](https://mlsanity.com/paper/2308.10014/) (2023)
- [Black Box Variational Inference with a Deterministic Objective: Faster, More Accurate, and Even More Black Box](https://mlsanity.com/paper/2304.05527/) (2023)
- [Variational Inference with Gaussian Score Matching](https://mlsanity.com/paper/2307.07849/) (2023)
- [Variational Hamiltonian Monte Carlo via Score Matching](https://mlsanity.com/paper/1602.02219/) (2016)
- [Boosting Black Box Variational Inference](https://mlsanity.com/paper/1806.02185/) (2018)
- [A trust-region method for stochastic variational inference with applications to streaming data](https://mlsanity.com/paper/1505.07649/) (2015)
- [A Framework for Improving the Reliability of Black-box Variational Inference](https://mlsanity.com/paper/2203.15945/) (2022)
- [Sample Average Approximation for Black-Box VI](https://mlsanity.com/paper/2304.06803/) (2023)

## Built on by (most cited first, 22 of 22)

- [PaperBench: Evaluating AI's Ability to Replicate AI Research](https://mlsanity.com/paper/2504.01848/) (2025)
- [Variational Inference for Uncertainty Quantification: an Analysis of Trade-offs](https://mlsanity.com/paper/2403.13748/) (2024)
- [EigenVI: score-based variational inference with orthogonal function expansions](https://mlsanity.com/paper/2410.24054/) (2024)
- [Variational Inference in Location-Scale Families: Exact Recovery of the Mean and Correlation Matrix](https://mlsanity.com/paper/2410.11067/) (2024)
- [Batch, match, and patch: low-rank approximations for score-based variational inference](https://mlsanity.com/paper/2410.22292/) (2024)
- [Variational Inference with Mixtures of Isotropic Gaussians](https://mlsanity.com/paper/2506.13613/) (2025)
- [Disentangling impact of capacity, objective, batchsize, estimators, and step-size on flow VI](https://mlsanity.com/paper/2412.08824/) (2024)
- [HiRAS: A Hierarchical Multi-Agent Framework for Paper-to-Code Generation and Execution](https://mlsanity.com/paper/2604.17745/) (2026)
- [Towards Understanding Gradient Dynamics of the Sliced-Wasserstein Distance via Critical Point Analysis](https://mlsanity.com/paper/2502.06525/) (2025)
- [Optimization Guarantees for Square-Root Natural-Gradient Variational Inference](https://mlsanity.com/paper/2507.07853/) (2025)
- [Generalized Guarantees for Variational Inference in the Presence of Even and Elliptical Symmetry](https://mlsanity.com/paper/2511.01064/) (2025)
- [Fisher meets Feynman: score-based variational inference with a product of experts](https://mlsanity.com/paper/2510.21598/) (2025)
- [A Kernel Approach for Semi-implicit Variational Inference](https://mlsanity.com/paper/2601.12023/) (2026)
- [Large-scale Score-based Variational Posterior Inference for Bayesian Deep Neural Networks](https://mlsanity.com/paper/2602.05873/) (2026)
- [Scalable Mean-Field Variational Inference via Preconditioned Primal-Dual Optimization](https://mlsanity.com/paper/2602.07632/) (2026)
- [How to Approximate Inference with Subtractive Mixture Models](https://mlsanity.com/paper/2604.16714/) (2026)
- [ATLAS: Adapting Trajectory Lengths and Step-Size for Hamiltonian Monte Carlo](https://mlsanity.com/paper/2410.21587/) (2024)
- [SGD for Variational Inference: Tackling Unbounded Variance via Preconditioning and Dynamic Batching](https://mlsanity.com/paper/2605.07531/) (2026)
- [On the Approximation Complexity of Matrix Product Operator (MPO) Born Machines](https://mlsanity.com/paper/2605.11471/) (2026)
- [Large-scale empirical tuning and comparison of default optimizers for variational inference](https://mlsanity.com/paper/2606.07841/) (2026)
- [QuanVI: Score-based Variational Inference via Quantum Maximally Mixed States](https://mlsanity.com/paper/2609.39164/) (2026)
- [SoftServe: A Scalable Quasi-Newton Method for Deep Learning](https://mlsanity.com/paper/2610.02182/) (2026)

## Abstract (arXiv)

Most leading implementations of black-box variational inference (BBVI) are based on optimizing a stochastic evidence lower bound (ELBO). But such approaches to BBVI often converge slowly due to the high variance of their gradient estimates and their sensitivity to hyperparameters. In this work, we propose batch and match (BaM), an alternative approach to BBVI based on a score-based divergence. Notably, this score-based divergence can be optimized by a closed-form proximal update for Gaussian variational families with full covariance matrices. We analyze the convergence of BaM when the target distribution is Gaussian, and we prove that in the limit of infinite batch size the variational parameter updates converge exponentially quickly to the target mean and covariance. We also evaluate the performance of BaM on Gaussian and non-Gaussian target distributions that arise from posterior inference in hierarchical and deep generative models. In these experiments, we find that BaM typically converges in fewer (and sometimes significantly fewer) gradient evaluations than leading implementations of BBVI based on ELBO maximization.