# Black Box Variational Inference with a Deterministic Objective: Faster, More Accurate, and Even More Black Box

arXiv: https://arxiv.org/abs/2304.05527 · HTML: https://mlsanity.com/paper/2304.05527/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 47 references that are in the map.
- First posted on arXiv 11 Apr 2023.

## Builds on (most cited first, 18 of 18)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Black Box Variational Inference](https://mlsanity.com/paper/1401.0118/) (2013)
- [Monte Carlo Gradient Estimation in Machine Learning](https://mlsanity.com/paper/1906.10652/) (2019)
- [Automatic Differentiation Variational Inference](https://mlsanity.com/paper/1603.00788/) (2016)
- [Importance Weighting and Variational Inference](https://mlsanity.com/paper/1808.09034/) (2018)
- [A Swiss Army Infinitesimal Jackknife](https://mlsanity.com/paper/1806.00550/) (2018)
- [Covariances, Robustness, and Variational Bayes](https://mlsanity.com/paper/1709.02536/) (2017)
- [An Automatic Finite-Sample Robustness Metric: When Can Dropping a Little Data Make a Big Difference?](https://mlsanity.com/paper/2011.14999/) (2020)
- [Validated Variational Inference via Practical Posterior Error Bounds](https://mlsanity.com/paper/1910.04102/) (2019)
- [Pathfinder: Parallel quasi-Newton variational inference](https://mlsanity.com/paper/2108.03782/) (2021)
- [Robust, Accurate Stochastic Optimization for Variational Inference](https://mlsanity.com/paper/2009.00666/) (2020)
- [Linear Response Methods for Accurate Covariance Estimates from Mean Field Variational Bayes](https://mlsanity.com/paper/1506.04088/) (2015)
- [Divide and Couple: Using Monte Carlo Variational Objectives for Posterior Approximation](https://mlsanity.com/paper/1906.10115/) (2019)
- [The Shrinkage-Delinkage Trade-off: An Analysis of Factorized Gaussian Approximations for Variational Inference](https://mlsanity.com/paper/2302.09163/) (2023)
- [A Framework for Improving the Reliability of Black-box Variational Inference](https://mlsanity.com/paper/2203.15945/) (2022)
- [Evaluating Sensitivity to the Stick-Breaking Prior in Bayesian Nonparametrics](https://mlsanity.com/paper/2107.03584/) (2021)
- [Sparse Bayesian Lasso via a Variable-Coefficient ℓ1 Penalty](https://mlsanity.com/paper/2211.05089/) (2022)

## Built on by (most cited first, 17 of 17)

- [Amortized Variational Inference: When and Why?](https://mlsanity.com/paper/2307.11018/) (2023)
- [On the Convergence of Black-Box Variational Inference](https://mlsanity.com/paper/2305.15349/) (2023)
- [Batch and match: black-box variational inference with a score-based divergence](https://mlsanity.com/paper/2402.14758/) (2024)
- [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)
- [Sample Average Approximation for Black-Box VI](https://mlsanity.com/paper/2304.06803/) (2023)
- [Streamlining Prediction in Bayesian Deep Learning](https://mlsanity.com/paper/2411.18425/) (2024)
- [FlowVAT: Normalizing Flow Variational Inference with Affine-Invariant Tempering](https://mlsanity.com/paper/2505.10466/) (2025)
- [VISA: Variational Inference with Sequential Sample-Average Approximations](https://mlsanity.com/paper/2403.09429/) (2024)
- [Nearly Dimension-Independent Convergence of Mean-Field Black-Box Variational Inference](https://mlsanity.com/paper/2505.21721/) (2025)
- [Large-scale Score-based Variational Posterior Inference for Bayesian Deep Neural Networks](https://mlsanity.com/paper/2602.05873/) (2026)
- [VaSST: Variational Inference for Symbolic Regression using Soft Symbolic Trees](https://mlsanity.com/paper/2602.23561/) (2026)
- [Fitting Large Nonlinear Mixed Effects Models Using Variational Expectation Maximization](https://mlsanity.com/paper/2604.26160/) (2026)
- [AutoGD: Automatic Learning Rate Selection for Gradient Descent](https://mlsanity.com/paper/2510.09923/) (2025)
- [SGD for Variational Inference: Tackling Unbounded Variance via Preconditioning and Dynamic Batching](https://mlsanity.com/paper/2605.07531/) (2026)
- [Corrected Integrated Laplace Approximation for Bayesian Inference in Latent Gaussian Models](https://mlsanity.com/paper/2605.20345/) (2026)
- [Large-scale empirical tuning and comparison of default optimizers for variational inference](https://mlsanity.com/paper/2606.07841/) (2026)

## Abstract (arXiv)

Automatic differentiation variational inference (ADVI) offers fast and easy-to-use posterior approximation in multiple modern probabilistic programming languages. However, its stochastic optimizer lacks clear convergence criteria and requires tuning parameters. Moreover, ADVI inherits the poor posterior uncertainty estimates of mean-field variational Bayes (MFVB). We introduce "deterministic ADVI" (DADVI) to address these issues. DADVI replaces the intractable MFVB objective with a fixed Monte Carlo approximation, a technique known in the stochastic optimization literature as the "sample average approximation" (SAA). By optimizing an approximate but deterministic objective, DADVI can use off-the-shelf second-order optimization, and, unlike standard mean-field ADVI, is amenable to more accurate posterior covariances via linear response (LR). In contrast to existing worst-case theory, we show that, on certain classes of common statistical problems, DADVI and the SAA can perform well with relatively few samples even in very high dimensions, though we also show that such favorable results cannot extend to variational approximations that are too expressive relative to mean-field ADVI. We show on a variety of real-world problems that DADVI reliably finds good solutions with default settings (unlike ADVI) and, together with LR covariances, is typically faster and more accurate than standard ADVI.