Stochastic Gradient Variational Inference with Price's Gradient Estimator from Bures-Wasserstein to Parameter Space

In variational inference (VI), the practitioner approximates a high-dimensional distribution $π$ with a simple surrogate one, often a (product) Gaussian distribution. However, in many cases of practical interest, Gaussian distributions might not capture the correct radial profile of $π$, resulting in poor coverage. In this work, we approach the VI problem from the perspective of optimizing over these radial profiles. Our algorithm radVI is a cheap, effective add-on to many existing VI schemes, such as Gaussian (mean-field) VI and Laplace approximation. We provide theoretical convergence guarantees for our algorithm, owing to recent developments in optimization over the Wasserstein space--the space of probability distributions endowed with the Wasserstein distance--and new regularity properties of radial transport maps in the style of Caffarelli (2000).

Black-box variational inference (BBVI) is a methodology for posterior approximation that relies on stochastic optimization. In practice, the stochastic optimizers underpinning BBVI generally require extensive problem-specific tuning, which undermines its promise as a truly "black box" inference algorithm. However, over the past decade, many new adaptive stochastic optimization algorithms have been developed that reduce or remove entirely the need for tuning. In this work, we investigate this new collection of adaptive methods in the context of BBVI, with the goal of establishing the current state of the art in tuning-free optimization-based inference. In particular, we present a large-scale empirical evaluation of 56 stochastic gradient-based optimization algorithms applied to 1092 Bayesian inference optimization problems, involving over 550,000 individual optimization runs and 15 core-years of compute. The optimization algorithms we evaluate are chosen to represent a wide spectrum of recent approaches and the benchmark problems are chosen to span a range of difficulty, with posterior target dimension 1-10^4, condition number 1-10^8, and a range of variational families. Our results show that no single method dominates, but running a selection of 5 algorithms suffices to reliably get close to the best-possible observed performance. We thus provide a strong baseline for applications where expert tuning is not possible and for comparison when developing new stochastic optimization algorithms.

Black-Box Variational Inference (BBVI) typically relies on Stochastic Gradient Descent (SGD) to optimize the Evidence Lower Bound (ELBO). However, the stochastic gradients in BBVI inherently exhibit unbounded variance, violating standard assumptions and instead satisfying the weaker Blum-Gladyshev (BG) condition, where variance grows quadratically with distance from the optimum. In this paper, we bridge the gap between stochastic optimization theory and the practical instances of BBVI. Focusing on the broad elliptic location-scale family of parameterized distributions, we offer two main contributions. First, we prove the existence of an ELBO solution, a foundational property usually assumed a priori in the literature. Second, we establish comprehensive convergence guarantees spanning finite-time and asymptotic regimes for Minibatch Projected SGD (PSGD) equipped with dynamic batching and preconditioning under the BG condition. Our theoretical framework demonstrates that dynamic batching combined with preconditioning systematically enables rigorous guarantees even in complex settings. We illustrate our theoretical findings with numerical results, highlighting the efficacy of our approach for modern inference tasks.