Stability of Finite-Batch Particle Mean-Field Variational Inference Beyond Strong Convexity

2026-08-11

AI Summary

This work addresses the stability of finite-batch particle-based mean-field variational inference when the target potential is not strongly convex. Viewing the algorithm as a fully discrete stochastic approximation of projected Wasserstein dynamics, the authors characterize convergence behavior by quantifying curvature deficiencies. They establish a non-asymptotic Wasserstein stability bound that disentangles contributions from initialization error, empirical approximation, and finite-batch drift error, and formulate a dimension-free condition on drift sensitivity. Key technical tools include Wasserstein metric analysis, static comparison couplings, and control of projected drift bias. Theoretically, they prove that particle iterates remain stable within an \(O(\sqrt{\beta/\alpha})\) neighborhood of the target. High-dimensional experiments on smooth non-convex benchmarks further reveal that explicit schemes require modification to ensure stability when the drift exhibits polynomial growth.

Abstract

We study the implementable finite-batch particle algorithm for mean-field variational inference as a fully discrete stochastic approximation of the projected Wasserstein dynamics. The target potential is globally smooth but need not be strongly convex. The departure from contractivity is quantified by the curvature defect \[ \mathfrak d_α(x,y) = \bigl[α\|x-y\|^2- \langle\nabla V(x)-\nabla V(y),x-y\rangle\bigr]_+, \] which is the additive loss in the one-step Euler contraction estimate. We prove a non-asymptotic Wasserstein stability bound that separates initialization, product-empirical approximation, finite-batch drift error, time discretization, and the defects accumulated along the coupled trajectories. Under the uniform bound $\mathfrak d_α\leqβ$, the particle iterates remain within $O(\sqrt{β/α})$ of any MFVI minimizer, up to explicit errors in the particle number, batch size, and step size. The proof uses a stationary comparison array whose population law is an MFVI minimizer but whose particle-level law is a random product empirical measure, and it controls the resulting projected-drift discrepancy explicitly. We also give coordinatewise defect estimates and structural conditions for dimension-independent projected-drift sensitivity, construct an arbitrary-dimensional smooth nonconvex benchmark with a closed-form MFVI minimizer, and explain why polynomially growing drifts require a modification of the untamed explicit scheme.

Problem

• mean-field variational inference
• finite-batch particle algorithm
• non-convex optimization
• Wasserstein stability
• curvature defect

Innovation

• mean-field variational inference
• finite-batch particle algorithm
• Wasserstein stability
• curvature defect
• non-convex optimization