Scalable Mean-Field Variational Inference via Preconditioned Primal-Dual Optimization

• URL: http://arxiv.org/abs/2602.07632v2
• Date: Tue, 10 Feb 2026 05:36:49 GMT
• Title: Scalable Mean-Field Variational Inference via Preconditioned Primal-Dual Optimization
• Authors: Jinhua Lyu, Tianmin Yu, Ying Ma, Naichen Shi,
• Abstract summary: We develop a novel primal-dual algorithm based on an augmented Lagrangian formulation, termed primal-dual variational inference (PD-VI)<n>PD-VI jointly updates global and local variational parameters in the evidence lower bound in a scalable manner.<n>We establish convergence guarantees for both PD-VI and P$2$D-VI under properly chosen constant step size.
• Score: 7.193011407502236
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this work, we investigate the large-scale mean-field variational inference (MFVI) problem from a mini-batch primal-dual perspective. By reformulating MFVI as a constrained finite-sum problem, we develop a novel primal-dual algorithm based on an augmented Lagrangian formulation, termed primal-dual variational inference (PD-VI). PD-VI jointly updates global and local variational parameters in the evidence lower bound in a scalable manner. To further account for heterogeneous loss geometry across different variational parameter blocks, we introduce a block-preconditioned extension, P$^2$D-VI, which adapts the primal-dual updates to the geometry of each parameter block and improves both numerical robustness and practical efficiency. We establish convergence guarantees for both PD-VI and P$^2$D-VI under properly chosen constant step size, without relying on conjugacy assumptions or explicit bounded-variance conditions. In particular, we prove $O(1/T)$ convergence to a stationary point in general settings and linear convergence under strong convexity. Numerical experiments on synthetic data and a real large-scale spatial transcriptomics dataset demonstrate that our methods consistently outperform existing stochastic variational inference approaches in terms of convergence speed and solution quality.

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