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

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.

An entropy formula for the Deep Linear Network

We study the Riemannian geometry of the Deep Linear Network (DLN) as a foundation for a thermodynamic description of the learning process. The main tools are the use of group actions to analyze overparametrization and the use of Riemannian submersion from the space of parameters to the space of observables. The foliation of the balanced manifold in the parameter space by group orbits is used to define and compute a Boltzmann entropy. We also show that the Riemannian geometry on the space of observables defined in [2] is obtained by Riemannian submersion of the balanced manifold. The main technical step is an explicit construction of an orthonormal basis for the tangent space of the balanced manifold using the theory of Jacobi matrices.

Riemannian Langevin Monte Carlo schemes for sampling PSD matrices with fixed rank

This paper introduces two explicit schemes to sample matrices from Gibbs distributions on $\mathcal S^{n,p}_+$, the manifold of real positive semi-definite (PSD) matrices of size $n\times n$ and rank $p$. Given an energy function $\mathcal E:\mathcal S^{n,p}_+\to \mathbb{R}$ and certain Riemannian metrics $g$ on $\mathcal S^{n,p}_+$, these schemes rely on an Euler-Maruyama discretization of the Riemannian Langevin equation (RLE) with Brownian motion on the manifold. We present numerical schemes for RLE under two fundamental metrics on $\mathcal S^{n,p}_+$: (a) the metric obtained from the embedding of $\mathcal S^{n,p}_+ \subset \mathbb{R}^{n\times n} $; and (b) the Bures-Wasserstein metric corresponding to quotient geometry. We also provide examples of energy functions with explicit Gibbs distributions that allow numerical validation of these schemes.