Dynamics of coordinate ascent variational inference: A case study in 2D Ising models

TL;DR

This paper analyzes the convergence behavior of coordinate ascent variational inference algorithms for the 2D Ising model, revealing stability in convex regimes and oscillations in non-convex regimes, with insights from dynamical systems and MCMC.

Contribution

It provides a detailed dynamical systems analysis of coordinate ascent algorithms for the Ising model, highlighting differences between sequential and parallel versions and introducing parameter expansion to improve convergence.

Findings

Sequential algorithm converges in convex regimes.

Parallel algorithm exhibits oscillations in non-convex regimes.

Parameter expansion enlarges convergence regime.

Abstract

Variational algorithms have gained prominence over the past two decades as a scalable computational environment for Bayesian inference. In this article, we explore tools from the dynamical systems literature to study convergence of coordinate ascent algorithms for mean field variational inference. Focusing on the Ising model defined on two nodes, we fully characterize the dynamics of the sequential coordinate ascent algorithm and its parallel version. We observe that in the regime where the objective function is convex, both the algorithms are stable and exhibit convergence to the unique fixed point. Our analyses reveal interesting {\em discordances} between these two versions of the algorithm in the region when the objective function is non-convex. In fact, the parallel version exhibits a periodic oscillatory behavior which is absent in the sequential version. Drawing intuition from the Markov chain Monte Carlo literature, we {\em empirically} show that a parameter expansion of the Ising model, popularly called as the Edward--Sokal coupling, leads to an enlargement of the regime of convergence to the global optima.