Quantifying Variational Approximation for the Log-Partition Function

Abstract: Variational approximation, such as mean-field (MF) and tree-reweighted (TRW), provide a computationally efficient approximation of the log-partition function for a generic graphical model. TRW provably provides an upper bound, but the approximation ratio is generally not quantified. As the primary contribution of this work, we provide an approach to quantify the approximation ratio through the property of the underlying graph structure. Specifically, we argue that (a variant of) TRW produces an estimate that is within factor $\frac{1}{\sqrt{\kappa(G)}}$ of the true log-partition function for any discrete pairwise graphical model over graph $G$, where $\kappa(G) \in (0,1]$ captures how far $G$ is from tree structure with $\kappa(G) = 1$ for trees and $2/N$ for the complete graph over $N$ vertices. As a consequence, the approximation ratio is $1$ for trees, $\sqrt{(d+1)/2}$ for any graph with maximum average degree $d$, and $\stackrel{\beta\to\infty}{\approx} 1+1/(2\beta)$ for graphs with girth (shortest cycle) at least $\beta \log N$. In general, $\kappa(G)$ is the solution of a max-min problem associated with $G$ that can be evaluated in polynomial time for any graph. Using samples from the uniform distribution over the spanning trees of G, we provide a near linear-time variant that achieves an approximation ratio equal to the inverse of square-root of minimal (across edges) effective resistance of the graph. We connect our results to the graph partition-based approximation method and thus provide a unified perspective. Keywords: variational inference, log-partition function, spanning tree polytope, minimum effective resistance, min-max spanning tree, local inference