Maximum Likelihood Learning With Arbitrary Treewidth via Fast-Mixing Parameter Sets

TL;DR

This paper introduces a new approach for maximum likelihood learning in high-treewidth graphical models by constraining parameters to a set of fast-mixing distributions, enabling efficient and theoretically guaranteed gradient approximation.

Contribution

It establishes that constraining parameters to fast-mixing sets allows for provably efficient MCMC-based gradient estimation in maximum likelihood learning.

Findings

Gradient descent with sampling approximates MLE within the fast-mixing set.

Effort is polynomial in 1/epsilon for unregularized solutions.

Regularization improves convergence to quadratic effort in 1/epsilon.

Abstract

Inference is typically intractable in high-treewidth undirected graphical models, making maximum likelihood learning a challenge. One way to overcome this is to restrict parameters to a tractable set, most typically the set of tree-structured parameters. This paper explores an alternative notion of a tractable set, namely a set of "fast-mixing parameters" where Markov chain Monte Carlo (MCMC) inference can be guaranteed to quickly converge to the stationary distribution. While it is common in practice to approximate the likelihood gradient using samples obtained from MCMC, such procedures lack theoretical guarantees. This paper proves that for any exponential family with bounded sufficient statistics, (not just graphical models) when parameters are constrained to a fast-mixing set, gradient descent with gradients approximated by sampling will approximate the maximum likelihood solution inside the set with high-probability. When unregularized, to find a solution epsilon-accurate in log-likelihood requires a total amount of effort cubic in 1/epsilon, disregarding logarithmic factors. When ridge-regularized, strong convexity allows a solution epsilon-accurate in parameter distance with effort quadratic in 1/epsilon. Both of these provide of a fully-polynomial time randomized approximation scheme.