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

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 commonin practice to approximatethe likelihoodgradientusing samples obtained from MCMC, such procedureslack 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 ϵ-accuratein log-likelihoodrequiresa total amountof effortcubic in 1/ϵ, disregarding logarithmic factors. When ridge-regularized, strong convexity allows asolutionϵ-accurate in parameter distance with effort quadratic in 1/ϵ. Bothof these provide of a fully-polynomial time randomized approximation scheme.