Information Theoretic Properties of Markov Random Fields, and their Algorithmic Applications

TL;DR

This paper explores the information-theoretic properties of Markov random fields, introduces a game-theoretic approach to lower bounds, and presents nearly optimal algorithms for learning high-order interactions in these models.

Contribution

It generalizes mutual information lower bounds to arbitrary Markov random fields and provides efficient algorithms with near-optimal sample complexity for learning them.

Findings

Algorithms for learning Markov random fields with bounded degree and high-order interactions.

Sample complexity is nearly information-theoretically optimal.

Running time is nearly optimal under standard hardness conjectures.

Abstract

Markov random fields area popular model for high-dimensional probability distributions. Over the years, many mathematical, statistical and algorithmic problems on them have been studied. Until recently, the only known algorithms for provably learning them relied on exhaustive search, correlation decay or various incoherence assumptions. Bresler gave an algorithm for learning general Ising models on bounded degree graphs. His approach was based on a structural result about mutual information in Ising models. Here we take a more conceptual approach to proving lower bounds on the mutual information through setting up an appropriate zero-sum game. Our proof generalizes well beyond Ising models, to arbitrary Markov random fields with higher order interactions. As an application, we obtain algorithms for learning Markov random fields on bounded degree graphs on $n$ nodes with $r$-order interactions in $n^r$ time and $\log n$ sample complexity. The sample complexity is information theoretically optimal up to the dependence on the maximum degree. The running time is nearly optimal under standard conjectures about the hardness of learning parity with noise.