Greedy Learning of Markov Network Structure

TL;DR

This paper introduces a greedy, computationally efficient algorithm for learning the structure of discrete graphical models, with theoretical guarantees on sample complexity and applicability to various graph types.

Contribution

It presents a new greedy algorithm for Markov network structure learning with proven sample complexity bounds, extending classical methods to graphs with cycles.

Findings

Algorithm reduces empirical conditional entropy to find neighborhoods.

Sample complexity characterized as a function of graph parameters.

Special case analysis for Ising models and tree graphs.

Abstract

We propose a new yet natural algorithm for learning the graph structure of general discrete graphical models (a.k.a. Markov random fields) from samples. Our algorithm finds the neighborhood of a node by sequentially adding nodes that produce the largest reduction in empirical conditional entropy; it is greedy in the sense that the choice of addition is based only on the reduction achieved at that iteration. Its sequential nature gives it a lower computational complexity as compared to other existing comparison-based techniques, all of which involve exhaustive searches over every node set of a certain size. Our main result characterizes the sample complexity of this procedure, as a function of node degrees, graph size and girth in factor-graph representation. We subsequently specialize this result to the case of Ising models, where we provide a simple transparent characterization of sample complexity as a function of model and graph parameters. For tree graphs, our algorithm is the same as the classical Chow-Liu algorithm, and in that sense can be considered the extension of the same to graphs with cycles.