On Learning Discrete Graphical Models Using Greedy Methods

TL;DR

This paper introduces a greedy algorithm for learning the structure of high-dimensional discrete graphical models, demonstrating improved sample complexity over existing methods and requiring milder conditions for accurate recovery.

Contribution

It provides a theoretical analysis of a greedy approach for structure learning, showing it outperforms convex methods in sample efficiency and relaxes certain assumptions.

Findings

Achieves graph recovery with sample size n = Omega(d^2 log p)

Requires milder conditions than irrepresentability assumptions

Validated through numerical simulations

Abstract

In this paper, we address the problem of learning the structure of a pairwise graphical model from samples in a high-dimensional setting. Our first main result studies the sparsistency, or consistency in sparsity pattern recovery, properties of a forward-backward greedy algorithm as applied to general statistical models. As a special case, we then apply this algorithm to learn the structure of a discrete graphical model via neighborhood estimation. As a corollary of our general result, we derive sufficient conditions on the number of samples n, the maximum node-degree d and the problem size p, as well as other conditions on the model parameters, so that the algorithm recovers all the edges with high probability. Our result guarantees graph selection for samples scaling as n = Omega(d^2 log(p)), in contrast to existing convex-optimization based algorithms that require a sample complexity of \Omega(d^3 log(p)). Further, the greedy algorithm only requires a restricted strong convexity condition which is typically milder than irrepresentability assumptions. We corroborate these results using numerical simulations at the end.