Model Selection Through Sparse Maximum Likelihood Estimation

TL;DR

This paper introduces two scalable algorithms for sparse maximum likelihood estimation of Gaussian and binary graphical models, enabling analysis of large-scale problems with thousands of nodes.

Contribution

The paper presents novel block coordinate descent and Nesterov's first order algorithms that significantly improve scalability for sparse graphical model estimation.

Findings

Algorithms successfully handle problems with over a thousand nodes.

Effective in estimating sparse Gaussian and binary graphical models.

Validated on synthetic, gene expression, and voting record datasets.

Abstract

We consider the problem of estimating the parameters of a Gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse. Our approach is to solve a maximum likelihood problem with an added l_1-norm penalty term. The problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes. We present two new algorithms for solving problems with at least a thousand nodes in the Gaussian case. Our first algorithm uses block coordinate descent, and can be interpreted as recursive l_1-norm penalized regression. Our second algorithm, based on Nesterov's first order method, yields a complexity estimate with a better dependence on problem size than existing interior point methods. Using a log determinant relaxation of the log partition function (Wainwright & Jordan (2006)), we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case. We test our algorithms on synthetic data, as well as on gene expression and senate voting records data.