On the Sample Complexity of Learning Bayesian Networks

TL;DR

This paper analyzes the sample complexity of learning Bayesian networks using the MDL principle, providing bounds on the number of samples needed for accurate learning and discussing implications for improving learning efficiency.

Contribution

It derives explicit sample complexity bounds for MDL-based Bayesian network learning and explores how these bounds can be used to enhance learning speed.

Findings

Sample complexity is polynomial in the inverse of the error threshold.

Number of samples needed is sub-linear in the confidence parameter.

Constants depend on the complexity of the target distribution.

Abstract

In recent years there has been an increasing interest in learning Bayesian networks from data. One of the most effective methods for learning such networks is based on the minimum description length (MDL) principle. Previous work has shown that this learning procedure is asymptotically successful: with probability one, it will converge to the target distribution, given a sufficient number of samples. However, the rate of this convergence has been hitherto unknown. In this work we examine the sample complexity of MDL based learning procedures for Bayesian networks. We show that the number of samples needed to learn an epsilon-close approximation (in terms of entropy distance) with confidence delta is O((1/epsilon)^(4/3)log(1/epsilon)log(1/delta)loglog (1/delta)). This means that the sample complexity is a low-order polynomial in the error threshold and sub-linear in the confidence bound. We also discuss how the constants in this term depend on the complexity of the target distribution. Finally, we address questions of asymptotic minimality and propose a method for using the sample complexity results to speed up the learning process.