Learning bounded-degree polytrees with known skeleton

TL;DR

This paper presents an efficient algorithm for learning bounded-degree polytrees with known skeletons, providing finite-sample guarantees and matching lower bounds, thus advancing the understanding of high-dimensional Bayesian network learning.

Contribution

It introduces a polynomial-time algorithm for learning d-polytrees with known skeletons and establishes nearly tight sample complexity bounds.

Findings

Algorithm achieves polynomial time and sample complexity.

Finite-sample guarantees are established for learning polytrees.

Sample complexity lower bounds show near-optimality.

Abstract

We establish finite-sample guarantees for efficient proper learning of bounded-degree polytrees, a rich class of high-dimensional probability distributions and a subclass of Bayesian networks, a widely-studied type of graphical model. Recently, Bhattacharyya et al. (2021) obtained finite-sample guarantees for recovering tree-structured Bayesian networks, i.e., 1-polytrees. We extend their results by providing an efficient algorithm which learns $d$-polytrees in polynomial time and sample complexity for any bounded $d$ when the underlying undirected graph (skeleton) is known. We complement our algorithm with an information-theoretic sample complexity lower bound, showing that the dependence on the dimension and target accuracy parameters are nearly tight.