Exact Asymptotics for Learning Tree-Structured Graphical Models with Side Information: Noiseless and Noisy Samples

TL;DR

This paper derives exact asymptotics for learning the structure of homogeneous Ising tree models with side information, improving previous large deviation results and extending to noisy samples, with strong experimental validation.

Contribution

It provides refined asymptotic analysis for structure learning of Ising trees with side info, including noisy data, surpassing prior large deviation bounds.

Findings

Exact asymptotics match experimental results for small sample sizes

Improved error exponents over previous large deviation bounds

Extension of results to noisy sample scenarios

Abstract

Given side information that an Ising tree-structured graphical model is homogeneous and has no external field, we derive the exact asymptotics of learning its structure from independently drawn samples. Our results, which leverage the use of probabilistic tools from the theory of strong large deviations, refine the large deviation (error exponents) results of Tan, Anandkumar, Tong, and Willsky [IEEE Trans. on Inform. Th., 57(3):1714--1735, 2011] and strictly improve those of Bresler and Karzand [Ann. Statist., 2020]. In addition, we extend our results to the scenario in which the samples are observed in random noise. In this case, we show that they strictly improve on the recent results of Nikolakakis, Kalogerias, and Sarwate [Proc. AISTATS, 1771--1782, 2019]. Our theoretical results demonstrate keen agreement with experimental results for sample sizes as small as that in the hundreds.