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

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 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. Our theoretical results demonstrate keen agreement with experimental results for sample sizes as small as that in the hundreds.

Near-Optimal Learning of Tree-Structured Distributions by Chow-Liu

“…Chow and Wagner [CW73] showed that the Chow-Liu algorithm consistently recovers structure, meaning that if the samples are generated by a T -structured distribution for a tree T , then it recovers T with probability approaching 1 as the number of samples tends to infinity. More recent works [TATW11,TTZ20] have used large-deviation theory to study the error exponent K P of T -structured distributions P , where:…”