Near-Optimal Learning of Tree-Structured Distributions by Chow-Liu
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We provide finite sample guarantees for the classical Chow-Liu algorithm (IEEE Trans. Inform. Theory, 1968) to learn a tree-structured graphical model of a distribution. For a distribution P on Σ^n and a tree T on n nodes, we say T is an ε-approximate tree for P if there is a T-structured distribution Q such that D(P || Q) is at most ε more than the best possible tree-structured distribution for P. We show that if P itself is tree-structured, then the Chow-Liu algorithm with the plug-in estimator for mutual information with O(|Σ|^3 nε^-1) i.i.d. samples outputs an ε-approximate tree for P with constant probability. In contrast, for a general P (which may not be tree-structured), Ω(n^2ε^-2) samples are necessary to find an ε-approximate tree. Our upper bound is based on a new conditional independence tester that addresses an open problem posed by Canonne, Diakonikolas, Kane, and Stewart (STOC, 2018): we prove that for three random variables X,Y,Z each over Σ, testing if I(X; Y | Z) is 0 or ≥ε is possible with O(|Σ|^3/ε) samples. Finally, we show that for a specific tree T, with O (|Σ|^2nε^-1) samples from a distribution P over Σ^n, one can efficiently learn the closest T-structured distribution in KL divergence by applying the add-1 estimator at each node.
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