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

• URL: http://arxiv.org/abs/2011.04144v2
• Date: Thu, 22 Jul 2021 06:37:12 GMT
• Title: Near-Optimal Learning of Tree-Structured Distributions by Chow-Liu
• Authors: Arnab Bhattacharyya, Sutanu Gayen, Eric Price, N. V. Vinodchandran
• Abstract summary: We provide finite sample guarantees for the classical ChowLiu algorithm (IEEE Trans.Inform.Theory, 1968) We show that for a specific tree $T$, with $widetildeO (|Sigma|2nvarepsilon-1)$ samples from a distribution $P$ over $Sigman$, one can efficiently learn the closest KL divergence.
• Score: 14.298220510927695
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $\Sigma^n$ and a tree $T$ on $n$ nodes, we say $T$ is an $\varepsilon$-approximate tree for $P$ if there is a $T$-structured distribution $Q$ such that $D(P\;||\;Q)$ is at most $\varepsilon$ 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 $\widetilde{O}(|\Sigma|^3 n\varepsilon^{-1})$ i.i.d.~samples outputs an $\varepsilon$-approximate tree for $P$ with constant probability. In contrast, for a general $P$ (which may not be tree-structured), $\Omega(n^2\varepsilon^{-2})$ samples are necessary to find an $\varepsilon$-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 $\Sigma$, testing if $I(X; Y \mid Z)$ is $0$ or $\geq \varepsilon$ is possible with $\widetilde{O}(|\Sigma|^3/\varepsilon)$ samples. Finally, we show that for a specific tree $T$, with $\widetilde{O} (|\Sigma|^2n\varepsilon^{-1})$ samples from a distribution $P$ over $\Sigma^n$, 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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