Near-optimal learning of tree-structured distributions by Chow-Liu

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 treestructured 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 2 ε -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| 2 nε -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. * Höffgen's capped the empirical probabilities away from 0 and 1 and then used a plug-in estimator for entropy/MI. † In total variation distance rather than KL ‡ This result (for TV distance) was also claimed in the appendix of [CDKS20], but the analysis there appears to incomplete [Can20].