We establish finite-sample guarantees for efficient proper learning of bounded-degree polytrees, a rich class of high-dimensional probability distributions and a subclass of Bayesian networks, a widely-studied type of graphical model. Recently, Bhattacharyya et al. (2021) obtained finite-sample guarantees for recovering tree-structured Bayesian networks, i.e., 1-polytrees. We extend their results by providing an efficient algorithm which learns $d$-polytrees in polynomial time and sample complexity for any bounded $d$ when the underlying undirected graph (skeleton) is known. We complement our algorithm with an information-theoretic sample complexity lower bound, showing that the dependence on the dimension and target accuracy parameters are nearly tight.

翻译：我们建立了高效正确学习有界度多树图的有限样本保证，这类图结构是一类丰富的高维概率分布，也是广泛研究的图模型——贝叶斯网络的一个子类。近期，Bhattacharyya等人（2021）获得了恢复树结构贝叶斯网络（即1-多树图）的有限样本保证。我们通过提供一个高效算法扩展了他们的结果，该算法在已知底层无向图（骨架）的情况下，对任何有界度d，能在多项式时间和样本复杂度内学习d-多树图。我们为算法补充了一个信息论样本复杂度下界，证明对维度和目标精度参数的依赖几乎是紧的。