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：我们为高效适当学习有界度多叉树建立了有限样本保证，这是一类丰富的高维概率分布，也是广泛研究的图模型——贝叶斯网络的子类。近期，Bhattacharyya等人（2021年）获得了恢复树结构贝叶斯网络（即1-多叉树）的有限样本保证。我们将他们的结果推广，提供了一种高效算法，当底层无向图（骨架）已知时，该算法能在多项式时间和样本复杂度内学习任意有界度$d$的$d$-多叉树。我们还通过信息论样本复杂度下界对算法进行了补充，表明对维度和目标精度参数的依赖几乎是紧的。