Learnability of Parameter-Bounded Bayes Nets

Arnab Bhattacharyya, Davin Choo, Sutanu Gayen, Dimitrios Myrisiotis

Keywords: Bayes nets, Bayes net parameters, learning, promise problems, search problems, NP-hardness, Turing reductions, sample complexity, total variation distance, TV distance

Abstract:

Bayes nets are extensively used in practice to efficiently represent joint probability distributions over a set of random variables and capture dependency relations.
In a seminal paper, Chickering et al. (JMLR 2004) showed that given a distribution P, that is defined as the marginal distribution of a Bayes net, it is NP-hard to decide whether there is a parameter-bounded Bayes net that represents P.
They called this problem LEARN.
In this work, we extend the NP-hardness result of LEARN and prove the NP-hardness of a promise search variant of LEARN, whereby the Bayes net in question is guaranteed to exist and one is asked to find such a Bayes net.
We complement our hardness result with a positive result about the sample complexity that is sufficient to recover a parameter-bounded Bayes net that is close (in TV distance) to a given distribution P, represented by some parameter-bounded Bayes net, thereby generalizing a degree-bounded sample complexity result of Brustle et al. (EC 2020).

Submission Number: 1