Learning Juntas under Markov Random Fields

We give an algorithm for learning $O(\log n)$ juntas in polynomial-time with respect to Markov Random Fields (MRFs) in a smoothed analysis framework, where only the external field has been randomly perturbed. This is a broad generalization of the work of Kalai and Teng, who gave an algorithm that succeeded with respect to smoothed *product* distributions (i.e., MRFs whose dependency graph has no edges). Our algorithm has two phases: (1) an unsupervised structure learning phase and (2) a greedy supervised learning algorithm. This is the first example where algorithms for learning the structure of undirected graphical models have downstream applications to supervised learning.