We give an algorithm for learning the structure of an undirected graphical model that has essentially optimal sample complexity and running time.  We make no assumptions on the structure of the graphical model.  For Ising models, this subsumes and improves on all prior work.  For general t-wise MRFs, these are the first results of their kind.

Our approach is new and uses a multiplicative-weight update algorithm.  Our algorithm-- Sparsitron-- is easy to implement (has only one parameter) and holds in the online setting. It also gives the first provably efficient solution to the problem of learning sparse Generalized Linear Models (GLMs).