# Learning Graphical Models Using Multiplicative Weights

## TL;DR

This paper introduces a simple multiplicative-weight update algorithm for learning undirected graphical models, achieving near-optimal sample complexity and quadratic runtime for Ising models, and extending to general alphabets and t-wise MRFs.

## Contribution

It presents the first efficient algorithm for learning Ising models over general alphabets and introduces the Sparsitron, a versatile, easy-to-implement method with applications to sparse GLMs.

## Key findings

- Achieves nearly optimal sample complexity for Ising models.
- Provides the first efficient algorithm for general alphabet Ising models.
- Introduces the Sparsitron algorithm with online and GLM applications.

## Abstract

We give a simple, multiplicative-weight update algorithm for learning undirected graphical models or Markov random fields (MRFs). The approach is new, and for the well-studied case of Ising models or Boltzmann machines, we obtain an algorithm that uses a nearly optimal number of samples and has quadratic running time (up to logarithmic factors), subsuming and improving on all prior work. Additionally, we give the first efficient algorithm for learning Ising models over general alphabets. Our main application is an algorithm for learning the structure of t-wise MRFs with nearly-optimal sample complexity (up to polynomial losses in necessary terms that depend on the weights) and running time that is $n^{O(t)}$. In addition, given $n^{O(t)}$ samples, we can also learn the parameters of the model and generate a hypothesis that is close in statistical distance to the true MRF. All prior work runs in time $n^{\Omega(d)}$ for graphs of bounded degree d and does not generate a hypothesis close in statistical distance even for t=3. We observe that our runtime has the correct dependence on n and t assuming the hardness of learning sparse parities with noise. Our algorithm--the Sparsitron-- is easy to implement (has only one parameter) and holds in the on-line setting. Its analysis applies a regret bound from Freund and Schapire's classic Hedge algorithm. It also gives the first solution to the problem of learning sparse Generalized Linear Models (GLMs).