# Learning Restricted Boltzmann Machines with Arbitrary External Fields

## TL;DR

This paper introduces an optimal-sample-complexity algorithm for learning ferromagnetic and antiferromagnetic Restricted Boltzmann Machines with arbitrary external fields, overcoming previous limitations to consistent external fields.

## Contribution

It presents the first algorithm capable of learning RBMs with arbitrary external fields, utilizing a new structural property and covariance-based neighborhood construction.

## Key findings

- Algorithm has optimal dependence on dimension for sample complexity and runtime.
- Successfully learns RBMs with arbitrary external fields, extending prior work.
- Relies on covariance properties even with arbitrary external fields.

## Abstract

We study the problem of learning graphical models with latent variables. We give the first algorithm for learning locally consistent (ferromagnetic or antiferromagnetic) Restricted Boltzmann Machines (or RBMs) with {\em arbitrary} external fields. Our algorithm has optimal dependence on dimension in the sample complexity and run time however it suffers from a sub-optimal dependency on the underlying parameters of the RBM. Prior results have been established only for {\em ferromagnetic} RBMs with {\em consistent} external fields (signs must be same)\cite{bresler2018learning}. The proposed algorithm strongly relies on the concavity of magnetization which does not hold in our setting. We show the following key structural property: even in the presence of arbitrary external field, for any two observed nodes that share a common latent neighbor, the covariance is high. This enables us to design a simple greedy algorithm that maximizes covariance to iteratively build the neighborhood of each vertex.