From Boltzmann Machines to Neural Networks and Back Again

TL;DR

This paper advances the theoretical understanding of learning Restricted Boltzmann Machines by establishing new connections to neural networks, providing nearly optimal algorithms under certain hardness assumptions, and exploring supervised RBMs.

Contribution

It introduces novel theoretical results linking RBMs to neural networks, and develops improved algorithms for supervised RBMs with distributional assumptions.

Findings

Nearly optimal learning results for RBMs under hardness conjectures

New connections between RBMs and two-layer neural networks

Improved algorithms for supervised RBMs with better runtime

Abstract

Graphical models are powerful tools for modeling high-dimensional data, but learning graphical models in the presence of latent variables is well-known to be difficult. In this work we give new results for learning Restricted Boltzmann Machines, probably the most well-studied class of latent variable models. Our results are based on new connections to learning two-layer neural networks under $\ell_{\infty}$ bounded input; for both problems, we give nearly optimal results under the conjectured hardness of sparse parity with noise. Using the connection between RBMs and feedforward networks, we also initiate the theoretical study of $supervised~RBMs$ [Hinton, 2012], a version of neural-network learning that couples distributional assumptions induced from the underlying graphical model with the architecture of the unknown function class. We then give an algorithm for learning a natural class of supervised RBMs with better runtime than what is possible for its related class of networks without distributional assumptions.