Why does deep and cheap learning work so well?

TL;DR

This paper explores how principles from physics like symmetry and locality contribute to the efficiency of deep learning, demonstrating that hierarchical data structures enable deep networks to outperform shallow ones in certain tasks.

Contribution

It introduces a theoretical framework linking physics-inspired properties to neural network efficiency and proves limitations of shallow networks in approximating certain functions.

Findings

Deep networks can efficiently approximate functions with physical properties.

Shallow networks require exponentially more neurons for certain operations.

Hierarchical data structures favor deep over shallow neural network architectures.

Abstract

We show how the success of deep learning could depend not only on mathematics but also on physics: although well-known mathematical theorems guarantee that neural networks can approximate arbitrary functions well, the class of functions of practical interest can frequently be approximated through "cheap learning" with exponentially fewer parameters than generic ones. We explore how properties frequently encountered in physics such as symmetry, locality, compositionality, and polynomial log-probability translate into exceptionally simple neural networks. We further argue that when the statistical process generating the data is of a certain hierarchical form prevalent in physics and machine-learning, a deep neural network can be more efficient than a shallow one. We formalize these claims using information theory and discuss the relation to the renormalization group. We prove various "no-flattening theorems" showing when efficient linear deep networks cannot be accurately approximated by shallow ones without efficiency loss, for example, we show that $n$ variables cannot be multiplied using fewer than 2^n neurons in a single hidden layer.