How DNNs break the Curse of Dimensionality: Compositionality and Symmetry Learning

TL;DR

This paper demonstrates that deep neural networks can efficiently learn compositional functions with bounded norms, enabling them to overcome the curse of dimensionality through symmetry and compositionality learning.

Contribution

The authors provide a theoretical framework showing how DNNs can learn compositional functions with bounded norms, breaking the curse of dimensionality, supported by empirical scaling laws.

Findings

DNNs can learn compositions of functions with bounded F1-norm efficiently.

A generalization bound combining compositionality and norm-based regularization is derived.

Empirical phase transitions occur depending on the complexity of the inner or outer functions.

Abstract

We show that deep neural networks (DNNs) can efficiently learn any composition of functions with bounded $F_{1}$-norm, which allows DNNs to break the curse of dimensionality in ways that shallow networks cannot. More specifically, we derive a generalization bound that combines a covering number argument for compositionality, and the $F_{1}$-norm (or the related Barron norm) for large width adaptivity. We show that the global minimizer of the regularized loss of DNNs can fit for example the composition of two functions $f^{*}=h\circ g$ from a small number of observations, assuming $g$ is smooth/regular and reduces the dimensionality (e.g. $g$ could be the quotient map of the symmetries of $f^{*}$), so that $h$ can be learned in spite of its low regularity. The measures of regularity we consider is the Sobolev norm with different levels of differentiability, which is well adapted to the $F_{1}$ norm. We compute scaling laws empirically and observe phase transitions depending on whether $g$ or $h$ is harder to learn, as predicted by our theory.