Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks

TL;DR

This paper surveys recent advances in high-dimensional function approximation, focusing on sparse polynomials and deep neural networks, highlighting theoretical limits, practical methods, and the gap between theory and practice.

Contribution

It introduces the concept of practical existence theory for DNNs, showing the existence of dimension-independent architectures with near-optimal generalization.

Findings

Sparse polynomial and DNN methods can efficiently learn high-dimensional functions.

There is a significant gap between DNN approximation theory and practical performance.

Practical existence theory suggests dimension-independent DNN architectures can achieve near-optimal errors.

Abstract

Learning approximations to smooth target functions of many variables from finite sets of pointwise samples is an important task in scientific computing and its many applications in computational science and engineering. Despite well over half a century of research on high-dimensional approximation, this remains a challenging problem. Yet, significant advances have been made in the last decade towards efficient methods for doing this, commencing with so-called sparse polynomial approximation methods and continuing most recently with methods based on Deep Neural Networks (DNNs). In tandem, there have been substantial advances in the relevant approximation theory and analysis of these techniques. In this work, we survey this recent progress. We describe the contemporary motivations for this problem, which stem from parametric models and computational uncertainty quantification; the relevant function classes, namely, classes of infinite-dimensional, Banach-valued, holomorphic functions; fundamental limits of learnability from finite data for these classes; and finally, sparse polynomial and DNN methods for efficiently learning such functions from finite data. For the latter, there is currently a significant gap between the approximation theory of DNNs and the practical performance of deep learning. Aiming to narrow this gap, we develop the topic of practical existence theory, which asserts the existence of dimension-independent DNN architectures and training strategies that achieve provably near-optimal generalization errors in terms of the amount of training data.