Proof of the Theory-to-Practice Gap in Deep Learning via Sampling Complexity bounds for Neural Network Approximation Spaces

TL;DR

This paper investigates the computational complexity of neural network approximation and integration, demonstrating a gap between theoretical approximation rates and practical algorithmic realizations in deep learning.

Contribution

It proves hardness results for neural network approximation and integration, confirming the theory-to-practice gap in deep learning.

Findings

Hardness results for neural network approximation and integration.

Confirmation of the theory-to-practice gap in deep learning.

Existence of theoretically achievable approximation rates.

Abstract

We study the computational complexity of (deterministic or randomized) algorithms based on point samples for approximating or integrating functions that can be well approximated by neural networks. Such algorithms (most prominently stochastic gradient descent and its variants) are used extensively in the field of deep learning. One of the most important problems in this field concerns the question of whether it is possible to realize theoretically provable neural network approximation rates by such algorithms. We answer this question in the negative by proving hardness results for the problems of approximation and integration on a novel class of neural network approximation spaces. In particular, our results confirm a conjectured and empirically observed theory-to-practice gap in deep learning. We complement our hardness results by showing that approximation rates of a comparable order of convergence are (at least theoretically) achievable.