Sharp Lower Bounds on Interpolation by Deep ReLU Neural Networks at Irregularly Spaced Data

TL;DR

This paper establishes sharp lower bounds on the number of parameters deep ReLU neural networks need to interpolate irregularly spaced data points, revealing limitations in their approximation capabilities at the embedding endpoint.

Contribution

It provides the first sharp lower bounds for interpolation by deep ReLU networks at irregularly spaced data, especially when data points are exponentially close.

Findings

Omega(N) parameters are necessary when data points are exponentially close.

Bit-extraction techniques for VC dimension bounds do not apply to irregular data.

Lower bounds on approximation rates for Sobolev spaces are derived.

Abstract

We study the interpolation power of deep ReLU neural networks. Specifically, we consider the question of how efficiently, in terms of the number of parameters, deep ReLU networks can interpolate values at $N$ datapoints in the unit ball which are separated by a distance $\delta$. We show that $\Omega(N)$ parameters are required in the regime where $\delta$ is exponentially small in $N$, which gives the sharp result in this regime since $O(N)$ parameters are always sufficient. This also shows that the bit-extraction technique used to prove lower bounds on the VC dimension cannot be applied to irregularly spaced datapoints. Finally, as an application we give a lower bound on the approximation rates that deep ReLU neural networks can achieve for Sobolev spaces at the embedding endpoint.