Near-Minimax Optimal Estimation With Shallow ReLU Neural Networks

TL;DR

This paper demonstrates that shallow ReLU neural networks achieve near-minimax optimal estimation rates for functions in a specific space, effectively overcoming the curse of dimensionality, unlike linear methods.

Contribution

The paper establishes the minimax optimality of shallow ReLU neural networks for a natural function space, showing they outperform linear methods in high dimensions.

Findings

Neural network estimators are minimax optimal up to logarithmic factors.

Linear methods suffer the curse of dimensionality in this setting.

Neural networks effectively break the curse of dimensionality for the studied function space.

Abstract

We study the problem of estimating an unknown function from noisy data using shallow ReLU neural networks. The estimators we study minimize the sum of squared data-fitting errors plus a regularization term proportional to the squared Euclidean norm of the network weights. This minimization corresponds to the common approach of training a neural network with weight decay. We quantify the performance (mean-squared error) of these neural network estimators when the data-generating function belongs to the second-order Radon-domain bounded variation space. This space of functions was recently proposed as the natural function space associated with shallow ReLU neural networks. We derive a minimax lower bound for the estimation problem for this function space and show that the neural network estimators are minimax optimal up to logarithmic factors. This minimax rate is immune to the curse of dimensionality. We quantify an explicit gap between neural networks and linear methods (which include kernel methods) by deriving a linear minimax lower bound for the estimation problem, showing that linear methods necessarily suffer the curse of dimensionality in this function space. As a result, this paper sheds light on the phenomenon that neural networks seem to break the curse of dimensionality.