Robust Nonparametric Regression with Deep Neural Networks

TL;DR

This paper develops robust deep neural network methods for nonparametric regression with heavy-tailed errors, providing non-asymptotic error bounds that mitigate the curse of dimensionality and outperform traditional least squares in heavy-tailed scenarios.

Contribution

It introduces a robust deep neural network regression framework with new error bounds that relax previous assumptions and handle heavy-tailed errors effectively.

Findings

Robust estimators outperform least squares with heavy-tailed errors.

Error bounds depend sub-linearly on predictor dimension d.

Method circumvents curse of dimensionality under relaxed assumptions.

Abstract

In this paper, we study the properties of robust nonparametric estimation using deep neural networks for regression models with heavy tailed error distributions. We establish the non-asymptotic error bounds for a class of robust nonparametric regression estimators using deep neural networks with ReLU activation under suitable smoothness conditions on the regression function and mild conditions on the error term. In particular, we only assume that the error distribution has a finite p-th moment with p greater than one. We also show that the deep robust regression estimators are able to circumvent the curse of dimensionality when the distribution of the predictor is supported on an approximate lower-dimensional set. An important feature of our error bound is that, for ReLU neural networks with network width and network size (number of parameters) no more than the order of the square of the dimensionality d of the predictor, our excess risk bounds depend sub-linearly on d. Our assumption relaxes the exact manifold support assumption, which could be restrictive and unrealistic in practice. We also relax several crucial assumptions on the data distribution, the target regression function and the neural networks required in the recent literature. Our simulation studies demonstrate that the robust methods can significantly outperform the least squares method when the errors have heavy-tailed distributions and illustrate that the choice of loss function is important in the context of deep nonparametric regression.