Mind the spikes: Benign overfitting of kernels and neural networks in fixed dimension

TL;DR

This paper investigates benign overfitting in fixed-dimensional settings, showing that estimator smoothness, rather than dimension, determines overfitting potential, and demonstrates how neural networks can be modified to overfit benignly.

Contribution

It establishes that benign overfitting depends on estimator derivatives in fixed dimension and introduces modifications to neural networks to achieve benign overfitting.

Findings

Benign overfitting is possible with large derivatives in fixed dimension.

Neural tangent kernels can be adjusted with high-frequency fluctuations to enable benign overfitting.

Experiments confirm neural networks can generalize well despite overfitting in low-dimensional data.

Abstract

The success of over-parameterized neural networks trained to near-zero training error has caused great interest in the phenomenon of benign overfitting, where estimators are statistically consistent even though they interpolate noisy training data. While benign overfitting in fixed dimension has been established for some learning methods, current literature suggests that for regression with typical kernel methods and wide neural networks, benign overfitting requires a high-dimensional setting where the dimension grows with the sample size. In this paper, we show that the smoothness of the estimators, and not the dimension, is the key: benign overfitting is possible if and only if the estimator's derivatives are large enough. We generalize existing inconsistency results to non-interpolating models and more kernels to show that benign overfitting with moderate derivatives is impossible in fixed dimension. Conversely, we show that rate-optimal benign overfitting is possible for regression with a sequence of spiky-smooth kernels with large derivatives. Using neural tangent kernels, we translate our results to wide neural networks. We prove that while infinite-width networks do not overfit benignly with the ReLU activation, this can be fixed by adding small high-frequency fluctuations to the activation function. Our experiments verify that such neural networks, while overfitting, can indeed generalize well even on low-dimensional data sets.