Harmless interpolation in regression and classification with structured features

TL;DR

This paper develops a general framework to understand when overparameterized models with structured features, such as kernels, can interpolate noisy data without harming test performance, extending previous results beyond independent features.

Contribution

It introduces a flexible framework for analyzing harmless interpolation in structured feature spaces, including bounded orthonormal systems, and clarifies differences between classification and regression performance.

Findings

Harmless interpolation occurs under specific Gram matrix conditions.

Results extend prior independent-feature analyses to more general feature structures.

Asymptotic separation between classification and regression performance is demonstrated.

Abstract

Overparametrized neural networks tend to perfectly fit noisy training data yet generalize well on test data. Inspired by this empirical observation, recent work has sought to understand this phenomenon of benign overfitting or harmless interpolation in the much simpler linear model. Previous theoretical work critically assumes that either the data features are statistically independent or the input data is high-dimensional; this precludes general nonparametric settings with structured feature maps. In this paper, we present a general and flexible framework for upper bounding regression and classification risk in a reproducing kernel Hilbert space. A key contribution is that our framework describes precise sufficient conditions on the data Gram matrix under which harmless interpolation occurs. Our results recover prior independent-features results (with a much simpler analysis), but they furthermore show that harmless interpolation can occur in more general settings such as features that are a bounded orthonormal system. Furthermore, our results show an asymptotic separation between classification and regression performance in a manner that was previously only shown for Gaussian features.