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

Harmless interpolation in regression and classification with structured features