Harmless interpolation in regression and classification with structured features
This work addresses the theoretical understanding of benign overfitting for researchers in machine learning, offering a more flexible analysis that extends beyond prior assumptions of independent or high-dimensional features.
The paper tackles the problem of understanding harmless interpolation in overparametrized models by providing a general framework for bounding risk in reproducing kernel Hilbert spaces, showing that harmless interpolation can occur with structured features like bounded orthonormal systems and revealing an asymptotic separation between classification and regression performance.
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.