Uniform Convergence of Interpolators: Gaussian Width, Norm Bounds and Benign Overfitting

We consider interpolation learning in high-dimensional linear regression with Gaussian data, and prove a generic uniform convergence guarantee on the generalization error of interpolators in an arbitrary hypothesis class in terms of the class's Gaussian width. Applying the generic bound to Euclidean norm balls recovers the consistency result of Bartlett et al. (2020) for minimum-norm interpolators, and confirms a prediction of Zhou et al. ( 2020 ) for near-minimal-norm interpolators in the special case of Gaussian data. We demonstrate the generality of the bound by applying it to the simplex, obtaining a novel consistency result for minimum 1 -norm interpolators (basis pursuit). Our results show how norm-based generalization bounds can explain and be used to analyze benign overfitting, at least in some settings. * These authors contributed equally. 1 Negrea et al. (2020) argue that Bartlett et al. (2020)'s proof technique is fundamentally based on uniform convergence of a surrogate predictor; Yang et al. (2021) study a closely related setting with a uniform convergence-type argument, but do not establish consistency. We discuss both papers in more detail in Section 4. 35th Conference on Neural Information Processing Systems (NeurIPS 2021).