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

We consider interpolation learning in high-dimensional linear regression with\nGaussian data, and prove a generic uniform convergence guarantee on the\ngeneralization error of interpolators in an arbitrary hypothesis class in terms\nof the class's Gaussian width. Applying the generic bound to Euclidean norm\nballs recovers the consistency result of Bartlett et al. (2020) for\nminimum-norm interpolators, and confirms a prediction of Zhou et al. (2020) for\nnear-minimal-norm interpolators in the special case of Gaussian data. We\ndemonstrate the generality of the bound by applying it to the simplex,\nobtaining a novel consistency result for minimum l1-norm interpolators (basis\npursuit). Our results show how norm-based generalization bounds can explain and\nbe used to analyze benign overfitting, at least in some settings.\n