%AZhou, L.%AKoehler, F.%ASur, P.%ASutherland, D.%ASrebro, N.%D2022%I
%K
%MOSTI ID: 10423426
%PMedium: X
%TA Non-Asymptotic Moreau Envelope Theory for High-Dimensional Generalized Linear Models
%XWe prove a new generalization bound that shows for any class of linear predictors in Gaussian space, the Rademacher complexity of the class and the training error under any continuous loss ℓ can control the test error under all Moreau envelopes of the loss ℓ . We use our finite-sample bound to directly recover the “optimistic rate” of Zhou et al. (2021) for linear regression with the square loss, which is known to be tight for minimal ℓ2-norm interpolation, but we also handle more general settings where the label is generated by a potentially misspecified multi-index model. The same argument can analyze noisy interpolation of max-margin classifiers through the squared hinge loss, and establishes consistency results in spiked-covariance settings. More generally, when the loss is only assumed to be Lipschitz, our bound effectively improves Talagrand’s well-known contraction lemma by a factor of two, and we prove uniform convergence of interpolators (Koehler et al. 2021) for all smooth, non-negative losses. Finally, we show that application of our generalization bound using localized Gaussian width will generally be sharp for empirical risk minimizers, establishing a non-asymptotic Moreau envelope theory
Country unknown/Code not availableOSTI-MSA