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- W4307206716 abstract "We 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 $ell$ can control the test error under all Moreau envelopes of the loss $ell$. 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 $ell_2$-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 for generalization that applies outside of proportional scaling regimes, handles model misspecification, and complements existing asymptotic Moreau envelope theories for M-estimation." @default.
- W4307206716 created "2022-10-30" @default.
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- W4307206716 date "2022-10-21" @default.
- W4307206716 modified "2023-10-14" @default.
- W4307206716 title "A Non-Asymptotic Moreau Envelope Theory for High-Dimensional Generalized Linear Models" @default.
- W4307206716 doi "https://doi.org/10.48550/arxiv.2210.12082" @default.
- W4307206716 hasPublicationYear "2022" @default.
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