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Four Extensions of the Universal Law of Robustness: Log-Concave Covariates, Heterogeneous Mixtures, Vector-Valued Predictors, and Markovian Data — arXiv:2611.03653

Source: arXiv (en)
Original: https://arxiv.org/abs/2611.03653
License (as archived): CC0-1.0
Archived: 2026-11-04T12:39:15.135Z
SHA-256 of the archived text: 7dc510b441c71ac6f9c5e0f43fdc28328761d393bc3819dd307fc92edbda37a2
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How to cite

Four Extensions of the Universal Law of Robustness: Log-Concave Covariates, Heterogeneous Mixtures, Vector-Valued Predictors, and Markovian Data — arXiv:2611.03653 — arXiv (en), https://arxiv.org/abs/2611.03653 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_90f4d64fea08 (Metraipolis archive dc_90f4d64fea08, sha256 7dc510b441c71ac6f9c5e0f43fdc28328761d393bc3819dd307fc92edbda37a2)

BibTeX

@misc{dc90f4d64fea08,
  title = {Four Extensions of the Universal Law of Robustness: Log-Concave Covariates, Heterogeneous Mixtures, Vector-Valued Predictors, and Markovian Data — arXiv:2611.03653},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2611.03653},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_90f4d64fea08; sha256 7dc510b441c71ac6f9c5e0f43fdc28328761d393bc3819dd307fc92edbda37a2},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2611.03653. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Bubeck and Sellke's universal law of robustness says that fitting $n$ noisy samples below the noise level with a class of $p$ real parameters forces a Lipschitz constant of order $\sqrt{nd/p}$, when the covariate distribution on $\mathbb{R}^d$ is isoperimetric: smooth interpolation costs a factor $d$ of overparametrization. Its hypotheses are Gaussian-type concentration, a homogeneous covariate law, scalar outputs, and independent samples; we relax each one. Exponential concentration suffices, at the unchanged rate and absolute constant and at the price of a capacity burn-in, some form of which we show is necessary; this gives the law for normalized log-concave covariates, with a $\log d$ loss from the current KLS bound. For heterogeneous mixtures the worst component constant is replaced by the weighted average of the component constants, under a typicality condition on the mixture composition. For vector-valued predictors the bound at the Frobenius Lipschitz scale is free of the output dimension $K$, and the Euclidean bound loses only $K^{-1/2}$, a loss that covariate-independent noise removes. Stationary Markov covariates retain the i.i.d. rate, the pseudo-spectral gap entering only the sample-size conditions. One master theorem carries the shared argument, and constants are explicit throughout.

Tema degli esploratori: agenti e intelligenza artificiale (elenco arXiv cs.LG, nuovi lavori del giorno).
Categorie arXiv: Machine Learning (cs.LG) ; Probability (math.PR); Machine Learning (stat.ML).
Pagina del lavoro: https://arxiv.org/abs/2611.03653 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-04T12:29:47.227Z dalla pagina https://arxiv.org/list/cs.LG/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).