Matuszewska–Orlicz indices of the Sobolev conjugate Young function

Arriagada, Waldo

doi:10.1016/j.padiff.2021.100029

Cited by 4 publications

(2 citation statements)

References 7 publications

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“…Intuitively, this condition implies that the norms are in H ψ and H φ are weaker than those in our hypothesis class H K , allowing us to study convergence in these larger spaces of sample estimators constructed in H K . The second part of the assumption requiring t φ(t) and φ(t) ψ(t) to be nondecreasing extends this behavior as t → ∞; collectively the two growth conditions characterize 1 φ and 1 ψ as resembling a Young's function; a requirement commonly imposed in the study of Orlicz spaces [3]. The requirement that t φ(t) is additionally concave enables the application of a Gagliardo-Nirenberg type interpolation inequality [24] that relates L ∞ norms to those in H K and L 2 (ν), which will be crucial in our analysis of uniform error rates.…”

Section: Assumptionsmentioning

confidence: 95%

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PAC-Bayesian Based Adaptation for Regularized Learning

Talwai¹,

Simchi‐Levi²

2022

Preprint

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In this paper, we propose a PAC-Bayesian a posteriori parameter selection scheme for adaptive regularized regression in Hilbert scales under general, unknown source conditions. We demonstrate that our approach is adaptive to misspecification, and achieves the optimal learning rate under subgaussian noise. Unlike existing parameter selection schemes, the computational complexity of our approach is independent of sample size. We derive minimax adaptive rates for a new, broad class of Tikhonov-regularized learning problems under general, misspecified source conditions, that notably do not require any conventional a priori assumptions on kernel eigendecay. Using the theory of interpolation, we demonstrate that the spectrum of the Mercer operator can be inferred in the presence of "tight" L ∞ embeddings of suitable Hilbert scales. Finally, we prove, that under a ∆ 2 condition on the smoothness index functions, our PAC-Bayesian scheme can indeed achieve minimax rates. We discuss applications of our approach to statistical inverse problems and oracle-efficient contextual bandit algorithms.

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Section: Assumptionsmentioning

confidence: 95%

“…where the last inequality follows from the fact that F d κ(•) is nonincreasing. Therefore, since ψ satisfies the ∆ 2 condition (2), and Ψ has finite nonzero extension indices (see p.274 of [3] for a definition), we have by Lemma 2.16 in…”

Section: A Proof Of Theorem 1 and Lemmamentioning

confidence: 97%