Optimal Recovery Meets Bayesian Learning: Where Worst-Case Bounds Pay Off
Abstract
Worst-case Optimal Recovery (OR) and Bayesian learning describe the same Gaussian-quadratic-Hilbert problems in two vocabularies. We sharpen the correspondence - the radius of information equals a nugget-optimized GP posterior variance and is attained by the posterior mean at a closed-form balance nugget - and measure, inside three published Bayesian systems, where the worst-case side pays. The ledger is two-sided: the losses instruct as much as the wins. Morozov calibration tracks a test-access oracle within $1.00−1.19\times$ where σ-blind rules fail, is $4.9−6.3\times$ more reproducible across noise draws (p=0.002-$0.004$), and is the only deployable rule whose selection survives a change of backend (1.36× against $12−30\times$ for the released weight, ML-II and GCV); tight certificates cover at the information-theoretic floor with no numerical slack. But on exchangeable data split-conformal beats the OR head on interval score, a water-filling prior adds nothing without an oracle noise hint, and under covariate shift the OR band keeps coverage on every dataset yet loses interval score to split-conformal, and to a feature-free constant band, on most cells; what pays is not shift but shift on a learnable target, which a training-free audit statistic predicts before any model is fitted. In Bayesian optimization the certified width is a validity floor whose scalar inflation we prove inert under a checkable margin condition and check at every step. Inertness is graded, not binary, and in the size of the inflation as much as in the objective: κ=2 is inert wherever κ=5 is and on more cells besides, while κ=5 moves half the Ackley seeds and every Griewank seed. Exploration is a shape problem, not a scale one. The design rule: match the guarantee tool to the data regime, and audit the regime first.
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BibTeX
@misc{verbii2026optimal,
title = {{Optimal Recovery Meets Bayesian Learning: Where Worst-Case Bounds Pay Off}},
author = {Gordei Verbii},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29622},
eprint = {2609.29622},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29622},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal Recovery Meets Bayesian Learning: Where Worst-Case Bounds Pay Off
AU - Gordei Verbii
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.29622
UR - https://arxiv.org/abs/2609.29622
AB - Worst-case Optimal Recovery (OR) and Bayesian learning describe the same Gaussian-quadratic-Hilbert problems in two vocabularies. We sharpen the correspondence - the radius of information equals a nugget-optimized GP posterior variance and is attained by the posterior mean at a closed-form balance nugget - and measure, inside three published Bayesian systems, where the worst-case side pays. The ledger is two-sided: the losses instruct as much as the wins. Morozov calibration tracks a test-access oracle within $1.00−1.19\times$ where σ-blind rules fail, is $4.9−6.3\times$ more reproducible across noise draws (p=0.002-$0.004$), and is the only deployable rule whose selection survives a change of backend (1.36× against $12−30\times$ for the released weight, ML-II and GCV); tight certificates cover at the information-theoretic floor with no numerical slack. But on exchangeable data split-conformal beats the OR head on interval score, a water-filling prior adds nothing without an oracle noise hint, and under covariate shift the OR band keeps coverage on every dataset yet loses interval score to split-conformal, and to a feature-free constant band, on most cells; what pays is not shift but shift on a learnable target, which a training-free audit statistic predicts before any model is fitted. In Bayesian optimization the certified width is a validity floor whose scalar inflation we prove inert under a checkable margin condition and check at every step. Inertness is graded, not binary, and in the size of the inflation as much as in the objective: κ=2 is inert wherever κ=5 is and on more cells besides, while κ=5 moves half the Ackley seeds and every Griewank seed. Exploration is a shape problem, not a scale one. The design rule: match the guarantee tool to the data regime, and audit the regime first.
ER -