High-Probability Minimax Adaptive Estimation in Besov Spaces via Online-to-Batch

TL;DR

This paper introduces a wavelet-based online learning algorithm that adaptively estimates functions in Besov spaces from noisy data, achieving minimax-optimal rates with high probability without tuning noise parameters.

Contribution

It proposes a novel online-to-batch method for adaptive minimax estimation in Besov spaces that automatically adjusts to unknown noise levels and provides high-probability guarantees.

Findings

Achieves minimax-optimal estimation rates in Besov spaces.

Provides high-probability adaptive regret bounds.

Eliminates the need for tuning noise variance parameters.

Abstract

We study nonparametric regression over Besov spaces from noisy observations under sub-exponential noise, aiming to achieve minimax-optimal guarantees on the integrated squared error that hold with high probability and adapt to the unknown noise level. To this end, we propose a wavelet-based online learning algorithm that dynamically adjusts to the observed gradient noise by adaptively clipping it at an appropriate level, eliminating the need to tune parameters such as the noise variance or gradient bounds. As a by-product of our analysis, we derive high-probability adaptive regret bounds that scale with the $\ell_1$-norm of the competitor. Finally, in the batch statistical setting, we obtain adaptive and minimax-optimal estimation rates for Besov spaces via a refined online-to-batch conversion. This approach carefully exploits the structure of the squared loss in combination with self-normalized concentration inequalities.