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

We study nonparametric regression over Besov spaces from noisy observations under sub-exponential noise. Our goal is to obtain minimax-optimal high-probability bounds for the integrated squared error while adapting to the unknown noise level and the regularity parameters of the underlying Besov class. We introduce a wavelet-based online learning algorithm that sequentially processes noisy gradients and adapts to the gradient noise through an adaptive clipping rule, thus avoiding 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, our method is the first to achieve high-probability minimax-optimal estimation rates over Besov spaces while adapting to all problem parameters, including the noise level. Our approach relies on a refined online-to-batch conversion and exploits the structure of the squared loss in combination with self-normalized concentration inequalities.