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.

翻译：我们研究在次指数噪声下从噪声观测中学习Besov空间上的非参数回归问题，旨在获得关于积分平方误差的极小极大最优保证，该保证以高概率成立并自适应于未知噪声水平。为此，我们提出一种基于小波的在线学习算法，该算法通过将观测到的梯度噪声自适应地裁剪至适当水平来动态调整，从而无需调整噪声方差或梯度边界等参数。作为分析的副产品，我们推导出与竞争者$\ell_1$范数成比例的高概率自适应遗憾界。最后，在批量统计设定中，通过精细的在线到批量转换，我们获得了Besov空间的自适应且极小极大最优的估计速率。该方法巧妙地结合平方损失的结构与自归一化集中不等式。