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

arXiv: https://arxiv.org/abs/2602.11747 · HTML: https://mlsanity.com/paper/2602.11747/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 32 references that are in the map.
- First posted on arXiv 12 Feb 2026.

## Builds on (most cited first, 18 of 18)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Coin Betting and Parameter-Free Online Learning](https://mlsanity.com/paper/1602.04128/) (2016)
- [Black-Box Reductions for Parameter-free Online Learning in Banach Spaces](https://mlsanity.com/paper/1802.06293/) (2018)
- [A Second-order Bound with Excess Losses](https://mlsanity.com/paper/1402.2044/) (2014)
- [Lipschitz and Comparator-Norm Adaptivity in Online Learning](https://mlsanity.com/paper/2002.12242/) (2020)
- [Impossible Tuning Made Possible: A New Expert Algorithm and Its Applications](https://mlsanity.com/paper/2102.01046/) (2021)
- [Second-order Quantile Methods for Experts and Combinatorial Games](https://mlsanity.com/paper/1502.08009/) (2015)
- [Parameter-free Mirror Descent](https://mlsanity.com/paper/2203.00444/) (2022)
- [Lipschitz Adaptivity with Multiple Learning Rates in Online Learning](https://mlsanity.com/paper/1902.10797/) (2019)
- [Online Learning Without Prior Information](https://mlsanity.com/paper/1703.02629/) (2017)
- [Optimal learning with Bernstein Online Aggregation](https://mlsanity.com/paper/1404.1356/) (2014)
- [Parameter-Free Online Convex Optimization with Sub-Exponential Noise](https://mlsanity.com/paper/1902.01500/) (2019)
- [Parameter-free Regret in High Probability with Heavy Tails](https://mlsanity.com/paper/2210.14355/) (2022)
- [Empirical entropy, minimax regret and minimax risk](https://mlsanity.com/paper/1308.1147/) (2013)
- [High-Probability Risk Bounds via Sequential Predictors](https://mlsanity.com/paper/2308.07588/) (2023)
- [Stochastic Online Convex Optimization. Application to probabilistic time series forecasting](https://mlsanity.com/paper/2102.00729/) (2021)
- [Optimal Recovery Meets Minimax Estimation](https://mlsanity.com/paper/2502.17671/) (2025)
- [Minimax Adaptive Online Nonparametric Regression over Besov Spaces](https://mlsanity.com/paper/2505.19741/) (2025)

## Abstract (arXiv)

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 ℓ1-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.