# Optimal Recovery Meets Minimax Estimation

arXiv: https://arxiv.org/abs/2502.17671 · HTML: https://mlsanity.com/paper/2502.17671/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 41 references that are in the map.
- First posted on arXiv 24 Feb 2025.

## Builds on (most cited first, 1 of 1)

- [High-probability minimax lower bounds](https://mlsanity.com/paper/2406.13447/) (2024)

## Built on by (most cited first, 4 of 4)

- [Towards Sharp Minimax Risk Bounds for Operator Learning](https://mlsanity.com/paper/2512.17805/) (2025)
- [A short tour of operator learning theory: Convergence rates, statistical limits, and open questions](https://mlsanity.com/paper/2603.00819/) (2026)
- [High-Probability Minimax Adaptive Estimation in Besov Spaces via Online-to-Batch](https://mlsanity.com/paper/2602.11747/) (2026)
- [Into the danger zone: stable extrapolation in high-dimensional function and operator learning](https://mlsanity.com/paper/2609.36709/) (2026)

## Abstract (arXiv)

A fundamental problem in statistics and machine learning is to estimate a function f from possibly noisy observations of its point samples. The goal is to design a numerical algorithm to construct an approximation hat f to f in a prescribed norm that asymptotically achieves the best possible error (as a function of the number m of observations and the variance σ2 of the noise). This problem has received considerable attention in both nonparametric statistics (noisy observations) and optimal recovery (noiseless observations). Quantitative bounds require assumptions on f, known as model class assumptions. Classical results assume that f is in the unit ball of a Besov space. In nonparametric statistics, the best possible performance of an algorithm for finding hat f is known as the minimax rate and has been studied in this setting under the assumption that the noise is Gaussian. In optimal recovery, the best possible performance of an algorithm is known as the optimal recovery rate and has also been determined in this setting. While one would expect that the minimax rate recovers the optimal recovery rate when the noise level σ tends to zero, it turns out that the current results on minimax rates do not carefully determine the dependence on σ and the limit cannot be taken. This paper handles this issue and determines the noise-level-aware (NLA) minimax rates for Besov classes when error is measured in an Lq-norm with matching upper and lower bounds. The end result is a reconciliation between minimax rates and optimal recovery rates. The NLA minimax rate continuously depends on the noise level and recovers the optimal recovery rate when σ tends to zero.