Optimal Recovery Meets Minimax Estimation

TL;DR

The paper addresses the problem of recovering a Besov-class function $f$ on a bounded domain from $m$ Gaussian-noise observations in the $L_q$-norm. It introduces noise-level-aware minimax rates, showing that the optimal error scales as $R_m(K,\sigma)_q \asymp m^{-{s}/{d}+(1/p-1/q)_+}+\min\{1,(\sigma^2/m)^{s/(2s+d)}\}$, and provides matching upper and lower bounds. A novel recovery algorithm based on piecewise polynomials with hard-thresholding avoids wavelets and boundary complications, and yields both high-probability and expectation guarantees that recover the optimal rate as $\sigma\to 0$, thereby bridging minimax estimation and optimal recovery. The results advance understanding of how estimation complexity transitions between high- and low-noise regimes and offer a practical, boundary-friendly method for Besov-function estimation in $L_q$.

Abstract

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 $σ^2$ 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 $L_q$-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.

Theorems & Definitions (31)

• Theorem 1
• Remark 1
• Theorem 2
• Theorem 3
• Remark 2
• Lemma 4
• Lemma 5
• proof
• Lemma 6
• proof