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arXiv · 2412.14744

Finite Sample Bounds for Non-Parametric Regression: Optimal Sample Efficiency and Space Complexity

Davide Maran·Marcello Restelli
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Abstract

We address the problem of learning an unknown smooth function and its derivatives from noisy pointwise evaluations under the supremum norm. While classical nonparametric regression provides a strong theoretical foundation, traditional kernel-based estimators often incur high computational costs and memory requirements that scale with the sample size, limiting their utility in real-time applications such as reinforcement learning. To overcome these challenges, we propose a parametric approach based on a finite-dimensional representation that achieves minimax-optimal uniform convergence rates. Our method enables lightweight inference without storing all samples in memory. We provide sharp finite-sample bounds under sub-Gaussian noise, derive second-order Bernstein-type guarantees, and prove matching lower bounds, thereby confirming the optimality of our approach in both estimation error and memory efficiency.

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Davide Maran, Marcello Restelli. 2026-03-09. Finite Sample Bounds for Non-Parametric Regression: Optimal Sample Efficiency and Space Complexity. https://arxiv.org/abs/2412.14744

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