提出轻量级非参数回归方法，实现最优精度与内存效率
Finite Sample Bounds for Non-Parametric Regression: Optimal Sample Efficiency and Space Complexity
- 用有限维参数表示替代传统核方法，降低存储开销
- 在子高斯噪声下达到最优的有限样本误差界
- 适合实时学习场景，如强化学习中的在线更新
我们研究在上确界范数下，从带噪声的点值观测中学习未知光滑函数及其导数的问题。经典非参数回归虽有良好理论基础，但传统基于核的方法常伴随随样本量增长的高计算成本和内存需求，限制了其在强化学习等实时应用中的使用。为此，我们提出一种基于有限维表示的参数化方法，实现了极小最大一致收敛速率。该方法支持轻量级推理，无需将所有样本存于内存中。我们在子高斯噪声条件下给出了紧致的有限样本界，推导出二阶伯恩斯坦型保证，并证明了匹配的下界，从而证实了该方法在估计误差与内存效率上的最优性。
原文摘要 · Abstract (English)
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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