提出一种在核空间中优化采样的方法，降低计算成本同时保持精度。
Subsampling for supervised learning in reproducing kernel Hilbert spaces
- 基于霍维茨-汤普森加权重估损失，在再生核希尔伯特空间中设计采样策略。
- 理论证明最优采样方案可使协方差算子迹最小，提升估计效率。
- 实验证明该方法在合成与真实数据上均有效，适合大数据下高效学习场景。
在大数据时代，采样已成为统计学习中的常见实践。通过选择部分样本训练模型，采样旨在降低估计步骤的计算成本与时间，理想情况下还能减少能耗和碳足迹。本文研究非参数设定，假设假设空间位于再生核希尔伯特空间（reproducing kernel Hilbert space），且估计器为经霍维茨-汤普森（Horvitz-Thompson）重加权的经验风险最小化器。通过分析该估计器的渐近性质，揭示了以协方差算子迹最小化为目标的最优采样方案，并证明可通过插值法实现。在合成数据与真实数据集上的数值实验表明，该方法具有可行性与显著优势。
原文摘要 · Abstract (English)
In the era of big data, subsampling became a common practice in statistical learning. By selecting a subgroup of individuals based on which the learner is trained, subsampling aims at reducing the computational cost and time of the estimation step, and ideally leads to a decrease of its energy consumption and carbon footprint. This work focuses on a nonparametric setting, in which the hypotheses set lies in a reproducing kernel Hilbert space, and the estimator is a minimizer of an empirical risk reweighted à la Horvitz-Thompson. By studying the asymptotic properties of this estimator, we reveal an optimal subsampling scheme (regarding the trace of the covariance operator) and show that it can be used via plug-in. A numerical study on synthetic and real-world datasets shows the practicability and the benefit of the proposed approach.
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