鲁棒线性预测：均匀集中、快速速率和模型误设的分析
Robust Linear Predictions: Analyses of Uniform Concentration, Fast Rates and Model Misspecification
AI总结：
提出希尔伯特空间上统一鲁棒线性预测框架，不要求异常点分布假设，实现误差率匹配最优已知速率，并可在附加条件下达到快速速率。
摘要：
过去一个世纪，线性预测问题在相当广义的框架下得到了广泛研究。鲁棒统计文献的最新进展使我们能够通过均值中位数（MoM）的视角分析经典线性模型的鲁棒版本。将这些方法零碎地组合可能导致临时性程序，且支撑每个单独贡献的受限理论结论可能不再有效。为了连贯地应对这些挑战，在本研究中，我们提供了一个统一的鲁棒框架，该框架包括希尔伯特空间上的广泛线性预测问题，并耦合了一类通用的损失函数。值得注意的是，我们不需要对异常数据点（$\mathcal{O}$）的分布或正常数据点（$\mathcal{I}$）支撑集的紧性做任何假设。在对偶范数的温和条件下，我们证明对于误设水平 $\epsilon$，这些估计量实现了 $O(\max\left\{|\mathcal{O}|^{1/2}n^{-1/2}, |\mathcal{I}|^{1/2}n^{-1} \right\}+\epsilon)$ 的误差率，与文献中最优已知速率相匹配。该速率略慢于经典速率 $O(n^{-1/2})$，表明我们需要在误差率方面付出代价以获得鲁棒估计。此外，我们证明在附加假设下，该速率可以改进以实现所谓的“快速速率”。
英文摘要：
The problem of linear predictions has been extensively studied for the past century under pretty generalized frameworks. Recent advances in the robust statistics literature allow us to analyze robust versions of classical linear models through the prism of Median of Means (MoM). Combining these approaches in a piecemeal way might lead to ad-hoc procedures, and the restricted theoretical conclusions that underpin each individual contribution may no longer be valid. To meet these challenges coherently, in this study, we offer a unified robust framework that includes a broad variety of linear prediction problems on a Hilbert space, coupled with a generic class of loss functions. Notably, we do not require any assumptions on the distribution of the outlying data points ($\mathcal{O}$) nor the compactness of the support of the inlying ones ($\mathcal{I}$). Under mild conditions on the dual norm, we show that for misspecification level $ε$, these estimators achieve an error rate of $O(\max\left\{|\mathcal{O}|^{1/2}n^{-1/2}, |\mathcal{I}|^{1/2}n^{-1} \right\}+ε)$, matching the best-known rates in literature. This rate is slightly slower than the classical rates of $O(n^{-1/2})$, indicating that we need to pay a price in terms of error rates to obtain robust estimates. Additionally, we show that this rate can be improved to achieve so-called "fast rates" under additional assumptions.
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