论文标题
强大的线性预测：均匀浓度，快速速率和模型错误指定的分析
Robust Linear Predictions: Analyses of Uniform Concentration, Fast Rates and Model Misspecification
论文作者
论文摘要
在相当广义的框架下，过去一个世纪的线性预测问题已经进行了广泛的研究。强大的统计文献的最新进展使我们能够通过平均值（MOM）的棱镜分析经典线性模型的强大版本。以零碎的方式将这些方法结合起来可能会导致临时程序，并且限制的理论结论是，基础每个人的贡献可能不再有效。为了一致地应对这些挑战，在这项研究中，我们提供了一个统一的健壮框架，其中包括希尔伯特空间上的各种线性预测问题，再加上一般的损失功能类别。值得注意的是，我们不需要关于偏远数据点（$ \ MATHCAL {O} $）的分布的任何假设，也不需要对Inlying的数据（$ \ Mathcal {i} $）的紧凑性。在双重规范的轻度条件下，我们表明，对于错误规定级别$ε$，这些估计器的错误率为$ O（\ max \ left \ {| \ Mathcal {o} |^{1/2} n^{1/2} n^{ - 1/1/2}，|文献中最著名的率。此速率比$ o（n^{ - 1/2}）$的经典费率稍慢，这表明我们需要按错误率支付价格才能获得可靠的估计。此外，我们表明可以提高此速率以在其他假设下实现所谓的“快速率”。
The problem of linear predictions has been extensively studied for the past century under pretty generalized frameworks. Recent advances in the robust statistics literature allow us to analyze robust versions of classical linear models through the prism of Median of Means (MoM). Combining these approaches in a piecemeal way might lead to ad-hoc procedures, and the restricted theoretical conclusions that underpin each individual contribution may no longer be valid. To meet these challenges coherently, in this study, we offer a unified robust framework that includes a broad variety of linear prediction problems on a Hilbert space, coupled with a generic class of loss functions. Notably, we do not require any assumptions on the distribution of the outlying data points ($\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.
