Generalized Resilience and Robust Statistics
Generalized Resilience and Robust Statistics
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DOI：
10.1214/22-aos2186
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发表时间：
2019-09
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影响因子：
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通讯作者：
Banghua Zhu;Jiantao Jiao;J. Steinhardt
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文献类型：
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作者：
Banghua Zhu;Jiantao Jiao;J. Steinhardt
稳健统计传统上关注的是离群值，即总变化距离的扰动。然而，数据集可能会以许多其他方式损坏，例如系统测量误差和丢失协变量。我们将稳健统计方法推广到考虑任意Wasserstein距离下的扰动，并证明了当分布的总体统计量在某一族友好扰动下是稳健的时，稳健估计是可能的。这推广了以前在有异常值的均值估计的特殊情况下使用的称为弹性的性质。我们证明了广义弹性性质在矩或超压缩条件下成立。即使在全变差的情况下，这些文献中用于均值估计、回归和协方差估计的条件也包含在内；由此产生的分析在总体极限和有限样本率方面简化了这些已知结果，有时还改进了它们。我们的稳健估计基于最小距离泛函(Donoho and Liu，1988)，该泛函在与扰动相关的偏差下投影到一组分布上。我们提出了两种设计具有良好有限采样率的MD估计量的方法：弱化偏差和扩展分布集。我们还介绍了与高等人的联系。(2019年)S最近分析了生成性对抗性网络的稳健估计。
Robust statistics traditionally focuses on outliers, or perturbations in total variation distance. However, a dataset could be corrupted in many other ways, such as systematic measurement errors and missing covariates. We generalize the robust statistics approach to consider perturbations under any Wasserstein distance, and show that robust estimation is possible whenever a distribution's population statistics are robust under a certain family of friendly perturbations. This generalizes a property called resilience previously employed in the special case of mean estimation with outliers. We justify the generalized resilience property by showing that it holds under moment or hypercontractive conditions. Even in the total variation case, these subsume conditions in the literature for mean estimation, regression, and covariance estimation; the resulting analysis simplifies and sometimes improves these known results in both population limit and finite-sample rate. Our robust estimators are based on minimum distance (MD) functionals (Donoho and Liu, 1988), which project onto a set of distributions under a discrepancy related to the perturbation. We present two approaches for designing MD estimators with good finite-sample rates: weakening the discrepancy and expanding the set of distributions. We also present connections to Gao et al. (2019)'s recent analysis of generative adversarial networks for robust estimation.