Mean embeddings provide an extremely flexible and powerful tool in machine learning and statistics to represent probability distributions and define a semi-metric (MMD, maximum mean discrepancy; also called N-distance or energy distance), with numerous successful applications. The representation is constructed as the expectation of the feature map defined by a kernel. As a mean, its classical empirical estimator, however, can be arbitrary severely affected even by a single outlier in case of unbounded features. To the best of our knowledge, unfortunately even the consistency of the existing few techniques trying to alleviate this serious sensitivity bottleneck is unknown. In this paper, we show how the recently emerged principle of median-of-means can be used to design estimators for kernel mean embedding and MMD with excessive resistance properties to outliers, and optimal sub-Gaussian deviation bounds under mild assumptions.
Generalization Bounds in the Presence of Outliers: a Median-of-Means Study
Università degli Studi di Milano（米兰大学）；Télécom Paris, Institut Polytechnique de Paris（巴黎综合理工学院电信学院）
与经验平均值相反，均值中位数（MoM）是平方可积r. v. $Z$的平均值$θ$的估计量，即使$Z$不表现出亚高斯尾部行为，也可以围绕其建立精确的非渐近置信界限。由于它在重尾数据上实现了高置信度，MoM在机器学习中找到了各种应用，用于设计对非典型观测不敏感的训练程序。最近，一项新的工作正在试图描述和利用MoM处理损坏数据的能力。在这种情况下，目前的工作提出了一个一般性的研究MoM的浓度属性下的污染制度，提供了一个清晰的理解的影响的离群值的比例和选择的块的数量。该分析扩展到（多样本）$U$ -统计，即观察元组的平均值，由于诱导的依赖性而引起额外的挑战。最后，我们表明后面的界限可以以简单的方式用于推导污染环境中成对学习的泛化保证，并提出了一种计算可证明可靠的决策函数的算法。
Convex Clustering Redefined: Robust Learning with the Median of Means Estimator
Indian Statistical Institute, Kolkata（印度统计研究院，科希马）；Department of Statistics and Data Science, University of Texas at Austin（德克萨斯大学奥斯汀分校统计与数据科学系）；Electronics and Communication Sciences, Indian Statistical Institute, Kolkata（印度统计研究院，科希马电子与通信科学）；Department of Statistics, University of Oxford（牛津大学统计系）；Department of Statistics, University of Michigan（密歇根大学统计系）
Tracking the Median of Gradients with a Stochastic Proximal Point Method
Technical University of Munich and Inria Paris, ENS, PSL Research University（慕尼黑技术大学和巴黎Inria、ENS、PSL研究大学）；Université Paris Cité and Sorbonne Université, CNRS（巴黎国立大学和索邦大学、CNRS）；Inria, CNRS, ENS, PSL Research University Paris（Inria、CNRS、ENS、PSL研究大学巴黎）；CCM, Flatiron Institute, Simons Foundation（CCM、Flatiron研究所、Simons基金会）
SNAP: A Self-Consistent Agreement Principle with Application to Robust Computation
Faculty of Mathematics and Computer Science, University of Münster（数学与计算机科学学院，穆斯特大学）；Institute for Disaster Management and Reconstruction, Sichuan University-Hongkong Polytechnic University（灾害管理与重建研究所，四川大学-香港理工大学）
Robust and Efficient Aggregation for Distributed Learning
Imperial College London（帝国理工学院）；Technische Universität Darmstadt（达姆施塔特工业大学）；Department of Electrical and Electronic Engineering, Imperial College London（帝国理工学院电子与电气工程系）；Signal Processing Group, Technische Universität Darmstadt（达姆施塔特工业大学信号处理组）
