We present a unified technique for sequential estimation of convex divergences between distributions, including integral probability metrics like the kernel maximum mean discrepancy, $φ$-divergences like the Kullback-Leibler divergence, and optimal transport costs, such as powers of Wasserstein distances. This is achieved by observing that empirical convex divergences are (partially ordered) reverse submartingales with respect to the exchangeable filtration, coupled with maximal inequalities for such processes. These techniques appear to be complementary and powerful additions to the existing literature on both confidence sequences and convex divergences. We construct an offline-to-sequential device that converts a wide array of existing offline concentration inequalities into time-uniform confidence sequences that can be continuously monitored, providing valid tests or confidence intervals at arbitrary stopping times. The resulting sequential bounds pay only an iterated logarithmic price over the corresponding fixed-time bounds, retaining the same dependence on problem parameters (like dimension or alphabet size if applicable). These results are also applicable to more general convex functionals -- like the negative differential entropy, suprema of empirical processes, and V-Statistics -- and to more general processes satisfying a key leave-one-out property.
Global Sequential Testing for Multi-Stream Auditing
Mathematical Institute for Data Science (MINDS), Johns Hopkins University（数据科学数学研究所（MINDS），约翰霍普金斯大学）；Department of Biomedical Engineering, Johns Hopkins University（生物医学工程系，约翰霍普金斯大学）；Amazon Responsible AI（亚马逊负责任人工智能）
Reference-Null Calibrated Thresholds for E-Processes with Applications to Conformal Martingales
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；Institute of Statistics and Big Data, Renmin University of China（中国人民大学统计与大数据研究院）
