We present a unified technique for sequential estimation of convex divergences between distributions, including integral probability metrics like the kernel maximum mean discrepancy, $\varphi$-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.

翻译：我们提出了一种统一的序贯估计技术，用于估计分布之间的凸散度，包括积分概率度量（如核最大均值差异）、φ-散度（如KL散度）以及最优传输成本（如Wasserstein距离的幂）。该技术通过观察到经验凸散度关于可交换滤流是（偏序）逆鞅，并结合此类过程的极大不等式来实现。这些方法似乎是对现有置信序列与凸散度文献的有力补充。我们构建了一个离线性到序贯性的转换机制，可将大量现有的离线集中不等式转化为可连续监测的时间一致置信序列，从而在任意停止时间提供有效的检验或置信区间。所得到的序贯界仅需付出相对于固定时间界的迭代对数代价，同时保持对问题参数（如适用时的维度或字母集大小）的相同依赖性。这些结果同样适用于更一般的凸泛函（如负微分熵、经验过程上确界及V-统计量），以及满足关键留一性质的一般过程。