A confidence sequence (CS) is a sequence of confidence intervals that is valid at arbitrary data-dependent stopping times. These are useful in applications like A/B testing, multi-armed bandits, off-policy evaluation, election auditing, etc. We present three approaches to constructing a confidence sequence for the population mean, under the minimal assumption that only an upper bound $\sigma^2$ on the variance is known. While previous works rely on light-tail assumptions like boundedness or subGaussianity (under which all moments of a distribution exist), the confidence sequences in our work are able to handle data from a wide range of heavy-tailed distributions. The best among our three methods -- the Catoni-style confidence sequence -- performs remarkably well in practice, essentially matching the state-of-the-art methods for $\sigma^2$-subGaussian data, and provably attains the $\sqrt{\log \log t/t}$ lower bound due to the law of the iterated logarithm. Our findings have important implications for sequential experimentation with unbounded observations, since the $\sigma^2$-bounded-variance assumption is more realistic and easier to verify than $\sigma^2$-subGaussianity (which implies the former). We also extend our methods to data with infinite variance, but having $p$-th central moment ($1<p<2$).

翻译：置信序列（CS）是一组在任意数据依赖的停止时间下均有效的置信区间序列，在A/B测试、多臂老虎机、离线策略评估、选举审计等应用中具有重要价值。本文提出三种在仅已知方差上界$\sigma^2$的最小假设下构建总体均值置信序列的方法。以往研究依赖轻尾假设（如有界性或次高斯性，这些假设保证分布所有矩存在），而本文构建的置信序列能够处理来自广泛重尾分布的数据。三种方法中性能最优的卡托尼型置信序列在实践中表现卓越，其对$\sigma^2$-次高斯型数据的效果与最先进方法相当，并可通过迭代对数定律证明达到$\sqrt{\log \log t/t}$的渐近下界。由于$\sigma^2$有界方差假设比$\sigma^2$-次高斯性（后者蕴含前者）更符合实际且易于验证，该发现对处理无界观测值的序贯实验具有重要参考意义。我们还将方法推广到具有有限$p$阶中心矩（$1<p<2$）但方差无限的数据场景。