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：置信序列是指对任意依赖于数据的停止时间，均有效的置信区间序列。该工具在A/B测试、多臂赌博机、离线策略评估、选举审计等应用中具有重要价值。本文提出三种构建总体均值置信序列的方法，其最小假设条件为仅已知方差上界$\sigma^2$。现有研究通常依赖有界性或次高斯性等轻尾假设（该假设下分布所有矩均存在），而本文提出的置信序列能够有效处理各类重尾分布数据。三种方法中表现最优的"卡托尼式置信序列"在实践中效果显著，不仅基本达到$\sigma^2$-次高斯数据最优方法的性能水平，还能基于重对数律证明其收敛速度达到$\sqrt{\log \log t/t}$的下界。由于$\sigma^2$有界方差假设比$\sigma^2$-次高斯性（包含前者作为特例）更贴近实际且易于验证，本研究成果对无界观测值的序贯实验具有重要启示。此外，我们将方法推广至方差无限但存在$p$阶中心矩（$1<p<2$）的数据场景。