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）是一族在任意数据依赖停时下均保持有效性的置信区间序列，在A/B测试、多臂老虎机、离策略评估、选举审计等领域具有重要应用。本文在仅已知方差上界$\sigma^2$的最小假设条件下，提出三种构建总体均值置信序列的方法。现有研究通常依赖轻尾假设（如有界性、次高斯性，此时分布所有矩存在），而本文方法可处理来自广泛重尾分布族的数据。三种方法中性能最优的卡托尼式置信序列在实际应用中表现卓越，不仅与面向$\sigma^2$-次高斯数据的先进方法效果相当，而且依据迭代对数定律理论证实达到$\sqrt{\log \log t/t}$下界。本发现对含无界观测值的序贯实验具有重要启示，因为$\sigma^2$-有界方差假设相比$\sigma^2$-次高斯性（蕴含前者）更符合实际且更易验证。此外，我们将方法推广至具有$p$阶中心矩（$1<p<2$）的无限方差数据场景。