Catoni-style confidence sequences for heavy-tailed mean estimation

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