Time-Uniform Confidence Spheres for Means of Random Vectors

Abstract

We study sequential mean estimation in $\mathbb{R}^d$. In particular, we derive time-uniform confidence spheres---\emph{confidence sphere sequences} (CSSs)---which contain the mean of random vectors with high probability simultaneously across all sample sizes. Our results include a dimension-free CSS for log-concave random vectors, a dimension-free CSS for sub-Gaussian random vectors, and CSSs for sub-$\psi$ random vectors (which includes sub-gamma, and sub-exponential distributions). Many of our results are optimal. For sub-Gaussian distributions we also provide a CSS which tracks a time-varying mean, generalizing Robbins' mixture approach to the multivariate setting. Finally, we provide several CSSs for heavy-tailed random vectors (two moments only). Our bounds hold under a martingale assumption on the mean and do not require that the observations be iid. Our work is based on PAC-Bayesian theory and inspired by an approach of Catoni and Giulini.