🤖 AI Summary
This paper addresses the sequential estimation of the mean of high-dimensional random vectors, aiming to construct time-uniform confidence spheres (CSS)—a sequence of confidence regions that simultaneously cover the true mean with high probability for all sample sizes. Methodologically, it introduces the first dimension-free CSS construction framework, integrating PAC-Bayesian theory, the Catoni–Giulini exponential martingale technique, generalized mixture priors, and adaptive truncation—applicable to non-i.i.d., non-stationary sequences. Key contributions include: (1) a unified treatment of log-concave, sub-Gaussian, sub-ψ, and heavy-tailed distributions with only second moments; (2) a multivariate, time-varying extension of Robbins’ mixture for dynamic mean tracking, yielding tight coverage under sub-Gaussianity; and (3) distribution-free guarantees with optimal or near-optimal bounds, ensuring strong robustness and broad applicability without i.i.d. assumptions.
📝 Abstract
We study sequential mean estimation in $mathbb{R}^d$. In particular, we derive time-uniform confidence spheres -- 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, sub-Poisson, 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.
