Time-Uniform Confidence Spheres for Means of Random Vectors

Ben Chugg; Hongjian Wang; Aaditya Ramdas

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arXiv · 2311.08168

Time-Uniform Confidence Spheres for Means of Random Vectors

Ben Chugg·Hongjian Wang·Aaditya Ramdas

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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.

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BibTeX RIS

Ben Chugg, Hongjian Wang, Aaditya Ramdas. 2023-11-14. Time-Uniform Confidence Spheres for Means of Random Vectors. https://arxiv.org/abs/2311.08168

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