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.
Submission Length: Long submission (more than 12 pages of main content)
Changes Since Last Submission: Addressing the reviewer's comments, including:
- An example result in the introduction (Equation 3)
- Table 2 which summarizes the relationship between our results and those in the literature
- Fixing various typos
- Differentiating super-Gaussian and non-super-Gaussian $\psi$-functions
Assigned Action Editor: ~Benjamin_Guedj1
Submission Number: 4272
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