# Time-Uniform Confidence Spheres for Means of Random Vectors

arXiv: https://arxiv.org/abs/2311.08168 · HTML: https://mlsanity.com/paper/2311.08168/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 82 references that are in the map.
- First posted on arXiv 14 Nov 2023.

## Builds on (most cited first, 25 of 25)

- [User-friendly introduction to PAC-Bayes bounds](https://mlsanity.com/paper/2110.11216/) (2021)
- [Time-uniform, nonparametric, nonasymptotic confidence sequences](https://mlsanity.com/paper/1810.08240/) (2018)
- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [Estimating means of bounded random variables by betting](https://mlsanity.com/paper/2010.09686/) (2020)
- [Mean estimation and regression under heavy-tailed distributions--a survey](https://mlsanity.com/paper/1906.04280/) (2019)
- [A Primer on PAC-Bayesian Learning](https://mlsanity.com/paper/1901.05353/) (2019)
- [Improved Optimistic Algorithms for Logistic Bandits](https://mlsanity.com/paper/2002.07530/) (2020)
- [A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Norm](https://mlsanity.com/paper/1902.03736/) (2019)
- [Robust Estimation via Robust Gradient Estimation](https://mlsanity.com/paper/1802.06485/) (2018)
- [Mixture Martingales Revisited with Applications to Sequential Tests and Confidence Intervals](https://mlsanity.com/paper/1811.11419/) (2018)
- [Sub-sampled Cubic Regularization for Non-convex Optimization](https://mlsanity.com/paper/1705.05933/) (2017)
- [Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Model](https://mlsanity.com/paper/1506.00691/) (2015)
- [Sub-Gaussian estimators of the mean of a random vector](https://mlsanity.com/paper/1702.00482/) (2017)
- [Robust and Heavy-Tailed Mean Estimation Made Simple, via Regret Minimization](https://mlsanity.com/paper/2007.15839/) (2020)
- [Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression](https://mlsanity.com/paper/1804.02605/) (2018)
- [Statistical Query Algorithms for Mean Vector Estimation and Stochastic Convex Optimization](https://mlsanity.com/paper/1512.09170/) (2015)
- [Time-uniform central limit theory and asymptotic confidence sequences](https://mlsanity.com/paper/2103.06476/) (2021)
- [A unified recipe for deriving (time-uniform) PAC-Bayes bounds](https://mlsanity.com/paper/2302.03421/) (2023)
- [PAC-Bayes Generalisation Bounds for Heavy-Tailed Losses through Supermartingales](https://mlsanity.com/paper/2210.00928/) (2022)
- [Anomaly Detection by Robust Statistics](https://mlsanity.com/paper/1707.09752/) (2017)
- [Martingale Methods for Sequential Estimation of Convex Functionals and Divergences](https://mlsanity.com/paper/2103.09267/) (2021)
- [Catoni-style confidence sequences for heavy-tailed mean estimation](https://mlsanity.com/paper/2202.01250/) (2022)
- [On the near-optimality of betting confidence sets for bounded means](https://mlsanity.com/paper/2310.01547/) (2023)
- [Rapid Regression Detection in Software Deployments through Sequential Testing](https://mlsanity.com/paper/2205.14762/) (2022)
- [Huber-Robust Confidence Sequences](https://mlsanity.com/paper/2301.09573/) (2023)

## Built on by (most cited first, 6 of 6)

- [Efficient Sequential Evaluation of Large Language Models](https://mlsanity.com/paper/2607.17409/) (2026)
- [Sharp Matrix Empirical Bernstein Inequalities](https://mlsanity.com/paper/2411.09516/) (2024)
- [A variational approach to dimension-free self-normalized concentration](https://mlsanity.com/paper/2508.06483/) (2025)
- [On the Tightness and Computational Tractability of Higher-Dimensional Confidence Sequences](https://mlsanity.com/paper/2610.03727/) (2026)
- [GAAVI: Global Asymptotic Anytime Valid Inference for the Conditional Mean Function](https://mlsanity.com/paper/2602.08096/) (2026)
- [The Cost of Adaptivity: Matching Lower Bounds Across Learning Problems](https://mlsanity.com/paper/2608.08826/) (2026)

## Abstract (arXiv)

We study sequential mean estimation in ℝd. 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.