mlsanityData to 7 Oct 2026

Time-Uniform Confidence Spheres for Means of Random Vectors

arXiv 2311.08168 ↗First posted 14 Nov 2023math.ST cs.IT math.ITcited by 6

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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.

Key facts. Cited by 6 papers in the map (data through 7 Oct 2026). Builds on 25 of its 82 references that are in the map. First posted on arXiv 14 Nov 2023.

Builds on

25 of 82 references.

References · 25

  1. 2021User-friendly introduction to PAC-Bayes boundscited by 234
  2. 2018Time-uniform, nonparametric, nonasymptotic confidence sequencescited by 220
  3. 2016Robust Estimators in High Dimensions without the Computational Intractabilitycited by 209
  4. 2020Estimating means of bounded random variables by bettingcited by 164
  5. 2019Mean estimation and regression under heavy-tailed distributions--a surveycited by 133
  6. 2019A Primer on PAC-Bayesian Learningcited by 130
  7. 2020Improved Optimistic Algorithms for Logistic Banditscited by 117
  8. 2019A Short Note on Concentration Inequalities for Random Vectors with SubGaussian Normcited by 105
  9. 2018Robust Estimation via Robust Gradient Estimationcited by 76
  10. 2018Mixture Martingales Revisited with Applications to Sequential Tests and Confidence Intervalscited by 76
  11. 2017Sub-sampled Cubic Regularization for Non-convex Optimizationcited by 63
  12. 2015Robust Covariance and Scatter Matrix Estimation under Huber's Contamination Modelcited by 47
13 more
  1. 2017Sub-Gaussian estimators of the mean of a random vectorcited by 47
  2. 2020Robust and Heavy-Tailed Mean Estimation Made Simple, via Regret Minimizationcited by 44
  3. 2018Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regressioncited by 42
  4. 2015Statistical Query Algorithms for Mean Vector Estimation and Stochastic Convex Optimizationcited by 41
  5. 2021Time-uniform central limit theory and asymptotic confidence sequencescited by 29
  6. 2023A unified recipe for deriving (time-uniform) PAC-Bayes boundscited by 25
  7. 2022PAC-Bayes Generalisation Bounds for Heavy-Tailed Losses through Supermartingalescited by 23
  8. 2017Anomaly Detection by Robust Statisticscited by 22
  9. 2021Martingale Methods for Sequential Estimation of Convex Functionals and Divergencescited by 17
  10. 2022Catoni-style confidence sequences for heavy-tailed mean estimationcited by 17
  11. 2023On the near-optimality of betting confidence sets for bounded meanscited by 9
  12. 2022Rapid Regression Detection in Software Deployments through Sequential Testingcited by 7
  13. 2023Huber-Robust Confidence Sequencescited by 7

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 1; 2025: 1; 2026: 4.

6 papers.

  1. 2026Efficient Sequential Evaluation of Large Language Modelscited by 3
  2. 2024Sharp Matrix Empirical Bernstein Inequalitiescited by 2
  3. 2025A variational approach to dimension-free self-normalized concentrationcited by 2
  4. 2026On the Tightness and Computational Tractability of Higher-Dimensional Confidence Sequencescited by 1
  5. 2026GAAVI: Global Asymptotic Anytime Valid Inference for the Conditional Mean Functioncited by 0
  6. 2026The Cost of Adaptivity: Matching Lower Bounds Across Learning Problemscited by 0