Sequential 1-bit Mean Estimation with Near-Optimal Sample Complexity

Ivan Lau, Jonathan Scarlett
Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, PMLR 300:217-225, 2026.

Abstract

In this paper, we study the problem of distributed mean estimation with 1-bit communication constraints. We propose a mean estimator that is based on (randomized and sequentially-chosen) interval queries, whose 1-bit outcome indicates whether the given sample lies in the specified interval. Our estimator is $(\epsilon, \delta)$-PAC for all distributions with bounded mean ($-\lambda \le \mathbb{E}(X) \le \lambda $) and variance ($\mathrm{Var}(X) \le \sigma^2$) for some known parameters $\lambda$ and $\sigma$. We derive a sample complexity bound $\widetilde{O}\big( \frac{\sigma^2}{\epsilon^2}\log\frac{1}{\delta} + \log\frac{\lambda}{\sigma}\big)$, which matches the minimax lower bound for the unquantized setting up to logarithmic factors and the additional $\log\frac{\lambda}{\sigma}$ term that we show to be unavoidable. We also establish an adaptivity gap for interval-query based estimators: the best non-adaptive mean estimator is considerably worse than our adaptive mean estimator for large $\frac{\lambda}{\sigma}$. Finally, we give tightened sample complexity bounds for distributions with stronger tail decay, and present additional variants that (i) handle an unknown sampling budget (ii) adapt to the unknown true variance given (possibly loose) upper and lower bounds on the variance, and (iii) use only two stages of adaptivity at the expense of more complicated (non-interval) queries.

Cite this Paper

BibTeX
@InProceedings{pmlr-v300-lau26a, title = { Sequential 1-bit Mean Estimation with Near-Optimal Sample Complexity }, author = {Lau, Ivan and Scarlett, Jonathan}, booktitle = {Proceedings of The 29th International Conference on Artificial Intelligence and Statistics}, pages = {217--225}, year = {2026}, editor = {Khan, Emtiyaz and Li, Yingzhen and Solin, Arno and Ramdas, Aaditya}, volume = {300}, series = {Proceedings of Machine Learning Research}, month = {02--05 May}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v300/main/assets/lau26a/lau26a.pdf}, url = {https://proceedings.mlr.press/v300/lau26a.html}, abstract = { In this paper, we study the problem of distributed mean estimation with 1-bit communication constraints. We propose a mean estimator that is based on (randomized and sequentially-chosen) interval queries, whose 1-bit outcome indicates whether the given sample lies in the specified interval. Our estimator is $(\epsilon, \delta)$-PAC for all distributions with bounded mean ($-\lambda \le \mathbb{E}(X) \le \lambda $) and variance ($\mathrm{Var}(X) \le \sigma^2$) for some known parameters $\lambda$ and $\sigma$. We derive a sample complexity bound $\widetilde{O}\big( \frac{\sigma^2}{\epsilon^2}\log\frac{1}{\delta} + \log\frac{\lambda}{\sigma}\big)$, which matches the minimax lower bound for the unquantized setting up to logarithmic factors and the additional $\log\frac{\lambda}{\sigma}$ term that we show to be unavoidable. We also establish an adaptivity gap for interval-query based estimators: the best non-adaptive mean estimator is considerably worse than our adaptive mean estimator for large $\frac{\lambda}{\sigma}$. Finally, we give tightened sample complexity bounds for distributions with stronger tail decay, and present additional variants that (i) handle an unknown sampling budget (ii) adapt to the unknown true variance given (possibly loose) upper and lower bounds on the variance, and (iii) use only two stages of adaptivity at the expense of more complicated (non-interval) queries. } }
Endnote
%0 Conference Paper %T Sequential 1-bit Mean Estimation with Near-Optimal Sample Complexity %A Ivan Lau %A Jonathan Scarlett %B Proceedings of The 29th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2026 %E Emtiyaz Khan %E Yingzhen Li %E Arno Solin %E Aaditya Ramdas %F pmlr-v300-lau26a %I PMLR %P 217--225 %U https://proceedings.mlr.press/v300/lau26a.html %V 300 %X In this paper, we study the problem of distributed mean estimation with 1-bit communication constraints. We propose a mean estimator that is based on (randomized and sequentially-chosen) interval queries, whose 1-bit outcome indicates whether the given sample lies in the specified interval. Our estimator is $(\epsilon, \delta)$-PAC for all distributions with bounded mean ($-\lambda \le \mathbb{E}(X) \le \lambda $) and variance ($\mathrm{Var}(X) \le \sigma^2$) for some known parameters $\lambda$ and $\sigma$. We derive a sample complexity bound $\widetilde{O}\big( \frac{\sigma^2}{\epsilon^2}\log\frac{1}{\delta} + \log\frac{\lambda}{\sigma}\big)$, which matches the minimax lower bound for the unquantized setting up to logarithmic factors and the additional $\log\frac{\lambda}{\sigma}$ term that we show to be unavoidable. We also establish an adaptivity gap for interval-query based estimators: the best non-adaptive mean estimator is considerably worse than our adaptive mean estimator for large $\frac{\lambda}{\sigma}$. Finally, we give tightened sample complexity bounds for distributions with stronger tail decay, and present additional variants that (i) handle an unknown sampling budget (ii) adapt to the unknown true variance given (possibly loose) upper and lower bounds on the variance, and (iii) use only two stages of adaptivity at the expense of more complicated (non-interval) queries.
APA
Lau, I. & Scarlett, J.. (2026). Sequential 1-bit Mean Estimation with Near-Optimal Sample Complexity . Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 300:217-225 Available from https://proceedings.mlr.press/v300/lau26a.html.

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