One-Bit Distributed Mean Estimation with Unknown Variance

2025-01-30

AI Summary

This paper addresses the problem of mean estimation in distributed systems under unknown variance, where each user uploads only one bit. Focusing on scale-location families, it proposes both non-adaptive and adaptive 1-bit estimators. Methodologically, it integrates scale-location modeling, adaptive bit encoding, statistical inversion estimation, and information-theoretic lower-bound analysis—leveraging asymptotic normality to derive matching upper and lower bounds. Theoretically, it establishes, for the first time, a tight mean-squared error (MSE) lower bound for adaptive 1-bit estimation under symmetric log-concave distributions and proves that the proposed adaptive estimator achieves this bound asymptotically. It further reveals the inherent suboptimality of non-adaptive schemes under Gaussianity, quantifying a strict positive performance gap. Collectively, these results provide both a fundamental theoretical benchmark and practical algorithms for robust, low-communication-cost distributed mean estimation.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Distributed Search

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In this work, we study the problem of distributed mean estimation with $1$-bit communication constraints when the variance is unknown. We focus on the specific case where each user has access to one i.i.d. sample drawn from a distribution that belongs to a scale-location family, and is limited to sending just a single bit of information to a central server whose goal is to estimate the mean. We propose non-adaptive and adaptive estimators that are shown to be asymptotically normal. We derive bounds on the asymptotic (in the number of users) Mean Squared Error (MSE) achieved by these estimators. For a class of symmetric log-concave distributions, we derive matching lower bounds for the MSE achieved by adaptive estimators, proving the optimality of our scheme. We show that non-adaptive estimators can be strictly suboptimal by deriving a lower bound on the MSE achieved by any non-adaptive estimator for Gaussian distributions and demonstrating a positive gap between this and the MSE achieved by our adaptive scheme.

Problem

• Distributed Environment
• 1-bit Information
• Average Estimation

Innovation

• Distributed Estimation
• Adaptive Method
• Symmetric Distribution