List Estimation

2026-03-26

AI Summary

This work addresses the k-list estimation problem, which seeks to generate k candidate estimates from a single observation so as to minimize the expected squared error between the true parameter and its nearest candidate. By establishing an equivalence between optimal k-list estimation and fixed-rate k-point vector quantization of the posterior distribution, and leveraging high-rate asymptotic analysis together with small-ball probability techniques, the authors prove that under standard regularity conditions, a centralized estimator achieves a decay rate of \(k^{-2/d}\). This rate is unattainable by any decentralized single-agent MMSE estimator if its error density is bounded at the origin, and performance degrades further if the density vanishes there. The theoretical findings are validated in Gaussian models and confirmed numerically, demonstrating that centralized k-list estimation can asymptotically match or even surpass the performance of symmetric decentralized MMSE methods using k independent observations.

Technology Category

• Intelligent Robots: State Estimation
• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Classical estimation outputs a single point estimate of an unknown $d$-dimensional vector from an observation. In this paper, we study \emph{$k$-list estimation}, in which a single observation is used to produce a list of $k$ candidate estimates and performance is measured by the expected squared distance from the true vector to the closest candidate. We compare this centralized setting with a symmetric decentralized MMSE benchmark in which $k$ agents observe conditionally i.i.d.\ measurements and each agent outputs its own MMSE estimate. On the centralized side, we show that optimal $k$-list estimation is equivalent to fixed-rate $k$-point vector quantization of the posterior distribution and, under standard regularity conditions, admits an exact high-rate asymptotic expansion with explicit constants and decay rate $k^{-2/d}$. On the decentralized side, we derive lower bounds in terms of the small-ball behavior of the single-agent MMSE error; in particular, when the conditional error density is bounded near the origin, the benchmark distortion cannot decay faster than order $k^{-2/d}$. We further show that if the error density vanishes at the origin, then the decentralized benchmark is provably unable to match the centralized $k^{-2/d}$ exponent, whereas the centralized estimator retains that scaling. Gaussian specializations yield explicit formulas and numerical experiments corroborate the predicted asymptotic behavior. Overall, the results show that, in the scaling with $k$, one observation combined with $k$ carefully chosen candidates can be asymptotically as effective as -- and in some regimes strictly better than -- this MMSE-based decentralized benchmark with $k$ independent observations.

Problem

• list estimation
• vector quantization
• MMSE
• decentralized estimation
• asymptotic analysis

Innovation

• k-list estimation
• vector quantization
• MMSE
• asymptotic analysis
• decentralized estimation