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List Estimation

Nikola Zlatanov, Amin Gohari, Farzad Shahrivari, Mikhail Rudakov

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.

List Estimation

Abstract

Classical estimation outputs a single point estimate of an unknown -dimensional vector from an observation. In this paper, we study \emph{-list estimation}, in which a single observation is used to produce a list of 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 agents observe conditionally i.i.d.\ measurements and each agent outputs its own MMSE estimate. On the centralized side, we show that optimal -list estimation is equivalent to fixed-rate -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 . 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 . We further show that if the error density vanishes at the origin, then the decentralized benchmark is provably unable to match the centralized 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 , one observation combined with carefully chosen candidates can be asymptotically as effective as -- and in some regimes strictly better than -- this MMSE-based decentralized benchmark with independent observations.

Paper Structure

This paper contains 25 sections, 9 theorems, 102 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Choice of performance metric
  3. Contributions
  4. Related works
  5. Notation
  6. Paper structure
  7. Problem Formulation and Distortion Definitions
  8. Centralized $k$-List Estimation
  9. Decentralized MMSE Benchmark
  10. Modeling remarks and scope
  11. Main Results in General Dimension
  12. Centralized $k$-List Estimation
  13. Decentralized MMSE Benchmark
  14. Exponent-Level Comparison in General Dimension
  15. Gaussian Specialization
  16. ...and 10 more sections

Key Result

Theorem 1

Assume the following conditions hold: Then the best-candidate distortion of the centralized $k$-list estimator satisfies where $G_d$ is the Zador--Gersho constant for squared-error vector quantization in $\mathbb{R}^d$1056490720541. In particular, if $\mathbb{E}[\mathcal{J}(\bm Y)]\in(0,\infty)$, then

Figures (3)

  • Figure 1: Empirical estimates of the centralized distortion $D_1(k)$ and the decentralized MMSE benchmark distortion $D_2(k)$, together with the theoretical lower bound on $D_2(k)$ and the high-rate prediction for $D_1(k)$, for $d=1$, $\sigma_X=1$, and $\sigma_N\in\{0.2,1,5\}$.
  • Figure 2: Empirical estimates of the centralized distortion $D_1(k)$ and the decentralized MMSE benchmark distortion $D_2(k)$, together with the theoretical lower bound on $D_2(k)$ and the high-rate prediction for $D_1(k)$, for $d=4$, $\sigma_X=1$, and $\sigma_N\in\{0.2,1,5\}$.
  • Figure 3: Empirical estimates of the centralized distortion $D_1(k)$ and the decentralized MMSE benchmark distortion $D_2(k)$, together with the theoretical lower bound on $D_2(k)$ and the high-rate prediction for $D_1(k)$, for $d=10$, $\sigma_X=1$, and $\sigma_N\in\{0.2,1,5\}$.

Theorems & Definitions (19)

  • Remark 1: Why MMSE may be suboptimal for the min-of-$k$ objective
  • Remark 2: Conservative comparison
  • Remark 3: Degenerate side information
  • Remark 4: Sanity check: $k=1$
  • Theorem 1
  • Corollary 1: Scalar specialization of Theorem \ref{['thm_2a']} ($d=1$)
  • Theorem 2
  • Corollary 2: Bounded joint error density near the origin
  • Remark 5: Non-uniform density bounds
  • Corollary 3
  • ...and 9 more