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Minimum mean-squared error estimation with bandit feedback

Ayon Ghosh, L. A. Prashanth, Dipayan Sen, Aditya Gopalan

TL;DR

This work addresses the problem of estimating the MSE ψ(A) for selecting informative subsets A of size m from a K-dimensional Gaussian with unknown covariance. It introduces a non-adaptive sample-average estimator and a regression-based adaptive estimator, deriving exponential concentration bounds and highlighting the superior performance of the regression-based approach. Framing the problem as bandits with bandit feedback, the authors develop a variant of successive elimination to identify the MSE-optimal subset with fixed confidence, and establish a minimax lower bound on sample complexity for the m = 2 case to characterize fundamental limits. The combination of efficient MSE estimation, adaptive subset selection, and theoretical lower bounds advances understanding of information-efficient covariance-based subset selection in correlated environments, with potential applications to sensor placement and regional monitoring. The results rely on concentration inequalities for Gaussian-derived statistics and information-theoretic analyses of Gaussian distributions to connect estimator accuracy with sampling requirements.

Abstract

We consider the problem of sequentially learning to estimate, in the mean squared error (MSE) sense, a Gaussian $K$-vector of unknown covariance by observing only $m < K$ of its entries in each round. We propose two MSE estimators, and analyze their concentration properties. The first estimator is non-adaptive, as it is tied to a predetermined $m$-subset and lacks the flexibility to transition to alternative subsets. The second estimator, which is derived using a regression framework, is adaptive and exhibits better concentration bounds in comparison to the first estimator. We frame the MSE estimation problem with bandit feedback, where the objective is to find the MSE-optimal subset with high confidence. We propose a variant of the successive elimination algorithm to solve this problem. We also derive a minimax lower bound to understand the fundamental limit on the sample complexity of this problem.

Minimum mean-squared error estimation with bandit feedback

TL;DR

This work addresses the problem of estimating the MSE ψ(A) for selecting informative subsets A of size m from a K-dimensional Gaussian with unknown covariance. It introduces a non-adaptive sample-average estimator and a regression-based adaptive estimator, deriving exponential concentration bounds and highlighting the superior performance of the regression-based approach. Framing the problem as bandits with bandit feedback, the authors develop a variant of successive elimination to identify the MSE-optimal subset with fixed confidence, and establish a minimax lower bound on sample complexity for the m = 2 case to characterize fundamental limits. The combination of efficient MSE estimation, adaptive subset selection, and theoretical lower bounds advances understanding of information-efficient covariance-based subset selection in correlated environments, with potential applications to sensor placement and regional monitoring. The results rely on concentration inequalities for Gaussian-derived statistics and information-theoretic analyses of Gaussian distributions to connect estimator accuracy with sampling requirements.

Abstract

We consider the problem of sequentially learning to estimate, in the mean squared error (MSE) sense, a Gaussian -vector of unknown covariance by observing only of its entries in each round. We propose two MSE estimators, and analyze their concentration properties. The first estimator is non-adaptive, as it is tied to a predetermined -subset and lacks the flexibility to transition to alternative subsets. The second estimator, which is derived using a regression framework, is adaptive and exhibits better concentration bounds in comparison to the first estimator. We frame the MSE estimation problem with bandit feedback, where the objective is to find the MSE-optimal subset with high confidence. We propose a variant of the successive elimination algorithm to solve this problem. We also derive a minimax lower bound to understand the fundamental limit on the sample complexity of this problem.
Paper Structure (12 sections, 8 theorems, 78 equations, 1 figure)

This paper contains 12 sections, 8 theorems, 78 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-adaptive MSE estimation
  4. Efficient MSE estimation
  5. Bandits with MSE objective: Upper bound
  6. Bandits with MSE objective: Lower Bound
  7. Convergence proofs
  8. Proof of Proposition \ref{['prop:nonadaptive-mse-conc']}
  9. Proof of Proposition \ref{['prop:adaptive-mse-conc']}
  10. Proof of Theorem \ref{['thm:lb']}
  11. Proof of Theorem \ref{['thm:se']}
  12. Conclusions

Key Result

Proposition 1

Assume ass:variance and ass:eigenvalue. Let $n_{AA},n_{A'A},n_{AA'}, n_{A'A'}$ denote the number of samples used to form $\widehat{\Sigma}_{AA}, \widehat{\Sigma}_{A'A}, \widehat{\Sigma}_{AA'}$, and $\widehat{\Sigma}_{A'A'}$, respectively. Set the projection parameter $\zeta$ in eq:sigma-hat-plus as where $\delta \in (0,1)$ denotes the confidence width. Let $n'=\min\left(n_{AA}, n_{A'A'}, n_{AA'},

Figures (1)

  • Figure 1: Operational flow of successive elimination for correlated bandits.

Theorems & Definitions (18)

  • Proposition 1: MSE concentration: Non-adaptive case
  • proof
  • Proposition 2: MSE concentration: Adaptive case
  • proof
  • Remark 1: Comparison with the non-adaptive case
  • Theorem 1: Sample complexity bound
  • proof
  • Theorem 2: Lower bound
  • proof
  • Remark 2: Comparison to upper bound
  • ...and 8 more