Minimum mean-squared error estimation with bandit feedback

TL;DR

This paper addresses the challenge of sequentially estimating the covariance of a Gaussian vector with limited observations, proposing adaptive estimators and algorithms to identify the optimal subset with high confidence.

Contribution

It introduces two MSE estimators, analyzes their concentration, and develops a bandit-based algorithm with theoretical guarantees for subset selection.

Findings

The adaptive estimator outperforms the non-adaptive one in concentration bounds.

The proposed algorithm effectively identifies the MSE-optimal subset with high confidence.

A minimax lower bound characterizes the fundamental sample complexity limits.

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.