# Admissibility of solution estimators for stochastic optimization

## TL;DR

This paper investigates the admissibility of the sample average estimator in stochastic optimization, showing it is admissible for linear problems over compact sets and for certain constrained quadratic problems, extending known results.

## Contribution

It demonstrates the conditions under which the sample average estimator is admissible in stochastic optimization, including constrained quadratic problems, expanding understanding beyond unconstrained cases.

## Key findings

- Sample average estimator is admissible for linear functions over compact sets.
- Admissibility holds for quadratic problems with box constraints in dimensions 3 and 4.
- Stein's paradox applies to unconstrained problems in dimensions ≥3, but not with certain constraints.

## Abstract

We look at stochastic optimization problems through the lens of statistical decision theory. In particular, we address admissibility, in the statistical decision theory sense, of the natural sample average estimator for a stochastic optimization problem (which is also known as the empirical risk minimization (ERM) rule in learning literature). It is well known that for some simple stochastic optimization problems, the sample average estimator may not be admissible. This is known as {\em Stein's paradox} in the statistics literature. We show in this paper that for optimizing stochastic linear functions over compact sets, the sample average estimator is admissible. Moreover, we study problems with convex quadratic objectives subject to box constraints. Stein's paradox holds when there are no constraints and the dimension of the problem is at least three. We show that in the presence of box constraints, admissibility is recovered for dimensions 3 and 4.