Ordinal Optimisation and the Offline Multiple Noisy Secretary Problem

TL;DR

This paper analyzes a variant of the secretary problem with noisy observations and multiple offline selections, using ordinal optimisation to derive success probabilities and bounds, with applications in stochastic simulation and optimisation.

Contribution

It introduces a novel formulation of the offline multiple noisy secretary problem using ordinal optimisation and copula-based analysis, including analytic bounds for success probability.

Findings

Success probability depends only on the underlying copula.

Derived an analytic lower bound for Gaussian copulas.

Provided sample size guarantees for high success probability.

Abstract

We study the success probability for a variant of the secretary problem, with noisy observations and multiple offline selection. Our formulation emulates, and is motivated by, problems involving noisy selection arising in the disciplines of stochastic simulation and simulation-based optimisation. In addition, we employ the philosophy of ordinal optimisation - involving an ordinal selection rule, and a percentile notion of goal softening for the success probability. As a result, it is shown that the success probability only depends on the underlying copula of the problem. Other general properties for the success probability are also presented. Specialising to the case of Gaussian copulas, we also derive an analytic lower bound for the success probability, which may then be inverted to find sufficiently large sample sizes that guarantee a high success probability arbitrarily close to one.