Convergence Rate Analysis for Optimal Computing Budget Allocation Algorithms

TL;DR

This paper analyzes the convergence rates of two OCBA algorithms in ordinal optimization, demonstrating their optimal performance in probability, opportunity cost, and regret measures, thus broadening their applicability.

Contribution

It provides the first convergence rate analysis for OCBA algorithms and extends their optimality to cumulative regret, enhancing their theoretical foundation.

Findings

Both OCBA algorithms achieve optimal convergence rates for probability of correct selection.

The algorithms also attain optimal rates under expected opportunity cost.

With minor modifications, they reach optimal convergence under cumulative regret.

Abstract

Ordinal optimization (OO) is a widely-studied technique for optimizing discrete-event dynamic systems (DEDS). It evaluates the performance of the system designs in a finite set by sampling and aims to correctly make ordinal comparison of the designs. A well-known method in OO is the optimal computing budget allocation (OCBA). It builds the optimality conditions for the number of samples allocated to each design, and the sample allocation that satisfies the optimality conditions is shown to asymptotically maximize the probability of correct selection for the best design. In this paper, we investigate two popular OCBA algorithms. With known variances for samples of each design, we characterize their convergence rates with respect to different performance measures. We first demonstrate that the two OCBA algorithms achieve the optimal convergence rate under measures of probability of correct selection and expected opportunity cost. It fills the void of convergence analysis for OCBA algorithms. Next, we extend our analysis to the measure of cumulative regret, a main measure studied in the field of machine learning. We show that with minor modification, the two OCBA algorithms can reach the optimal convergence rate under cumulative regret. It indicates the potential of broader use of algorithms designed based on the OCBA optimality conditions.