The (Surprising) Rate Optimality of Greedy Procedures for Large-Scale Ranking and Selection

Large-scale ranking and selection (R&S), which aims to select the best alternative with the largest mean performance from a finite set of alternatives, has emerged as an important research topic in simulation optimization. Ideal large-scale R&S procedures should be rate optimal, i.e., the total sample size required to deliver an asymptotically non-zero probability of correct selection (PCS) grows at the minimal rate (linear rate) in the number of alternatives. Surprisingly, we discover that the na\"ive greedy procedure that keeps sampling the alternative with the largest running average performs strikingly well and appears rate optimal. To understand this discovery, we develop a new boundary-crossing perspective and prove that the greedy procedure is indeed rate optimal. We further show that the derived PCS lower bound is asymptotically tight for the slippage configuration of means with a common variance. Moreover, we propose the explore-first greedy (EFG) procedure and its enhanced version ($\mbox{EFG}^+$ procedure) by adding an exploration phase to the na\"ive greedy procedure. Both procedures are proven to be rate optimal and consistent. Last, we conduct extensive numerical experiments to empirically understand the performance of our greedy procedures in solving large-scale R&S problems.