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

Ranking and selection (R&S), which aims to select the best alternative with the largest mean performance from a finite set of alternatives, is a classic research topic in simulation optimization. Recently, considerable attention has turned towards the large-scale variant of the R&S problem which involves a large number of alternatives. Ideal large-scale R&S procedures should be sample optimal, i.e., the total sample size required to deliver an asymptotically non-zero probability of correct selection (PCS) grows at the minimal order (linear order) in the number of alternatives, but not many procedures in the literature are sample optimal. Surprisingly, we discover that the na\"ive greedy procedure, which keeps sampling the alternative with the largest running average, performs strikingly well and appears sample optimal. To understand this discovery, we develop a new boundary-crossing perspective and prove that the greedy procedure is indeed sample 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 (EFG$^+$ procedure) by adding an exploration phase to the na\"ive greedy procedure. Both procedures are proven to be sample 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.