Ranking and selection (R&S) conventionally aims to select the unique best alternative with the largest mean performance from a finite set of alternatives. However, for better supporting decision making, it may be more informative to deliver a small menu of alternatives whose mean performances are among the top $m$. Such problem, called optimal subset selection (OSS), is generally more challenging to address than the conventional R&S. This challenge becomes even more significant when the number of alternatives is considerably large. Thus, the focus of this paper is on addressing the large-scale OSS problem. To achieve this goal, we design a top-$m$ greedy selection mechanism that keeps sampling the current top $m$ alternatives with top $m$ running sample means and propose the explore-first top-$m$ greedy (EFG-$m$) procedure. Through an extended boundary-crossing framework, we prove that the EFG-$m$ procedure is both sample optimal and consistent in terms of the probability of good selection, confirming its effectiveness in solving large-scale OSS problem. Surprisingly, we also demonstrate that the EFG-$m$ procedure enables to achieve an indifference-based ranking within the selected subset of alternatives at no extra cost. This is highly beneficial as it delivers deeper insights to decision-makers, enabling more informed decision-makings. Lastly, numerical experiments validate our results and demonstrate the efficiency of our procedures.

翻译：排序与选择（R&S）的传统目标是从有限备选方案中选出具有最大均值性能的唯一最优方案。然而，为更好地支持决策制定，提供一个包含均值性能位于前 $m$ 位的少量备选方案菜单可能更具信息价值。此类问题称为最优子集选择（OSS），通常比传统的 R&S 问题更具挑战性。当备选方案数量极大时，这一挑战尤为显著。因此，本文聚焦于解决大规模 OSS 问题。为实现此目标，我们设计了一种前 $m$ 贪心选择机制，该机制持续对当前具有前 $m$ 位运行样本均值的备选方案进行采样，并提出了探索优先的前 $m$ 贪心（EFG-$m$）算法。通过扩展的边界穿越框架，我们证明了 EFG-$m$ 算法在良好选择概率方面同时具有样本最优性和一致性，从而证实了其解决大规模 OSS 问题的有效性。令人惊讶的是，我们还证明了 EFG-$m$ 算法能够在不增加额外成本的情况下，在所选备选方案子集内实现基于无差异性的排序。这具有显著优势，因为它能为决策者提供更深入的洞察，支持更明智的决策制定。最后，数值实验验证了我们的结果，并证明了所提算法的高效性。