Tight Lower Bounds for the Multi-Secretary Problem via Bellman Certificates

2026-07-02

AI Summary

This work investigates the additive regret lower bound for the multiple-secretary problem under support gaps, focusing on whether an extra logarithmic factor is unavoidable for distributions with bounded density. By constructing a Bellman certificate—a relaxed feasible solution to the Bellman recursion—the authors transform the lower-bound analysis into an explicit construction problem. They establish, for the first time, that when the capacity lies at a critical scale, the optimal regret for a mixture of two well-separated uniform distributions is at least on the order of $(\log T)^2$. This result not only confirms the tightness of the $(\log T)^2$-order lower bound but also uncovers the fundamental mechanism by which support gaps amplify regret, thereby resolving a long-standing theoretical question in this area.

Abstract

This paper studies additive regret in the multi-secretary problem, defined as the gap between the expected offline prophet reward and the reward of the best online policy. Prior work established \(O(\log T)\) regret for bounded-density distributions with connected support and \(O((\log T)^2)\) upper bounds for bounded-density distributions with support gaps. It was unknown whether the extra logarithmic factor is necessary even in the one-resource model. We prove that it is necessary. For a mixture of two separated uniform distributions at the critical capacity, the optimal regret grows at least on the order of \((\log T)^2\). Thus the existing \(O((\log T)^2)\) upper bounds for bounded-density gapped instances, including those implied by network revenue management models with continuous rewards, are tight in this simplest specialization. The same framework also yields a matching lower bound for gapped distributions whose gap-facing densities vanish near the support edges; this companion result is given in the appendix. The proofs use Bellman certificates: feasible solutions to a relaxation of the exact Bellman recursion. This framework converts lower bounds into explicit certificate constructions and identifies why support gaps permit larger regret.

Problem

• multi-secretary problem
• additive regret
• support gaps
• lower bounds
• bounded-density distributions

Innovation

• multi-secretary problem
• additive regret
• Bellman certificates
• support gaps
• lower bounds