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

arXiv:2607.02150v1 Announce Type: cross 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.

Source: arXiv cs.LG | 2026-07-03