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

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.

• The paper demonstrates that additive regret in the multi-secretary problem scales as Θ((log T)²) under bounded-density gapped distributions.
• It introduces a novel framework using Bellman certificates to derive lower bounds via duality over the Bellman recursion.
• The results clarify conditions where online algorithms incur unavoidable regret, guiding algorithm design in revenue management.

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

Introduction and Problem Context

The paper "Tight Lower Bounds for the Multi-Secretary Problem via Bellman Certificates" (2607.02150) addresses the additive regret in the multi-secretary problem—a canonical instance of online resource allocation where a decision-maker observes $T$ i.i.d. values $V_1, \ldots, V_T\sim F$ and is permitted to select up to $k$ of them, aiming to maximize the sum of accepted values. The online policy is evaluated via additive regret, defined as the expected gap between the reward attained by the best online policy and that of a "prophet" who selects the top $k$ values with full hindsight.

Prior literature established that, under mild regularity conditions (connected support, density bounded away from zero), the regret scales as $O(\log T)$ [Bray 2024], while for distributions with support gaps but bounded densities, the best known upper bound was $O((\log T)^2)$ [Besbes et al. 2024, Jiang et al. 2025]. However, whether this log-squared factor was intrinsic or an artifact of analysis remained unresolved, even in the basic single-resource, two-uniform distribution case.

Main Results and Contributions

This work establishes matching $\Omega((\log T)^2)$ lower bounds for additive regret in the multi-secretary problem with bounded-density gaps in the support, specifically for a two-segment uniform distribution mixture at the critical capacity regime. This result closes the gap with previously known upper bounds, demonstrating their tightness.

The contributions are three-fold:

• Sharp lower bound in the gapped uniform mixture: For a two-uniform distribution with a gap, regret at critical capacity ($k_T=\lfloor qT\rfloor$, $q$ being the upper support probability mass) satisfies

$\mathrm{Regret}(T, k_T; F) \geq c (\log T)^2$

for an explicit $V_1, \ldots, V_T\sim F$0 and all large $V_1, \ldots, V_T\sim F$1.

• Framework for lower bounds via Bellman certificates: Unlike previous indistinguishability constructions, this work introduces explicit Bellman certificates—feasible solutions to relaxations of the Bellman recursion for the regret process—that yield lower bounds by duality and allow transparent decomposition of sources of regret.
• Extension to vanishing-density edges ($V_1, \ldots, V_T\sim F$2): The approach also yields optimal lower bounds for gapped distributions where edge densities vanish as a power-law ($V_1, \ldots, V_T\sim F$3distance$V_1, \ldots, V_T\sim F$4), confirming the necessity of an extra log factor in regret in shift-from-critical capacity regimes in these settings.

Technical Approach: Bellman Certificate Method

The standard dynamic programming (Bellman) recursions for both the online and prophet benchmarks produce a recursion for their difference (the regret function). The key insight is to construct an explicit non-negative function $V_1, \ldots, V_T\sim F$5 over the $V_1, \ldots, V_T\sim F$6 state space satisfying a one-sided Bellman certificate inequality:

$V_1, \ldots, V_T\sim F$7

where $V_1, \ldots, V_T\sim F$8 is the first difference, and $V_1, \ldots, V_T\sim F$9 combines order-statistic (Jensen) and finite difference terms, with convexity of the residual $k$0 from the value distribution $k$1 playing a central role.

This construction exploits that in the presence of a support gap, the operating threshold for selection lands in an interval where the regret operator is affine (flat), so curvature (which would limit regret growth) appears only near the edges. The certificate is engineered so that for capacities $k$2 near $k$3, both the mean and perturbed thresholds remain in the gap, suppressing the finite-difference penalty. However, moderate-deviation fluctuations ensure that, with polynomial probability, the prophet's marginal value may land near the gap edges, where curvature is present and can be exploited to certify unavoidable regret. This mechanism enables a feasible certificate of height $k$4 at the critical state.

Numerical and Analytical Results

The lower bound is shown to be tight:

• For the two-uniform mixture at critical capacity, regret is $k$5.
• For gapped, bounded-density power-law edge distributions, the framework recovers the necessity of the extra log factor in the shifted capacity regime.
• The certificate is constructed and analyzed via precise central and moderate deviation analysis, carefully bounding all drift, source (Jensen), and perturbation terms.

No scenarios (including the single-resource specialization or network revenue management with continuous rewards) allow improvement on the order of the upper bound: $k$6 is unavoidable under support gaps with bounded density. This validates the sharpness of prior polylogarithmic regret results for network revenue management and online linear programming in contexts lacking strong-convexity or degeneracy-avoidance assumptions [Jiang et al. 2025].

Implications and Future Directions

This work fundamentally clarifies the limits of online selection and allocation under mild distributional degeneracy. The Bellman-certificate dual framework is generally applicable and structurally unifies upper and lower bounding techniques for online versus offline (prophet) gaps, providing a precise tool for identifying when and why polylogarithmic regret may be achieved—or not.

Practically, the result identifies precise circumstances under which log-squared regret is unimprovable, enabling algorithm designers and practitioners in revenue management and online allocation to distinguish tractable from intractable parametric regimes. Theoretically, it suggests fertile ground for extending Bellman certificate techniques to richer combinatorial constraints (matroids, multi-resource settings), nontrivial consumption structure, or broader classes of dynamical online decision-making.

Anticipated future developments include:

• Application of Bellman certificate lower bounds to high-dimensional network resource allocation settings;
• Exploration of tight regret bounds under information and feedback constraints;
• Extension to adversarial settings and contexts with partial or distributional ambiguity.

Conclusion

This paper decisively proves that, in the multi-secretary problem with bounded-density gapped support, the additive regret must scale as $k$7 at critical capacity. The Bellman-certificate approach introduced provides a robust, transparent, and versatile tool for deriving lower bounds for prophet inequalities and regret minimization, clarifying the inherent performance limits of online allocation algorithms without degeneracy or structure-rich distributional assumptions. These insights will inform both the theory and practice of online decision-making, and the framework is broadly applicable across related stochastic selection and allocation problems.