The Secretary Problem with Predicted Additive Gap

TL;DR

This work studies the secretary problem under a weak predictive signal: the additive gap $c_k = w_1-w_k$ between the top weight and the $k$-th best. It introduces a threshold-based online algorithm and proves that with exact gap information, one can achieve at least $0.4\,w_1$, beating the classic $1/\mathrm{e}$ barrier; it further develops robustness-consistency trade-offs so the same approach remains competitive even when the gap prediction is imperfect. A strengthened variant handles bounded prediction error, yielding $E[ALG] \ge 0.4\,w_1 - 2\epsilon$, and practical correlations are demonstrated via simulations showing consistent improvements over the classical secretary policy. The results reveal that knowing a weak, additive property of the top values can yield constant-factor improvements and that the proposed RC framework offers favorable robustness while retaining favorable consistency when the prediction is accurate. The work opens avenues for extending these ideas to more general gaps and multi-selection settings, with implications for online decision making under weak predictive information.

Abstract

The secretary problem is one of the fundamental problems in online decision making; a tight competitive ratio for this problem of $1/\mathrm{e} \approx 0.368$ has been known since the 1960s. Much more recently, the study of algorithms with predictions was introduced: The algorithm is equipped with a (possibly erroneous) additional piece of information upfront which can be used to improve the algorithm's performance. Complementing previous work on secretary problems with prior knowledge, we tackle the following question: What is the weakest piece of information that allows us to break the $1/\mathrm{e}$ barrier? To this end, we introduce the secretary problem with predicted additive gap. As in the classical problem, weights are fixed by an adversary and elements appear in random order. In contrast to previous variants of predictions, our algorithm only has access to a much weaker piece of information: an \emph{additive gap} $c$. This gap is the difference between the highest and $k$-th highest weight in the sequence. Unlike previous pieces of advice, knowing an exact additive gap does not make the problem trivial. Our contribution is twofold. First, we show that for any index $k$ and any gap $c$, we can obtain a competitive ratio of $0.4$ when knowing the exact gap (even if we do not know $k$), hence beating the prevalent bound for the classical problem by a constant. Second, a slightly modified version of our algorithm allows to prove standard robustness-consistency properties as well as improved guarantees when knowing a range for the error of the prediction.

Figures (7)

• Figure 1: Choosing the optimal parameters $\tau$ and $1- \gamma$ for our analysis in \ref{['thm:RC']}: For a given level of robustness, what is the best consistency we can obtain with our analysis.
• Figure 2: Trade-off between robustness and consistency as a function of the time $1-\gamma$ for fixed choice of $\tau = 0.2$.
• Figure 3: Competitive ratios for weights based on (i). On the $x$-axis, we have the index $k$ from $2$ to $n$. The $y$-axis shows the competitive ratios.
• Figure 4: Competitive ratios for weights based on (ii)
• Figure 5: Competitive ratios for weights based on (iii)

Theorems & Definitions (20)

• Theorem 1: Theorem \ref{['Theorem:general_gap']}, simplified form
• Theorem 2: Theorem \ref{['thm:RC']}, simplified form
• Theorem 3: Theorem \ref{['Theorem:approx_gap']}, simplified form
• Theorem 4
• Lemma 1
• Lemma 2
• proof : Proof of Lemma \ref{['lemma:proof_case_1']}.
• proof : Proof of Lemma \ref{['lemma:proof_case_2']} (i).
• proof : Proof of Lemma \ref{['lemma:proof_case_2']} (ii).
• proof : Proof of Theorem \ref{['Theorem:general_gap']}.