The Secretary Problem with Predictions and a Chosen Order
Helia Karisani, Mohammadreza Daneshvaramoli, Hedyeh Beyhaghi, Mohammad Hajiesmaili, Cameron Musco
TL;DR
This work studies a learning-augmented secretary problem with predictions in two online models: ROSP (random arrival order) and COSP (chosen arrival order). It introduces a simple randomized algorithm that initially trusts predictions and hires the top-predicted candidate, but switches to a Dynkin-style threshold strategy when large prediction deviations are observed, using probabilistic rules with parameters $\gamma$ and $\delta$ to balance consistency and robustness. The main results give improved competitive guarantees: COSP has a lower bound of $\max\{0.262, \frac{1-\epsilon}{1+\epsilon}\}$ and ROSP a lower bound of $\max\{0.221, \frac{1-\epsilon}{1+\epsilon}\}$, both independent of the knowledge of $\epsilon$, and the paper also proves that in ROSP, randomized and deterministic algorithms are equivalent in power. These findings demonstrate that incorporating predictions together with control over arrival order yields performance closer to the classic secretary benchmark and provide a framework for tuning robustness via $\gamma$ and $\delta$.
Abstract
We study a learning-augmented variant of the secretary problem, recently introduced by Fujii and Yoshida (2023), in which the decision-maker has access to machine-learned predictions of candidate values. The central challenge is to balance consistency and robustness: when predictions are accurate, the algorithm should select a near-optimal secretary, while under inaccurate predictions it should still guarantee a bounded competitive ratio. We consider both the classical Random Order Secretary Problem (ROSP), where candidates arrive in a uniformly random order, and a more natural learning-augmented model in which the decision-maker may choose the arrival order based on predicted values. We call this model the Chosen Order Secretary Problem (COSP), capturing scenarios such as interview schedules set in advance. We propose a new randomized algorithm applicable to both ROSP and COSP. Our method switches from fully trusting predictions to a threshold-based rule once a large prediction deviation is detected. Let $ε\in [0,1]$ denote the maximum multiplicative prediction error. For ROSP, our algorithm achieves a competitive ratio of $\max\{0.221, (1-ε)/(1+ε)\}$, improving upon the prior bound of $\max\{0.215, (1-ε)/(1+ε)\}$. For COSP, we achieve $\max\{0.262, (1-ε)/(1+ε)\}$, surpassing the $0.25$ worst-case bound for prior approaches and moving closer to the classical secretary benchmark of $1/e \approx 0.368$. These results highlight the benefit of combining predictions with arrival-order control in online decision-making.