The Secretary Problem with Predictions and a Chosen Order

TL;DR

This paper introduces a learning-augmented secretary problem model with a new algorithm that balances trust in predictions and robustness, improving competitive ratios in both random and chosen order scenarios.

Contribution

It proposes a novel randomized algorithm for secretary problems that adaptively switches strategies based on prediction accuracy, enhancing performance bounds.

Findings

Achieves a competitive ratio of up to 0.221 in ROSP, improving previous bounds.

Attains a competitive ratio of up to 0.262 in COSP, surpassing prior results.

Demonstrates the advantage of combining machine learning predictions with arrival order control.

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 $\epsilon \in [0,1]$ denote the maximum multiplicative prediction error. For ROSP, our algorithm achieves a competitive ratio of $\max\{0.221, (1-\epsilon)/(1+\epsilon)\}$, improving upon the prior bound of $\max\{0.215, (1-\epsilon)/(1+\epsilon)\}$. For COSP, we achieve $\max\{0.262, (1-\epsilon)/(1+\epsilon)\}$, 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.