Robust and Consistent Ski Rental with Distributional Advice

TL;DR

This paper develops a framework for the ski rental problem that leverages full distributional advice to improve decision-making, ensuring robustness and consistency even with imperfect predictions.

Contribution

It introduces a systematic approach to incorporate distributional advice into ski rental algorithms, with new policies that balance robustness and consistency.

Findings

The optimal threshold-buy algorithm is derived under perfect distributional predictions.

The Clamp Policy maintains performance with large prediction errors and improves as predictions become accurate.

Experimental results show significant improvement in consistency over point-prediction methods while preserving robustness.

Abstract

The ski rental problem is a canonical model for online decision-making under uncertainty, capturing the fundamental trade-off between repeated rental costs and a one-time purchase. While classical algorithms focus on worst-case competitive ratios and recent "learning-augmented" methods leverage point-estimate predictions, neither approach fully exploits the richness of full distributional predictions while maintaining rigorous robustness guarantees. We address this gap by establishing a systematic framework that integrates distributional advice of unknown quality into both deterministic and randomized algorithms. For the deterministic setting, we formalize the problem under perfect distributional prediction and derive an efficient algorithm to compute the optimal threshold-buy day. We provide a rigorous performance analysis, identifying sufficient conditions on the predicted distribution under which the expected competitive ratio (ECR) matches the classic optimal randomized bound. To handle imperfect predictions, we propose the Clamp Policy, which restricts the buying threshold to a safe range controlled by a tunable parameter. We show that this policy is both robust, maintaining good performance even with large prediction errors, and consistent, approaching the optimal performance as predictions become accurate. For the randomized setting, we characterize the stopping distribution via a Water-Filling Algorithm, which optimizes expected cost while strictly satisfying robustness constraints. Experimental results across diverse distributions (Gaussian, geometric, and bi-modal) demonstrate that our framework improves consistency significantly over existing point-prediction baselines while maintaining comparable robustness.