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Robust and Consistent Ski Rental with Distributional Advice

Jihwan Kim, Chenglin Fan

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.

Robust and Consistent Ski Rental with Distributional Advice

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.

Paper Structure

This paper contains 84 sections, 11 theorems, 229 equations, 1 figure, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions and Overview
  3. Related Work
  4. Distributional Predictions.
  5. Organization
  6. Preliminaries and Problem Setup
  7. The Ski Rental Problem
  8. Distributional Predictions
  9. Algorithms and Randomization
  10. Performance Metrics
  11. Notation
  12. Perfect Prediction Setting
  13. Algorithm
  14. A Concrete Example with $1 < t^* < \infty$
  15. Performance Analysis
  16. ...and 69 more sections

Key Result

Theorem 3.1

If the optimal threshold satisfies $t^* \le b$, the $\mathrm{CR}(p)$ is bounded by:

Figures (1)

  • Figure 1: Consistency under prediction error. The x-axis is the Wasserstein perturbation budget $\eta$ used to generate $\widehat{p}$ from the true $p$; policies are computed from $\widehat{p}$, but the consistency is evaluated under $p$. The plot reports the mean over $25$ random transports per $\eta$.

Theorems & Definitions (26)

  • Theorem 3.1: Competitive Ratio for $t^* \le b$
  • Theorem 3.2: Competitive Ratio for $t^* > b$
  • Theorem 3.3
  • Definition 4.1: Clamp Policy
  • Lemma 4.2: Wasserstein Stability
  • Lemma 4.3: Distribution-Free Threshold Bound
  • Theorem 4.4: Robust-Consistent Bound
  • Corollary 4.5: Robustness and Consistency
  • Remark 4.6: Alternative Metric
  • Lemma 5.1: Robustness Constraints
  • ...and 16 more