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Learning-Augmented Ski Rental with Discrete Distributions: A Bayesian Approach

Bosun Kang, Hyejun Park, Chenglin Fan

TL;DR

<3-5 sentence high-level summary>This work introduces a discrete Bayesian framework for the ski rental problem that maintains an exact posterior over the unknown horizon and bases decisions on the expected remaining rental cost, seamlessly integrating prior knowledge and predictions. It provides prior-dependent competitive guarantees, robustness to misspecification, and extends to multiple predictions and contextual priors. Theoretical analyses illustrate when Bayesian decisions outperform classical algorithms, and extensive experiments confirm near-optimal performance under informative priors while preserving robust worst-case behavior. The approach offers a practical, extensible tool for online decision problems with imperfect predictions and dynamic prior information.

Abstract

We revisit the classic ski rental problem through the lens of Bayesian decision-making and machine-learned predictions. While traditional algorithms minimize worst-case cost without assumptions, and recent learning-augmented approaches leverage noisy forecasts with robustness guarantees, our work unifies these perspectives. We propose a discrete Bayesian framework that maintains exact posterior distributions over the time horizon, enabling principled uncertainty quantification and seamless incorporation of expert priors. Our algorithm achieves prior-dependent competitive guarantees and gracefully interpolates between worst-case and fully-informed settings. Our extensive experimental evaluation demonstrates superior empirical performance across diverse scenarios, achieving near-optimal results under accurate priors while maintaining robust worst-case guarantees. This framework naturally extends to incorporate multiple predictions, non-uniform priors, and contextual information, highlighting the practical advantages of Bayesian reasoning in online decision problems with imperfect predictions.

Learning-Augmented Ski Rental with Discrete Distributions: A Bayesian Approach

TL;DR

<3-5 sentence high-level summary>This work introduces a discrete Bayesian framework for the ski rental problem that maintains an exact posterior over the unknown horizon and bases decisions on the expected remaining rental cost, seamlessly integrating prior knowledge and predictions. It provides prior-dependent competitive guarantees, robustness to misspecification, and extends to multiple predictions and contextual priors. Theoretical analyses illustrate when Bayesian decisions outperform classical algorithms, and extensive experiments confirm near-optimal performance under informative priors while preserving robust worst-case behavior. The approach offers a practical, extensible tool for online decision problems with imperfect predictions and dynamic prior information.

Abstract

We revisit the classic ski rental problem through the lens of Bayesian decision-making and machine-learned predictions. While traditional algorithms minimize worst-case cost without assumptions, and recent learning-augmented approaches leverage noisy forecasts with robustness guarantees, our work unifies these perspectives. We propose a discrete Bayesian framework that maintains exact posterior distributions over the time horizon, enabling principled uncertainty quantification and seamless incorporation of expert priors. Our algorithm achieves prior-dependent competitive guarantees and gracefully interpolates between worst-case and fully-informed settings. Our extensive experimental evaluation demonstrates superior empirical performance across diverse scenarios, achieving near-optimal results under accurate priors while maintaining robust worst-case guarantees. This framework naturally extends to incorporate multiple predictions, non-uniform priors, and contextual information, highlighting the practical advantages of Bayesian reasoning in online decision problems with imperfect predictions.

Paper Structure

This paper contains 31 sections, 4 theorems, 29 equations, 3 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Technical Overview
  4. Notation.
  5. Units.
  6. Related Work
  7. Preliminaries
  8. Classical Ski Rental
  9. Discrete Bayesian Extension
  10. Computational Complexity
  11. Bayesian Decision Procedure
  12. Algorithm Description
  13. Algorithm Properties
  14. When Bayesian Outperforms Classical Algorithms
  15. Case 2: Geometric Prior with Fixed Support
  16. ...and 16 more sections

Key Result

Lemma 1

Under the log-concave prior distribution, the expected remaining rental cost $E_{\text{rent}}(t)$ is non-increasing in $t$:

Figures (3)

  • Figure 1: Robustness under prior misspecification. (a) Performance degradation across uncertainty regimes remains small even at high total variation (TV) distances. (b) Mean errors have the largest impact, while variance and model errors are negligible. (c) CR distribution remains stable across TV bins.
  • Figure 2: Competitive ratio under perfect prior knowledge. The Bayesian method achieves CR $\approx 1.02$ across all priors, significantly outperforming classical algorithms.
  • Figure 3: Comparison of multi-modal priors (left) and their survival functions (right). Despite strong multi-modality in the density, the Bayesian threshold depends only on the integrated survival mass $S(t)$, producing stable and coherent purchase decisions.

Theorems & Definitions (5)

  • Lemma 1: Monotonicity of Purchase Incentive
  • Theorem 1: Uniform Prior
  • Theorem 2: Truncated Geometric Prior
  • Theorem 3: Adaptive Regret Bound
  • Definition 1: Contextual Prior