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What is the Value of Censored Data? An Exact Analysis for the Data-driven Newsvendor

Rachitesh Kumar, Omar Mouchtaki

TL;DR

This analysis of the Kaplan-Meier policy shows that while demand censoring fundamentally limits what can be learned from passive sales data, just a small amount of targeted exploration at high inventory levels can substantially improve worst-case guarantees, enabling near-optimal performance even under heavy censoring.

Abstract

We study the offline data-driven newsvendor problem with censored demand data. In contrast to prior works where demand is fully observed, we consider the setting where demand is censored at the inventory level and only sales are observed; sales match demand when there is sufficient inventory, and equal the available inventory otherwise. We provide a general procedure to compute the exact worst-case regret of classical data-driven inventory policies, evaluated over all demand distributions. Our main technical result shows that this infinite-dimensional, non-convex optimization problem can be reduced to a finite-dimensional one, enabling an exact characterization of the performance of policies for any sample size and censoring levels. We leverage this reduction to derive sharp insights on the achievable performance of standard inventory policies under demand censoring. In particular, our analysis of the Kaplan-Meier policy shows that while demand censoring fundamentally limits what can be learned from passive sales data, just a small amount of targeted exploration at high inventory levels can substantially improve worst-case guarantees, enabling near-optimal performance even under heavy censoring. In contrast, when the point-of-sale system does not record stockout events and only reports realized sales, a natural and commonly used approach is to treat sales as demand. Our results show that policies based on this sales-as-demand heuristic can suffer severe performance degradation as censored data accumulates, highlighting how the quality of point-of-sale information critically shapes what can, and cannot, be learned offline.

What is the Value of Censored Data? An Exact Analysis for the Data-driven Newsvendor

TL;DR

This analysis of the Kaplan-Meier policy shows that while demand censoring fundamentally limits what can be learned from passive sales data, just a small amount of targeted exploration at high inventory levels can substantially improve worst-case guarantees, enabling near-optimal performance even under heavy censoring.

Abstract

We study the offline data-driven newsvendor problem with censored demand data. In contrast to prior works where demand is fully observed, we consider the setting where demand is censored at the inventory level and only sales are observed; sales match demand when there is sufficient inventory, and equal the available inventory otherwise. We provide a general procedure to compute the exact worst-case regret of classical data-driven inventory policies, evaluated over all demand distributions. Our main technical result shows that this infinite-dimensional, non-convex optimization problem can be reduced to a finite-dimensional one, enabling an exact characterization of the performance of policies for any sample size and censoring levels. We leverage this reduction to derive sharp insights on the achievable performance of standard inventory policies under demand censoring. In particular, our analysis of the Kaplan-Meier policy shows that while demand censoring fundamentally limits what can be learned from passive sales data, just a small amount of targeted exploration at high inventory levels can substantially improve worst-case guarantees, enabling near-optimal performance even under heavy censoring. In contrast, when the point-of-sale system does not record stockout events and only reports realized sales, a natural and commonly used approach is to treat sales as demand. Our results show that policies based on this sales-as-demand heuristic can suffer severe performance degradation as censored data accumulates, highlighting how the quality of point-of-sale information critically shapes what can, and cannot, be learned offline.
Paper Structure (22 sections, 14 theorems, 131 equations, 4 figures)

This paper contains 22 sections, 14 theorems, 131 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Related Literature
  4. Model
  5. Notation.
  6. Exact Performance Characterization
  7. General Reduction
  8. Analysis of Biased SAA
  9. Analysis of the Kaplan-Meier Policy
  10. Structural Insights
  11. On the Value of Exploration for Kaplan-Meier
  12. Performance Behavior of the BSAA Policy
  13. Sample Complexity for Different Censoring Levels
  14. Conclusion
  15. Proofs of Results in \ref{['sec:gen_reduction']}
  16. ...and 7 more sections

Key Result

Lemma 1

For any policy $\pi_{\bm{n}}$, any fixed design $\bm{x}$, and any demand distribution $F$, the expected regret satisfies where $q = c_u/(c_u+c_o)$ is the critical fractile.

Figures (4)

  • Figure 1: $\Psi^{\mathrm{BSAA}}_k(t)$ functions for $K=5$, $q=0.9$, and sample allocation $\bm n=(1,1,1,1,15)$.
  • Figure 2: KM policy, $q = 0.8$: effect of sample size $n$.
  • Figure 3: BSAA policy, $c_u = 0.8$: effect of sample size $n$.
  • Figure 4: Sample complexity. Minimal sample size required to achieve the target worst-case regret for the Kaplan-Meier policy, for different critical quantiles. The target regret for each curve is $25\%$ of the no-information regret.

Theorems & Definitions (26)

  • Lemma 1
  • Definition 1: Piecewise-separable policy
  • Theorem 1
  • Lemma 2: Action distribution of $\pi^{\mathrm{BSAA}}$
  • Corollary 1
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 2
  • Lemma 6: Piecewise-separability of the Kaplan--Meier policy
  • ...and 16 more