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Learning Decisions Offline from Censored Observations with ε-insensitive Operational Costs

Minxia Chen, Ke Fu, Teng Huang, Miao Bai

TL;DR

The paper tackles offline, data-driven decision-making when historical observations are censored and censoring indicators are unavailable. It introduces $\varepsilon$-insensitive operational costs to guide one-step learning without distributional assumptions. It develops three algorithms—LR-$\varepsilon$NVC, LR-$\varepsilon$NVC-R, and NN-$\varepsilon$NVC—with corresponding stability and generalization guarantees, and demonstrates superior out-of-sample performance on a multi-feature newsvendor problem, achieving cost savings up to $14.40\%$ and $12.21\%$ compared to two strong baselines. The results highlight the approach's practical value for learning decisions from censored offline data in operations management.

Abstract

Many important managerial decisions are made based on censored observations. Making decisions without adequately handling the censoring leads to inferior outcomes. We investigate the data-driven decision-making problem with an offline dataset containing the feature data and the censored historical data of the variable of interest without the censoring indicators. Without assuming the underlying distribution, we design and leverage ε-insensitive operational costs to deal with the unobserved censoring in an offline data-driven fashion. We demonstrate the customization of the ε-insensitive operational costs for a newsvendor problem and use such costs to train two representative ML models, including linear regression (LR) models and neural networks (NNs). We derive tight generalization bounds for the custom LR model without regularization (LR-εNVC) and with regularization (LR-εNVC-R), and a high-probability generalization bound for the custom NN (NN-εNVC) trained by stochastic gradient descent. The theoretical results reveal the stability and learnability of LR-εNVC, LR-εNVC-R and NN-εNVC. We conduct extensive numerical experiments to compare LR-εNVC-R and NN-εNVC with two existing approaches, estimate-as-solution (EAS) and integrated estimation and optimization (IEO). The results show that LR-εNVC-R and NN-εNVC outperform both EAS and IEO, with maximum cost savings up to 14.40% and 12.21% compared to the lowest cost generated by the two existing approaches. In addition, LR-εNVC-R's and NN-εNVC's order quantities are statistically significantly closer to the optimal solutions should the underlying distribution be known.

Learning Decisions Offline from Censored Observations with ε-insensitive Operational Costs

TL;DR

The paper tackles offline, data-driven decision-making when historical observations are censored and censoring indicators are unavailable. It introduces -insensitive operational costs to guide one-step learning without distributional assumptions. It develops three algorithms—LR-NVC, LR-NVC-R, and NN-NVC—with corresponding stability and generalization guarantees, and demonstrates superior out-of-sample performance on a multi-feature newsvendor problem, achieving cost savings up to and compared to two strong baselines. The results highlight the approach's practical value for learning decisions from censored offline data in operations management.

Abstract

Many important managerial decisions are made based on censored observations. Making decisions without adequately handling the censoring leads to inferior outcomes. We investigate the data-driven decision-making problem with an offline dataset containing the feature data and the censored historical data of the variable of interest without the censoring indicators. Without assuming the underlying distribution, we design and leverage ε-insensitive operational costs to deal with the unobserved censoring in an offline data-driven fashion. We demonstrate the customization of the ε-insensitive operational costs for a newsvendor problem and use such costs to train two representative ML models, including linear regression (LR) models and neural networks (NNs). We derive tight generalization bounds for the custom LR model without regularization (LR-εNVC) and with regularization (LR-εNVC-R), and a high-probability generalization bound for the custom NN (NN-εNVC) trained by stochastic gradient descent. The theoretical results reveal the stability and learnability of LR-εNVC, LR-εNVC-R and NN-εNVC. We conduct extensive numerical experiments to compare LR-εNVC-R and NN-εNVC with two existing approaches, estimate-as-solution (EAS) and integrated estimation and optimization (IEO). The results show that LR-εNVC-R and NN-εNVC outperform both EAS and IEO, with maximum cost savings up to 14.40% and 12.21% compared to the lowest cost generated by the two existing approaches. In addition, LR-εNVC-R's and NN-εNVC's order quantities are statistically significantly closer to the optimal solutions should the underlying distribution be known.
Paper Structure (34 sections, 10 theorems, 53 equations, 5 figures, 11 tables, 1 algorithm)

This paper contains 34 sections, 10 theorems, 53 equations, 5 figures, 11 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Decision-Making with Unobserved Censoring
  3. Newsvendor with Sales Data.
  4. Custom Pricing.
  5. Preventive Replacement.
  6. Model Overview
  7. Summary of Results and Contributions
  8. Literature Review
  9. Data-Driven Decision-Making under Uncertainty
  10. Censored Observations in Business Decision-Making
  11. $\varepsilon$-insensitive Loss Functions
  12. Offline Decision-Learning Algorithms with Censored Observations
  13. Custom $\varepsilon$-insensitive Operational Costs
  14. Newsvendor Ordering.
  15. Custom Pricing.
  16. ...and 19 more sections

Key Result

Proposition 1

The gradient with respect to the parameters $\theta^j$ in the LR model for the loss function $\mathcal{L}^{\varepsilon NV}\xspace_i(\boldsymbol{\theta}|(\boldsymbol{x}_i,s_i))$ is

Figures (5)

  • Figure 1: A Simple Neural Network
  • Figure 2: Out-of-Sample Average Newsvendor Costs
  • Figure 3: Mean Percentage Cost Savings of LR-$\varepsilon$NVC-R (NN-$\varepsilon$NVC) over LR-NVC (NN-NVC)
  • Figure 4: Out-of-Sample $\text{RMSE}^{\text{Q}}$
  • Figure C.1: QQ Plot of Residual Error for Store 10 with Normal Distribution $\mathcal{N}(0,46.57^2)$

Theorems & Definitions (11)

  • Proposition 1: Gradients of $\mathcal{L}^{\varepsilon NV}\xspace$ in LR-$\varepsilon$NVC
  • Proposition 2: Gradients of $\mathcal{L}^{\varepsilon NV}\xspace$ in NN-$\varepsilon$NVC
  • Lemma 1: Tight uniform bound on $\mathcal{L}^{\varepsilon NV}\xspace$
  • Lemma 2
  • Proposition 3: Uniform stability of LR-$\varepsilon$NVC
  • Theorem 1: Generalization bound for LR-$\varepsilon$NVC
  • Proposition 4: Uniform stability of LR-$\varepsilon$NVC-R
  • Theorem 2: Generalization bound for LR-$\varepsilon$NVC-R
  • Proposition 5: Bound on UAS for NN-$\varepsilon$NVC (SGD)
  • Theorem 3: Generalization bound for NN-$\varepsilon$NVC (SGD)
  • ...and 1 more