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Robust Data-driven Prescriptiveness Optimization

Mehran Poursoltani, Erick Delage, Angelos Georghiou

TL;DR

This paper addresses robustly leveraging side information in data-driven prescriptive optimization by introducing the distributionally robust coefficient of prescriptiveness (DRPCR). It casts maximizing prescriptiveness as a convex optimization and solves it with a bisection method that reduces to linear programs under nested CVaR ambiguity, with extensions to generalized nested sets. The authors demonstrate the approach on a contextual shortest-path problem under distribution shift, showing DRPCR yields superior out-of-sample performance and a distinctive regularization effect anchored to the SAA benchmark. The work provides a principled framework for robust prescriptive policies and highlights future directions to improve tractability and applicability across domains.

Abstract

The abundance of data has led to the emergence of a variety of optimization techniques that attempt to leverage available side information to provide more anticipative decisions. The wide range of methods and contexts of application have motivated the design of a universal unitless measure of performance known as the coefficient of prescriptiveness. This coefficient was designed to quantify both the quality of contextual decisions compared to a reference one and the prescriptive power of side information. To identify policies that maximize the former in a data-driven context, this paper introduces a distributionally robust contextual optimization model where the coefficient of prescriptiveness substitutes for the classical empirical risk minimization objective. We present a bisection algorithm to solve this model, which relies on solving a series of linear programs when the distributional ambiguity set has an appropriate nested form and polyhedral structure. Studying a contextual shortest path problem, we evaluate the robustness of the resulting policies against alternative methods when the out-of-sample dataset is subject to varying amounts of distribution shift.

Robust Data-driven Prescriptiveness Optimization

TL;DR

This paper addresses robustly leveraging side information in data-driven prescriptive optimization by introducing the distributionally robust coefficient of prescriptiveness (DRPCR). It casts maximizing prescriptiveness as a convex optimization and solves it with a bisection method that reduces to linear programs under nested CVaR ambiguity, with extensions to generalized nested sets. The authors demonstrate the approach on a contextual shortest-path problem under distribution shift, showing DRPCR yields superior out-of-sample performance and a distinctive regularization effect anchored to the SAA benchmark. The work provides a principled framework for robust prescriptive policies and highlights future directions to improve tractability and applicability across domains.

Abstract

The abundance of data has led to the emergence of a variety of optimization techniques that attempt to leverage available side information to provide more anticipative decisions. The wide range of methods and contexts of application have motivated the design of a universal unitless measure of performance known as the coefficient of prescriptiveness. This coefficient was designed to quantify both the quality of contextual decisions compared to a reference one and the prescriptive power of side information. To identify policies that maximize the former in a data-driven context, this paper introduces a distributionally robust contextual optimization model where the coefficient of prescriptiveness substitutes for the classical empirical risk minimization objective. We present a bisection algorithm to solve this model, which relies on solving a series of linear programs when the distributional ambiguity set has an appropriate nested form and polyhedral structure. Studying a contextual shortest path problem, we evaluate the robustness of the resulting policies against alternative methods when the out-of-sample dataset is subject to varying amounts of distribution shift.
Paper Structure (20 sections, 6 theorems, 53 equations, 3 figures, 2 tables, 5 algorithms)

This paper contains 20 sections, 6 theorems, 53 equations, 3 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Motivation for optimizing ${\mathcal{P}}$ and its robustification
  3. Robust Data-driven Prescriptiveness Optimization
  4. Convex formulation for DRPCR
  5. The nested CVaR ambiguity set $\mathcal{D}$
  6. Generalized nested ambiguity set $\mathcal{D}$
  7. Experiments
  8. Conclusion
  9. Proofs
  10. Proof of Lemma \ref{['thm:ratioInterval']}
  11. Proof of Lemma \ref{['thm:ERMreduction']}
  12. Proof of Proposition \ref{['thm:epigraph']}
  13. Proof of Proposition \ref{['thm:epigraph_reformulation']}
  14. Proof of Lemma \ref{['convergence_algorithm1']}
  15. Proof of Proposition \ref{['generalized_proposition2']}
  16. ...and 5 more sections

Key Result

Lemma 1

If $\bar{{{\boldsymbol x}}}\in{\mathcal{H}}$, then the optimal value of DRPCR is necessarily in the interval $[0,\,1]$.

Figures (3)

  • Figure 1: Shortest path problem: (a) statistics of the out-of-sample coefficient of prescriptiveness (lower values indicate worse performance). (b) statistics of $\mathbb{E}_{{\breve{F}}}[\|\boldsymbol{x^*}(\boldsymbol \zeta) - \boldsymbol{\hat{x}}\|_1]$ where $\breve F$ is the out-of-sample distribution (lower values reflect a closer proximity to the SAA solution).
  • Figure 2: Visualization of the basic (left) and accelerated (right) bisection algorithm. The blue squared brackets indicate the current estimated interval containing the optimal $\gamma^*$ and the red squared brackets indicate the interval in the next iterations. The right graph also visualizes the over and under estimators of $\psi(\gamma)$.
  • Figure 3: Shortest path problem (relaxed version): (a) statistics of the out-of-sample coefficient of prescriptiveness (lower values indicate worse performance). (b) statistics of $\mathbb{E}_{{\breve{F}}}[\|\boldsymbol{x^*}(\boldsymbol \zeta) - \boldsymbol{\hat{x}}\|_1]$ where $\breve F$ is the out-of-sample distribution (lower values reflect a closer proximity to the SAA solution).

Theorems & Definitions (6)

  • Lemma 1
  • Lemma 2
  • Proposition 1
  • Proposition 2
  • Lemma 3
  • Proposition 3