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Conformal Inverse Optimization

Bo Lin, Erick Delage, Timothy C. Y. Chan

TL;DR

This work addresses the failure mode of traditional inverse optimization when point estimates yield prescriptive decisions that misalign with human intuition. It introduces conformal IO, which learns an angular uncertainty set around a point estimate and solves a robust forward optimization to prescribe decisions, with probabilistic guarantees on coverage and bounded optimality gaps. Theoretical results establish conservative validity and asymptotic exactness of the learned uncertainty sets, along with explicit bounds on AOG and POG; empirical studies on shortest path and knapsack problems show substantial performance gains over classic IO. The approach enhances practical adoption by producing high-quality decisions that better reflect decision-maker perceptions, while maintaining tractable calibration and prediction steps. Overall, conformal IO integrates decision-data-driven uncertainty calibration with robust optimization to produce reliable, human-aligned prescriptions in data-driven optimization tasks.

Abstract

Inverse optimization has been increasingly used to estimate unknown parameters in an optimization model based on decision data. We show that such a point estimation is insufficient in a prescriptive setting where the estimated parameters are used to prescribe new decisions. The prescribed decisions may be low-quality and misaligned with human intuition and thus are unlikely to be adopted. To tackle this challenge, we propose conformal inverse optimization, which seeks to learn an uncertainty set for the unknown parameters and then solve a robust optimization model to prescribe new decisions. Under mild assumptions, we show that our method enjoys provable guarantees on solution quality, as evaluated using both the ground-truth parameters and the decision maker's perception of the unknown parameters. Our method demonstrates strong empirical performance compared to classic inverse optimization.

Conformal Inverse Optimization

TL;DR

This work addresses the failure mode of traditional inverse optimization when point estimates yield prescriptive decisions that misalign with human intuition. It introduces conformal IO, which learns an angular uncertainty set around a point estimate and solves a robust forward optimization to prescribe decisions, with probabilistic guarantees on coverage and bounded optimality gaps. Theoretical results establish conservative validity and asymptotic exactness of the learned uncertainty sets, along with explicit bounds on AOG and POG; empirical studies on shortest path and knapsack problems show substantial performance gains over classic IO. The approach enhances practical adoption by producing high-quality decisions that better reflect decision-maker perceptions, while maintaining tractable calibration and prediction steps. Overall, conformal IO integrates decision-data-driven uncertainty calibration with robust optimization to produce reliable, human-aligned prescriptions in data-driven optimization tasks.

Abstract

Inverse optimization has been increasingly used to estimate unknown parameters in an optimization model based on decision data. We show that such a point estimation is insufficient in a prescriptive setting where the estimated parameters are used to prescribe new decisions. The prescribed decisions may be low-quality and misaligned with human intuition and thus are unlikely to be adopted. To tackle this challenge, we propose conformal inverse optimization, which seeks to learn an uncertainty set for the unknown parameters and then solve a robust optimization model to prescribe new decisions. Under mild assumptions, we show that our method enjoys provable guarantees on solution quality, as evaluated using both the ground-truth parameters and the decision maker's perception of the unknown parameters. Our method demonstrates strong empirical performance compared to classic inverse optimization.
Paper Structure (35 sections, 16 theorems, 72 equations, 5 figures, 2 tables)

This paper contains 35 sections, 16 theorems, 72 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Preliminaries
  4. Problem Setup
  5. Assumptions
  6. Evaluation Metrics
  7. An Inverse Optimization Pipeline
  8. Robustifying the Inverse Optimization Pipeline
  9. Conformal Inverse Optimization
  10. Learning an Uncertainty Set
  11. Properties of Conformal IO
  12. Numerical Studies
  13. Conclusion and Future Work
  14. Omitted Statements and Proofs in Section \ref{['sec:prelim']}
  15. Poof of Lemma \ref{['lemma:2d_exp_asy_est']}
  16. ...and 20 more sections

Key Result

Lemma 1

In Example ex:2d, let $\bar{\boldsymbol{\theta}}_N$ denote an optimal solution to $\mathbf{IO}(\mathcal{D})$ with the sub-optimality loss eq:sub_opt_loss, we have $\mathbb{P}\left( \bar{\boldsymbol{\theta}}_N = \boldsymbol{\theta}_u \right) \rightarrow 1$ as $N \rightarrow \infty$, where $\boldsymbo

Figures (5)

  • Figure 1: Classic and conformal IO pipelines.
  • Figure 2: Illustration of Example \ref{['ex:2d']}. The gray areas are the feasible region $\mathcal{X}(u)$. The black arrows are the ground-truth parameter $\boldsymbol{\theta}^*$. The gray arrows are the extreme rays of $\boldsymbol{\Theta}$. The blue and green arrows are the point estimation $\bar{\boldsymbol{\theta}}$. The green area is the uncertainty set $\mathcal{C}(\bar{\boldsymbol{\theta}}, \alpha)$. The black circles are the optimal solution to $\mathbf{FO}(\boldsymbol{\theta}^*, u)$. The blue and green circles are the suggested decisions. Note that $\bar{\mathbf{x}}_\textrm{IO}$ may suggest any decisions on the facet of $x_1 + ux_2 \geq u$, which are omitted for clarity.
  • Figure 3: Empirical coverage achieved by the learned uncertainty set (error bar = range).
  • Figure 4: Performance profile of classic (blue) and conformal IO (green).
  • Figure 5: Percentage reduction in AOG and POG when using the conformal IO vs classic IO.

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Lemma 1
  • Proposition 1
  • Remark 1
  • Lemma 2
  • Remark 2: Optimality Conditions
  • Remark 3: Feasibility
  • ...and 17 more