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Addressing misspecification in contextual optimization

Omar Bennouna, Jiawei Zhang, Saurabh Amin, Asuman Ozdaglar

TL;DR

The paper addresses misspecification in contextual optimization by introducing the Consistent Integrated Learning and Optimization (CILO) surrogate. CILO aligns the surrogate with the target downstream loss $\ell_P$ while remaining tractable through a difference-of-convex (DC) structure and smoothing via Moreau envelopes, enabling global optimality for linear predictor classes. The authors prove generalization guarantees and a quantitative link between the surrogate and the true loss, and they provide an algorithmic procedure with a line search over $\beta$ to obtain near-optimal solutions under misspecification. Empirical results on synthetic data show that CILO robustly outperforms SPO$^+$ and SLO when the hypothesis class is misspecified, highlighting its practical impact for reliable decision-making under uncertainty. ${}$

Abstract

We study a linear contextual optimization problem where a decision maker has access to historical data and contextual features to learn a cost prediction model aimed at minimizing decision error. We adopt the predict-then-optimize framework for this analysis. Given that perfect model alignment with reality is often unrealistic in practice, we focus on scenarios where the chosen hypothesis set is misspecified. In this context, it remains unclear whether current contextual optimization approaches can effectively address such model misspecification. In this paper, we present a novel integrated learning and optimization approach designed to tackle model misspecification in contextual optimization. This approach offers theoretical generalizability, tractability, and optimality guarantees, along with strong practical performance. Our method involves minimizing a tractable surrogate loss that aligns with the performance value from cost vector predictions, regardless of whether the model is misspecified, and can be optimized in reasonable time. To our knowledge, no previous work has provided an approach with such guarantees in the context of model misspecification.

Addressing misspecification in contextual optimization

TL;DR

The paper addresses misspecification in contextual optimization by introducing the Consistent Integrated Learning and Optimization (CILO) surrogate. CILO aligns the surrogate with the target downstream loss while remaining tractable through a difference-of-convex (DC) structure and smoothing via Moreau envelopes, enabling global optimality for linear predictor classes. The authors prove generalization guarantees and a quantitative link between the surrogate and the true loss, and they provide an algorithmic procedure with a line search over to obtain near-optimal solutions under misspecification. Empirical results on synthetic data show that CILO robustly outperforms SPO and SLO when the hypothesis class is misspecified, highlighting its practical impact for reliable decision-making under uncertainty.

Abstract

We study a linear contextual optimization problem where a decision maker has access to historical data and contextual features to learn a cost prediction model aimed at minimizing decision error. We adopt the predict-then-optimize framework for this analysis. Given that perfect model alignment with reality is often unrealistic in practice, we focus on scenarios where the chosen hypothesis set is misspecified. In this context, it remains unclear whether current contextual optimization approaches can effectively address such model misspecification. In this paper, we present a novel integrated learning and optimization approach designed to tackle model misspecification in contextual optimization. This approach offers theoretical generalizability, tractability, and optimality guarantees, along with strong practical performance. Our method involves minimizing a tractable surrogate loss that aligns with the performance value from cost vector predictions, regardless of whether the model is misspecified, and can be optimized in reasonable time. To our knowledge, no previous work has provided an approach with such guarantees in the context of model misspecification.
Paper Structure (29 sections, 11 theorems, 92 equations, 3 figures, 1 algorithm)

This paper contains 29 sections, 11 theorems, 92 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem overview
  3. Our focus and contributions
  4. Quantiyfing the level of misspecification
  5. Previous approaches to Integrated Learning and Optimization
  6. Directly optimizing the target loss.
  7. Optimizing a surrogate loss.
  8. Our approach
  9. Rewriting $\ell_P$ under a generic assumption
  10. Introducing our surrogate loss function
  11. Technical approach
  12. Generalization performance
  13. Relationship between the optimality gaps of $\ell_P$ and $\ell_P^\beta$
  14. Optimizing our surrogate
  15. Landscape properties and smoothing.
  16. ...and 14 more sections

Key Result

Proposition 1

If the context variable x has a continuous probability distribution and the mapping $x\longmapsto \hat{c}_\theta (x)$ is a nonzero analytic function for any $\theta\neq 0$, then Assumption regularity holds for any polyhedron $W$ that satisfies $\hat{c}_{\theta}(x)^\top v\neq 0$ for any $\theta\in {\

Figures (3)

  • Figure 1: Intuition for assumption \ref{['marginal hypothesis']}: $\hat{c}_\theta(x)$ is less likely to have a direction included in the red cones, which represent nearly perpendicular directions to one of the faces of the polyhedron.
  • Figure 2: Construction of $\Tilde{\lambda}$: when $\lambda$ is close to the sets $\overline{V}_{P_n}^\beta$ and $V_{P_n}$, we construct a new approximately stationary point by moving away from $V_{P_n}$ orthogonally to its boundary.
  • Figure 3: SPO+ v.s. SLO v.s. our approach comparison for different levels of misspecification

Theorems & Definitions (19)

  • Example 1
  • Definition 1
  • Proposition 1
  • Definition 2
  • Example 2
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1
  • ...and 9 more