Invariant Predict-and-Combinatorial Optimization under Distribution Shifts

ICLR 2026 Conference Submission15933 Authors

Keywords: Combinatorial Optimization, Predict-and-Optimize, Generalization

Abstract: Machine learning has been well introduced to solve combinatorial optimization (CO) problems over the decade, while most works only consider the deterministic setting. Yet in real-world applications, decisions have often to be made in uncertain environments, which is typically reflected by the stochasticity of the coefficients of the problem at hand, considered as a special case of the more general and emerging "predict-and-optimize" (PnO) paradigm in the sense that the prediction and optimization are jointly learned and performed. In this paper, we consider the problem of learning to solve CO under the above uncertain setting and formulate it as "predict-and-combinatorial optimization" (PnCO), particularly in a challenging yet practical out-of-distribution (OOD) setting, where there is a distribution shift between training and testing CO instances. We propose the Invariant Predict-and-Combinatorial Optimization (Inv-PnCO) framework to alleviate this challenge. Inv-PnCO derives a learning objective that reduces the distance of distribution of solutions with the true distribution and uses a regularization term to learn invariant decision-oriented factors that are stable under various environments, thereby enhancing the generalizability of predictions and subsequent optimizations. We also provide a theoretical analysis of how the proposed loss reduces OOD error. The empirical evaluation across three distinct tasks on knapsack, visual shortest path planning, and traveling salesman problem covering array, image, and graph inputs underscores the efficacy of Inv-PnCO to enhance the generalizability, both for predict-then-optimize and predict-and-optimize approaches.

Primary Area: optimization

Submission Number: 15933