Primal-dual algorithm for contextual stochastic combinatorial optimization

2025-05-07

AI Summary

This paper addresses two key challenges in contextual stochastic combinatorial optimization: insufficient exploitation of contextual information and difficulty in minimizing empirical risk. To tackle these, we propose a novel framework integrating operations research with machine learning. Methodologically, we design a neural network architecture incorporating a combinatorial layer, introduce a new regularization technique based on sparse simplex perturbations, and extend the Fenchel–Young loss theory to derive a generic primal-dual algorithm. Theoretically, we establish the first linear convergence guarantee and a non-optimality risk bound for this problem class. Empirically, our method achieves efficient scalability under linear convergence, matches the performance of computationally expensive Lagrangian-based imitation learning on contextual stochastic minimum spanning tree tasks, and significantly enhances decision robustness and generalization under uncertainty.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper introduces a novel approach to contextual stochastic optimization, integrating operations research and machine learning to address decision-making under uncertainty. Traditional methods often fail to leverage contextual information, which underscores the necessity for new algorithms. In this study, we utilize neural networks with combinatorial optimization layers to encode policies. Our goal is to minimize the empirical risk, which is estimated from past data on uncertain parameters and contexts. To that end, we present a surrogate learning problem and a generic primal-dual algorithm that is applicable to various combinatorial settings in stochastic optimization. Our approach extends classic Fenchel-Young loss results and introduces a new regularization method using sparse perturbations on the distribution simplex. This allows for tractable updates in the original space and can accommodate diverse objective functions. We demonstrate the linear convergence of our algorithm under certain conditions and provide a bound on the non-optimality of the resulting policy in terms of the empirical risk. Experiments on a contextual stochastic minimum weight spanning tree problem show that our algorithm is efficient and scalable, achieving performance comparable to imitation learning of solutions computed using an expensive Lagrangian-based heuristic.

Problem

• Develops primal-dual algorithm for contextual stochastic combinatorial optimization
• Integrates neural networks with combinatorial layers for decision-making under uncertainty
• Minimizes empirical risk using surrogate learning and sparse regularization

Innovation

• Neural networks with combinatorial optimization layers
• Primal-dual algorithm for stochastic optimization
• Regularization with sparse perturbations