Generalization Guarantees for Learning Branch-and-Cut Policies in Integer Programming

2025-05-16

AI Summary

This paper establishes a rigorous generalization theory for learning Branch-and-Cut (B&C) policies in integer programming, focusing on three core sequential decisions—node selection, cut selection, and branching variable selection—each guided by parameterized scoring functions. Method: Leveraging the piecewise-polynomial structure of such scoring functions, we derive the first unified VC-dimension characterization for broad function classes—including linear policies and ReLU neural networks—and integrate sequential decision-theoretic generalization analysis with statistical learning theory to obtain tight sample complexity bounds. Contribution/Results: We prove that a finite number of training instances suffices to guarantee generalization of learned B&C policies to unseen problem instances. Furthermore, we extend this theoretical framework to generic sequential decision-making problems, thereby bridging a fundamental gap between classical operations research theory and modern machine learning practice.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Online Learning & Bandits
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• 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: Social networks and social learning

Abstract

Mixed-integer programming (MIP) provides a powerful framework for optimization problems, with Branch-and-Cut (B&C) being the predominant algorithm in state-of-the-art solvers. The efficiency of B&C critically depends on heuristic policies for making sequential decisions, including node selection, cut selection, and branching variable selection. While traditional solvers often employ heuristics with manually tuned parameters, recent approaches increasingly leverage machine learning, especially neural networks, to learn these policies directly from data. A key challenge is to understand the theoretical underpinnings of these learned policies, particularly their generalization performance from finite data. This paper establishes rigorous sample complexity bounds for learning B&C policies where the scoring functions guiding each decision step (node, cut, branch) have a certain piecewise polynomial structure. This structure generalizes the linear models that form the most commonly deployed policies in practice and investigated recently in a foundational series of theoretical works by Balcan et al. Such piecewise polynomial policies also cover the neural network architectures (e.g., using ReLU activations) that have been the focal point of contemporary practical studies. Consequently, our theoretical framework closely reflects the models utilized by practitioners investigating machine learning within B&C, offering a unifying perspective relevant to both established theory and modern empirical research in this area. Furthermore, our theory applies to quite general sequential decision making problems beyond B&C.

Problem

• Theoretical generalization guarantees for learned B&C policies
• Sample complexity bounds for piecewise polynomial scoring functions
• Unifying framework for ML in B&C and sequential decision-making

Innovation

• Learns B&C policies using piecewise polynomial models
• Generalizes linear models with neural networks
• Provides sample complexity bounds for generalization