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

TL;DR

This paper provides theoretical sample complexity bounds for learning decision policies in Branch-and-Cut algorithms for integer programming, encompassing neural networks and piecewise polynomial models, thus bridging theory and practice.

Contribution

It establishes the first rigorous generalization guarantees for learned B&C policies with piecewise polynomial scoring functions, including neural networks.

Findings

Sample complexity bounds are derived for piecewise polynomial policies.

The framework applies to neural network architectures like ReLU-based models.

The theory extends to general sequential decision-making problems.

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.