Generalization Guarantees for Learning Branch-and-Cut Policies in Integer Programming
Hongyu Cheng, Amitabh Basu
TL;DR
This work addresses the generalization of learned Branch-and-Cut policies for mixed-integer programming by formalizing scoring functions $f_k(s,a,\mathbf{w}^k)$ that guide node, cut, and branching decisions as piecewise polynomial functions of the learnable parameters. The authors develop a general sequential decision framework in which the overall cost $V(I,\mathbf{w})$ is shown to be piecewise constant when the scoring components possess a $(\Gamma,\gamma,\beta)$-structure, enabling rigorous pseudo-dimension bounds that extend beyond linear models to nonlinear networks such as ReLU-MLPs. They derive both worst-case and data-dependent sample complexity guarantees: for linear policies, $\mathrm{Pdim}(\mathcal{V}^L)=O(W M \sum_k \log \rho_k)$ with improvements over prior results, and for MLP-based scoring, explicit $\mathrm{Pdim}$ bounds that specialize to ReLU networks; data-dependent bounds via empirical Rademacher complexity further tie generalization to observed decision-space complexity $Q_{M,k}(I)$. These results provide principled, distribution-free guarantees for data-driven policy learning in B&C and demonstrate a unifying theory that encompasses modern neural architectures within a sequential decision-making context, with potential to extend to broader classes of optimization and CGF-based cuts.
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.