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

arXiv: https://arxiv.org/abs/2505.11636 · HTML: https://mlsanity.com/paper/2505.11636/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 47 references that are in the map.
- First posted on arXiv 16 May 2025.

## Builds on (most cited first, 24 of 24)

- [Exact Combinatorial Optimization with Graph Convolutional Neural Networks](https://mlsanity.com/paper/1906.01629/) (2019)
- [Nearly-tight VC-dimension and pseudodimension bounds for piecewise linear neural networks](https://mlsanity.com/paper/1703.02930/) (2017)
- [Reinforcement Learning for Integer Programming: Learning to Cut](https://mlsanity.com/paper/1906.04859/) (2019)
- [Learning To Cut By Looking Ahead: Cutting Plane Selection via Imitation Learning](https://mlsanity.com/paper/2206.13414/) (2022)
- [A PAC Approach to Application-Specific Algorithm Selection](https://mlsanity.com/paper/1511.07147/) (2015)
- [Learning to Select Cuts for Efficient Mixed-Integer Programming](https://mlsanity.com/paper/2105.13645/) (2021)
- [Sample Complexity of Tree Search Configuration: Cutting Planes and Beyond](https://mlsanity.com/paper/2106.04033/) (2021)
- [Learning-Theoretic Foundations of Algorithm Configuration for Combinatorial Partitioning Problems](https://mlsanity.com/paper/1611.04535/) (2016)
- [How much data is sufficient to learn high-performing algorithms? Generalization guarantees for data-driven algorithm design](https://mlsanity.com/paper/1908.02894/) (2019)
- [Generalization Bounds for Data-Driven Numerical Linear Algebra](https://mlsanity.com/paper/2206.07886/) (2022)
- [Structural Analysis of Branch-and-Cut and the Learnability of Gomory Mixed Integer Cuts](https://mlsanity.com/paper/2204.07312/) (2022)
- [A Study of Learning Search Approximation in Mixed Integer Branch and Bound: Node Selection in SCIP](https://mlsanity.com/paper/2007.03948/) (2020)
- [Provably tuning the ElasticNet across instances](https://mlsanity.com/paper/2207.10199/) (2022)
- [Improved Sample Complexity Bounds for Branch-and-Cut](https://mlsanity.com/paper/2111.11207/) (2021)
- [Generalization Guarantees for Multi-item Profit Maximization: Pricing, Auctions, and Randomized Mechanisms](https://mlsanity.com/paper/1705.00243/) (2017)
- [Generalization Bound and Learning Methods for Data-Driven Projections in Linear Programming](https://mlsanity.com/paper/2309.00203/) (2023)
- [Sample complexity of data-driven tuning of model hyperparameters in neural networks with structured parameter-dependent dual function](https://mlsanity.com/paper/2501.13734/) (2025)
- [Generalization in portfolio-based algorithm selection](https://mlsanity.com/paper/2012.13315/) (2020)
- [Sample Complexity of Algorithm Selection Using Neural Networks and Its Applications to Branch-and-Cut](https://mlsanity.com/paper/2402.02328/) (2024)
- [Sample Complexity of Learning Heuristic Functions for Greedy-Best-First and A* Search](https://mlsanity.com/paper/2205.09963/) (2022)
- [Learning Cut Generating Functions for Integer Programming](https://mlsanity.com/paper/2405.13992/) (2024)
- [An Abstract Model for Branch and Cut](https://mlsanity.com/paper/2111.09907/) (2021)
- [Learning complexity of gradient descent and conjugate gradient algorithms](https://mlsanity.com/paper/2412.13473/) (2024)
- [A geometric approach to cut-generating functions](https://mlsanity.com/paper/1701.06692/) (2017)

## Abstract (arXiv)

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.