Title: Generalization Guarantees for Learning Score-Based Branch-and-Cut Policies in Integer Programming
Venue: NeurIPS 2025
Accepted: yes
Keywords: Integer programming

branch-and-cut

cutting planes

branch-and-bound

sample complexity

learning theory
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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper contributes to a recent line of work on provable sample-complexity guarantees for learning to tune branch-and-cut. Prior work by Balcan et al. has focused on learning linear weights to combine predetermined heuristic scores such as efficacy, parallelism, etc. in order to choose between branch-and-cut actions (such as what cuts to add, what variable to branch on, what node to explore next, and so on). The present work builds and extends upon that line of work by learning scores that are piecewise polynomial functions of the parameter weights. This is better reflective of current practice that has used neural networks to map instances (the state of a branch-and-cut solve of an integer program) to desired actions (the next cut to add, etc.) directly. The bounds are obtained by analyzing the pseudodimension of the class of induced cost functions, and the authors do this for a general abstraction of a sequential decision process that generalizes branch-and-cut.\n\nSTRENGTHS AND WEAKNESSES:\nThis is a nice theory paper that contributes a missing piece to the recent generalization theory for learning cut selection policies. It provides a theoretical backing for various experimental papers. The proofs mostly seem to rely on by-now standard techniques, but that is ok. Some experiments showing the impact of parameter tuning and/or empirically testing the sample complexity would be nice, but not required for a theory paper.\n\nOne weakness is that the work could be seen as largely incremental to prior work, with no fundamentally new techniques introduced. But I also think that is OK.\n\nSee the question section for a few specific comments.\n\nQUESTIONS:\nIn [1] the authors also consider a general abstraction of tree search that generalizes branch-and-cut, and their main sample complexity results are for that general model of tree search. How does that model compare to the model of a sequential decision process that you consider in the present paper? (I see that the authors in [1] also only consider linear weightings of scores, but a comparison of the two models would nonetheless be useful.) Also, how does the model compare to the formal model of dynamic programming?\n\nDefinition 3.1 to me seems like a special case of the general notion of piecewise decomposability in [2]. Is that correct? If so there should probably be some mention of this.\n\nVery minor nitpick on notation: I think using k = 1, 2, 3 to denote node selection, cuts, and branching is awkward and you should just write something like A^s_{node}, A^s_{cut}, A^s_{branch} or something like that.\n\n\n[1] Balcan, Prasad, Sandholm, Vitercik. Improved sample complexity bounds for branch-and-cut. CP 2022.\n\n[2] Balcan, Deblasio, Dick, Kingsford, Sandholm, Vitercik. How Much Data Is Sufficient to Learn High-Performing Algorithms? Journal of the ACM.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-17"}

comments: SUMMARY:
This paper presents a rigorous theoretical framework to analyze the generalization performance of learned heuristic policies for Branch-and-Cut (B&C) algorithms in Mixed Integer Programming (MIP). It introduces sample complexity bounds for policies guided by piecewise polynomial scoring functions, a class that includes linear models and neural networks (e.g., MLPs with ReLU).

STRENGTHS AND WEAKNESSES:
The strength of this paper:

1. extends previous PAC-style algorithm configuration theory, yielding new pseudo-dimension and Rademacher complexity bounds for both worst-case and data-dependent settings. It bridges the gap between empirical methods and theoretical guarantees.

The weakness of this paper:

1. Lacks guidance on how to translate theory into practice (e.g., how to bound pseudo-dimension for specific networks in practice). Providing case studies may facilitate the reader's better understanding of the claims in this paper.

QUESTIONS:
(1) How sensitive are your generalization guarantees to the expert quality (e.g., strong branching vs. suboptimal heuristic)?
(2) How do the theoretical bounds scale in practice with real-world ReLU network sizes used in B&C solvers?

LIMITATIONS:
(1) Role of oracle quality in training (e.g., strong branching) could influence generalization.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-06-23

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper develops a theoretical framework for analyzing the generalization guarantees of learned policies for Branch and Cut (B&C) when solving mixed-integer programs (MIPs). B&C is one of the most widely used techniques for solving MIPs. Existing solvers primarily rely on heuristics for node selection, cut selection, and branching variable selection. Recently, there has been growing interest in learning B&C policies. This paper contributes to this line of work by providing sample complexity bounds for learning such policies. \n\n \n\nThe authors view B&C as a sequential decision-making process involving three types of actions at each step: node selection, cut selection, and branching variable selection. The goal is to learn a parametric scoring function that guides these actions. To analyze the difficulty of learning such policies, the authors derive sample complexity upper bounds for both linear and multilayer perceptron (MLP) scoring functions. \n\nThe framework presented in Section 3 provides a general analysis of imitation learning, with learning B&C policies treated as a specific application of imitation learning. Finally Section 4 provides bound for B&C from the bounds developed in Section 3.\n\nSTRENGTHS AND WEAKNESSES:\nI am not entirely familiar with the research topic. I am aware of some recent works which aim to learn policies for solving MIP problems. This paper, on the other hand, makes a strong theoretical contribution and provides a generic framework for sample complexity bounds for learning B&C policies\n\nWhile the theoretical framework is well-developed, the paper could benefit from a computational study or empirical experiments.\n\nQUESTIONS:\n- In the context of Branch-and-Cut, what exactly does the penalty function measure? Could you clarify its practical motivation—for example, how it reflects solver performance or computational cost? \n\n- How does the complexity analysis compare to the analysis provided in Theorem 5.1 of Cheng, Basu (NeurIPS 2024)?  \n\n- How might your theoretical findings impact  or strengthen existing learning-based B&C methods such as those proposed by Tang et al. (ICML,2024), Khalil el al (AAAI, 2016), Cheng, Basu (NeurIPS 2024)?\n\nLIMITATIONS:\nYES", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 4, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper provides a theoretical analysis of the sample complexity for learning heuristic policies within the Branch-and-Cut (B&C) framework for mixed-integer programming. The authors introduce a general model for sequential decision-making that abstracts the B&C process. Their central contribution is to show that if the scoring functions guiding decisions (e.g., node, cut, and variable selection) have a \"piecewise polynomial\" structure with respect to their learnable parameters, then the overall performance metric (e.g., B&C tree size) is a \"piecewise constant\" function of those same parameters. This structural insight allows them to derive novel generalization bounds, via pseudo-dimension and Rademacher complexity, for policies parameterized not only by traditional linear models but also, significantly, by modern neural networks with ReLU activations.\n\nSTRENGTHS AND WEAKNESSES:\nThough I checked the proofs, I am not an expert in statistical learning theory and may have overlooked some concepts\n\n# Strengths\n1. The primary strength of this paper is extending the formal analysis of learnable algorithm policies to the non-linear domain, specifically ReLU-based MLPs. This is highly relevant, as much of the recent empirical success in this area has come from such models. This work provides a much-needed theoretical foundation for these practices.\n\n2. The introduction of the $(\\Gamma,\\gamma,\\beta)$-structure is an elegant and powerful abstraction.\n\n3. The paper is mathematically rigorous. All major claims are stated formally as theorems, propositions, or lemmas, and are accompanied by detailed proofs in the appendix.\n\n# Weaknesses:\n\n1. The primary weakness is that the derived worst-case bounds (Propositions 3.8, 4.1, 4.2) depend on parameters that can be astronomically large, such as the maximum number of algorithm rounds \\(M\\) and the total number of network parameters \\(W\\). The dependency on \\(M\\) is particularly concerning, as it can lead to exponentially large terms in the complexity bounds. This raises questions about the practical utility of the bounds, as they may be too loose to provide meaningful sample size estimates for real-world applications.\n\n2. The analysis for neural networks is contingent on the activation functions being piecewise-polynomial. This is a strong restriction that excludes many popular and high-performing non-linearities (e.g.,  Sigmoid) and advanced architectures (e.g., GNNs, attention mechanisms). This limits the direct applicability of the results to a specific, albeit common, class of neural network architectures.\n\n3. No experiments are presented to gauge how the theoretical bounds correlate with actual B&C policy learning e.g., how many instances are needed in practice, or whether data-dependent Rademacher estimates are indeed small. This leaves a gap between theory and solver practice.\n\n## Minor\n4. The title, \"Generalization Guarantees for Learning Branch-and-Cut Policies,\" implies broad applicability. However, the concrete results for neural networks apply to a very specific class of models. This creates a potential mismatch between the perceived scope of the paper and the actual scope of the theoretical guarantees provided. While all theoretical work relies on assumptions, the restrictiveness here is significant enough that the title may be seen as over-promising.\n\nQUESTIONS:\nPlease see the above sections.\n\nQ1: How robust is the “piecewise-constant” nature of the overall cost function $V(I,w)$ to the choice of the penalty functions $P_k$? Specifically, if $P_k$ was not constant but a complex function modeling the time of an action (e.g., the time to solve an LP), would the core theoretical result (Theorem 3.3) still hold?\n\nQ2 : In Proposition 3.8 you deliberately use the weakened upper bound on $\\Gamma$ from Lemma 3.7, even though the proof of Lemma 3.7 yields a strictly tighter bound. Why not employ the tighter bound to obtain a sharper estimate of the Pdim in Proposition 3.8?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}