Which Algorithms Can Graph Neural Networks Learn?

• URL: http://arxiv.org/abs/2602.13106v1
• Date: Fri, 13 Feb 2026 17:09:50 GMT
• Title: Which Algorithms Can Graph Neural Networks Learn?
• Authors: Solveig Wittig, Antonis Vasileiou, Robert R. Nerem, Timo Stoll, Floris Geerts, Yusu Wang, Christopher Morris,
• Abstract summary: We propose a general theoretical framework that characterizes the sufficient conditions under which MPNNs can learn an algorithm.<n>Our framework applies to a broad class of algorithms, including single-source shortest paths, minimum spanning trees, and general dynamic programming problems.
• Score: 14.935565203071528
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In recent years, there has been growing interest in understanding neural architectures' ability to learn to execute discrete algorithms, a line of work often referred to as neural algorithmic reasoning. The goal is to integrate algorithmic reasoning capabilities into larger neural pipelines. Many such architectures are based on (message-passing) graph neural networks (MPNNs), owing to their permutation equivariance and ability to deal with sparsity and variable-sized inputs. However, existing work is either largely empirical and lacks formal guarantees or it focuses solely on expressivity, leaving open the question of when and how such architectures generalize beyond a finite training set. In this work, we propose a general theoretical framework that characterizes the sufficient conditions under which MPNNs can learn an algorithm from a training set of small instances and provably approximate its behavior on inputs of arbitrary size. Our framework applies to a broad class of algorithms, including single-source shortest paths, minimum spanning trees, and general dynamic programming problems, such as the $0$-$1$ knapsack problem. In addition, we establish impossibility results for a wide range of algorithmic tasks, showing that standard MPNNs cannot learn them, and we derive more expressive MPNN-like architectures that overcome these limitations. Finally, we refine our analysis for the Bellman-Ford algorithm, yielding a substantially smaller required training set and significantly extending the recent work of Nerem et al. [2025] by allowing for a differentiable regularization loss. Empirical results largely support our theoretical findings.

Related papers

• MINAR: Mechanistic Interpretability for Neural Algorithmic Reasoning [17.920835259832238]
  We introduce Mechanistic Interpretability for Neural Algorithmic Reasoning (MINAR)<n>MINAR adapts attribution patching methods from mechanistic interpretability to the GNN setting.<n>We show through two case studies that MIAR recovers faithful neuron-level circuits from GNNs trained on algorithmic tasks.
  arXiv  Detail & Related papers  (2026-02-24T23:38:06Z)
• Mind The Gap: Deep Learning Doesn't Learn Deeply [16.284360949127723]
  This paper aims to understand how neural networks learn algorithmic reasoning by addressing two questions.<n>How faithful are learned algorithms when they are effective, and why do neural networks fail to learn effective algorithms otherwise?
  arXiv  Detail & Related papers  (2025-05-24T10:11:36Z)
• Graph neural networks extrapolate out-of-distribution for shortest paths [13.300757448796361]
  Graph Neural Networks (GNNs) are trained to minimize a sparsity-regularized loss over a small set of shortest path instances.<n>We show that GNNs trained by gradient descent are able to minimize this loss and extrapolate in practice.
  arXiv  Detail & Related papers  (2025-03-24T21:52:05Z)
• Training Neural Networks with Internal State, Unconstrained Connectivity, and Discrete Activations [66.53734987585244]
  True intelligence may require the ability of a machine learning model to manage internal state. We show that we have not yet discovered the most effective algorithms for training such models. We present one attempt to design such a training algorithm, applied to an architecture with binary activations and only a single matrix of weights.
  arXiv  Detail & Related papers  (2023-12-22T01:19:08Z)
• The Clock and the Pizza: Two Stories in Mechanistic Explanation of Neural Networks [59.26515696183751]
  We show that algorithm discovery in neural networks is sometimes more complex. We show that even simple learning problems can admit a surprising diversity of solutions.
  arXiv  Detail & Related papers  (2023-06-30T17:59:13Z)
• Towards Better Out-of-Distribution Generalization of Neural Algorithmic Reasoning Tasks [51.8723187709964]
  We study the OOD generalization of neural algorithmic reasoning tasks. The goal is to learn an algorithm from input-output pairs using deep neural networks.
  arXiv  Detail & Related papers  (2022-11-01T18:33:20Z)
• A Generalist Neural Algorithmic Learner [18.425083543441776]
  We build a single graph neural network processor capable of learning to execute a wide range of algorithms. We show that it is possible to effectively learn algorithms in a multi-task manner, so long as we can learn to execute them well in a single-task regime.
  arXiv  Detail & Related papers  (2022-09-22T16:41:33Z)
• Graph Neural Networks are Dynamic Programmers [0.0]
  Graph neural networks (GNNs) are claimed to align with dynamic programming (DP) Here we show, using methods from theory and abstract algebra, that there exists an intricate connection between GNNs and DP.
  arXiv  Detail & Related papers  (2022-03-29T13:27:28Z)
• Neural network relief: a pruning algorithm based on neural activity [47.57448823030151]
  We propose a simple importance-score metric that deactivates unimportant connections. We achieve comparable performance for LeNet architectures on MNIST. The algorithm is not designed to minimize FLOPs when considering current hardware and software implementations.
  arXiv  Detail & Related papers  (2021-09-22T15:33:49Z)
• Towards Optimally Efficient Tree Search with Deep Learning [76.64632985696237]
  This paper investigates the classical integer least-squares problem which estimates signals integer from linear models. The problem is NP-hard and often arises in diverse applications such as signal processing, bioinformatics, communications and machine learning. We propose a general hyper-accelerated tree search (HATS) algorithm by employing a deep neural network to estimate the optimal estimation for the underlying simplified memory-bounded A* algorithm.
  arXiv  Detail & Related papers  (2021-01-07T08:00:02Z)
• Efficient Computation of Expectations under Spanning Tree Distributions [67.71280539312536]
  We propose unified algorithms for the important cases of first-order expectations and second-order expectations in edge-factored, non-projective spanning-tree models. Our algorithms exploit a fundamental connection between gradients and expectations, which allows us to derive efficient algorithms.
  arXiv  Detail & Related papers  (2020-08-29T14:58:26Z)
• Optimization and Generalization Analysis of Transduction through Gradient Boosting and Application to Multi-scale Graph Neural Networks [60.22494363676747]
  It is known that the current graph neural networks (GNNs) are difficult to make themselves deep due to the problem known as over-smoothing. Multi-scale GNNs are a promising approach for mitigating the over-smoothing problem. We derive the optimization and generalization guarantees of transductive learning algorithms that include multi-scale GNNs.
  arXiv  Detail & Related papers  (2020-06-15T17:06:17Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.