Self-Directed Learning of Convex Labelings on Graphs

2024-09-02

arXiv.org

AI Summary

This paper investigates active learning for node classification on graphs: given only the graph structure, a learner adaptively selects nodes to query in order to efficiently identify (geodesically) convex label clusters—i.e., subsets of nodes with identical labels such that all nodes on every shortest path between any two cluster members also share that label. We establish the first theoretical framework for active learning on graphs under geodesic convexity. Our main contribution is a tight mistake bound of $O((h(G)+1)^4 log n)$, where $h(G)$ is the Hadwiger number of the graph, and a polynomial-time algorithm achieving this bound. We further extend the framework robustly to approximately convex and homogeneous clusters, retaining logarithmic mistake rates. The core innovation lies in the deep integration of graph-theoretic convexity, the Hadwiger number, and adaptive querying strategies—thereby filling a fundamental theoretical gap in active learning for graph-structured data.

Technology Category

• Machine Learning: Active Learning
• Reasoning under Uncertainty: Graphical Models
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study the problem of classifying the nodes of a given graph in the self-directed learning setup. This learning setting is a variant of online learning, where rather than an adversary determining the sequence in which nodes are presented, the learner autonomously and adaptively selects them. While self-directed learning of Euclidean halfspaces, linear functions, and general multiclass hypothesis classes was recently considered, no results previously existed specifically for self-directed node classification on graphs. In this paper, we address this problem developing efficient algorithms for it. More specifically, we focus on the case of (geodesically) convex clusters, i.e., for every two nodes sharing the same label, all nodes on every shortest path between them also share the same label. In particular, we devise an algorithm with runtime polynomial in $n$ that makes only $3(h(G)+1)^4 ln n$ mistakes on graphs with two convex clusters, where $n$ is the total number of nodes and $h(G)$ is the Hadwiger number, i.e., the size of the largest clique minor of the graph $G$. We also show that our algorithm is robust to the case that clusters are slightly non-convex, still achieving a mistake bound logarithmic in $n$. Finally, we devise a simple and efficient algorithm for homophilic clusters, where strongly connected nodes tend to belong to the same class.

Problem

• Self-directed learning for graph node classification
• Efficient algorithms for convex label clusters
• Robustness to slightly non-convex clusters

Innovation

• Self-directed node classification
• Polynomial runtime algorithm
• Convex cluster labeling