Self-Directed Learning of Convex Labelings on Graphs

TL;DR

This paper introduces algorithms for self-directed node classification on graphs, focusing on convex clusters, with mistake bounds and robustness, advancing understanding of adaptive learning in graph structures.

Contribution

It develops the first efficient algorithms for self-directed graph node classification with convex and homophilic clusters, providing mistake bounds and robustness analysis.

Findings

Mistake bound of $3(h(G)+1)^4 \\ln n$ for two convex clusters.

Algorithm is polynomial-time and robust to slight non-convexity.

Simple algorithm for homophilic clusters with efficient performance.

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.