# Self-Directed Learning of Convex Labelings on Graphs

arXiv: https://arxiv.org/abs/2409.01428 · HTML: https://mlsanity.com/paper/2409.01428/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 62 references that are in the map.
- First posted on arXiv 2 Sep 2024.

## Builds on (most cited first, 10 of 10)

- [Large Scale Learning on Non-Homophilous Graphs: New Benchmarks and Strong Simple Methods](https://mlsanity.com/paper/2110.14446/) (2021)
- [A Trichotomy for Transductive Online Learning](https://mlsanity.com/paper/2311.06428/) (2023)
- [Active Learning on Trees and Graphs](https://mlsanity.com/paper/1301.5112/) (2013)
- [Exact Recovery of Clusters in Finite Metric Spaces Using Oracle Queries](https://mlsanity.com/paper/2102.00504/) (2021)
- [An Experimental Study of the Treewidth of Real-World Graph Data (Extended Version)](https://mlsanity.com/paper/1901.06862/) (2019)
- [Maximal Closed Set and Half-Space Separations in Finite Closure Systems](https://mlsanity.com/paper/2001.04417/) (2020)
- [The Dimension of Self-Directed Learning](https://mlsanity.com/paper/2402.13400/) (2024)
- [Convexity in complex networks](https://mlsanity.com/paper/1608.03402/) (2016)
- [Efficient Algorithms for Learning Monophonic Halfspaces in Graphs](https://mlsanity.com/paper/2405.00853/) (2024)
- [Structural Sparsity of Complex Networks: Bounded Expansion in Random Models and Real-World Graphs](https://mlsanity.com/paper/1406.2587/) (2014)

## Abstract (arXiv)

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.