Be Greedy: How Chromatic Number meets Regret Minimization in Graph Bandits.

Shreyas, SS and Saha, AN and Bhattacharyya, C (2020) Be Greedy: How Chromatic Number meets Regret Minimization in Graph Bandits. In: 35th Conference on Uncertainty in Artificial Intelligence, UAI 2019, 22-25 July 2019, Tel Aviv; Israel.

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Abstract

We study the classical linear bandit problem on graphs modelling arm rewards through an underlying graph structure G(V ,E) such that rewards of neighboring nodes are similar. Previous attempts along this line have primarily considered the arm rewards to be a smooth function over graph Laplacian, which however failed to characterize the inherent problem complexity in terms of the graph structure.We bridge this gap by showing a regret guarantee of Õ(χ(G)√T) 1 that scales only with the chromatic number of the complement graph χ(G), assuming the rewards to be a smooth function over a general class of graph embeddings—Orthonormal Representations. Our proposed algorithms yield a regret guarantee of Õ(r√T) for any general embedding of rank r. Moreover, if the rewards correspond to a minimum rank embedding, the regret boils down to Õ(χ(G)√T)–none of the existing works were able to bring out such influences of graph structures over arm rewards. Finally, noting that computing the above minimum rank embedding is NP-Hard, we also propose an alternative O(|V | + |E|) time computable embedding scheme—Greedy Embeddings—based on greedy graph coloring, with which our algorithms perform optimally on a large family of graphs, e.g. union of cliques, complement of k-colorable graphs, regular graphs, trees etc, and are also shown to outperform state-of-the-art methods on real datasets. Our findings open up new roads for exploiting graph structures on regret performance.

Item Type:	Conference Paper
Publication:	35th Conference on Uncertainty in Artificial Intelligence, UAI 2019
Publisher:	Association For Uncertainty in Artificial Intelligence (AUAI)
Additional Information:	cited By 0; Conference of 35th Conference on Uncertainty in Artificial Intelligence, UAI 2019 ; Conference Date: 22 July 2019 Through 25 July 2019; Conference Code:151391
Keywords:	Artificial intelligence; Embeddings; Graphic methods; Large dataset; Trees (mathematics), Complement graphs; K-colorable graphs; Neighboring nodes; Problem complexity; Regret minimization; Smooth functions; State-of-the-art methods; Underlying graphs, Linear algebra
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	15 Oct 2020 07:33
Last Modified:	08 Dec 2022 09:47
URI:	https://eprints.iisc.ac.in/id/eprint/65386

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