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Near-Optimal First-Order Regret for Bandits with Feedback Graphs

Summary

This paper studies first-order regret guarantees for adversarial bandits with a fixed known directed feedback graph. For strongly observable graphs, the authors bridge the gap between the minimax regret Θ̃(√(αT)) and the best existing first-order bound Õ(√((κ+1)L⋆)) by achieving an expected regret of Õ(√(αL⋆)+α), where α is the independence number and L⋆ is the cumulative loss of the best arm.

SourcearXiv Machine LearningAuthor: Mengxiao Zhang, Yuheng Zhang, Haipeng Luo, Paul Mineiro
Near-Optimal First-Order Regret for Bandits with Feedback Graphs
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[Submitted on 29 Oct 2026]

Title:Near-Optimal First-Order Regret for Bandits with Feedback Graphs

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Abstract:We study first-order regret guarantees for adversarial bandits with a fixed, known directed feedback graph. For strongly observable graphs, [ACBDK15] establishes a minimax regret of $\widetilde{\Theta}(\sqrt{\alpha T})$, where $\alpha$ denotes the independence number of the graph and $T$ denotes the horizon. On the other hand, the best existing first-order bound scales as $\widetilde{O}(\sqrt{(\kappa+1) L^\star})$ [LLZ20], where $\kappa$ denotes the clique partition number, satisfying $\kappa+1\geq\alpha$, and $L^\star$ denotes the cumulative loss of the best arm, leaving a gap between the graph dependence in worst-case and first-order guarantees. In this paper, we bridge this gap by obtaining an expected regret bound of $\widetilde{O}(\sqrt{\alpha L^\star}+\alpha)$. Our first result is an algorithm for the special case of self-aware graphs, in which every vertex has a self-loop. The algorithm combines the update of EXP3 [ACBFS02] with a sampling rule inspired by exploration by optimization [LS20]. More specifically, we construct a map whose fixed points yield sampling distributions that control the one-round regret remainder in terms of the learner's loss and the graph's independence number, which is crucial in deriving optimal first-order guarantees. We further show that a sufficiently accurate fixed point can be computed in polynomial time by a geometrically convergent iteration. Finally, we extend the algorithm to general strongly observable graphs and achieve the same $\widetilde{O}(\sqrt{\alpha L^\star}+\alpha)$ guarantee by adapting the two-level aggregation approach of [LLZ20].

Subjects:

Machine Learning (cs.LG); Machine Learning (stat.ML)

Cite as: arXiv:2610.40861 [cs.LG]

(or arXiv:2610.40861v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2610.40861

arXiv-issued DOI via DataCite (pending registration)

Submission history

From: Mengxiao Zhang [view email] [v1] Thu, 29 Oct 2026 18:37:56 UTC (42 KB)

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Key points

  • Existing first-order bound Õ(√((κ+1)L⋆)) has graph dependence κ+1 ≥ α, leaving a gap to the minimax Θ̃(√(αT)).
  • The authors propose an algorithm for self-aware graphs combining EXP3 updates with a sampling rule from exploration by optimization, using fixed points of a map to control per-round regret remainder.
  • A sufficiently accurate fixed point can be computed in polynomial time via a geometrically convergent iteration.
  • The algorithm is extended to general strongly observable graphs using a two-level aggregation approach, achieving the same Õ(√(αL⋆)+α) expected regret.

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