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We study the stochastic multi-armed bandit problem with the graph-basedfeedback structure introduced by Mannor and Shamir. We analyze the performanceof the two most prominent stochastic bandit algorithms, Thompson Sampling andUpper Confidence Bound (UCB), in the graph-based feedback setting. We show thatthese algorithms achieve regret guarantees that combine the graph structure andthe gaps between the means of the arm distributions. Surprisingly this holdsdespite the fact that these algorithms do not explicitly use the graphstructure to select arms; they observe the additional feedback but do notexplore based on it. Towards this result we introduce a "layering technique"highlighting the commonalities in the two algorithms.

Feedback graph regret bounds for Thompson Sampling and UCB