Symmetry arises in many optimization and decision-making problems, and has attracted considerable attention from the optimization community: By utilizing the existence of such symmetries, the process of searching for optimal solutions can be improved significantly. Despite its success in (offline) optimization, the utilization of symmetries has not been well examined within the online optimization settings, especially in the bandit literature. As such, in this paper we study the invariant Lipschitz bandit setting, a subclass of the Lipschitz bandits where the reward function and the set of arms are preserved under a group of transformations. We introduce an algorithm named \texttt{UniformMesh-N}, which naturally integrates side observations using group orbits into the \texttt{UniformMesh} algorithm (\cite{Kleinberg2005_UniformMesh}), which uniformly discretizes the set of arms. Using the side-observation approach, we prove an improved regret upper bound, which depends on the cardinality of the group, given that the group is finite. We also prove a matching regret's lower bound for the invariant Lipschitz bandit class (up to logarithmic factors). We hope that our work will ignite further investigation of symmetry in bandit theory and sequential decision-making theory in general.
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