Asymptotically-Optimal Gaussian Bandits with Side Observations

We study the problem of Gaussian bandits with general side information, as first introduced by Wu, Szepesv\'{a}ri, and Gy\"{o}rgy. In this setting, the play of an arm reveals information about other arms, according to an arbitrary {\em a priori} known {\em side information} matrix: each element of this matrix encodes the fidelity of the information that the row" arm reveals about thecolumn" arm. In the case of Gaussian noise, this model subsumes standard bandits, full-feedback, and graph-structured feedback as special cases. In this work, we first construct an LP-based asymptotic instance-dependent lower bound on the regret. The LP optimizes the cost (regret) required to reliably estimate the suboptimality gap of each arm. This LP lower bound motivates our main contribution: the first known asymptotically optimal algorithm for this general setting.