We study a variant of the multi-armed bandit problem where side information in the form of bounds on the mean of each arm is provided. We develop the novel non-optimistic Global Under-Explore (GLUE) algorithm which uses the provided mean bounds (across all the arms) to infer pseudo-variances for each arm, which in turn decide the rate of exploration for the arms. We analyze the regret of GLUE and prove regret upper bounds that are never worse than that of the standard UCB algorithm. Furthermore, we show that GLUE improves upon regret guarantees that exists in literature for structured bandit problems (both theoretically and empirically). Finally, we study the practical setting of learning adaptive interventions using prior data that has been confounded by unrecorded variables that affect rewards. We show that mean bounds can be inferred naturally from such logs and can thus be used to improve the learning process. We validate our findings through semi-synthetic experiments on data derived from real data sets.