We investigate the problem of bandits with expert advice when the experts are fixed and known distributions over the actions. Improving on previous analyses, we show that the regret in this setting is controlled by information-theoretic quantities that measure the similarity between experts. In some natural special cases, this allows us to obtain the first regret bound for EXP4 that can get arbitrarily close to zero if the experts are similar enough. While for a different algorithm, we provide another bound that describes the similarity between the experts in terms of the KL-divergence, and we show that this bound can be smaller than the one of EXP4 in some cases. Additionally, we provide lower bounds for certain classes of experts showing that the algorithms we analyzed are nearly optimal in some cases.
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