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At each instant of time we are required to sample a fixed number out of i.i.d, processes whose distributions belong to a family suitably parameterized by a real number . The objective is to maximize the long run total expected value of the samples. Following Lai and Robbins, the learning loss of a sampling scheme corresponding to a configuration of parameters is quantified by the regret . This is the difference between the maximum expected reward at time that could be achieved if were known and the expected reward actually obtained by the sampling scheme. We provide a lower bound for the regret associated with any uniformly good scheme, and construct a scheme which attains the lower bound for every configuration . The lower bound is given explicitly in terms of the Kullback-Liebler number between pairs of distributions. Part II of this paper considers the same problem …