  We consider the \mnk{classical} problem of a controller activating (or
sampling) sequentially from a finite number of $N \geq 2$ populations,
specified by unknown distributions. Over some time horizon, at each time $n =
1, 2, \ldots$, the controller wishes to select a population to sample, with the
goal of sampling from a population that optimizes some "score" function of its
distribution, e.g., maximizing the expected sum of outcomes or minimizing
variability. We define a class of \textit{Uniformly Fast (UF)} sampling
policies and show, under mild regularity conditions, that there is an
asymptotic lower bound for the expected total number of sub-optimal population
activations. Then, we provide sufficient conditions under which a UCB policy is
UF and asymptotically optimal, since it attains this lower bound. Explicit
solutions are provided for a number of examples of interest, including general
score functionals on unconstrained Pareto distributions (of potentially
infinite mean), and uniform distributions of unknown support. Additional
results on bandits of Normal distributions are also provided.
