  Consider a setting in which a policy maker assigns subjects to treatments,
observing each outcome before the next subject arrives. Initially, it is
unknown which treatment is best, but the sequential nature of the problem
permits learning about the effectiveness of the treatments. While the
multi-armed-bandit literature has shed much light on the situation when the
policy maker compares the effectiveness of the treatments through their mean,
much less is known about other targets. This is restrictive, because a cautious
decision maker may prefer to target a robust location measure such as a
quantile or a trimmed mean. Furthermore, socio-economic decision making often
requires targeting purpose specific characteristics of the outcome
distribution, such as its inherent degree of inequality, welfare or poverty. In
the present paper we introduce and study sequential learning algorithms when
the distributional characteristic of interest is a general functional of the
outcome distribution. Minimax expected regret optimality results are obtained
within the subclass of explore-then-commit policies, and for the unrestricted
class of all policies.
