  We consider a multi-armed bandit problem with covariates. Given a realization
of the covariate vector, instead of targeting the treatment with highest
conditional expectation, the decision maker targets the treatment which
maximizes a general functional of the conditional potential outcome
distribution, e.g., a conditional quantile, trimmed mean, or a socio-economic
functional such as an inequality, welfare or poverty measure. We develop
expected regret lower bounds for this problem, and construct a near minimax
optimal assignment policy.
