  We study the Combinatorial Pure Exploration problem with Continuous and
Separable reward functions (CPE-CS) in the stochastic multi-armed bandit
setting. In a CPE-CS instance, we are given several stochastic arms with
unknown distributions, as well as a collection of possible decisions. Each
decision has a reward according to the distributions of arms. The goal is to
identify the decision with the maximum reward, using as few arm samples as
possible. The problem generalizes the combinatorial pure exploration problem
with linear rewards, which has attracted significant attention in recent years.
In this paper, we propose an adaptive learning algorithm for the CPE-CS
problem, and analyze its sample complexity. In particular, we introduce a new
hardness measure called the consistent optimality hardness, and give both the
upper and lower bounds of sample complexity. Moreover, we give examples to
demonstrate that our solution has the capacity to deal with non-linear reward
functions.
