  We study the problem of stochastic combinatorial pure exploration (CPE),
where an agent sequentially pulls a set of single arms (a.k.a. a super arm) and
tries to find the best super arm. Among a variety of problem settings of the
CPE, we focus on the full-bandit setting, where we cannot observe the reward of
each single arm, but only the sum of the rewards. Although we can regard the
CPE with full-bandit feedback as a special case of pure exploration in linear
bandits, an approach based on linear bandits is not computationally feasible
since the number of super arms may be exponential. In this paper, we first
propose a polynomial-time bandit algorithm for the CPE under general
combinatorial constraints and provide an upper bound of the sample complexity.
Second, we design an approximation algorithm for the 0-1 quadratic maximization
problem, which arises in many bandit algorithms with confidence ellipsoids.
Based on our approximation algorithm, we propose novel bandit algorithms for
the top-k selection problem, and prove that our algorithms run in polynomial
time. Finally, we conduct experiments on synthetic and real-world datasets, and
confirm the validity of our theoretical analysis in terms of both the
computation time and the sample complexity.
