A classification bandit problem is a kind of pure exploration K-armed bandit problems in which a given set of K arms must be judged whether it contains at least L good arms or not with probability at least \(1-\delta \) for a given positive integer \(L(\le K)\) and \(\delta >0\) , by drawing as small number of arms as possible, where an arm is good if and only if its expected reward \(\mu \) is at least a given threshold \(\xi \) . To apply algorithms for this problem to more diverse real-world problems, we extend the problem with one dimensional rewards to that with multi-dimensional rewards by defining good arms as arms whose i-th dimensional expected reward \(\mu _i\) is at least given threshold \(\xi _i\) for all the dimensions i. We also extend P-Tracking algorithm, which is reported to be a best performer for the original one-dimensional-reward problem, to that for our multi-dimensional-reward problem. Our results using numerical simulations demonstrate the superiority of the extended P-Tracking algorithm in sample efficiency compared to extended other existing algorithms.