Polynomial-Time Algorithms for Multiple-Arm Identification with Full-Bandit Feedback

We study the problem of stochastic multiple-arm identification, where an agent sequentially explores a size-$k$ subset of arms (also known as a super arm) from given $n$ arms and tries to identify the best super arm. Most work so far has considered the semi-bandit setting, where the agent can observe the reward of each pulled arm or assumed each arm can be queried at each round. However, in real-world applications, it is costly or sometimes impossible to observe a reward of individual arms. In this study, we tackle the full-bandit setting, where only a noisy observation of the total sum of a super arm is given at each pull. Although our problem can be regarded as an instance of the best arm identification in linear bandits, a naive approach based on linear bandits is computationally infeasible since the number of super arms $K$ is exponential. To cope with this problem, we first design a polynomial-time approximation algorithm for a 0-1 quadratic programming problem arising in confidence ellipsoid maximization. Based on our approximation algorithm, we propose a bandit algorithm whose computation time is $O$(log $K$⁠), thereby achieving an exponential speedup over linear bandit algorithms. We provide a sample complexity upper bound that is still worst-case optimal. Finally, we conduct experiments on large-scale data sets with more than 10$10$ super arms, demonstrating the superiority of our algorithms in terms of both the computation time and the sample complexity.