Follow-the-Perturbed-Leader Nearly Achieves Best-of-Both-Worlds for the m-Set Semi-Bandit Problems

Jingxin Zhan, Yuchen Xin, Chenjie Sun, Zhihua Zhang

Keywords: multi-armed bandit, combinatorial semi-bandit, m-set semi-bandit, best-of-both-worlds, follow-the-perturbed-leader, Fréchet distribution

Abstract: We consider a common case of the combinatorial semi-bandit problem, the $m$-set semi-bandit, where the learner exactly selects $m$ arms from the total $d$ arms. In the adversarial setting, the best regret bound, known to be $\mathcal{O}(\sqrt{nmd})$ for time horizon $n$, is achieved by the well-known Follow-the-Regularized-Leader (FTRL) policy. However, this requires to explicitly compute the arm-selection probabilities via optimizing problems at each time step and sample according to them. This problem can be avoided by the Follow-the-Perturbed-Leader (FTPL) policy, which simply pulls the $m$ arms that rank among the $m$ smallest (estimated) loss with random perturbation. In this paper, we show that FTPL with a Fr\'echet perturbation also enjoys the near optimal regret bound $\mathcal{O}(\sqrt{nm}(\sqrt{d\log(d)}+m^{5/6}))$ in the adversarial setting and approaches best-of-both-world regret bounds, i.e., achieves a logarithmic regret for the stochastic setting. Moreover, our lower bounds show that the extra factors are unavoidable with our approach; any improvement would require a fundamentally different and more challenging method.

Primary Area: Optimization (e.g., convex and non-convex, stochastic, robust)

Submission Number: 17519