Follow-the-Perturbed-Leader Approaches Best-of-Both-Worlds for the m-Set Semi-Bandit Problems
This work provides a more efficient algorithm for combinatorial semi-bandit problems, benefiting researchers and practitioners in online learning, though it is incremental as it builds on existing FTPL methods.
The paper tackles the m-set semi-bandit problem by showing that Follow-the-Perturbed-Leader (FTPL) with Fréchet perturbation achieves a near-optimal regret bound of O(√nm(√d log(d) + m^{5/6})) in adversarial settings and logarithmic regret in stochastic settings, with lower bounds indicating these factors are unavoidable.
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 Fréchet 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.