Follow-the-Perturbed-Leader with Fréchet-type Tail Distributions:
  Optimality in Adversarial Bandits and Best-of-Both-Worlds
Jongyeong Lee Junya Honda Shinji Ito Min-hwan Oh
This paper studies the optimality of the Follow-the-Perturbed-Leader (FTPL)
policy in both adversarial and stochastic $K$-armed bandits. Despite the
widespread use of the Follow-the-Regularized-Leader (FTRL) framework with
various choices of regularization, the FTPL framework, which relies on random
perturbations, has not received much attention, despite its inherent
simplicity. In adversarial bandits, there has been conjecture that FTPL could
potentially achieve $\mathcal{O}(\sqrt{KT})$ regrets if perturbations follow a
distribution with a Fr\'{e}chet-type tail. Recent work by Honda et al. (2023)
showed that FTPL with Fr\'{e}chet distribution with shape $\alpha=2$ indeed
attains this bound and, notably logarithmic regret in stochastic bandits,
meaning the Best-of-Both-Worlds (BOBW) capability of FTPL. However, this result
only partly resolves the above conjecture because their analysis heavily relies
on the specific form of the Fr\'{e}chet distribution with this shape. In this
paper, we establish a sufficient condition for perturbations to achieve
$\mathcal{O}(\sqrt{KT})$ regrets in the adversarial setting, which covers,
e.g., Fr\'{e}chet, Pareto, and Student-$t$ distributions. We also demonstrate
the BOBW achievability of FTPL with certain Fr\'{e}chet-type tail
distributions. Our results contribute not only to resolving existing
conjectures through the lens of extreme value theory but also potentially offer
insights into the effect of the regularization functions in FTRL through the
mapping from FTPL to FTRL.