We consider regret minimization for Adversarial Markov Decision Processes (AMDPs), where the loss functions are changing over time and adversarially chosen, and the learner only observes the losses for the visited state-action pairs (i.e., bandit feedback). While there has been a surge of studies on this problem using Online-Mirror-Descent (OMD) methods, very little is known about the Follow-the-Perturbed-Leader (FTPL) methods, which are usually computationally more efficient and also easier to implement since it only requires solving an offline planning problem. Motivated by this, we take a closer look at FTPL for learning AMDPs, starting from the standard episodic finite-horizon setting. We find some unique and intriguing difficulties in the analysis and propose a workaround to eventually show that FTPL is also able to achieve near-optimal regret bounds in this case. More importantly, we then find two significant applications: First, the analysis of FTPL turns out to be readily generalizable to delayed bandit feedback with order-optimal regret, while OMD methods exhibit extra difficulties (Jin et al., 2022). Second, using FTPL, we also develop the first no-regret algorithm for learning communicating AMDPs in the infinite-horizon setting with bandit feedback and stochastic transitions. Our algorithm is efficient assuming access to an offline planning oracle, while even for the easier full-information setting, the only existing algorithm (Chandrasekaran and Tewari, 2021) is computationally inefficient.

We study the decoupled multi-armed bandit (MAB) problem, where the learner selects one arm for exploration and one arm for exploitation in each round. The loss of the explored arm is observed but not counted, while the loss of the exploited arm is incurred without being observed. We propose a policy within the Follow-the-Perturbed-Leader (FTPL) framework using Pareto perturbations. Our policy achieves (near-)optimal regret regardless of the environment, i.e., Best-of-Both-Worlds (BOBW): constant regret in the stochastic regime, improving upon the optimal bound of the standard MABs, and minimax optimal regret in the adversarial regime. Moreover, the practicality of our policy stems from avoiding both the convex optimization step required by the previous BOBW policy, Decoupled-Tsallis-INF (Rouyer & Seldin, 2020), and the resampling step that is typically necessary in FTPL. Consequently, it achieves substantial computational improvement, about $20$ times faster than Decoupled-Tsallis-INF, while also demonstrating better empirical performance in both regimes. Finally, we empirically show that our approach outperforms a pure exploration policy, and that naively combining a pure exploration with a standard exploitation policy is suboptimal.

This paper studies the optimality and complexity of Follow-the-Perturbed-Leader (FTPL) policy in size-invariant combinatorial semi-bandit problems. Recently, Honda et al. (2023) and Lee et al. (2024) showed that FTPL achieves Best-of-Both-Worlds (BOBW) optimality in standard multi-armed bandit problems with Fréchet-type distributions. However, the optimality of FTPL in combinatorial semi-bandit problems remains unclear. In this paper, we consider the regret bound of FTPL with geometric resampling (GR) in size-invariant semi-bandit setting, showing that FTPL respectively achieves $O\left(\sqrt{m^2 d^\frac{1}αT}+\sqrt{mdT}\right)$ regret with Fréchet distributions, and the best possible regret bound of $O\left(\sqrt{mdT}\right)$ with Pareto distributions in adversarial setting. Furthermore, we extend the conditional geometric resampling (CGR) to size-invariant semi-bandit setting, which reduces the computational complexity from $O(d^2)$ of original GR to $O\left(md\left(\log(d/m)+1\right)\right)$ without sacrificing the regret performance of FTPL.

We consider regret minimization for Adversarial Markov Decision Processes (AMDPs), where the loss functions are changing over time and adversarially chosen, and the learner only observes the losses for the visited state-action pairs (i.e., bandit feedback). While there has been a surge of studies on this problem using Online-Mirror-Descent (OMD) methods, very little is known about the Follow-the-Perturbed-Leader (FTPL) methods, which are usually computationally more efficient and also easier to implement since it only requires solving an offline planning problem. Motivated by this, we take a closer look at FTPL for learning AMDPs, starting from the standard episodic finite-horizon setting. We find some unique and intriguing difficulties in the analysis and propose a workaround to eventually show that FTPL is also able to achieve near-optimal regret bounds in this case. More importantly, we then find two significant applications: First, the analysis of FTPL turns out to be readily generalizable to delayed bandit feedback with order-optimal regret, while OMD methods exhibit extra difficulties (Jin et al., 2022).