Shinji Ito

Zachary Chase, Shinji Ito, Idan Mehalel

We determine the minimax optimal expected regret in the classic non-stochastic multi-armed bandit with expert advice problem, by proving a lower bound that matches the upper bound of Kale (2014). The two bounds determine the minimax optimal expected regret to be $Θ\left( \sqrt{T K \log (N/K) } \right)$, where $K$ is the number of arms, $N$ is the number of experts, and $T$ is the time horizon.

Jongyeong Lee, Junya Honda, Shinji Ito et al.

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 Fréchet-type tail. Recent work by Honda et al. (2023) showed that FTPL with Fréchet distribution with shape $α=2$ 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 Fréchet 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., Fréchet, Pareto, and Student-$t$ distributions. We also demonstrate the BOBW achievability of FTPL with certain Fréchet-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.

Canzhe Zhao, Shinji Ito, Shuai Li

Heavy-tailed bandits have been extensively studied since the seminal work of \citet{Bubeck2012BanditsWH}. In particular, heavy-tailed linear bandits, enabling efficient learning with both a large number of arms and heavy-tailed noises, have recently attracted significant attention \citep{ShaoYKL18,XueWWZ20,ZhongHYW21,Wang2025heavy,tajdini2025improved}. However, prior studies focus almost exclusively on stochastic regimes, with few exceptions limited to the special case of heavy-tailed multi-armed bandits (MABs) \citep{Huang0H22,ChengZ024,Chen2024uniINF}. In this work, we propose a general framework for adversarial heavy-tailed bandit problems, which performs follow-the-regularized-leader (FTRL) over the loss estimates shifted by a bonus function. Via a delicate setup of the bonus function, we devise the first FTRL-type best-of-both-worlds (BOBW) algorithm for heavy-tailed MABs, which does not require the truncated non-negativity assumption and achieves an $\widetilde{O}(T^{\frac{1}{\varepsilon}})$ worst-case regret in the adversarial regime as well as an $\widetilde{O}(\log T)$ gap-dependent regret in the stochastic regime. We then extend our framework to the linear case, proposing the first algorithm for adversarial heavy-tailed linear bandits with finite arm sets. This algorithm achieves an $\widetilde{O}(d^{\frac{1}{2}}T^{\frac{1}{\varepsilon}})$ regret, matching the best-known worst-case regret bound in stochastic regimes. Moreover, we propose a general data-dependent learning rate, termed \textit{heavy-tailed noise aware stability-penalty matching} (HT-SPM). We prove that HT-SPM guarantees BOBW regret bounds for general heavy-tailed bandit problems once certain conditions are satisfied. By using HT-SPM and, in particular, a variance-reduced linear loss estimator, we obtain the first BOBW result for heavy-tailed linear bandits.

Kazuma Shimizu, Junya Honda, Shinji Ito et al.

This paper discusses the revenue management (RM) problem to maximize revenue by pricing items or services. One challenge in this problem is that the demand distribution is unknown and varies over time in real applications such as airline and retail industries. In particular, the time-varying demand has not been well studied under scenarios of unknown demand due to the difficulty of jointly managing the remaining inventory and estimating the demand. To tackle this challenge, we first introduce an episodic generalization of the RM problem motivated by typical application scenarios. We then propose a computationally efficient algorithm based on posterior sampling, which effectively optimizes prices by solving linear programming. We derive a Bayesian regret upper bound of this algorithm for general models where demand parameters can be correlated between time periods, while also deriving a regret lower bound for generic algorithms. Our empirical study shows that the proposed algorithm performs better than other benchmark algorithms and comparably to the optimal policy in hindsight. We also propose a heuristic modification of the proposed algorithm, which further efficiently learns the pricing policy in the experiments.