This paper presents two algorithms, AdaOFUL and VARA, for online sequential decision-making in the presence of heavy-tailed rewards with only finite variances. For linear stochastic bandits, we address the issue of heavy-tailed rewards by modifying the adaptive Huber regression and proposing AdaOFUL. AdaOFUL achieves a state-of-the-art regret bound of $\widetilde{\mathcal{O}}\big(d\big(\sum_{t=1}^T \nu_{t}^2\big)^{1/2}+d\big)$ as if the rewards were uniformly bounded, where $\nu_{t}^2$ is the observed conditional variance of the reward at round $t$, $d$ is the feature dimension, and $\widetilde{\mathcal{O}}(\cdot)$ hides logarithmic dependence. Building upon AdaOFUL, we propose VARA for linear MDPs, which achieves a tighter variance-aware regret bound of $\widetilde{\mathcal{O}}(d\sqrt{H\mathcal{G}^*K})$. Here, $H$ is the length of episodes, $K$ is the number of episodes, and $\mathcal{G}^*$ is a smaller instance-dependent quantity that can be bounded by other instance-dependent quantities when additional structural conditions on the MDP are satisfied. Our regret bound is superior to the current state-of-the-art bounds in three ways: (1) it depends on a tighter instance-dependent quantity and has optimal dependence on $d$ and $H$, (2) we can obtain further instance-dependent bounds of $\mathcal{G}^*$ under additional structural conditions on the MDP, and (3) our regret bound is valid even when rewards have only finite variances, achieving a level of generality unmatched by previous works. Overall, our modified adaptive Huber regression algorithm may serve as a useful building block in the design of algorithms for online problems with heavy-tailed rewards.

该论文提出了两个算法 - AdaOFUL 和 VARA，以解决在有限方差的重尾回报情况下的在线顺序决策问题。这些算法可以应用于线性随机赌博机和线性马尔可夫决策过程，并且新的算法优于之前的算法。其中AdaOFUL和算法Huber回归适应进行对于重尾回报的处理，VARA则提供了更紧的可变方差回报较紧的方案。

具有线性功能逼近的重尾奖励方差感知健壮增强学习