We consider adversarial multi-armed bandit problems where the learner is allowed to observe losses of a number of arms beside the arm that it actually chose. We study the case where all non-chosen arms reveal their loss with a fixed but unknown probability $r$, independently of each other and the action of the learner. We propose two algorithms that work for different ranges of $r$. We show that after $T$ rounds in a bandit problem with $N$ arms, the expected regret of our first algorithm is $O(\sqrt{(T /r) \log N })$ whenever $r\ge(\log T)/(2N)$, while our second algorithm achieves a regret of $O(\sqrt{(T/r) \log (N+T)})$ for smaller values of $r$. We also give a quick estimation procedure that decides the range of~$r$. All our bounds are within logarithmic factors of the best achievable performance of any algorithm that is even allowed to know~$r$.

翻译：我们考虑对抗性多臂赌博机问题，其中学习器除了观测其实际选择的臂之外，还允许观测若干其他臂的损失。我们研究所有未被选择的臂以固定但未知的概率 $r$ 独立于彼此以及学习器的动作而泄露其损失的情形。我们提出了两种适用于不同 $r$ 取值范围的算法。我们证明，在一个包含 $N$ 个臂的赌博机问题中经历 $T$ 轮后，当 $r\ge(\log T)/(2N)$ 时，我们的第一种算法的期望遗憾为 $O(\sqrt{(T /r) \log N })$；而对于较小的 $r$ 值，我们的第二种算法实现了 $O(\sqrt{(T/r) \log (N+T)})$ 的遗憾。我们还给出了一种快速估计算法，用于确定 $r$ 的范围。我们的所有界与即使允许知晓 $r$ 的任何算法所能达到的最优性能相比，仅差对数因子。