Stochastic Multi-armed Bandits in Constant Space

We consider the stochastic bandit problem in the sublinear space setting, where one cannot record the win-loss record for all $K$ arms. We give an algorithm using $O(1)$ words of space with regret $$\sum_{i=1}^{K}\frac{1}{\Delta_i}\log \frac{\Delta_i}{\Delta}\log T$$ where $\Delta_i$ is the gap between the best arm and arm $i$ and $\Delta$ is the gap between the best and the second-best arms. If the rewards are bounded away from $0$ and $1$, this is within an $O(\log 1/\Delta)$ factor of the optimum regret possible without space constraints.