Fast UCB-type algorithms for stochastic bandits with heavy and super heavy symmetric noise

In this study, we propose a new method for constructing UCB-type algorithms for stochastic multi-armed bandits based on general convex optimization methods with an inexact oracle. We derive the regret bounds corresponding to the convergence rates of the optimization methods. We propose a new algorithm Clipped-SGD-UCB and show, both theoretically and empirically, that in the case of symmetric noise in the reward, we can achieve an $O(\log T\sqrt{KT\log T})$ regret bound instead of $O\left(T^{\frac{1}{1+\alpha }}{K}^{\frac{\alpha }{1+\alpha }}\right)$ for the case when the reward distribution satisfies ${\mathbb{E}}_{X\in D}[\mathrm{\mid }X{\mathrm{\mid }}^{1+\alpha }]\le {\sigma }^{1+\alpha }$ ($\alpha \in (0,1])$, i.e. perform better than it is assumed by the general lower bound for bandits with heavy-tails. Moreover, the same bound holds even when the reward distribution does not have the expectation, that is, when $\alpha <0$.