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 $\mathbb{ E}_{ X \in D}[|X|^{ 1+\alpha}] \leq \sigma^{ 1+\alpha}$ ($\alpha \in (0, 1])$, 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 $\alpha<0$.
Categories: cs.LG math.OC stat.ML