  Boltzmann exploration is widely used in reinforcement learning to provide a
trade-off between exploration and exploitation. Recently, in (Cesa-Bianchi et
al., 2017) it has been shown that pure Boltzmann exploration does not perform
well from a regret perspective, even in the simplest setting of stochastic
multi-armed bandit (MAB) problems. In this paper, we show that a simple
modification to Boltzmann exploration, motivated by a variation of the standard
doubling trick, achieves $O(K\log^{1+\alpha} T)$ regret for a stochastic MAB
problem with $K$ arms, where $\alpha>0$ is a parameter of the algorithm. This
improves on the result in (Cesa-Bianchi et al., 2017), where an algorithm
inspired by the Gumbel-softmax trick achieves $O(K\log^2 T)$ regret. We also
show that our algorithm achieves $O(\beta(G) \log^{1+\alpha} T)$ regret in
stochastic MAB problems with graph-structured feedback, without knowledge of
the graph structure, where $\beta(G)$ is the independence number of the
feedback graph. Additionally, we present extensive experimental results on real
datasets and applications for multi-armed bandits with both traditional bandit
feedback and graph-structured feedback. In all cases, our algorithm performs as
well or better than the state-of-the-art.
