Efficient-UCBV: An Almost Optimal Algorithm using Variance Estimates

We propose a novel variant of the UCB algorithm (referred to as Efficient-UCB-Variance (EUCBV)) for minimizing cumulative regret in the stochastic multi-armed bandit (MAB) setting. EUCBV incorporates the arm elimination strategy proposed in UCB-Improved citepauer2010ucb, while taking into account the variance estimates to compute the arms' confidence bounds, similar to UCBV citepaudibert2009exploration. Through a theoretical analysis we establish that EUCBV incurs a gap-dependent regret bound of scriptsize Oleft( dfracKσ2max log (TDelta2 /K)Deltaright) after T trials, where Delta is the minimal gap between optimal and sub-optimal arms; the above bound is an improvement over that of existing state-of-the-art UCB algorithms (such as UCB1, UCB-Improved, UCBV, MOSS). Further, EUCBV incurs a gap-independent regret bound of scriptsize Oleft(√KTright) which is an improvement over that of UCB1, UCBV and UCB-Improved, while being comparable with that of MOSS and OCUCB. Through an extensive numerical study we show that EUCBV significantly outperforms the popular UCB variants (like MOSS, OCUCB, etc.) as well as Thompson sampling and Bayes-UCB algorithms.