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We propose a novel variant of the UCB algorithm (referred to asEfficient-UCB-Variance (EUCBV)) for minimizing cumulative regret in thestochastic multi-armed bandit (MAB) setting. EUCBV incorporates the armelimination strategy proposed in UCB-Improved \citep{auer2010ucb}, while takinginto account the variance estimates to compute the arms' confidence bounds,similar to UCBV \citep{audibert2009exploration}. Through a theoretical analysiswe establish that EUCBV incurs a \emph{gap-dependent} regret bound of{\scriptsize $O\left( \dfrac{K\sigma^2_{\max} \log (T\Delta^2/K)}{\Delta}\right)$} after $T$ trials, where $\Delta$ is the minimal gapbetween optimal and sub-optimal arms; the above bound is an improvement overthat of existing state-of-the-art UCB algorithms (such as UCB1, UCB-Improved,UCBV, MOSS). Further, EUCBV incurs a \emph{gap-independent} regret bound of{\scriptsize $O\left(\sqrt{KT}\right)$} which is an improvement over that ofUCB1, UCBV and UCB-Improved, while being comparable with that of MOSS andOCUCB. Through an extensive numerical study we show that EUCBV significantlyoutperforms the popular UCB variants (like MOSS, OCUCB, etc.) as well asThompson sampling and Bayes-UCB algorithms.

Efficient-UCBV: An Almost Optimal Algorithm using Variance Estimates