Abstract: Inspired by the Reward-Biased Maximum Likelihood Estimate method of adaptive control, we propose RBMLE -- a novel family of learning algorithms for stochastic multi-armed bandits (SMABs). For a broad range of SMABs including both the parametric Exponential Family as well as the non-parametric sub-Gaussian/Exponential family, we show that RBMLE yields an index policy. To choose the bias-growth rate $\alpha(t)$ in RBMLE, we reveal the nontrivial interplay between $\alpha(t)$ and the regret bound that generally applies in both the Exponential Family as well as the sub-Gaussian/Exponential family bandits. To quantify the finite-time performance, we prove that RBMLE attains order-optimality by adaptively estimating the unknown constants in the expression of $\alpha(t)$ for Gaussian and sub-Gaussian bandits. Extensive experiments demonstrate that the proposed RBMLE achieves empirical regret performance competitive with the state-of-the-art methods, while being more computationally efficient and scalable in comparison to the best-performing ones among them.