How humans achieve long-term goals in an uncertain environment, via repeated trials and noisy observations, is an important problem in cognitive science. We investigate this behavior in the context of a multi-armed bandit task. We compare human behavior to a variety of models that vary in their representational and computational complexity. Our result shows that subjects' choices, on a trial-to-trial basis, are best captured by a "forgetful" Bayesian iterative learning model in combination with a partially myopic decision policy known as Knowledge Gradient. This model accounts for subjects' trial-by-trial choice better than a number of other previously proposed models, including optimal Bayesian learning and risk minimization, epsilon-greedy and win-stay-lose-shift. It has the added benefit of being closest in performance to the optimal Bayesian model than all the other heuristic models that have the same computational complexity (all are significantly less complex than the optimal model). These results constitute an advancement in the theoretical understanding of how humans negotiate the tension between exploration and exploitation in a noisy, imperfectly known environment.