Before revision:  We analyze the theoretical properties of the related operator, showing that it is able to converge to the global optimal value function , with a ratio faster than the traditional Bellman Optimality Operator. Furthermore, based on this new operator, we derive new model-free RL algorithms named Greedy Multi-Step Q Learning (and Greedy Multi-step DQN) . Experiments reveal that the proposed methods are reliable, easy to implement , and achieve state-of-the-art performance on a series of standard benchmark datasets .
After revision:  We analyze the theoretical properties of the related operator, showing that it is able to converge to the global optimal value function , with a ratio faster than the classical operator. Besides, it leads to a new multi-step off-policy algorithm, which is capable of safely utilizing any off-policy data collected by the arbitrary policy . Experiments reveal that the proposed methods are reliable, easy to implement , and achieve state-of-the-art performance on a series of standard benchmark datasets .