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In recent years, the deep reinforcement learning (DRL) algorithms have been developed rapidly and have achieved excellent performance in many challenging tasks. However, due to the complexity of network structure and a large amount of network parameters, the training of deep network is time-consuming, and consequently, the learning efficiency of DRL is limited. In this paper, aiming to speed up the learning process of DRL agent, we propose a novel approximate policy-based accelerated (APA) algorithm from the viewpoint of the error analysis of approximate policy iteration reinforcement learning algorithms. The proposed APA is proven to be convergent even with a more aggressive learning rate, making the DRL agent have a faster learning speed. Furthermore, to combine the accelerated algorithm with deep Q-network (DQN), Double DQN and deep deterministic policy gradient (DDPG), we proposed three novel DRL algorithms: APA-DQN, APA-Double DQN, and APA-DDPG, which demonstrates the adaptability of the accelerated algorithm with DRL algorithms. We have tested the proposed algorithms on both discrete-action and continuous-action tasks. Their superior performance demonstrates their great potential in the practical applications. [ABSTRACT FROM AUTHOR]