Title: Efficient Exploration through Bayesian Deep Q-Networks. Abstract: We propose Bayesian Deep Q-Networks (BDQN), a principled and a practical Deep Reinforcement Learning (DRL) algorithm for Markov decision processes (MDP). It combines Thompson sampling with deep-Q networks (DQN). Thompson sampling ensures more efficient exploration-exploitation tradeoff in high dimensions. It is typically carried out through posterior sampling over the model parameters, which makes it computationally expensive. To overcome this limitation, we directly incorporate uncertainty over the value (Q) function. Further, we only introduce randomness in the last layer (i.e. the output layer) of the DQN and use independent Gaussian priors on the weights. This allows us to efficiently carry out Thompson sampling through Gaussian sampling and Bayesian Linear Regression (BLR), which has fast closed-form updates. The rest of the layers of the Q network are trained through back propagation, as in a standard DQN. We apply our method to a wide range of Atari games in Arcade Learning Environments and compare BDQN to a powerful baseline: the double deep Q-network (DDQN). Since BDQN carries out more efficient exploration, it is able to reach higher rewards substantially faster: in less than 5M±1M samples for almost half of the games to reach DDQN scores while a typical run of DDQN is 50-200M. We also establish theoretical guarantees for the special case when the feature representation is fixed and not learnt. We show that the Bayesian regret is bounded by O􏰒(d \sqrt(N)) after N time steps for a d-dimensional feature map, and this bound is shown to be tight up-to logarithmic factors. To the best of our knowledge, this is the first Bayesian theoretical guarantee for Markov Decision Processes (MDP) beyond the tabula rasa setting.