  Deep Q-Learning is an important reinforcement learning algorithm, which
involves training a deep neural network, called Deep Q-Network (DQN), to
approximate the well-known Q-function. Although wildly successful under
laboratory conditions, serious gaps between theory and practice as well as a
lack of formal guarantees prevent its use in the real world. Adopting a
dynamical systems perspective, we provide a theoretical analysis of a popular
version of Deep Q-Learning under realistic and verifiable assumptions. More
specifically, we prove an important result on the convergence of the algorithm,
characterizing the asymptotic behavior of the learning process. Our result
sheds light on hitherto unexplained properties of the algorithm and helps
understand empirical observations, such as performance inconsistencies even
after training. Unlike previous theories, our analysis accommodates state
Markov processes with multiple stationary distributions. In spite of the focus
on Deep Q-Learning, we believe that our theory may be applied to understand
other deep learning algorithms
