  Deep Q-Learning (DQL), a family of temporal difference algorithms for
control, employs three techniques collectively known as the `deadly triad' in
reinforcement learning: bootstrapping, off-policy learning, and function
approximation. Prior work has demonstrated that together these can lead to
divergence in Q-learning algorithms, but the conditions under which divergence
occurs are not well-understood. In this note, we give a simple analysis based
on a linear approximation to the Q-value updates, which we believe provides
insight into divergence under the deadly triad. The central point in our
analysis is to consider when the leading order approximation to the deep-Q
update is or is not a contraction in the sup norm. Based on this analysis, we
develop an algorithm which permits stable deep Q-learning for continuous
control without any of the tricks conventionally used (such as target networks,
adaptive gradient optimizers, or using multiple Q functions). We demonstrate
that our algorithm performs above or near state-of-the-art on standard MuJoCo
benchmarks from the OpenAI Gym.
