  In reinforcement learning an agent interacts with the environment by taking
actions and observing the next state and reward. When sampled
probabilistically, these state transitions, rewards, and actions can all induce
randomness in the observed long-term return. Traditionally, reinforcement
learning algorithms average over this randomness to estimate the value
function. In this paper, we build on recent work advocating a distributional
approach to reinforcement learning in which the distribution over returns is
modeled explicitly instead of only estimating the mean. That is, we examine
methods of learning the value distribution instead of the value function. We
give results that close a number of gaps between the theoretical and
algorithmic results given by Bellemare, Dabney, and Munos (2017). First, we
extend existing results to the approximate distribution setting. Second, we
present a novel distributional reinforcement learning algorithm consistent with
our theoretical formulation. Finally, we evaluate this new algorithm on the
Atari 2600 games, observing that it significantly outperforms many of the
recent improvements on DQN, including the related distributional algorithm C51.
