  TD-learning is a foundation reinforcement learning (RL) algorithm for value
prediction. Critical to the accuracy of value predictions is the quality of
state representations. In this work, we consider the question: how does
end-to-end TD-learning impact the representation over time? Complementary to
prior work, we provide a set of analysis that sheds further light on the
representation dynamics under TD-learning. We first show that when the
environments are reversible, end-to-end TD-learning strictly decreases the
value approximation error over time. Under further assumptions on the
environments, we can connect the representation dynamics with spectral
decomposition over the transition matrix. This latter finding establishes
fitting multiple value functions from randomly generated rewards as a useful
auxiliary task for representation learning, as we empirically validate on both
tabular and Atari game suites.
