  Temporal difference learning with linear function approximation is a popular
method to obtain a low-dimensional approximation of the value function of a
policy in a Markov Decision Process. We give a new interpretation of this
method in terms of a splitting of the gradient of an appropriately chosen
function. As a consequence of this interpretation, convergence proofs for
gradient descent can be applied almost verbatim to temporal difference
learning. Beyond giving a new, fuller explanation of why temporal difference
works, our interpretation also yields improved convergence times. We consider
the setting with $1/\sqrt{T}$ step-size, where previous comparable finite-time
convergence time bounds for temporal difference learning had the multiplicative
factor $1/(1-\gamma)$ in front of the bound, with $\gamma$ being the discount
factor. We show that a minor variation on TD learning which estimates the mean
of the value function separately has a convergence time where $1/(1-\gamma)$
only multiplies an asymptotically negligible term.
