Temporal Difference Learning as Gradient Splitting

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 provide an 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 fuller explanation of why temporal difference works, this 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.