  Gradient temporal difference (Gradient TD) algorithms are a popular class of
stochastic approximation (SA) algorithms used for policy evaluation in
reinforcement learning. Here, we consider Gradient TD algorithms with an
additional heavy ball momentum term and provide choice of step size and
momentum parameter that ensures almost sure convergence of these algorithms
asymptotically. In doing so, we decompose the heavy ball Gradient TD iterates
into three separate iterates with different step sizes. We first analyze these
iterates under one-timescale SA setting using results from current literature.
However, the one-timescale case is restrictive and a more general analysis can
be provided by looking at a three-timescale decomposition of the iterates. In
the process, we provide the first conditions for stability and convergence of
general three-timescale SA. We then prove that the heavy ball Gradient TD
algorithm is convergent using our three-timescale SA analysis. Finally, we
evaluate these algorithms on standard RL problems and report improvement in
performance over the vanilla algorithms.
