  We consider the emphatic temporal-difference (TD) algorithm, ETD($\lambda$),
for learning the value functions of stationary policies in a discounted, finite
state and action Markov decision process. The ETD($\lambda$) algorithm was
recently proposed by Sutton, Mahmood, and White to solve a long-standing
divergence problem of the standard TD algorithm when it is applied to
off-policy training, where data from an exploratory policy are used to evaluate
other policies of interest. The almost sure convergence of ETD($\lambda$) has
been proved in our recent work under general off-policy training conditions,
but for a narrow range of diminishing stepsize. In this paper we present
convergence results for constrained versions of ETD($\lambda$) with constant
stepsize and with diminishing stepsize from a broad range. Our results
characterize the asymptotic behavior of the trajectory of iterates produced by
those algorithms, and are derived by combining key properties of ETD($\lambda$)
with powerful convergence theorems from the weak convergence methods in
stochastic approximation theory. For the case of constant stepsize, in addition
to analyzing the behavior of the algorithms in the limit as the stepsize
parameter approaches zero, we also analyze their behavior for a fixed stepsize
and bound the deviations of their averaged iterates from the desired solution.
These results are obtained by exploiting the weak Feller property of the Markov
chains associated with the algorithms, and by using ergodic theorems for weak
Feller Markov chains, in conjunction with the convergence results we get from
the weak convergence methods. Besides ETD($\lambda$), our analysis also applies
to the off-policy TD($\lambda$) algorithm, when the divergence issue is avoided
by setting $\lambda$ sufficiently large.
