  In this paper we introduce the idea of improving the performance of
parametric temporal-difference (TD) learning algorithms by selectively
emphasizing or de-emphasizing their updates on different time steps. In
particular, we show that varying the emphasis of linear TD($\lambda$)'s updates
in a particular way causes its expected update to become stable under
off-policy training. The only prior model-free TD methods to achieve this with
per-step computation linear in the number of function approximation parameters
are the gradient-TD family of methods including TDC, GTD($\lambda$), and
GQ($\lambda$). Compared to these methods, our _emphatic TD($\lambda$)_ is
simpler and easier to use; it has only one learned parameter vector and one
step-size parameter. Our treatment includes general state-dependent discounting
and bootstrapping functions, and a way of specifying varying degrees of
interest in accurately valuing different states.
