  Off-policy multi-step reinforcement learning algorithms consist of
conservative and non-conservative algorithms: the former actively cut traces,
whereas the latter do not. Recently, Munos et al. (2016) proved the convergence
of conservative algorithms to an optimal Q-function. In contrast,
non-conservative algorithms are thought to be unsafe and have a limited or no
theoretical guarantee. Nonetheless, recent studies have shown that
non-conservative algorithms empirically outperform conservative ones. Motivated
by the empirical results and the lack of theory, we carry out theoretical
analyses of Peng's Q($\lambda$), a representative example of non-conservative
algorithms. We prove that it also converges to an optimal policy provided that
the behavior policy slowly tracks a greedy policy in a way similar to
conservative policy iteration. Such a result has been conjectured to be true
but has not been proven. We also experiment with Peng's Q($\lambda$) in complex
continuous control tasks, confirming that Peng's Q($\lambda$) often outperforms
conservative algorithms despite its simplicity. These results indicate that
Peng's Q($\lambda$), which was thought to be unsafe, is a theoretically-sound
and practically effective algorithm.
