Abstract

In this work, we take a fresh look at some old and new algorithms for off-policy, return-based reinforcement learning. Expressing these in a common form, we derive a novel algorithm, Retrace( $\lambda$ ), with three desired properties: (1) it has low variance; (2) it safely uses samples collected from any behaviour policy, whatever its degree of “off-policyness”; and (3) it is efficient as it makes the best use of samples collected from near on-policy behaviour policies. We analyze the contractive nature of the related operator under both off-policy policy evaluation and control settings and derive online sample-based algorithms. We believe this is the first return-based off-policy control algorithm converging a.s. to $Q^{*}$ without the GLIE assumption (Greedy in the Limit with Infinite Exploration). As a corollary, we prove the convergence of Watkins’ Q( $\lambda$ ), which was an open problem since 1989. We illustrate the benefits of Retrace( $\lambda$ ) on a standard suite of Atari 2600 games.