Off-policy learning based on weighted importance sampling with linear computational complexity

Importance sampling is an essential component of model-free off-policy learning algorithms. Weighted importance sampling (WIS) is generally considered superior to ordinary importance sampling but, when combined with function approximation, it has hitherto required computational complexity that is O(n2) or more in the number of features. In this paper we introduce new off-policy learning algorithms that obtain the benefits of WIS with O(n) computational complexity. Our algorithms maintain for each component of the parameter vector a measure of the extent to which that component has been used in previous examples. This measure is used to determine component-wise step sizes, merging the ideas of stochastic gradient descent and sample averages. We present our main WIS-based algorithm first in an intuitive acausal form (the forward view) and then derive a causal algorithm using eligibility traces that is equivalent but more efficient (the backward view). In three small experiments, our algorithms performed significantly better than prior O(n) algorithms for off-policy policy evaluation. We also show that our adaptive step-size technique can also improve the performance of on-policy algorithms such as TD(λ) and true online TD(λ).