Policy gradient (PG) estimators for softmax policies are ineffective with sub-optimally saturated initialization, which happens when the density concentrates on a sub-optimal action. Sub-optimal policy saturation may arise from bad policy initialization or sudden changes in the environment that occur after the policy has already converged, and softmax PG estimators require a large number of updates to recover an effective policy. This severe issue causes high sample inefficiency and poor adaptability to new situations. To mitigate this problem, we propose a novel policy gradient estimator for softmax policies that utilizes the bias in the critic estimate and the noise present in the reward signal to escape the saturated regions of the policy parameter space. Our analysis and experiments, conducted on bandits and classical MDP benchmarking tasks, show that our estimator is more robust to policy saturation.

We study off-policy evaluation (OPE) from multiple logging policies, each generating a dataset of fixed size, i.e., stratified sampling. Previous work noted that in this setting the ordering of the variances of different importance sampling estimators is instance-dependent, which brings up a dilemma as to which importance sampling weights to use. In this paper, we resolve this dilemma by finding the OPE estimator for multiple loggers with minimum variance for any instance, i.e., the efficient one. In particular, we establish the efficiency bound under stratified sampling and propose an estimator achieving this bound when given consistent $q$-estimates. To guard against misspecification of $q$-functions, we also provide a way to choose the control variate in a hypothesis class to minimize variance. Extensive experiments demonstrate the benefits of our methods' efficiently leveraging of the stratified sampling of off-policy data from multiple loggers.