  We consider a regularized expected reward optimization problem in the
non-oblivious setting that covers many existing problems in reinforcement
learning (RL). In order to solve such an optimization problem, we apply and
analyze the classical stochastic proximal gradient method. In particular, the
method has shown to admit an $O(\epsilon^{-4})$ sample complexity to an
$\epsilon$-stationary point, under standard conditions. Since the variance of
the classical stochastic gradient estimator is typically large which slows down
the convergence, we also apply an efficient stochastic variance-reduce proximal
gradient method with an importance sampling based ProbAbilistic Gradient
Estimator (PAGE). To the best of our knowledge, the application of this method
represents a novel approach in addressing the general regularized reward
optimization problem. Our analysis shows that the sample complexity can be
improved from $O(\epsilon^{-4})$ to $O(\epsilon^{-3})$ under additional
conditions. Our results on the stochastic (variance-reduced) proximal gradient
method match the sample complexity of their most competitive counterparts under
similar settings in the RL literature.
