A general sample complexity analysis of vanilla policy gradient

We adapt recent tools developed for the analysis of Stochastic Gradient Descent (SGD) in non-convex optimization to obtain convergence guarantees and sample complexities for the vanilla policy gradient (PG) -- REINFORCE and GPOMDP. Our only assumptions are that the expected return is smooth w.r.t. the policy parameters and that the second moment of its gradient satisfies a certain \emph{ABC assumption}. The ABC assumption allows for the second moment of the gradient to be bounded by $A\geq 0$ times the suboptimality gap, $B \geq 0$ times the norm of the full batch gradient and an additive constant $C \geq 0$, or any combination of aforementioned. We show that the ABC assumption is more general than the commonly used assumptions on the policy space to prove convergence to a stationary point. We provide a single convergence theorem under the ABC assumption, and show that, despite the generality of the ABC assumption, we recover the $\widetilde{\mathcal{O}}(\epsilon^{-4})$ sample complexity of PG. Our convergence theorem also affords greater flexibility in the choice of hyper parameters such as the step size and places no restriction on the batch size $m$. Even the single trajectory case (i.e., $m=1$) fits within our analysis. We believe that the generality of the ABC assumption may provide theoretical guarantees for PG to a much broader range of problems that have not been previously considered.