#### Sample Complexity of Policy Gradient Finding Second-Order Stationary Points

##### Long Yang, Qian Zheng, Gang Pan

The goal of policy-based reinforcement learning (RL) is to search the maximal point of its objective. However, due to the inherent non-concavity of its objective, convergence to a first-order stationary point (FOSP) can not guarantee the policy gradient methods finding a maximal point. A FOSP can be a minimal or even a saddle point, which is undesirable for RL. Fortunately, if all the saddle points are \emph{strict}, all the second-order stationary points (SOSP) are exactly equivalent to local maxima. Instead of FOSP, we consider SOSP as the convergence criteria to character the sample complexity of policy gradient. Our result shows that policy gradient converges to an $(\epsilon,\sqrt{\epsilon\chi})$-SOSP with probability at least $1-\widetilde{\mathcal{O}}(\delta)$ after the total cost of $\mathcal{O}\left(\dfrac{\epsilon^{-\frac{9}{2}}}{(1-\gamma)\sqrt\chi}\log\dfrac{1}{\delta}\right)$, where $\gamma\in(0,1)$. Our result improves the state-of-the-art result significantly where it requires $\mathcal{O}\left(\dfrac{\epsilon^{-9}\chi^{\frac{3}{2}}}{\delta}\log\dfrac{1}{\epsilon\chi}\right)$. Our analysis is based on the key idea that decomposes the parameter space $\mathbb{R}^p$ into three non-intersected regions: non-stationary point, saddle point, and local optimal region, then making a local improvement of the objective of RL in each region. This technique can be potentially generalized to extensive policy gradient methods.

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