The Sufficiency of Off-policyness and Soft Clipping: PPO is insufficient according to an Off-policy Measure

Many policy gradient methods optimize the objective, $\max_{\pi}E_{\pi}[A_{\pi_{old}}(s,a)]$, where $A_{\pi_{old}}$ is the advantage function of the old policy. The objective is not feasible to be directly optimized because we don't have samples for the new policy yet. Thus the importance sampling (IS) ratio arises, giving an IS corrected objective or the CPI objective, $\max_{\pi}E_{\pi_{old}}[\frac{\pi(s,a)}{\pi_{old}(s,a)}A_{\pi_{old}}(s,a)]$. However, optimizing this objective is still problematic due to extremely large IS ratios that can cause algorithms to fail catastrophically. Thus PPO uses a surrogate objective, and seeks an approximation to the solution in two clipped policy spaces, $\Pi_{\epsilon}^+=\{\pi; \frac{\pi(s,a)}{\pi_{old}(s,a)} \le 1+\epsilon \}$ (for non-negative Advantages) and $\Pi_{\epsilon}^-=\{\pi; \frac{\pi(s,a)}{\pi_{old}(s,a)}\ge 1-\epsilon \}$ (for negative Advantages), where $\epsilon>0$ is a small number. One question that drives this paper is, {\em How grounded is this hypothesis that $\Pi_{\epsilon}^+$ and $\Pi_{\epsilon}^-$ contain good enough policies?} {\bfseries Does there exist better policies outside of them?} Using a novel surrogate objective that employs the sigmoid function resulting in an interesting way of exploration, we found that there indeed exists much better policies out of these two policy spaces; In addition, the better policies found are located very far from them. Results also show that our P3O {\em \bfseries maximizes the CPI objective better} than PPO during training.