Revisit Policy Optimization in Matrix Form
This work provides a reformulation for RL researchers, but it appears incremental as it builds on existing notation and frameworks.
The paper tackles the problem of policy optimization in reinforcement learning by disentangling policy and environment dynamics using matrix notation, showing that this approach can generalize policy gradient and TRPO methods and potentially extend to model-based RL.
In tabular case, when the reward and environment dynamics are known, policy evaluation can be written as $\bm{V}_{\bmπ} = (I - γP_{\bmπ})^{-1} \bm{r}_{\bmπ}$, where $P_{\bmπ}$ is the state transition matrix given policy ${\bmπ}$ and $\bm{r}_{\bmπ}$ is the reward signal given ${\bmπ}$. What annoys us is that $P_{\bmπ}$ and $\bm{r}_{\bmπ}$ are both mixed with ${\bmπ}$, which means every time when we update ${\bmπ}$, they will change together. In this paper, we leverage the notation from \cite{wang2007dual} to disentangle ${\bmπ}$ and environment dynamics which makes optimization over policy more straightforward. We show that policy gradient theorem \cite{sutton2018reinforcement} and TRPO \cite{schulman2015trust} can be put into a more general framework and such notation has good potential to be extended to model-based reinforcement learning.