The authors present an iterative approach for optimizing policies with guaranteed monotonic improvement.
TRPO is similar to natural policy gradient methods and can be applied effectively in optimization of large nonlinear policies.
\cite{KakadeL02} gave monotonic improvement guarantees for mixture of policies $\pi_{new}(a|s)=(1-\alpha)\pi_{old}(a|s) + \alpha\pi'(a|s)$ where $\pi'=\mathrm{arg}\max_{\pi'}L_{\pi_{old}}(\pi')$ is the approximated expected return of a policy $\pi'$ in terms of the advantage over $\pi_{old}$, as $\eta(\pi_{new})\geq L_{\pi_{old}}(\pi_{new}) - \frac{2\epsilon\gamma}{(1-\gamma)^2}\alpha^2$ with $\eta$ the true expected return and $\epsilon$ the maximally expected advantage.
The authors extend this approach to be applicable for all stochastic policy classes by replacing $\alpha$ with a distance measure between two policies $\pi_{new}$ and $\pi_{old}$.
As distance measure they use the maximal Kullback–Leibler divergence $D_{KL}^{\max}(\pi_{new},\pi_{old})$ and show that $\eta(\pi_{new})\geq L_{\pi_{old}}(\pi_{new}) -CD_{KL}^{\max}(\pi_{new},\pi_{old})$, with $C= \frac{4\epsilon\gamma}{(1-\gamma)^2}$.
From this follows, that one is guaranteed to improve the true objective $\eta$ when performing the following maximaization $\mathrm{maximize}_\pi\left[L_{\pi_{old}}(\pi)-CD_{KL}^{\max}(\pi,\pi_{old})\right]$. In practice however $C$ would only allow for small steps. Thus constraining $\mathrm{maximize}_\pi L_{\pi_{old}}(\pi)$ subject to $D_{KL}^{\max}(\pi,\pi_{old}) \leq \delta$ allows for larger steps in a **Trust Region**
Due to the large number of constraints this problem is impractical to solve, which is why the authors replace the maximum KL divergence with approximated average KL.
TRPO then works as follows:
1. Use a rollout procedure to collet a set of state-action-pairs wit Monte Carlo estimates of their $Q$-Values
2. Average over the samples to construct the estimate objective $L_{\pi}$ as well as the constraint
3. Approximately solve the constrained optimization problem to update the policy parameters.
They use the conjugate gradient algorithm followed by a linesearch.
Their experiments support the claim that TRPO is able to effectively optimize large nonlinear policies.