\begin{algorithm}[H]
\caption{Model-based deterministic policy REINFORCE} \label{alg: det reinforce}
\begin{algorithmic} [1]
\State Initialize deterministic policy $\mu_\theta$.
\State Choose a batch size $K$, a gradient based optimization algorithm, a corresponding learning rate $\lambda > 0$, a time step size $\Delta t$ and a stopping criterion.
\Repeat
\State Simulate $K$ trajectories by running the policy in the environment's dynamics.
\State Estimate the policy gradient $\nabla_\theta J(\mu_\theta)$ by
\begin{equation*}
\frac{1}{K} \sum\limits_{k=1}^K \sum\limits_{t=0}^{T^{(k)} -1} \Bigl(- \Delta t \, \mu_\theta(s_t^{(k)}) \cdot \nabla_\theta \mu_\theta(s_t^{(k)}) + G_0(\tau; \theta) \eta_{t+1} \cdot \nabla_\theta \mu_\theta(s_t^{(k)}) \Bigr).
\end{equation*}
\State Update the parameters $\theta$ based on the optimization algorithm.
\Until{stopping criterion is fulfilled.}
\end{algorithmic}
\label{alg: reinforce deterministic policy}
\end{algorithm}

\begin{algorithm}
[H]
\caption{Model-based deterministic policy REINFORCE} \begin{algorithmic}
[1]
\State Initialize deterministic policy $\mu_\theta$.
\State Choose a batch size $K$, a gradient based optimization algorithm, a corresponding learning rate $\lambda > 0$, a time step size $\Delta t$ and a stopping criterion.
\Repeat
\State Simulate $K$ trajectories by running the policy in the environment's dynamics.
\State Estimate the policy gradient $\nabla_\theta J(\mu_\theta)$ by
\begin{equation*}
\frac{1}{K} \sum\limits_{k=1}^K \sum\limits_{t=0}^{T^{(k)} -1} \Bigl(- \Delta t \, \mu_\theta(s_t^{(k)}) \cdot \nabla_\theta \mu_\theta(s_t^{(k)}) + G_0(\tau; \theta) \eta_{t+1} \cdot \nabla_\theta \mu_\theta(s_t^{(k)}) \Bigr).
\end{equation*}
\State Update the parameters $\theta$ based on the optimization algorithm.
\Until{stopping criterion is fulfilled.}
\end{algorithmic}
\end{algorithm}
{
    "title": "connecting stochastic optimal control and reinforcement learning",
    "id": "2211.02474",
    "abstract": "in this paper the connection between stochastic optimal control and reinforcement learning is investigated. our main motivation is to apply importance sampling to sampling rare events which can be reformulated as an optimal control problem. by using a parameterised approach the optimal control problem becomes a stochastic optimization problem which still raises some open questions regarding how to tackle the scalability to high-dimensional problems and how to deal with the intrinsic metastability of the system. to explore new methods we link the optimal control problem to reinforcement learning since both share the same underlying framework, namely a markov decision process (mdp). for the optimal control problem we show how the mdp can be formulated. in addition we discuss how the stochastic optimal control problem can be interpreted in the framework of reinforcement learning. at the end of the article we present the application of two different reinforcement learning algorithms to the optimal control problem and a comparison of the advantages and disadvantages of the two algorithms.",
    "categories": "math.oc",
    "doi": "",
    "created": "2022-11-04",
    "updated": "2024-02-15",
    "authors": [
        "jannes quer",
        "enric ribera borrell"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2211.02474"
}
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