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\begin{document}
\begin{preview}
    \begin{algorithm}[H]
        \begin{algorithmic}
        \Require
        \Statex Sates $\mathcal{X} = \{1, \dots, n_x\}$
        \Statex Actions $\mathcal{A} = \{1, \dots, n_a\},\qquad A: \mathcal{X} \Rightarrow \mathcal{A}$
        \Statex Reward function $R: \mathcal{X} \times \mathcal{A} \rightarrow \mathbb{R}$
        \Statex Black-box (probabilistic) transition function $T: \mathcal{X} \times \mathcal{A} \rightarrow \mathcal{X}$
        \Statex Learning rate $\alpha \in [0, 1]$, typically $\alpha = 0.1$
        \Statex Discounting factor $\gamma \in [0, 1]$
        \Statex $\lambda \in [0, 1]$: Trade-off between TD and MC
        \Procedure{QLearning}{$\mathcal{X}$, $A$, $R$, $T$, $\alpha$, $\gamma$, $\lambda$}
            \State Initialize $Q: \mathcal{X} \times \mathcal{A} \rightarrow \mathbb{R}$ arbitrarily
            \State Initialize $M: \mathcal{X} \times \mathcal{A} \rightarrow \mathcal{X} \times \mathbb{R}$ arbitrarily \Comment{Model}
            \While{$Q$ is not converged}
                \State Select $s \in \mathcal{X}$ arbitrarily
                \State $a \gets \pi(s)$
                \State $r \gets R(s, a)$
                \State $s' \gets T(s, a)$ \Comment{Receive the new state}
                \State $Q(s, a) \gets (1 - \alpha) \cdot Q(s, a) + \alpha \cdot (r + \gamma \cdot \max_{a'} Q(s, a'))$
                \State $M(s, a) \gets (s', r)$
                \For{$i$ in range $1, \dots, N$}
                    \State Select $(\tilde{s}, \tilde{a}) \in \mathcal{X} \times \mathcal{A}$ arbitrarily
                    \State $(s', r) \gets M(\tilde{x}, \tilde{a})$
                    \State $Q(\tilde{s}, \tilde{a}) \gets (1 - \alpha) \cdot Q(\tilde{s}, \tilde{a}) + \alpha \cdot (r + \gamma \cdot \max_{a'} Q(s', a'))$
                \EndFor
                \State Calculate $\pi$ based on $Q$ (e.g. $\varepsilon$-greedy)
            \EndWhile
            \Return $Q$
        \EndProcedure
        \end{algorithmic}
    \caption{Dyna-Q: Learn function $Q: \mathcal{X} \times \mathcal{A} \rightarrow \mathbb{R}$}
    \label{alg:dyna-q}
    \end{algorithm}
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\end{document}
