Policy-Iteration.tex

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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 Cost function $g: \mathcal{X} \times \mathcal{A} \rightarrow \mathbb{R}$
        \Statex Transition probabilities $f$, $F$
        \Statex $\alpha \in (0, 1)$
        \Procedure{PolicyIteration}{$\mathcal{X}$, $A$, $g$, $f$, $F$, $\alpha$}
            \State Initialize $\pi$ arbitrarily
            \While{$\pi$ is not converged}
                \State $J \gets$ solve system of linear equations $(I - \alpha \cdot F(\pi)) \cdot J = g(\pi)$

                \For{$x \in \mathcal{X}$}
                    \For{$a \in A(x)$}
                        \State $Q(x, a) \gets g(x, a) + \alpha \sum_{j=1}^{n_x} f_{xj}(a) \cdot J(j)$
                    \EndFor
                \EndFor
                \For{$x \in \mathcal{X}$}
                    \State $\pi(x) \gets \arg \min_a \{Q(x, a)\}$
                \EndFor
            \EndWhile
            \Return $\pi$
        \EndProcedure
        \end{algorithmic}
    \caption{Policy Iteration: Learning a policy $\pi: \mathcal{X} \rightarrow \mathcal{A}$}
    \label{alg:policy-iteration}
    \end{algorithm}
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\end{document}