dynamic-programming.tex

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\usepackage{algorithm}
\usepackage[noend]{algpseudocode}

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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 Horizon $N \in \mathbb{N}_{\geq 1}$
        \Statex Discounting factor $\alpha \in [0, 1]$
        \Procedure{DynamicProgramming}{$\mathcal{X}$, $A$, $g$, $N$, $\alpha$}
            \State $J_N(x) \gets g_N(x) \quad \forall x \in \mathcal{X}$
            \For{$k = N - 1$ to $0$}
                \For{$x \in \mathcal{X}$}
                    \For{$a \in A(x)$}
                        \State $Q(x, a) \gets g_k(x, a) + \alpha \sum_{j=1}^{n_x} f_{xj}(a) \cdot J_{k+1}(j)$
                    \EndFor

                    \State $J_k (x) \gets \min_a (Q_k(x,a))$
                    \State $\pi_k(x) \gets \arg \min_a (Q_k(x, a))$
                \EndFor
            \EndFor
            \Return $\pi_{0:N-1}$
        \EndProcedure
        \end{algorithmic}
    \caption{Dynamic Programming}
    \label{alg:dynamic-programming: Learn a strategy}
    \end{algorithm}
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\end{document}