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NIPS_VarianceMinization
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VMETD的更新公式和收敛性证明添加进去了

parent afbe69ae
Showing with 470 additions and 252 deletions
main/appendix.tex
...@@ -577,74 +577,179 @@ the parameter $\theta$ converges to $A^{-1}b$. ...@@ -577,74 +577,179 @@ the parameter $\theta$ converges to $A^{-1}b$.
\end{algorithmic} \end{algorithmic}
\end{algorithm} \end{algorithm}
\begin{algorithm}[t]
\caption{VMGTD algorithm with linear function approximation in the off-policy setting}
\label{alg:algorithm 3}
\begin{algorithmic}
\STATE {\bfseries Input:} $\theta_{0}$, $u_0$, $\omega_{0}$, $\gamma
$, learning rate $\alpha_t$, $\zeta_t$ and $\beta_t$, behavior policy $\mu$ and target policy $\pi$
\REPEAT
\STATE For any episode, initialize $\theta_{0}$ arbitrarily, $u_t$ and $\omega_{0}$ to $0$, $\gamma \in (0,1]$, and $\alpha_t$, $\zeta_t$ and $\beta_t$ are constant.\\
\textbf{Output}: $\theta^*$.\\
\FOR{$t=0$ {\bfseries to} $T-1$}
\STATE Take $A_t$ from $S_t$ according to $\mu$, and arrive at $S_{t+1}$\\
\STATE Observe sample ($S_t$,$R_{t+1}$,$S_{t+1}$) at time step $t$ (with their corresponding state feature vectors)\\
\STATE $\delta_t = R_{t+1}+\gamma\theta_t^{\top}\phi_{t+1}-\theta_t^{\top}\phi_t$
\STATE $\rho_{t} \leftarrow \frac{\pi(A_t | S_t)}{\mu(A_t | S_t)}$
\STATE $\theta_{t+1}\leftarrow \theta_{t}+\alpha_t \rho_t[\phi_t - \gamma \phi_{t+1}]\phi^{\top}_{t} u_t$
\STATE $u_{t+1}\leftarrow u_{t}+\zeta_t[\rho_t(\delta_t-\omega_t) \phi_t - u_t]$
\STATE $\omega_{t+1}\leftarrow \omega_{t}+\beta_t \rho_t(\delta_t-\omega_t)$
\STATE $S_t=S_{t+1}$
\ENDFOR
\UNTIL{terminal episode}
\end{algorithmic}
\end{algorithm}
\begin{algorithm}[t] \begin{algorithm}[t]
\caption{VMGTD2 algorithm with linear function approximation in the off-policy setting} \caption{VMETD algorithm with linear function approximation in the off-policy setting}
\label{alg:algorithm 4} \label{alg:algorithm 5}
\begin{algorithmic} \begin{algorithmic}
\STATE {\bfseries Input:} $\theta_{0}$, $u_0$, $\omega_{0}$, $\gamma \STATE {\bfseries Input:} $\theta_{0}$, $u_0$, $\omega_{0}$, $\gamma
$, learning rate $\alpha_t$, $\zeta_t$ and $\beta_t$, behavior policy $\mu$ and target policy $\pi$ $, learning rate $\alpha_t$, $\zeta_t$ and $\beta_t$, behavior policy $\mu$ and target policy $\pi$
\REPEAT \REPEAT
\STATE For any episode, initialize $\theta_{0}$ arbitrarily, $u_t$ and $\omega_{0}$ to $0$, $\gamma \in (0,1]$, and $\alpha_t$, $\zeta_t$ and $\beta_t$ are constant.\\ \STATE For any episode, initialize $\theta_{0}$ arbitrarily, $u_t$ to $1$ and $\omega_{0}$ to $0$, $\gamma \in (0,1]$, and $\alpha_t$, $\zeta_t$ and $\beta_t$ are constant.\\
\textbf{Output}: $\theta^*$.\\ \textbf{Output}: $\theta^*$.\\
\FOR{$t=0$ {\bfseries to} $T-1$} \FOR{$t=0$ {\bfseries to} $T-1$}
\STATE Take $A_t$ from $S_t$ according to $\mu$, and arrive at $S_{t+1}$\\ \STATE Take $A_t$ from $S_t$ according to $\mu$, and arrive at $S_{t+1}$\\
\STATE Observe sample ($S_t$,$R_{t+1}$,$S_{t+1}$) at time step $t$ (with their corresponding state feature vectors)\\ \STATE Observe sample ($S_t$,$R_{t+1}$,$S_{t+1}$) at time step $t$ (with their corresponding state feature vectors)\\
\STATE $\delta_t = R_{t+1}+\gamma\theta_t^{\top}\phi_{t+1}-\theta_t^{\top}\phi_t$ \STATE $\delta_t = R_{t+1}+\gamma\theta_t^{\top}\phi_{t+1}-\theta_t^{\top}\phi_t$
\STATE $\rho_{t} \leftarrow \frac{\pi(A_t | S_t)}{\mu(A_t | S_t)}$ \STATE $\rho_{t} \leftarrow \frac{\pi(A_t | S_t)}{\mu(A_t | S_t)}$
\STATE $\theta_{t+1}\leftarrow \theta_{t}+\alpha_t \rho_t[\phi_t - \gamma \phi_{t+1}]\phi^{\top}_{t} u_t$ \STATE $F_{t}\leftarrow \gamma \rho_t F_{t-1} +1$
\STATE $u_{t+1}\leftarrow u_{t}+\zeta_t[\rho_t(\delta_t-\omega_t) - \phi^{\top}_{t} u_t] \phi_t$ \STATE $\theta_{t+1}\leftarrow \theta_{t}+\alpha_t (F_t \rho_t\delta_t-\omega_t)\phi_t$
\STATE $\omega_{t+1}\leftarrow \omega_{t}+\beta_t \rho_t(\delta_t-\omega_t)$ \STATE $\omega_{t+1}\leftarrow \omega_{t}+\beta_t (F_t \rho_t\delta_t-\omega_t)$
\STATE $S_t=S_{t+1}$ \STATE $S_t=S_{t+1}$
\ENDFOR \ENDFOR
\UNTIL{terminal episode} \UNTIL{terminal episode}
\end{algorithmic} \end{algorithmic}
\end{algorithm} \end{algorithm}
% \begin{algorithm}[t] \subsection{Proof of VMETD convergence}
% \caption{VMETD algorithm with linear function approximation in the off-policy setting} \label{proofVMETD}
% \label{alg:algorithm 5} VMETD's $\theta$ by the following update:
% \begin{algorithmic} \begin{equation}
% \STATE {\bfseries Input:} $\theta_{0}$, $u_0$, $\omega_{0}$, $\gamma \begin{split}
% $, learning rate $\alpha_t$, $\zeta_t$ and $\beta_t$, behavior policy $\mu$ and target policy $\pi$ \theta_{k+1}&\leftarrow \theta_k+\alpha_k F_k \rho_k (R_{k+1}+\gamma \theta_k^{\top}\phi_{k+1}-\theta_k^{\top}\phi_k)\phi_k -\alpha_k \omega_{k+1}\phi_k\\
% \REPEAT &=\theta_k+\alpha_k F_k \rho_k (R_{k+1}+\gamma \theta_k^{\top}\phi_{k+1}-\theta_k^{\top}\phi_k)\phi_k -\alpha_k \mathbb{E}_{\mu}[F_k \rho_k \delta_k]\phi_k\\
% \STATE For any episode, initialize $\theta_{0}$ arbitrarily, $u_t$ to $1$ and $\omega_{0}$ to $0$, $\gamma \in (0,1]$, and $\alpha_t$, $\zeta_t$ and $\beta_t$ are constant.\\ &= \theta_k+\alpha_k \{\underbrace{(F_k\rho_kR_{k+1}-\mathbb{E}_{\mu}[F_k\rho_k R_{k+1}])\phi_k}_{\textbf{b}_{\text{VMETD},k}}
% \textbf{Output}: $\theta^*$.\\ -\underbrace{(F_k\rho_k\phi_k(\phi_k-\gamma\phi_{k+1})^{\top}-\phi_k\mathbb{E}_{\mu}[F_k\rho_k (\phi_k-\gamma\phi_{k+1})]^{\top})}_{\textbf{A}_{\text{VMETD},k}}\theta_k\}
% \FOR{$t=0$ {\bfseries to} $T-1$} \end{split}
% \STATE Take $A_t$ from $S_t$ according to $\mu$, and arrive at $S_{t+1}$\\ \end{equation}
% \STATE Observe sample ($S_t$,$R_{t+1}$,$S_{t+1}$) at time step $t$ (with their corresponding state feature vectors)\\ \begin{equation}
% \STATE $\delta_t = R_{t+1}+\gamma\theta_t^{\top}\phi_{t+1}-\theta_t^{\top}\phi_t$ \begin{split}
% \STATE $\rho_{t} \leftarrow \frac{\pi(A_t | S_t)}{\mu(A_t | S_t)}$ \textbf{A}_{\text{VMETD}}&=\lim_{k \rightarrow \infty} \mathbb{E}[\textbf{A}_{\text{VMETD},k}]\\
% \STATE $\theta_{t+1}\leftarrow \theta_{t}+\alpha_t u_t \rho_t(\delta_t-\omega_t)\phi_t$ &= \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[F_k \rho_k \phi_k (\phi_k - \gamma \phi_{k+1})^{\top}]- \lim_{k\rightarrow \infty} \mathbb{E}_{\mu}[ \phi_k]\mathbb{E}_{\mu}[F_k \rho_k (\phi_k - \gamma \phi_{k+1})]^{\top}\\
% \STATE $u_{t+1}\leftarrow \gamma \rho_t u_t +1$ &= \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[\underbrace{\phi_k}_{X}\underbrace{F_k \rho_k (\phi_k - \gamma \phi_{k+1})^{\top}}_{Y}]- \lim_{k\rightarrow \infty} \mathbb{E}_{\mu}[ \phi_k]\mathbb{E}_{\mu}[F_k \rho_k (\phi_k - \gamma \phi_{k+1})]^{\top}\\
% \STATE $\omega_{t+1}\leftarrow \omega_{t}+\beta_t \rho_t(\delta_t-\omega_t)$ &= \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[F_k \rho_k \phi_k (\phi_k - \gamma \phi_{k+1})^{\top}]- \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[ \phi_k]\lim_{k \rightarrow \infty}\mathbb{E}_{\mu}[F_k \rho_k (\phi_k - \gamma \phi_{k+1})]^{\top}\\
% \STATE $S_t=S_{t+1}$ &=\sum_{s} f(s) \phi(s)(\phi(s) - \gamma \sum_{s'}[\textbf{P}_{\pi}]_{ss'}\phi(s'))^{\top} - \sum_{s} d_{\mu}(s) \phi(s) * \sum_{s} f(s)(\phi(s) - \gamma \sum_{s'}[\textbf{P}_{\pi}]_{ss'}\phi(s'))^{\top} \\
% \ENDFOR &={\bm{\Phi}}^{\top} \textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi}) \bm{\Phi} - {\bm{\Phi}}^{\top} \textbf{d}_{\mu} \textbf{f}^{\top} (\textbf{I} - \gamma \textbf{P}_{\mu}) \bm{\Phi} \\
% \UNTIL{terminal episode} &={\bm{\Phi}}^{\top} (\textbf{F} - \textbf{d}_{\mu} \textbf{f}^{\top}) (\textbf{I} - \gamma \textbf{P}_{\pi}){\bm{\Phi}} \\
% \end{algorithmic} &={\bm{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{f}^{\top} (\textbf{I} - \gamma \textbf{P}_{\pi})){\bm{\Phi}} \\
% \end{algorithm} &={\bm{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} ){\bm{\Phi}} \\
\end{split}
\end{equation}
\begin{proof}
Any matrix $\bm{\text{M}}$ is positive definite if and only if
the symmetric matrix $\bm{\text{S}}=\bm{\text{M}}+\bm{\text{M}}^{\top}$ is positive definite.
Any symmetric real matrix $\bm{\text{S}}$ is positive definite if the absolute values of
its diagonal entries are greater than the sum of the absolute values of the corresponding
off-diagonal entries.
\begin{equation}
\label{rowsum}
\begin{split}
(\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} )\textbf{1}
&=\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})\textbf{1}-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} \textbf{1}\\
&=\textbf{F}(\textbf{1}-\gamma \textbf{P}_{\pi} \textbf{1})-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} \textbf{1}\\
&=(1-\gamma)\textbf{F}\textbf{1}-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} \textbf{1}\\
&=(1-\gamma)\textbf{f}-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} \textbf{1}\\
&=(1-\gamma)\textbf{f}-\textbf{d}_{\mu} \\
&=(1-\gamma)(\textbf{I}-\gamma\textbf{P}_{\pi}^{\top})^{-1}\textbf{d}_{\mu}-\textbf{d}_{\mu} \\
&=(1-\gamma)[(\textbf{I}-\gamma\textbf{P}_{\pi}^{\top})^{-1}-\textbf{I}]\textbf{d}_{\mu} \\
&=(1-\gamma)[\sum_{t=0}^{\infty}(\gamma\textbf{P}_{\pi}^{\top})^{t}-\textbf{I}]\textbf{d}_{\mu} \\
&=(1-\gamma)[\sum_{t=1}^{\infty}(\gamma\textbf{P}_{\pi}^{\top})^{t}]\textbf{d}_{\mu} > 0 \\
\end{split}
\end{equation}
\begin{equation}
\label{columnsum}
\begin{split}
\textbf{1}^{\top}(\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} )
&=\textbf{1}^{\top}\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{1}^{\top}\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} \\
&=\textbf{d}_{\mu}^{\top}-\textbf{1}^{\top}\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} \\
&=\textbf{d}_{\mu}^{\top}- \textbf{d}_{\mu}^{\top} \\
&=0
\end{split}
\end{equation}
(\ref{rowsum}) and (\ref{columnsum}) show that the matrix $\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top}$ of
diagonal entries are positive and its off-diagonal entries are negative. So its each row sum plus the corresponding column sum is positive.
The proof is given above
\end{proof}
% \begin{equation}
% F_k = \gamma \rho_{k-1} F_{k-1} + 1,
% \end{equation}
% \begin{equation}
% \rho_{k} \leftarrow \frac{\pi(A_k | S_k)}{\mu(A_k | S_k)},
% \end{equation}
% \begin{equation}
% \theta_{k+1}= \alpha_k F_k \rho_k (r_{k+1}+\gamma \theta_k^{\top}\phi_{k}'-\theta_k^{\top}\phi_k)\phi_k.
% \end{equation}
% ETD(0)' \textbf{A} matrix is:
% \begin{equation}
% \textbf{A}_{\text{ETD}} = {\bm{\Phi}}^{\top} \textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi}) \bm{\Phi},
% \end{equation}
% where \textbf{F} is a diagonal matrix with diagonal elements $f(s) = d_{\mu}(s) \lim_{k\rightarrow \infty }\mathbb{E}_{\mu}[F_k|S_k=s]$,
% which we assume exists. As we show later, the vector $\textbf{f} \in \mathbb{R}^N$ with components
% $[\textbf{f}]_s = f(s)$ can be written as
% \begin{equation}
% \begin{array}{ccl}
% \textbf{f}&=& \textbf{d}_\mu + \gamma \textbf{P}^{\top}_{\pi} \textbf{d}_\mu + (\gamma \textbf{P}^{\top}_{\pi} \textbf{d}_\mu)^{2} + \cdots \\
% &=&(\textbf{I} - \gamma \textbf{P}^{\top}_{\pi})^{-1} \textbf{d}_\mu.
% \end{array}
% \end{equation}
% The key matrix is $\textbf{F}(\textbf{I} - \gamma \textbf{P}_{\pi})$, and the vector of its column sums is
% \begin{equation}
% \begin{array}{ccl}
% \textbf{1}^{\top} \textbf{F}(\textbf{I} - \gamma \textbf{P}_{\pi})&=& \textbf{f}^{\top}(\textbf{I} - \gamma \textbf{P}_{\pi}) \\
% &=&\textbf{d}^{\top}_{\mu}(\textbf{I} - \gamma \textbf{P}_{\pi})^{-1} (\textbf{I} - \gamma \textbf{P}_{\pi}) \\
% &=&\textbf{d}^{\top}_{\mu},
% \end{array}
% \end{equation}
% all components of which are positive. Thus, the key matrix and the $\textbf{A}_{\text{ETD}}$ matrix are positive
% definite and the algorithm is stable.
% VMETD by the following update:
% \begin{equation}
% \theta_{k+1}= \alpha_k F_k \rho_k (r_{k+1}+\gamma \theta_k^{\top}\phi_{k}'-\theta_k^{\top}\phi_k - \mathbb{E}_{\mu}[F_k \rho_k \delta_k])\phi_k.
% \end{equation}
% % VMETD' \textbf{A} matrix is:
% % \begin{equation}
% % \begin{array}{ccl}
% % \textbf{A}_{\text{VMETD}}&=&\lim_{k \rightarrow \infty} \mathbb{E}[\textbf{A}_{\text{VMETD},k}]\\
% % &=& \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[F_k \rho_k \phi_k (\phi_k - \gamma \phi'_{k} - \mathbb{E}_{\mu}[\phi_k - \gamma \phi'_{k}])^{\top}]\\
% % &=&\sum_{s} d_{\mu}(s)\lim_{k \rightarrow \infty}\mathbb{E}_{\mu}[F_k \rho_k \phi_k (\phi_k - \gamma \phi'_{k} - \mathbb{E}_{\mu}[\phi_k - \gamma \phi'_{k}])^{\top}|S_k = s] \\
% % &=&\sum_{s} d_{\mu}(s)\lim_{k \rightarrow \infty}\mathbb{E}_{\mu}[F_k|S_k = s]\mathbb{E}_{\mu}[\rho_k \phi_k (\phi_k - \gamma \phi'_{k} - \mathbb{E}_{\mu}[\phi_k - \gamma \phi'_{k}])^{\top}|S_k = s] \\
% % &=&\sum_{s} f(s)\mathbb{E}_{\mu}[\rho_t \phi_t (\phi_t - \gamma \phi'_{t} - \mathbb{E}_{\mu}[\phi_t - \gamma \phi'_{t}])^{\top}|S_t = s] \\
% % &=&\sum_{s} f(s)\mathbb{E}_{\mu}[\rho_t \phi_t (\phi_t - \gamma \phi'_{t})^{\top}|S_t = s] - \sum_{s} f(s)\mathbb{E}_{\mu}[\rho_t \phi_t \mathbb{E}_{\mu}[\phi_t - \gamma \phi'_{t}]^{\top}|S_t = s] \\
% % &=&\sum_{s} f(s)\mathbb{E}_{\pi}[\phi_t (\phi_t - \gamma \phi'_{t})^{\top}|S_t = s] - \sum_{s} f(s)\mathbb{E}_{\pi}[\phi_t |S_t = s]\mathbb{E}_{\mu}[\phi_t - \gamma \phi'_{t}]^{\top} \\
% % &=&\sum_{s} f(s)(\mathbb{E}_{\pi}[\phi_t (\phi_t - \gamma \phi'_{t})^{\top}|S_t = s] - \mathbb{E}_{\pi}[\phi_t |S_t = s]\mathbb{E}_{\mu}[\phi_t - \gamma \phi'_{t}]^{\top}) \\
% % &=&\sum_{s} f(s) \phi(s) (\phi(s) - \gamma \sum_{s'}[\textbf{P}_{\pi}]_{ss'}\phi(s') - \sum_{s} d_{\mu}(s)(\phi(s) - \gamma \sum_{s'}[\textbf{P}_{\mu}]_{ss'}\phi(s')))^{\top}\\
% % &=&{\bm{\Phi}}^{\top} \textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi}) \bm{\Phi} - {\bm{\Phi}}^{\top} \textbf{F}\textbf{E}_\mu
% % \end{array}
% % \end{equation}
% % where $\textbf{E}_\mu \in \mathbb{R}^{N \times d }$ and $\textbf{E}_\mu$'s every row has elements equal to $\mathbb{E}_{\mu}[\phi_t - \gamma \phi'_{t}]^{\top}$.
% \begin{equation}
% \begin{array}{ccl}
% \textbf{A}_{\text{VMETD}}&=&\lim_{k \rightarrow \infty} \mathbb{E}[\textbf{A}_{\text{VMETD},k}]\\
% &=& \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[F_k \rho_k \phi_k (\phi_k - \gamma \phi'_{k})^{\top}]- \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[F_k \rho_k \phi_k]\mathbb{E}_{\mu}[F_k \rho_k \phi_k - \gamma \phi'_{k}]^{\top}\\
% &=& \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[F_k \rho_k \phi_k (\phi_k - \gamma \phi'_{k})^{\top}]- \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[F_k \rho_k \phi_k]\lim_{k \rightarrow \infty}\mathbb{E}_{\mu}[F_k \rho_k \phi_k - \gamma \phi'_{k}]^{\top}\\
% &=&\sum_{s} f(s) \phi(s)(\phi(s) - \gamma \sum_{s'}[\textbf{P}_{\pi}]_{ss'}\phi(s'))^{\top} - \sum_{s} f(s) \phi(s) * \sum_{s} f(s)(\phi(s) - \gamma \sum_{s'}[\textbf{P}_{\pi}]_{ss'}\phi(s'))^{\top} \\
% &=&{\bm{\Phi}}^{\top} \textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi}) \bm{\Phi} - {\bm{\Phi}}^{\top} \textbf{f} \textbf{f}^{\top} (\textbf{I} - \gamma \textbf{P}_{\mu}) \bm{\Phi} \\
% &=&{\bm{\Phi}}^{\top} (\textbf{F} - \textbf{f} \textbf{f}^{\top}) (\textbf{I} - \gamma \textbf{P}_{\pi}){\bm{\Phi}} \\
% \end{array}
% \end{equation}
% The key matrix is $\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi}) - \textbf{f} \textbf{d}_{\mu}^{\top} (\textbf{I} - \gamma \textbf{P}_{\mu})$, and the vector of its column sums is
% \begin{equation}
% \begin{array}{ccl}
% \textbf{1}^{\top}(\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi}) - \textbf{f} \textbf{d}_{\mu}^{\top} (\textbf{I} - \gamma \textbf{P}_{\mu}))&=& \textbf{d}^{\top}_{\mu} - \textbf{d}^{\top}_{\mu}(\textbf{I} - \gamma \textbf{P}_{\mu}) \textbf{1}^{\top}\textbf{f}\\
% &=&\textbf{d}^{\top}_{\mu} - \textbf{d}^{\top}_{\mu}(1 - \gamma)\textbf{1}^{\top}\textbf{f} \\
% &=&\textbf{d}^{\top}_{\mu} - \textbf{d}^{\top}_{\mu} \textbf{1}^{\top} (\textbf{f} - \gamma \textbf{f}),
% \end{array}
% \end{equation}
\section{Experimental details} \section{Experimental details}
\label{experimentaldetails} \label{experimentaldetails}
......
main/motivation.tex
...@@ -97,7 +97,7 @@ stochastic gradient descent: ...@@ -97,7 +97,7 @@ stochastic gradient descent:
\end{equation} \end{equation}
where $\delta_k$ is the TD error as follows: where $\delta_k$ is the TD error as follows:
\begin{equation} \begin{equation}
\delta_k = r+\gamma \delta_k = r_{k+1}+\gamma
\theta_k^{\top}\phi_{k}'-\theta_k^{\top}\phi_k. \theta_k^{\top}\phi_{k}'-\theta_k^{\top}\phi_k.
\label{delta} \label{delta}
\end{equation} \end{equation}
...@@ -204,4 +204,41 @@ and ...@@ -204,4 +204,41 @@ and
\end{equation} \end{equation}
where $\delta_{k}$ is (\ref{deltaQ}) and $A^{*}_{k+1}={\arg \max}_{a}(\theta_{k}^{\top}\phi(s_{k+1},a))$. where $\delta_{k}$ is (\ref{deltaQ}) and $A^{*}_{k+1}={\arg \max}_{a}(\theta_{k}^{\top}\phi(s_{k+1},a))$.
This paper also introduces an additional parameter $\omega$ into the GTD and GTD2 algorithms. For details, please refer to the appendix. \subsection{Variance Minimization ETD Learning: VMETD}
\ No newline at end of file VMETD by the following update:
% \begin{equation}
% \delta_{t}= R_{t+1}+\gamma \theta_t^{\top}\phi_{t+1}-\theta_t^{\top}\phi_t.
% \end{equation}
\begin{equation}
\rho_{k} \leftarrow \frac{\pi(A_k | S_k)}{\mu(A_k | S_k)}
\end{equation}
\begin{equation}
\label{fvmetd}
F_k \leftarrow \gamma \rho_{k-1}F_{k-1}+1,
\end{equation}
\begin{equation}
\label{omegavmetd}
\omega_{k+1} \leftarrow \omega_k+\beta_k(F_k \rho_k \delta_k - \omega_k),
\end{equation}
\begin{equation}
\label{thetavmetd}
\theta_{k+1}\leftarrow \theta_k+\alpha_k F_k \rho_k (R_{k+1}+\gamma \theta_k^{\top}\phi_{k+1}-\theta_k^{\top}\phi_k)\phi_k -\alpha_k \omega_{k+1}\phi_k,
\end{equation}
where $\mu$ is behavior policy and $\pi$ is target policy,
$F_t$ is a scalar variable,
$F_0=1$, $\omega$ is used to estimate $\mathbb{E}[\delta]$, i.e., $\omega \doteq \mathbb{E}[\delta]$, and
$\textbf{F}$ is a diagonal matrix with diagonal elements
$f(s)\dot{=}d_{\mu}(s)\lim_{t\rightarrow \infty}\mathbb{E}_{\mu}[F_k|S_k=s]$,
which we assume exists.
The vector $\textbf{f}\in \mathbb{R}^N$ with components
$[\textbf{f}]_s\dot{=}f(s)$ can be written as
\begin{equation}
\begin{split}
\textbf{f}&=\textbf{d}_{\mu}+\gamma \textbf{P}_{\pi}^{\top}\textbf{d}_{\mu}+(\gamma \textbf{P}_{\pi}^{\top})^2\textbf{d}_{\mu}+\ldots\\
&=(\textbf{I}-\gamma\textbf{P}_{\pi}^{\top})^{-1}\textbf{d}_{\mu}.
\end{split}
\end{equation}
main/theory.tex
...@@ -82,4 +82,31 @@ Please refer to the appendix \ref{proofcorollary4_2} for detailed proof process. ...@@ -82,4 +82,31 @@ Please refer to the appendix \ref{proofcorollary4_2} for detailed proof process.
Then the parameter vector $\theta_k$ converges with probability one Then the parameter vector $\theta_k$ converges with probability one
to $A^{-1}b$. to $A^{-1}b$.
\end{theorem} \end{theorem}
Please refer to the appendix \ref{proofth2} for detailed proof process. Please refer to the appendix \ref{proofth2} for detailed proof process.
\ No newline at end of file
\begin{theorem}
\label{theorem3}(Convergence of VMETD).
In the case of off-policy learning, consider the iterations (\ref{omegavmetd}) and (\ref{thetavmetd}) with (\ref{delta}) of VMETD.
Let the step-size sequences $\alpha_k$ and $\beta_k$, $k\geq 0$ satisfy in this case $\alpha_k,\beta_k>0$, for all $k$,
$
\sum_{k=0}^{\infty}\alpha_k=\sum_{k=0}^{\infty}\beta_k=\infty,
$
$
\sum_{k=0}^{\infty}\alpha_k^2<\infty,
$
$
\sum_{k=0}^{\infty}\beta_k^2<\infty,
$
and
$
\alpha_k = o(\beta_k).
$
Assume that $(\phi_k,r_k,\phi_k')$ is an i.i.d. sequence with
uniformly bounded second moments, where $\phi_k$ and $\phi'_{k}$ are sampled from the same Markov chain.
Let $A = \mathrm{Cov}(\phi,\phi-\gamma\phi')$,
$b=\mathrm{Cov}(r,\phi)$.
Assume that matrix $A$ is non-singular.
Then the parameter vector $\theta_k$ converges with probability one
to $A^{-1}b$.
\end{theorem}
Please refer to the appendix \ref{proofVMETD} for detailed proof process.
\ No newline at end of file
neurips_2024.aux
...@@ -61,55 +61,46 @@ ...@@ -61,55 +61,46 @@
\newlabel{deltaSarsa}{{8}{4}{Variance Minimization TD Learning: VMTD}{equation.3.8}{}} \newlabel{deltaSarsa}{{8}{4}{Variance Minimization TD Learning: VMTD}{equation.3.8}{}}
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[]\T1/ptm/m/n/10 (+20) Three ran-dom walk ex-per-i-ments: the $\OML/cmm/m/it/10 $ \T1/ptm/m/n/10 (+20) val-ues for all al-go-rithms are in the range of []\T1/ptm/m/n/10 (+20) Three ran-dom walk ex-per-i-ments: the $\OML/cmm/m/it/10 $ \T1/ptm/m/n/10 (+20) val-ues for all al-go-rithms are in the range of
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neurips_2024.pdf
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neurips_2024.tex
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\usepackage{mathtools} \usepackage{mathtools}
\usepackage{amsthm} \usepackage{amsthm}
\usepackage{tikz} \usepackage{tikz}
\usepackage{bm}
\usepackage{esvect}
\usepackage{multirow}
\theoremstyle{plain} \theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section] \newtheorem{theorem}{Theorem}[section]
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