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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 87 additions and 19 deletions
main/motivation.tex
......@@ -97,7 +97,7 @@ stochastic gradient descent:
\end{equation}
where $\delta_k$ is the TD error as follows:
\begin{equation}
\delta_k = r+\gamma
\delta_k = r_{k+1}+\gamma
\theta_k^{\top}\phi_{k}'-\theta_k^{\top}\phi_k.
\label{delta}
\end{equation}
......@@ -204,4 +204,41 @@ and
\end{equation}
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.
\ No newline at end of file
\subsection{Variance Minimization ETD Learning: VMETD}
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.
Then the parameter vector $\theta_k$ converges with probability one
to $A^{-1}b$.
\end{theorem}
Please refer to the appendix \ref{proofth2} for detailed proof process.
\ No newline at end of file
Please refer to the appendix \ref{proofth2} for detailed proof process.
\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.out
......@@ -4,17 +4,19 @@
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neurips_2024.pdf
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neurips_2024.synctex.gz
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neurips_2024.tex
......@@ -42,6 +42,9 @@
\usepackage{mathtools}
\usepackage{amsthm}
\usepackage{tikz}
\usepackage{bm}
\usepackage{esvect}
\usepackage{multirow}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
......
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