From 9c95a8b39d25ed951905734fa9e010abf23b10b6 Mon Sep 17 00:00:00 2001
From: GongYu <1356681720@qq.com>
Date: Mon, 12 Aug 2024 17:16:50 +0800
Subject: [PATCH] 新版本

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 AAAI控制实验图/2-state.pdf                                | Bin 0 -> 160430 bytes
 AAAI控制实验图/7-state.pdf                                | Bin 0 -> 160612 bytes
 AAAI控制实验图/acrobot.pdf                                | Bin 0 -> 171121 bytes
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 AAAI控制实验图/cl.pdf                                     | Bin 0 -> 171847 bytes
 AAAI控制实验图/cliff_walking.svg                          | 51353 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diff --git a/AnonymousSubmission/anonymous-submission-latex-2025.log b/AnonymousSubmission/anonymous-submission-latex-2025.log
index 3f0e3ba..425f249 100644
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diff --git a/AnonymousSubmission/anonymous-submission-latex-2025.pdf b/AnonymousSubmission/anonymous-submission-latex-2025.pdf
index f8e0690..c98e913 100644
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diff --git a/AnonymousSubmission/anonymous-submission-latex-2025.synctex.gz b/AnonymousSubmission/anonymous-submission-latex-2025.synctex.gz
index 7afce30..d688c22 100644
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diff --git a/AnonymousSubmission/main/theory.tex b/AnonymousSubmission/main/theory.tex
index cbaec98..ad429e5 100644
--- a/AnonymousSubmission/main/theory.tex
+++ b/AnonymousSubmission/main/theory.tex
@@ -302,7 +302,7 @@ and the VMTDC algorithm, and also presents a corollary on the convergence rate o
     Assume that $(\bm{\bm{\phi}}_k,r_k,\bm{\bm{\phi}}_k')$ is an i.i.d. sequence with
     uniformly bounded second moments, where $\bm{\bm{\phi}}_k$ and $\bm{\bm{\phi}}'_{k}$ are sampled from the same Markov chain.
     Let $\textbf{A}_{\textbf{VMETD}} ={\bm{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} ){\bm{\Phi}}$,
-    $\bm{b}_{\textbf{VMETD}}=\bm{\Phi}^{\top}(\textbf{F}-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top})\textbf{r}_{\pi}$.
+    $\bm{b}_{\textbf{VMETD}}=\bm{\Phi}^{\top}(\textbf{F}-\textbf{d}_{\mu} \textbf{f}^{\top})\textbf{r}_{\pi}$.
     Assume that matrix $A$ is  non-singular. 
     Then the parameter vector $\bm{\theta}_k$ converges with probability one 
     to $\textbf{A}_{\textbf{VMETD}}^{-1}\bm{b}_{\textbf{VMETD}}$.
diff --git a/Apendix/anonymous-submission-latex-2024.aux b/Apendix/anonymous-submission-latex-2024.aux
index ef4b422..20866e9 100644
--- a/Apendix/anonymous-submission-latex-2024.aux
+++ b/Apendix/anonymous-submission-latex-2024.aux
@@ -35,21 +35,20 @@
 \citation{sutton2016emphatic}
 \newlabel{odetheta}{{A-17}{4}}
 \newlabel{rowsum}{{A-20}{4}}
+\citation{baird1995residual,sutton2009fast}
+\citation{baird1995residual,sutton2009fast,maei2011gradient}
+\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
+\newlabel{bairdexample}{{1}{5}}
 \newlabel{columnsum}{{A-21}{5}}
 \newlabel{odethetafinal}{{A-22}{5}}
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-\newlabel{randomwalk}{{\caption@xref {randomwalk}{ on input line 754}}{6}}
-\newlabel{boyanchain}{{\caption@xref {boyanchain}{ on input line 777}}{6}}
+\newlabel{experimentaldetails}{{B}{5}}
 \bibdata{aaai24}
-\bibcite{borkar1997stochastic}{{1}{1997}{{Borkar}}{{}}}
-\bibcite{borkar2000ode}{{2}{2000}{{Borkar and Meyn}}{{}}}
-\bibcite{hirsch1989convergent}{{3}{1989}{{Hirsch}}{{}}}
-\bibcite{sutton2009fast}{{4}{2009}{{Sutton et~al.}}{{Sutton, Maei, Precup, Bhatnagar, Silver, Szepesv{\'a}ri, and Wiewiora}}}
-\bibcite{sutton2016emphatic}{{5}{2016}{{Sutton, Mahmood, and White}}{{}}}
-\newlabel{lrofways}{{6}{7}}
-\gdef \@abspage@last{7}
+\bibcite{baird1995residual}{{1}{1995}{{Baird et~al.}}{{}}}
+\bibcite{borkar1997stochastic}{{2}{1997}{{Borkar}}{{}}}
+\bibcite{borkar2000ode}{{3}{2000}{{Borkar and Meyn}}{{}}}
+\bibcite{hirsch1989convergent}{{4}{1989}{{Hirsch}}{{}}}
+\bibcite{maei2011gradient}{{5}{2011}{{Maei}}{{}}}
+\bibcite{sutton2009fast}{{6}{2009}{{Sutton et~al.}}{{Sutton, Maei, Precup, Bhatnagar, Silver, Szepesv{\'a}ri, and Wiewiora}}}
+\bibcite{sutton2016emphatic}{{7}{2016}{{Sutton, Mahmood, and White}}{{}}}
+\newlabel{lrofways}{{1}{6}}
+\gdef \@abspage@last{6}
diff --git a/Apendix/anonymous-submission-latex-2024.bbl b/Apendix/anonymous-submission-latex-2024.bbl
index 8bd13dd..c409b1b 100644
--- a/Apendix/anonymous-submission-latex-2024.bbl
+++ b/Apendix/anonymous-submission-latex-2024.bbl
@@ -1,6 +1,11 @@
-\begin{thebibliography}{5}
+\begin{thebibliography}{7}
 \providecommand{\natexlab}[1]{#1}
 
+\bibitem[{Baird et~al.(1995)}]{baird1995residual}
+Baird, L.; et~al. 1995.
+\newblock Residual algorithms: Reinforcement learning with function approximation.
+\newblock In \emph{Proc. 12th Int. Conf. Mach. Learn.}, 30--37.
+
 \bibitem[{Borkar(1997)}]{borkar1997stochastic}
 Borkar, V.~S. 1997.
 \newblock Stochastic approximation with two time scales.
@@ -16,6 +21,11 @@ Hirsch, M.~W. 1989.
 \newblock Convergent activation dynamics in continuous time networks.
 \newblock \emph{Neural Netw.}, 2(5): 331--349.
 
+\bibitem[{Maei(2011)}]{maei2011gradient}
+Maei, H.~R. 2011.
+\newblock \emph{Gradient temporal-difference learning algorithms}.
+\newblock Ph.D. thesis, University of Alberta.
+
 \bibitem[{Sutton et~al.(2009)Sutton, Maei, Precup, Bhatnagar, Silver, Szepesv{\'a}ri, and Wiewiora}]{sutton2009fast}
 Sutton, R.; Maei, H.; Precup, D.; Bhatnagar, S.; Silver, D.; Szepesv{\'a}ri, C.; and Wiewiora, E. 2009.
 \newblock Fast gradient-descent methods for temporal-difference learning with linear function approximation.
diff --git a/Apendix/anonymous-submission-latex-2024.blg b/Apendix/anonymous-submission-latex-2024.blg
index 117d52a..020924d 100644
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diff --git a/Apendix/anonymous-submission-latex-2024.log b/Apendix/anonymous-submission-latex-2024.log
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-
-Underfull \hbox (badness 10000) in paragraph at lines 861--878
-\OT1/ptm/m/n/10 of $\OML/cmm/m/it/10 ^^P$ \OT1/ptm/m/n/10 is $\OMS/cmsy/m/n/10 f\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 05\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 3\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 4\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 5\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 6\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 7\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 8\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 9\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 3\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 4\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 5\OMS/cmsy/m/n/10 g$\OT1/ptm/m/n/10 . For the VMTD al-go-
- []
-
-
-Underfull \hbox (badness 10000) in paragraph at lines 861--878
-\OT1/ptm/m/n/10 rithm, the range of $\OML/cmm/m/it/10 $ \OT1/ptm/m/n/10 is $\OMS/cmsy/m/n/10 f\OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 05\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 1\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 2\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 3\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 4\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 5\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 6\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 7\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 8\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 0\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 9\OML/cmm/m/it/10 ; \OT1/cmr/m/n/10 1\OML/cmm/m/it/10 :\OT1/cmr/m/n/10 0\OMS/cmsy/m/n/10 g$\OT1/ptm/m/n/10 , and the range of $\OML/cmm/m/it/10 ^^L$ \OT1/ptm/m/n/10 is
- []
-
-(./anonymous-submission-latex-2024.bbl) [7] (./anonymous-submission-latex-2024.aux)
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diff --git a/Apendix/anonymous-submission-latex-2024.pdf b/Apendix/anonymous-submission-latex-2024.pdf
index abfc15c..48159e7 100644
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diff --git a/Apendix/anonymous-submission-latex-2024.synctex.gz b/Apendix/anonymous-submission-latex-2024.synctex.gz
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diff --git a/Apendix/anonymous-submission-latex-2024.tex b/Apendix/anonymous-submission-latex-2024.tex
index b2a2c0e..3d18dca 100644
--- a/Apendix/anonymous-submission-latex-2024.tex
+++ b/Apendix/anonymous-submission-latex-2024.tex
@@ -187,7 +187,45 @@
 \onecolumn
 \appendix
 \section{Relevant proofs}
-\subsection{Proof of Theorem 2}
+% \subsection{VMTD}
+% \begin{equation}
+%     \begin{array}{ccl}
+%         \text{VBE}(\bm{\theta})&=&\mathbb{E}[(\mathbb{E}[\delta|s]-\kappa \mathbb{E}[\mathbb{E}[\delta|s]])^2].
+%     \end{array}
+% \end{equation}
+
+% semi-gradient:
+% \begin{equation}
+%     \begin{array}{ccl}
+%         0&=&\mathbb{E}[\mathbb{E}[\delta|s]-\kappa \mathbb{E}[\mathbb{E}[\delta|s]](\bm{\phi} - \kappa\mathbb{E}[\bm{\phi}])]\\
+%         &=&\mathbb{E}[\delta \phi] - (2\kappa - \kappa^{2})\mathbb{E}[\delta]\mathbb{E}[\phi].
+%     \end{array}
+% \end{equation}
+
+% or
+% \begin{equation}
+%     \begin{array}{ccl}
+%         0&=&\mathbb{E}[\delta \phi] - \kappa\mathbb{E}[\delta]\mathbb{E}[\phi].
+%     \end{array}
+% \end{equation}
+
+% Therefore:
+% \begin{equation}
+%     \begin{array}{ccl}
+%         \textbf{A}_{\text{VMTD}}&=&{\bm{\Phi}}^{\top} (\textbf{D}_{\mu}-(2\kappa - \kappa^{2})\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} )(\textbf{I} - \gamma\textbf{P}_{\pi}){\bm{\Phi}}.
+%     \end{array}
+% \end{equation}
+
+% or
+% \begin{equation}
+%     \begin{array}{ccl}
+%         \textbf{A}_{\text{VMTD}}&=&{\bm{\Phi}}^{\top} (\textbf{D}_{\mu}-\kappa\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} )(\textbf{I} - \gamma\textbf{P}_{\pi}){\bm{\Phi}}.
+%     \end{array}
+% \end{equation}
+
+
+
+\subsection{Proof of Theorem 1}
 \label{proofth2}
 \begin{proof}
 The proof is similar to that given by \cite{sutton2009fast} for TDC, but it is based on multi-time-scale stochastic approximation.
@@ -197,9 +235,9 @@ For the VMTDC algorithm, a new one-step linear TD solution is defined as:
     0=\mathbb{E}[(\bm{\phi} - \gamma \bm{\phi}' - \mathbb{E}[\bm{\phi} - \gamma \bm{\phi}'])\bm{\phi}^\top]\mathbb{E}[\bm{\phi} \bm{\phi}^{\top}]^{-1}\mathbb{E}[(\delta -\mathbb{E}[\delta])\bm{\phi}]=\textbf{A}^{\top}\textbf{C}^{-1}(-\textbf{A}\bm{\theta}+\bm{b}).
 \end{equation*}
 The matrix $\textbf{A}^{\top}\textbf{C}^{-1}\textbf{A}$ is positive definite. Thus, the  VMTD's solution is
-$\bm{\theta}_{\text{VMTDC}}=\bm{\theta}_{\text{VMTD}}=\textbf{A}^{-1}\bm{b}$.
+$\bm{\theta}_{\text{VMTDC}}=\textbf{A}^{-1}\bm{b}$.
 
-First, note that recursion (11) and (12) can be rewritten as, respectively, 
+First, note that recursion (5) and (6) can be rewritten as, respectively, 
 \begin{equation*}
     \bm{\theta}_{k+1}\leftarrow \bm{\theta}_k+\zeta_k \bm{x}(k),
 \end{equation*}
@@ -214,7 +252,7 @@ where
     \bm{y}(k)=\frac{\zeta_k}{\beta_k}[\delta_{k}-\omega_k - \bm{\phi}^{\top}_k \bm{u}_k]\bm{\phi}_k.
 \end{equation*}
 
-Recursion (11) can also be rewritten as
+Recursion (5) can also be rewritten as
 \begin{equation*}
     \bm{\theta}_{k+1}\leftarrow \bm{\theta}_k+\beta_k z(k),
 \end{equation*}
@@ -225,10 +263,10 @@ where
 
 Due to the settings of step-size schedule 
 $\alpha_k = o(\zeta_k)$, $\zeta_k = o(\beta_k)$, $\bm{x}(k)\rightarrow 0$, $\bm{y}(k)\rightarrow 0$, $z(k)\rightarrow 0$ almost surely as $k\rightarrow 0$.
-That is that the increments in iteration (13) are uniformly larger than
-those in (12) and  the increments in iteration (12) are uniformly larger than
-those in (11), thus (13) is the fastest recursion, (12) is the second fast recursion and (11) is the slower recursion.
-Along the fastest time scale, iterations of (11), (12) and (13)
+That is that the increments in iteration (7) are uniformly larger than
+those in (6) and  the increments in iteration (6) are uniformly larger than
+those in (5), thus (7) is the fastest recursion, (6) is the second fast recursion and (5) is the slower recursion.
+Along the fastest time scale, iterations of (5), (6) and (7)
 are associated to ODEs system as follows:
 \begin{equation}
     \dot{\bm{\theta}}(t) = 0,
@@ -248,9 +286,9 @@ and $\bm{u}(t)\equiv \bm{u}$ when viewed from the fastest timescale.
 By the Hirsch lemma \cite{hirsch1989convergent}, it follows that
 $||\bm{\theta}_k-\bm{\theta}||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
 $\bm{\theta}$ that depends on the initial condition $\bm{\theta}_0$ of recursion
-(11) and $||\bm{u}_k-\bm{u}||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
+(5) and $||\bm{u}_k-\bm{u}||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
 $u$ that depends on the initial condition $u_0$ of recursion
-(12). Thus, the ODE pair (\ref{thetavmtdcFastest})-(ref{omegavmtdcFastest})
+(6). Thus, the ODE pair (\ref{thetavmtdcFastest})-(ref{omegavmtdcFastest})
 can be written as 
 \begin{equation}
     \dot{\omega}(t)=\mathbb{E}[\delta_t|\bm{u},\bm{\theta}]-\omega(t).
@@ -276,7 +314,7 @@ associated strict Liapunov function. Then,
 the origin of (\ref{omegavmtdcInfty}) is a globally asymptotically stable
 equilibrium.
 
-Consider now the recursion (13).
+Consider now the recursion (7).
 Let
 $M_{k+1}=(\delta_k-\omega_k)
 -\mathbb{E}[(\delta_k-\omega_k)|\mathcal{F}(k)]$,
@@ -301,8 +339,8 @@ conditions on the step-size sequences $\alpha_k$,$\zeta_k$, $\beta_k$. Thus,
 by Theorem 2.2 of \cite{borkar2000ode} we obtain that
 $||\omega_k-\omega^*||\rightarrow 0$ almost surely as $k\rightarrow \infty$.
 
-Consider now the second time scale recursion (12).
-Based on the above analysis, (12) can be rewritten as
+Consider now the second time scale recursion (6).
+Based on the above analysis, (6) can be rewritten as
 % \begin{equation*}
 %     \bm{u}_{k+1}\leftarrow u_{k}+\zeta_{k}[\delta_{k}-\mathbb{E}[\delta_k|\bm{u}_k,\bm{\theta}_k] - \bm{\phi}^{\top} (s_k) \bm{u}_k]\bm{\phi}(s_k).
 % \end{equation*}
@@ -319,9 +357,9 @@ when viewed from the second fast timescale.
 By the Hirsch lemma \cite{hirsch1989convergent}, it follows that
 $||\bm{\theta}_k-\bm{\theta}||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
 $\bm{\theta}$ that depends on the initial condition $\bm{\theta}_0$ of recursion
-(11). 
+(5). 
 
-Consider now the recursion (12).
+Consider now the recursion (6).
 Let
 $N_{k+1}=((\delta_k-\mathbb{E}[\delta_k]) - \bm{\phi}_k \bm{\phi}^{\top}_k \bm{u}_k) -\mathbb{E}[((\delta_k-\mathbb{E}[\delta_k]) - \bm{\phi}_k \bm{\phi}^{\top}_k \bm{u}_k)|\mathcal{I} (k)]$,
 where $\mathcal{I}(k)=\sigma(\bm{u}_l,\bm{\theta}_l,l\leq k;\bm{\phi}_s,\bm{\phi}_s',r_s,s<k)$, 
@@ -361,8 +399,8 @@ conditions on the step-size sequences $\alpha_k$,$\zeta_k$, $\beta_k$. Thus,
 by Theorem 2.2 of \cite{borkar2000ode} we obtain that
 $||\bm{u}_k-\bm{u}^*||\rightarrow 0$ almost surely as $k\rightarrow \infty$.
 
-Consider now the slower timescale recursion (11). In the light of the above,
-(11) can be rewritten as 
+Consider now the slower timescale recursion (5). In the light of the above,
+(5) can be rewritten as 
 \begin{equation}
     \bm{\theta}_{k+1} \leftarrow \bm{\theta}_{k} + \alpha_k (\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k]) \bm{\phi}_k\\
     - \alpha_k \gamma\bm{\phi}'_{k}(\bm{\phi}^{\top}_k \textbf{C}^{-1}\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k])\bm{\phi}|\bm{\theta}_k]).
@@ -387,7 +425,7 @@ c_3(1+||\bm{\theta}_k||^2), k\geq 0
 for some constant $c_3 \geq 0$, again beacuse $\bm{\phi}_k$, $r_k$, and $\bm{\phi}_k'$   have
 uniformly bounded second moments, it can be seen that for some constant.
 
-Consider now the following ODE associated with (11):
+Consider now the following ODE associated with (5):
 \begin{equation}
     \dot{\bm{\theta}}(t) = (\textbf{I} - \mathbb{E}[\gamma \bm{\phi}' \bm{\phi}^{\top}]\textbf{C}^{-1})\mathbb{E}[(\delta -\mathbb{E}[\delta|\bm{\theta}(t)]) \bm{\phi}|\bm{\theta}(t)].
     \label{thetavmtdcSlowerFinal}
@@ -431,7 +469,7 @@ In the slower time scale,
 the parameter $\bm{\theta}$ converges to $\textbf{A}^{-1}\bm{b}$.
 \end{proof}
 
-\subsection{Proof of Theorem 3}
+\subsection{Proof of Theorem 2}
 \label{proofVMETD}
 \begin{proof}
     \label{th1proof}   
@@ -452,9 +490,9 @@ The  VMTD's solution is
         \end{equation*}
         Due to the settings of step-size schedule $\alpha_k = o(\beta_k)$,
         $\bm{\xi}(k)\rightarrow 0$ almost surely as $k\rightarrow\infty$. 
-        That is the increments in iteration (20) are uniformly larger than
-        those in (19), thus (20) is the faster recursion.
-        Along the faster time scale, iterations of (20) and (19)
+        That is the increments in iteration (13) are uniformly larger than
+        those in (12), thus (13) is the faster recursion.
+        Along the faster time scale, iterations of (12) and (13)
         are associated to ODEs system as follows:
         \begin{equation}
         \dot{\bm{\theta}}(t) = 0,
@@ -469,7 +507,7 @@ The  VMTD's solution is
         By the Hirsch lemma \cite{hirsch1989convergent}, it follows that
         $||\bm{\theta}_k-\bm{\theta}||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
         $\bm{\theta}$ that depends on the initial condition $\bm{\theta}_0$ of recursion
-        (19).
+        (12).
         Thus, the ODE pair (\ref{thetaFast})-(\ref{omegaFast}) can be written as
         \begin{equation}
         \dot{\omega}(t)=\mathbb{E}_{\mu}[F_t\rho_t\delta_t|\bm{\theta}]-\omega(t).
@@ -495,7 +533,7 @@ The  VMTD's solution is
         equilibrium.
         
         
-        Consider now the recursion (20).
+        Consider now the recursion (13).
         Let
         $M_{k+1}=(F_k\rho_k\delta_k-\omega_k)
         -\mathbb{E}_{\mu}[(F_k\rho_k\delta_k-\omega_k)|\mathcal{F}(k)]$,
@@ -520,8 +558,8 @@ The  VMTD's solution is
         by Theorem 2.2 of \cite{borkar2000ode} we obtain that
         $||\omega_k-\omega^*||\rightarrow 0$ almost surely as $k\rightarrow \infty$.
         
-        Consider now the slower time scale recursion (19).
-        Based on the above analysis, (19) can be rewritten as 
+        Consider now the slower time scale recursion (12).
+        Based on the above analysis, (12) can be rewritten as 
         % \begin{equation*}
         % \bm{\theta}_{k+1}\leftarrow
         % \bm{\theta}_{k}+\alpha_k(F_k\rho_k\delta_k-\mathbb{E}_{\mu}[F_k\rho_k\delta_k|\bm{\theta}_k])\bm{\phi}_k.
@@ -578,7 +616,7 @@ The  VMTD's solution is
         c_2(1+||\bm{\theta}_k||^2).
         \end{equation*}
         
-        Consider now the following ODE associated with (19):
+        Consider now the following ODE associated with (12):
         \begin{equation}
         \begin{array}{ccl}
         \dot{\bm{\theta}}(t)&=&-\textbf{A}_{\text{VMETD}}\bm{\theta}(t)+\bm{b}_{\text{VMETD}}.
@@ -606,7 +644,7 @@ The  VMTD's solution is
                 &= \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[\bm{\phi}_kF_k\rho_kR_{k+1}]- \lim_{k\rightarrow \infty} \mathbb{E}_{\mu}[  \bm{\phi}_k]\mathbb{E}_{\mu}[\bm{\phi}_k]\mathbb{E}_{\mu}[F_k\rho_kR_{k+1}]\\ 
                 &= \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[\bm{\phi}_kF_k\rho_kR_{k+1}]- \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[ \bm{\phi}_k]\lim_{k \rightarrow \infty}\mathbb{E}_{\mu}[F_k\rho_kR_{k+1}]\\  
                 &=\sum_{s} f(s) \bm{\phi}(s)r_{\pi} - \sum_{s} d_{\mu}(s) \bm{\phi}(s) * \sum_{s} f(s)r_{\pi}  \\
-                &=\bm{\bm{\Phi}}^{\top}(\textbf{F}-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top})\textbf{r}_{\pi} \\
+                &=\bm{\bm{\Phi}}^{\top}(\textbf{F}-\textbf{d}_{\mu} \textbf{f}^{\top})\textbf{r}_{\pi} \\
             \end{split}
         \end{equation}
         Let $\vec{h}(\bm{\theta}(t))$ be the driving vector field of the ODE
@@ -670,178 +708,154 @@ The  VMTD's solution is
     \end{proof}
 
 
-\section{Mathematical Analysis}
-\label{mathematicalanalysis}
-Table \ref{keymatrices} shows the key matrices of various algorithms.
-Table \ref{minimumeigenvalues} shows minimum eigenvalues $\frac{1}{2}\lambda_{\min}(\textbf{A} + \textbf{A}^\top)$ of various algorithms on several examples.
-\begin{table}[htb]
-    \centering
-       \caption{Key matrices of various algorithms.}
-       \label{keymatrices}
-       {
-       \begin{tabular}{cccc}
-           \toprule
-           Algorithm&Key matrix $\textbf{A}$&{Positive definite}&{$\bm{b}$}\\\midrule
-           On-policy TD&$\bm{\Phi}^{\top}\textbf{D}_{\pi}(\textbf{I}-\gamma
-         \textbf{P}_{\pi})\bm{\Phi}$&$\checkmark$&$\bm{b}_{\text{on}}=\bm{\Phi}^{\top}\textbf{D}_{\pi}\textbf{r}_{\pi}$\\
-         On-policy VMTD&${\bm{\Phi}}^{\top}(\textbf{D}_{\pi}-\textbf{d}_{\pi} \textbf{d}_{\pi}^{\top} )(\textbf{I} - \gamma\textbf{P}_{\pi}){\bm{\Phi}}$
-           &$\checkmark$&$\bm{\Phi}^{\top}(\textbf{D}_{\pi}-\textbf{d}_{\pi} \textbf{d}_{\pi}^{\top})\textbf{r}_{\pi}$\\
-         \midrule
-         Off-policy TD&$\textbf{A}_{\text{off}}={\bm{\Phi}}^{\top}\textbf{D}_{\mu}(\textbf{I}-\gamma
-         \textbf{P}_{\pi}){\bm{\Phi}}$&$\times$&$\bm{b}_{\text{off}}=\bm{\Phi}^{\top}\textbf{D}_{\mu}\textbf{r}_{\pi}$\\
-           TDC& $\textbf{A}_{\text{off}}^{\top}\textbf{C}^{-1}\textbf{A}_{\text{off}}$&$\checkmark$&$\textbf{A}_{\text{off}}^{\top}\textbf{C}^{-1}\bm{b}_{\text{off}}$
-           \\
-           ETD& ${\bm{\Phi}}^{\top}\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi}){\bm{\Phi}}$
-           &$\checkmark$&$\bm{\Phi}^{\top}\textbf{F}\textbf{r}_{\pi}$\\
-           \midrule
-           Off-policy VMTD&$\textbf{A}_{\text{VM}}={\bm{\Phi}}^{\top} (\textbf{D}_{\mu}-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} )(\textbf{I} - \gamma\textbf{P}_{\pi}){\bm{\Phi}}$
-           &$\times$&$\bm{b}_{\text{VM}}=\bm{\Phi}^{\top}(\textbf{D}_{\mu}-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top})\textbf{r}_{\pi}$\\
-           VMTDC& $\textbf{A}_{\text{VM}}^{\top}\textbf{C}^{-1}\textbf{A}_{\text{VM}}$&$\checkmark$&$\textbf{A}_{\text{VM}}^{\top}\textbf{C}^{-1}\bm{b}_{\text{VM}}$
-           \\
-           VMETD& ${\bm{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} ){\bm{\Phi}}$
-           &$\checkmark$&$\bm{\Phi}^{\top}(\textbf{F}-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top})\textbf{r}_{\pi}$\\
-           \bottomrule
-       \end{tabular}
-       }
-\end{table}
-\begin{table}[htb]
-    \centering
-    \caption{Minimum eigenvalues $\frac{1}{2}\lambda_{\min}(\textbf{A} + \textbf{A}^\top)$ of various algorithms on several examples.}
-    \label{minimumeigenvalues}
-    {
-    \begin{tabular}{ccccccc}
-        \toprule
-        \multirow{2}{*}{Algorithm}&\multirow{2}{*}{2-state} & \multirow{2}{*}{Baird's}&\multicolumn{3}{c}{Random Walk}&\multirow{2}{*}{Boyan Chain}
-            \\\cmidrule{4-6} %\cline{3-4}
-        &&&Tabular & Inverted & Dependent& \\
-        \midrule
-        TD& $-0.2$ & $-0.791$&$0.018$&$0.017$&$0.07$&$0.024$\\
-        VMTD & & & & & &  \\
-        \midrule
-        TDC&$0.016$ & $-0.002$&$0.002$&$0.007$&$0.011$&$0.002$\\
-        VMTDC & & & & & &  \\
-        \midrule
-        ETD&$\textbf{3.4}$& $-2.82e-16$&$\textbf{0.195}$&$\textbf{0.165}$&$\textbf{0.76}$&$\textbf{0.245}$\\
-        VMETD & & & & & &  \\
-        \bottomrule
-    \end{tabular}
-    }
-\end{table}
-\begin{table}[htb]
-    \label{bairdcounterexample}
-    \centering
-    \caption{Baird's Counterexample.}
-    {
-    \begin{tabular}{ccc}
-        \toprule
-        \multirow{2}{*}{Algorithm}&\multicolumn{2}{c}{Baird's Counterexample}
-            \\\cmidrule{2-3} 
-        &Two State & Seven State \\
-        \midrule
-        TD& [-0.2]& [-7.91e-01, 1.07e-16, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 7.48e-01]\\
-        VMTD & [0.25]& [-2.25e-01, -3.45e-17, 5.711e-01, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 2.87e+00] \\
-        \midrule
-        TDC&[0.016]&[-0.0024, 0.0078, 0.5714, 0.5714, 0.5714, 0.5714, 0.5729, 1.7682]\\
-        VMTDC &[0.025] &[-1.55e-03, -3.15e-04, 3.56e-01, 5.47e-01, 5.70e-01, 5.72e-01, 5.83e-01, 5.98e-01]   \\
-        \midrule
-        ETD& [3.4]& [-2.82e-16, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 7.40e-01, 3.72e+00]\\
-        VMETD &[1.15] &[-2.25e-01, -3.45e-17, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 2.86e+00]  \\
-        \bottomrule
-    \end{tabular}
-    }
-\end{table}
-\begin{table}[htb]
-    \label{randomwalk}
-    \centering
-    \caption{Random Walk.}
-    {
-    \begin{tabular}{cccc}
-        \toprule
-        \multirow{2}{*}{Algorithm}&\multicolumn{3}{c}{Random Walk}
-            \\\cmidrule{2-4} 
-        &Tabular & Inverted & Dependent \\
-        \midrule
-        TD&[0.018,0.074,0.183,0.260,0.464] &[0.017,0.044,0.065,0.083,0.116] &[0.070,0.113,0.138]\\
-        VMTD & [1.62e-17,0.073,0.180,0.260,0.463]& [-2.07e-17,0.018,0.045,0.065,0.115] &[0.022,0.115,0.116]\\
-        \midrule
-        TDC&[0.002,0.046,0.240,0.364,0.769]&[0.007,0.012,0.060,0.091,0.192]&[0.011,0.065,0.182]\\
-        VMTDC & [6.74e-17,0.044,0.240,0.364,0.769]& [-7.248e-18,0.011,0.060,0.091,0.192] & [0.0008,0.062,0.182]\\
-        \midrule
-        ETD&[0.195,0.669,1.712,2.765,4.660] & [0.165,0.420,0.678,0.820,1.168]&[0.760,1.084,1.394]\\
-        VMETD &[-3.40e-04,0.664,1.69,2.76,4.65] & [-0.001,0.167,0.423,0.689,1.163]&[0.221,1.043,1.293] \\
-        \bottomrule
-    \end{tabular}
-    }
-\end{table}
-\begin{table}[htb]
-    \label{boyanchain}
-    \centering
-       \caption{Boyan Chain.}
-       {
-       \begin{tabular}{cc}
-           \toprule
-           Algorithm&eigenvalues\\
-           \midrule
-           TD&[0.024495,0.054,0.065,0.135]\\
-         VMTD&[2.70e-18,0.053,0.065,0.135]\\
-         \midrule
-           TDC&[0.002,0.058,0.067,0.153] \\
-           VMTDC& [-1.40e-18,0.057,0.067,0.153]\\
-           \midrule
-           ETD& [0.245,0.540,0.647,1.352]\\
-           VMETD&[1.57e-17,0.529,0.647,1.352] \\
-           \bottomrule
-       \end{tabular}
-       }
-\end{table}
+% \section{Mathematical Analysis}
+% \label{mathematicalanalysis}
+% Table \ref{keymatrices} shows the key matrices of various algorithms.
+% Table \ref{minimumeigenvalues} shows minimum eigenvalues $\frac{1}{2}\lambda_{\min}(\textbf{A} + \textbf{A}^\top)$ of various algorithms on several examples.
+% \begin{table}[htb]
+%     \centering
+%        \caption{Key matrices of various algorithms.}
+%        \label{keymatrices}
+%        {
+%        \begin{tabular}{cccc}
+%            \toprule
+%            Algorithm&Key matrix $\textbf{A}$&{Positive definite}&{$\bm{b}$}\\\midrule
+%            On-policy TD&$\bm{\Phi}^{\top}\textbf{D}_{\pi}(\textbf{I}-\gamma
+%          \textbf{P}_{\pi})\bm{\Phi}$&$\checkmark$&$\bm{b}_{\text{on}}=\bm{\Phi}^{\top}\textbf{D}_{\pi}\textbf{r}_{\pi}$\\
+%          On-policy VMTD&${\bm{\Phi}}^{\top}(\textbf{D}_{\pi}-\textbf{d}_{\pi} \textbf{d}_{\pi}^{\top} )(\textbf{I} - \gamma\textbf{P}_{\pi}){\bm{\Phi}}$
+%            &$\checkmark$&$\bm{\Phi}^{\top}(\textbf{D}_{\pi}-\textbf{d}_{\pi} \textbf{d}_{\pi}^{\top})\textbf{r}_{\pi}$\\
+%          \midrule
+%          Off-policy TD&$\textbf{A}_{\text{off}}={\bm{\Phi}}^{\top}\textbf{D}_{\mu}(\textbf{I}-\gamma
+%          \textbf{P}_{\pi}){\bm{\Phi}}$&$\times$&$\bm{b}_{\text{off}}=\bm{\Phi}^{\top}\textbf{D}_{\mu}\textbf{r}_{\pi}$\\
+%            TDC& $\textbf{A}_{\text{off}}^{\top}\textbf{C}^{-1}\textbf{A}_{\text{off}}$&$\checkmark$&$\textbf{A}_{\text{off}}^{\top}\textbf{C}^{-1}\bm{b}_{\text{off}}$
+%            \\
+%            ETD& ${\bm{\Phi}}^{\top}\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi}){\bm{\Phi}}$
+%            &$\checkmark$&$\bm{\Phi}^{\top}\textbf{F}\textbf{r}_{\pi}$\\
+%            \midrule
+%            Off-policy VMTD&$\textbf{A}_{\text{VMTD}}={\bm{\Phi}}^{\top} (\textbf{D}_{\mu}-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} )(\textbf{I} - \gamma\textbf{P}_{\pi}){\bm{\Phi}}$
+%            &$\times$&$\bm{b}_{\text{VMTD}}=\bm{\Phi}^{\top}(\textbf{D}_{\mu}-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top})\textbf{r}_{\pi}$\\
+%            VMTDC& $\textbf{A}_{\text{VMTD}}^{\top}\textbf{C}^{-1}\textbf{A}_{\text{VMTD}}$&$\checkmark$&$\textbf{A}_{\text{VMTD}}^{\top}\textbf{C}^{-1}\bm{b}_{\text{VMTD}}$
+%            \\
+%            VMETD& ${\bm{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} ){\bm{\Phi}}$
+%            &$\checkmark$&$\bm{\Phi}^{\top}(\textbf{F}-\textbf{d}_{\mu} \textbf{f}^{\top})\textbf{r}_{\pi}$\\
+%            \bottomrule
+%        \end{tabular}
+%        }
+% \end{table}
+% \begin{table}[htb]
+%     \centering
+%     \caption{Minimum eigenvalues $\frac{1}{2}\lambda_{\min}(\textbf{A} + \textbf{A}^\top)$ of various algorithms on several examples.}
+%     \label{minimumeigenvalues}
+%     {
+%     \begin{tabular}{ccccccc}
+%         \toprule
+%         \multirow{2}{*}{Algorithm}&\multirow{2}{*}{2-state} & \multirow{2}{*}{Baird's}&\multicolumn{3}{c}{Random Walk}&\multirow{2}{*}{Boyan Chain}
+%             \\\cmidrule{4-6} %\cline{3-4}
+%         &&&Tabular & Inverted & Dependent& \\
+%         \midrule
+%         TD& $-0.2$ & $-0.791$&$0.018$&$0.017$&$0.07$&$0.024$\\
+%         VMTD & & & & & &  \\
+%         \midrule
+%         TDC&$0.016$ & $-0.002$&$0.002$&$0.007$&$0.011$&$0.002$\\
+%         VMTDC & & & & & &  \\
+%         \midrule
+%         ETD&$\textbf{3.4}$& $-2.82e-16$&$\textbf{0.195}$&$\textbf{0.165}$&$\textbf{0.76}$&$\textbf{0.245}$\\
+%         VMETD & & & & & &  \\
+%         \bottomrule
+%     \end{tabular}
+%     }
+% \end{table}
+% \begin{table}[htb]
+%     \label{bairdcounterexample}
+%     \centering
+%     \caption{Baird's Counterexample.}
+%     {
+%     \begin{tabular}{ccc}
+%         \toprule
+%         \multirow{2}{*}{Algorithm}&\multicolumn{2}{c}{Baird's Counterexample}
+%             \\\cmidrule{2-3} 
+%         &Two State & Seven State \\
+%         \midrule
+%         TD& [-0.2]& [-7.91e-01, 1.07e-16, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 7.48e-01]\\
+%         VMTD & [0.25]& [-2.25e-01, -3.45e-17, 5.711e-01, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 2.87e+00] \\
+%         \midrule
+%         TDC&[0.016]&[-0.0024, 0.0078, 0.5714, 0.5714, 0.5714, 0.5714, 0.5729, 1.7682]\\
+%         VMTDC &[0.025] &[-1.55e-03, -3.15e-04, 3.56e-01, 5.47e-01, 5.70e-01, 5.72e-01, 5.83e-01, 5.98e-01]   \\
+%         \midrule
+%         ETD& [3.4]& [-2.82e-16, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 7.40e-01, 3.72e+00]\\
+%         VMETD &[1.15] &[-2.25e-01, -3.45e-17, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 5.71e-01, 2.86e+00]  \\
+%         \bottomrule
+%     \end{tabular}
+%     }
+% \end{table}
+% \begin{table}[htb]
+%     \label{randomwalk}
+%     \centering
+%     \caption{Random Walk.}
+%     {
+%     \begin{tabular}{cccc}
+%         \toprule
+%         \multirow{2}{*}{Algorithm}&\multicolumn{3}{c}{Random Walk}
+%             \\\cmidrule{2-4} 
+%         &Tabular & Inverted & Dependent \\
+%         \midrule
+%         TD&[0.018,0.074,0.183,0.260,0.464] &[0.017,0.044,0.065,0.083,0.116] &[0.070,0.113,0.138]\\
+%         VMTD & [1.62e-17,0.073,0.180,0.260,0.463]& [-2.07e-17,0.018,0.045,0.065,0.115] &[0.022,0.115,0.116]\\
+%         \midrule
+%         TDC&[0.002,0.046,0.240,0.364,0.769]&[0.007,0.012,0.060,0.091,0.192]&[0.011,0.065,0.182]\\
+%         VMTDC & [6.74e-17,0.044,0.240,0.364,0.769]& [-7.248e-18,0.011,0.060,0.091,0.192] & [0.0008,0.062,0.182]\\
+%         \midrule
+%         ETD&[0.195,0.669,1.712,2.765,4.660] & [0.165,0.420,0.678,0.820,1.168]&[0.760,1.084,1.394]\\
+%         VMETD &[-3.40e-04,0.664,1.69,2.76,4.65] & [-0.001,0.167,0.423,0.689,1.163]&[0.221,1.043,1.293] \\
+%         \bottomrule
+%     \end{tabular}
+%     }
+% \end{table}
+% \begin{table}[htb]
+%     \label{boyanchain}
+%     \centering
+%        \caption{Boyan Chain.}
+%        {
+%        \begin{tabular}{cc}
+%            \toprule
+%            Algorithm&eigenvalues\\
+%            \midrule
+%            TD&[0.024495,0.054,0.065,0.135]\\
+%          VMTD&[2.70e-18,0.053,0.065,0.135]\\
+%          \midrule
+%            TDC&[0.002,0.058,0.067,0.153] \\
+%            VMTDC& [-1.40e-18,0.057,0.067,0.153]\\
+%            \midrule
+%            ETD& [0.245,0.540,0.647,1.352]\\
+%            VMETD&[1.57e-17,0.529,0.647,1.352] \\
+%            \bottomrule
+%        \end{tabular}
+%        }
+% \end{table}
 
 \section{Experimental details}
 \label{experimentaldetails}
-The feature matrices corresponding to three random walks are shown below respectively:
-\begin{equation*}
-    \Phi_{tabular}=\left[ 
-    \begin{array}{ccccc}
-    1 & 0& 0& 0& 0\\
-    0 & 1& 0& 0& 0\\
-    0 & 0& 1& 0& 0\\
-    0 & 0& 0& 1& 0\\
-    0 & 0& 0& 0& 1
-    \end{array}\right]
-    \end{equation*}
-    \begin{equation*}
-    \Phi_{inverted}=\left[ 
-    \begin{array}{ccccc}
-    0 & \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\\
-    \frac{1}{2} & 0& \frac{1}{2}& \frac{1}{2}& \frac{1}{2}\\
-    \frac{1}{2} & \frac{1}{2}& 0& \frac{1}{2}& \frac{1}{2}\\
-    \frac{1}{2} & \frac{1}{2}& \frac{1}{2}& 0& \frac{1}{2}\\
-    \frac{1}{2} & \frac{1}{2}& \frac{1}{2}& \frac{1}{2}& 0
-    \end{array}\right]
-    \end{equation*}
-    \begin{equation*}
-    \Phi_{dependent}=\left[ 
-    \begin{array}{ccccc}
-    1 & 0& 0\\
-    \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}& 0\\
-    \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}}& \frac{1}{\sqrt{3}}\\
-    0 & \frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}\\
-    0 & 0& 1
-    \end{array}\right]
-    \end{equation*}
-
-Three random walk experiments: the $\alpha$ values for 
-all algorithms are in the range of $\{0.008, 0.015, 0.03, 0.06, 0.12, 0.25, 0.5\}$. For the TDC algorithm, 
-the range of the ratio $\frac{\zeta}{\alpha}$ is $\{\frac{1}{512}, \frac{1}{256}, \frac{1}{128}, \frac{1}{64}, \frac{1}{32}, \frac{1}{16}, \frac{1}{8}, \frac{1}{4}, \frac{1}{2}, 1, 2\}$. For the VMTD algorithm, 
-the range of the ratio $\frac{\beta}{\alpha}$ is $\{\frac{1}{512}, \frac{1}{256}, \frac{1}{128}, \frac{1}{64}, \frac{1}{32}, \frac{1}{16}, \frac{1}{8}, \frac{1}{4}, \frac{1}{2}, 1, 2\}$. It can be observed from 
-the update formula of VMTDC that when $\zeta$ takes a very small value, 
-the VMTDC update tends to be similar to VMTD update. Similarly, 
-when $\beta$ takes a very small value, the VMTDC update tends to be 
-similar to TDC update. Through experiments, it was found that 
-setting $\zeta$ to a small value makes VMTDC updates approach VMTD 
-updates, resulting in better performance. Therefore, for the VMTDC 
-algorithm, the range of $\frac{\beta}{\alpha}$ ratio is $\{\frac{1}{512}, \frac{1}{256}, \frac{1}{128}, \frac{1}{64}, \frac{1}{32}, \frac{1}{16}, \frac{1}{8}, \frac{1}{4}, \frac{1}{2}, 1, 2\}$, and the range of 
-$\zeta$ is $\{0.1, 0.01, 0.001, 0.0001, 0.00001\}$. The learning curves in Figure \ref{Evaluation_full} correspond to the optimal 
-parameters.
+The 2-state counterexample and the 7-state counterexample 
+are well-known off-policy experimental environments. The 2-state 
+counterexample is relatively simple, so next, I'll provide a detailed 
+description of the 7-state counterexample environment.
+
+\textbf{Baird's off-policy counterexample:} This task is well known as a
+counterexample, in which TD diverges \cite{baird1995residual,sutton2009fast}. As
+shown in Figure \ref{bairdexample}, reward for each transition is zero. Thus the true values are zeros for all states and for any given policy. The behaviour policy
+chooses actions represented by solid lines with a probability of $\frac{1}{7}$
+and actions represented by dotted lines with a probability of $\frac{6}{7}$. The
+target policy is expected to choose the solid line with more probability than $\frac{1}{7}$,
+and it chooses the solid line with probability of $1$ in this paper.
+ The discount factor $\gamma =0.99$, and the feature matrix is
+defined in Appendix \ref{experimentaldetails} \cite{baird1995residual,sutton2009fast,maei2011gradient}.
+\begin{figure}
+    \begin{center}
+    \input{pic/BairdExample.tex}
+    \caption{7-state version of Baird's off-policy counterexample.}
+    \label{bairdexample}
+    \end{center}
+\end{figure}
 
 The feature matrix of 7-state version of Baird's off-policy counterexample is
 defined as follow:
@@ -858,21 +872,14 @@ defined as follow:
 \end{array}\right]
 \end{equation*}
 
-7-state version of Baird's off-policy counterexample: 
-for TD algorithm, $\alpha$ is set to 0.1. For the TDC algorithm, the range of 
-$\alpha$ is $\{0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0\}$, 
-and the range of 
-$\zeta$ is $\{0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5\}$. 
-For the VMTD algorithm, the range of 
-$\alpha$ is $\{0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0\}$, 
-and the range of 
-$\beta$ is $\{0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5\}$. Through experiments, it was found 
-that setting $\zeta$ to a small value makes VMTDC updates approach VMTD 
-updates, resulting in better performance. Therefore, for the VMTDC 
-algorithm, The range of values for $\alpha$ and $\beta$ is the same as that of VMTD  
-and the range of $\zeta$ 
-is $\{0.1, 0.01, 0.001, 0.0001, 0.00001\}$. 
-The learning curves in Figure \ref{Complete_full} correspond to the optimal parameters.
+2-state version of Baird's off-policy counterexample: All learning rates follow linear learning rate decay.
+For TDC algorithm, $\frac{\alpha_k}{\zeta_k}=5$ and $\alpha_0 = 0.1$.For ETD algorithm, $\alpha_0 = 0.1$.
+For VMTDC algorithm, $\frac{\alpha_k}{\zeta_k}=5$, $\frac{\alpha_k}{\omega_k}=4$,and $\alpha_0 = 0.1$.For ETD algorithm, $\frac{\alpha_k}{\omega_k}=4$ and $\alpha_0 = 0.1$.
+
+7-state version of Baird's off-policy counterexample: All learning rates follow linear learning rate decay.
+For TDC algorithm, $\frac{\alpha_k}{\zeta_k}=3$ and $\alpha_0 = 0.1$.For ETD algorithm, $\alpha_0 = 0.1$.
+For VMTDC algorithm, $\frac{\alpha_k}{\zeta_k}=3$, $\frac{\alpha_k}{\omega_k}=4$,and $\alpha_0 = 0.1$.For ETD algorithm, $\frac{\alpha_k}{\omega_k}=4$ and $\alpha_0 = 0.1$.
+
 For all policy evaluation experiments, each experiment 
 is independently run 100 times.
 
@@ -880,6 +887,29 @@ For the four control experiments: The learning rates for each
 algorithm in all experiments are shown in Table \ref{lrofways}.
 For all control experiments, each experiment is independently run 50 times.
 
+\textbf{Maze}:  The learning agent should find a shortest path from the upper
+left corner to the lower right corner. 
+ In each state,
+there are four alternative actions: $up$, $down$, $left$, and $right$, which
+takes the agent deterministically to the corresponding neighbour state,
+except when a movement is blocked by an obstacle or the edge
+of the maze. Rewards are $-1$ in all transitions until the
+agent reaches the goal state.
+The discount factor $\gamma=0.99$, and states $s$ are represented by tabular
+features.The maximum number of moves in the game is set to 1000.
+ \begin{figure}
+\centering
+\includegraphics[scale=0.35]{pic/maze_13_13.pdf} 
+\caption{Maze.}
+\end{figure}
+
+\textbf{The other three control environments}: Cliff Walking, Mountain Car, and Acrobot are 
+selected from the gym official website and correspond to the following 
+versions: ``CliffWalking-v0'', ``MountainCar-v0'' and ``Acrobot-v1''. 
+For specific details, please refer to the gym official website.
+The maximum number of steps for the Mountain Car environment is set to 1000, 
+while the default settings are used for the other two environments. In  Mountain car and Acrobot, features are generated by tile coding.
+
 \begin{table*}[htb]
     \centering
     \caption{Learning rates ($lr$) of four control experiments.}
@@ -888,13 +918,15 @@ For all control experiments, each experiment is independently run 50 times.
         \hline
         \multicolumn{1}{c|}{\diagbox{algorithms($lr$)}{envs}} &Maze &Cliff walking &Mountain Car &Acrobot \\
         \hline
-         Sarsa($\alpha$)&$0.1$ &$0.1$ &$0.1$ &$0.1$ \\
-         GQ(0)($\alpha,\zeta$)&$0.1,0.003$ &$0.1,0.004$ &$0.1,0.01$ &$0.1,0.01$ \\
-         VMSarsa($\alpha,\beta$)&$0.1,0.001$ &$0.1,\text{1e-4}$ &$0.1,\text{1e-4}$ &$0.1,\text{1e-4}$ \\
-         VMGQ(0)($\alpha,\zeta,\beta$)&$0.1,0.001,0.001$ &$0.1,0.005,\text{1e-4}$ &$0.1,\text{5e-4},\text{1e-4}$ &$0.1,\text{5e-4},\text{1e-4}$ \\
-         AC($lr_{\text{actor}},lr_{\text{critic}}$)&$0.01,0.1$ &$0.01,0.01$ &$0.01,0.05$ &$0.01,0.05$ \\
-         Q-learning($\alpha$)&$0.1$ &$0.1$ &$0.1$ &$0.1$ \\
-         VMQ($\alpha,\beta$)&$0.1,0.001$ &$0.1,\text{1e-4}$ &$0.1,\text{1e-4}$ &$0.1,\text{1e-4}$ \\
+        %  Sarsa($\alpha$)&$0.1$ &$0.1$ &$0.1$ &$0.1$ \\
+         GQ($\alpha,\zeta$)&$0.1,0.003$ &$0.1,0.004$ &$0.1,0.01$ &$0.1,0.01$ \\
+         EQ($\alpha$)&$0.006$ &$0.005$ &$0.001$ &$0.0005$ \\
+        %  VMSarsa($\alpha,\beta$)&$0.1,0.001$ &$0.1,\text{1e-4}$ &$0.1,\text{1e-4}$ &$0.1,\text{1e-4}$ \\
+         VMGQ($\alpha,\zeta,\beta$)&$0.1,0.001,0.001$ &$0.1,0.005,\text{1e-4}$ &$0.1,\text{5e-4},\text{1e-4}$ &$0.1,\text{5e-4},\text{1e-4}$ \\
+         VMEQ($\alpha,\beta$)&$0.001,0.0005$ &$0.005,0.0001$ &$0.001,0.0001$ &$0.0005,0.0001$ \\
+        %  AC($lr_{\text{actor}},lr_{\text{critic}}$)&$0.01,0.1$ &$0.01,0.01$ &$0.01,0.05$ &$0.01,0.05$ \\
+        %  Q-learning($\alpha$)&$0.1$ &$0.1$ &$0.1$ &$0.1$ \\
+        %  VMQ($\alpha,\beta$)&$0.1,0.001$ &$0.1,\text{1e-4}$ &$0.1,\text{1e-4}$ &$0.1,\text{1e-4}$ \\
          \hline
     \end{tabular}
     \label{lrofways}
diff --git a/Apendix/pic/BairdExample.tex b/Apendix/pic/BairdExample.tex
new file mode 100644
index 0000000..af5a22e
--- /dev/null
+++ b/Apendix/pic/BairdExample.tex
@@ -0,0 +1,69 @@
+\resizebox{7cm}{4.4cm}{
+\begin{tikzpicture}[smooth]
+\node[coordinate] (origin) at (0.3,0) {};
+\node[coordinate] (num7) at (3,0) {};
+\node[coordinate] (num1) at (1,2.5) {};
+\path (num7) ++ (-10:0.5cm) node (num7_bright1) [coordinate] {};
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+\path (origin) ++ (90:3cm) node (origin_above) [coordinate] {};
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+\path (num1) ++ (90:0.5cm) node (num1_a) [coordinate] {};
+\path (num1) ++ (-90:0.3cm) node (num1_b) [coordinate] {};
+
+\path (num1) ++ (0:1cm) node (num2) [coordinate] {};
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+\path (num5_a) ++ (0:1cm) node (num6_a) [coordinate] {};
+\path (num5_b) ++ (0:1cm) node (num6_b) [coordinate] {};
+
+
+%\draw[->](0,0) -- (1,1);
+%\draw[dashed,line width = 0.03cm] (0,0) -- (1,1);
+ %\fill (0.5,0.5) circle (0.5);
+ %\draw[shape=circle,fill=white,draw=black] (a) at (num7) {7};
+
+ 
+\draw[dashed,line width = 0.03cm,xshift=3cm] plot[tension=0.06]
+coordinates{(num7) (origin) (origin_above) (origin_aright)}; 
+
+\draw[->,>=stealth,line width = 0.02cm,xshift=3cm] plot[tension=0.5]
+coordinates{(num7) (num7_bright1) (num7_bright2)(num7_bright4) (num7_bright3)};
+ 
+\node[line width = 0.02cm,shape=circle,fill=white,draw=black] (g) at (num7) {7};
+
+
+
+\draw[<->,>=stealth,dashed,line width = 0.03cm,] (num1) -- (num1_a) ;
+\node[line width = 0.02cm,shape=circle,fill=white,draw=black] (a) at (num1_b) {1};
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+\node[line width = 0.02cm,shape=circle,fill=white,draw=black] (d) at (num4_b) {4};
+\draw[<->,>=stealth,dashed,line width = 0.03cm,] (num5) -- (num5_a) ;
+\node[line width = 0.02cm,shape=circle,fill=white,draw=black] (e) at (num5_b) {5};
+\draw[<->,>=stealth,dashed,line width = 0.03cm,] (num6) -- (num6_a) ;
+\node[line width = 0.02cm,shape=circle,fill=white,draw=black] (f) at (num6_b) {6};
+
+\draw[->,>=stealth,line width = 0.02cm] (a)--(g);
+\draw[->,>=stealth,line width = 0.02cm] (b)--(g);
+\draw[->,>=stealth,line width = 0.02cm] (c)--(g);
+\draw[->,>=stealth,line width = 0.02cm] (d)--(g);
+\draw[->,>=stealth,line width = 0.02cm] (e)--(g);
+\draw[->,>=stealth,line width = 0.02cm] (f)--(g);
+\end{tikzpicture}
+}
+
diff --git a/Apendix/pic/maze_13_13.pdf b/Apendix/pic/maze_13_13.pdf
new file mode 100644
index 0000000..cda62be
Binary files /dev/null and b/Apendix/pic/maze_13_13.pdf differ
diff --git a/NEW_aaai/aaai25.bib b/NEW_aaai/aaai25.bib
new file mode 100644
index 0000000..08e4dae
--- /dev/null
+++ b/NEW_aaai/aaai25.bib
@@ -0,0 +1,1138 @@
+@inproceedings{langley00,
+ author    = {P. Langley},
+ title     = {Crafting Papers on Machine Learning},
+ year      = {2000},
+ pages     = {1207--1216},
+ editor    = {Pat Langley},
+ booktitle     = {Proceedings of the 17th International Conference
+              on Machine Learning (ICML 2000)},
+ address   = {Stanford, CA},
+ publisher = {Morgan Kaufmann}
+}
+
+@TechReport{mitchell80,
+  author = 	 "T. M. Mitchell",
+  title = 	 "The Need for Biases in Learning Generalizations",
+  institution =  "Computer Science Department, Rutgers University",
+  year = 	 "1980",
+  address =	 "New Brunswick, MA",
+}
+
+@phdthesis{kearns89,
+  author = {M. J. Kearns},
+  title =  {Computational Complexity of Machine Learning},
+  school = {Department of Computer Science, Harvard University},
+  year =   {1989}
+}
+
+@Book{MachineLearningI,
+  editor = 	 "R. S. Michalski and J. G. Carbonell and T.
+		  M. Mitchell",
+  title = 	 "Machine Learning: An Artificial Intelligence
+		  Approach, Vol. I",
+  publisher = 	 "Tioga",
+  year = 	 "1983",
+  address =	 "Palo Alto, CA"
+}
+
+@Book{DudaHart2nd,
+  author =       "R. O. Duda and P. E. Hart and D. G. Stork",
+  title =        "Pattern Classification",
+  publisher =    "John Wiley and Sons",
+  edition =      "2nd",
+  year =         "2000"
+}
+
+@misc{anonymous,
+  title= {Suppressed for Anonymity},
+  author= {Author, N. N.},
+  year= {2021}
+}
+
+@InCollection{Newell81,
+  author =       "A. Newell and P. S. Rosenbloom",
+  title =        "Mechanisms of Skill Acquisition and the Law of
+                  Practice", 
+  booktitle =    "Cognitive Skills and Their Acquisition",
+  pages =        "1--51",
+  publisher =    "Lawrence Erlbaum Associates, Inc.",
+  year =         "1981",
+  editor =       "J. R. Anderson",
+  chapter =      "1",
+  address =      "Hillsdale, NJ"
+}
+
+
+@Article{Samuel59,
+  author = 	 "A. L. Samuel",
+  title = 	 "Some Studies in Machine Learning Using the Game of
+		  Checkers",
+  journal =	 "IBM Journal of Research and Development",
+  year =	 "1959",
+  volume =	 "3",
+  number =	 "3",
+  pages =	 "211--229"
+}
+@inproceedings{langley00,
+ author    = {P. Langley},
+ title     = {Crafting Papers on Machine Learning},
+ year      = {2000},
+ pages     = {1207--1216},
+ editor    = {Pat Langley},
+ booktitle     = {Proceedings of the 17th International Conference
+              on Machine Learning (ICML 2000)},
+ address   = {Stanford, CA},
+ publisher = {Morgan Kaufmann}
+}
+
+@TechReport{mitchell80,
+  author = 	 "T. M. Mitchell",
+  title = 	 "The Need for Biases in Learning Generalizations",
+  institution =  "Computer Science Department, Rutgers University",
+  year = 	 "1980",
+  address =	 "New Brunswick, MA",
+}
+
+@phdthesis{kearns89,
+  author = {M. J. Kearns},
+  title =  {Computational Complexity of Machine Learning},
+  school = {Department of Computer Science, Harvard University},
+  year =   {1989}
+}
+
+@Book{MachineLearningI,
+  editor = 	 "R. S. Michalski and J. G. Carbonell and T.
+		  M. Mitchell",
+  title = 	 "Machine Learning: An Artificial Intelligence
+		  Approach, Vol. I",
+  publisher = 	 "Tioga",
+  year = 	 "1983",
+  address =	 "Palo Alto, CA"
+}
+
+@Book{DudaHart2nd,
+  author =       "R. O. Duda and P. E. Hart and D. G. Stork",
+  title =        "Pattern Classification",
+  publisher =    "John Wiley and Sons",
+  edition =      "2nd",
+  year =         "2000"
+}
+
+@misc{anonymous,
+  title= {Suppressed for Anonymity},
+  author= {Author, N. N.},
+  year= {2021}
+}
+
+@InCollection{Newell81,
+  author =       "A. Newell and P. S. Rosenbloom",
+  title =        "Mechanisms of Skill Acquisition and the Law of
+                  Practice", 
+  booktitle =    "Cognitive Skills and Their Acquisition",
+  pages =        "1--51",
+  publisher =    "Lawrence Erlbaum Associates, Inc.",
+  year =         "1981",
+  editor =       "J. R. Anderson",
+  chapter =      "1",
+  address =      "Hillsdale, NJ"
+}
+
+
+@Article{Samuel59,
+  author = 	 "A. L. Samuel",
+  title = 	 "Some Studies in Machine Learning Using the Game of
+		  Checkers",
+  journal =	 "IBM Journal of Research and Development",
+  year =	 "1959",
+  volume =	 "3",
+  number =	 "3",
+  pages =	 "211--229"
+}
+
+@book{em:86,
+  editor  = "Engelmore, Robert and Morgan, Anthony",
+  title   = "Blackboard Systems",
+  year    = 1986,
+  address = "Reading, Mass.",
+  publisher = "Addison-Wesley",
+}
+@inproceedings{dalal2018finite,
+  title={Finite sample analyses for TD (0) with function approximation},
+  author={Dalal, Gal and Szorenyi, Balazs and Thoppe, Gugan and Mannor, Shie},
+  booktitle={Proceedings of the Thirty-Second AAAI Conference on Artificial Intelligence and Thirtieth Innovative Applications of Artificial Intelligence Conference and Eighth AAAI Symposium on Educational Advances in Artificial Intelligence},
+  pages={6144--6160},
+  year={2018}
+}
+@inproceedings{xu2019reanalysis,
+  title={Reanalysis of Variance Reduced Temporal Difference Learning},
+  author={Xu, Tengyu and Wang, Zhe and Zhou, Yi and Liang, Yingbin},
+  booktitle={International Conference on Learning Representations},
+  year={2019}
+}
+@inproceedings{c:83,
+  author  = "Clancey, William J.",
+  year    = 1983,
+  title   = "{Communication, Simulation, and Intelligent
+Agents: Implications of Personal Intelligent Machines
+for Medical Education}",
+  booktitle="Proceedings of the Eighth International Joint Conference on Artificial Intelligence {(IJCAI-83)}", 
+  pages   = "556-560",
+  address = "Menlo Park, Calif",
+  publisher = "{IJCAI Organization}",
+}
+@inproceedings{c:84,
+  author  = "Clancey, William J.",
+  year    = 1984,
+  title   = "{Classification Problem Solving}",
+  booktitle = "Proceedings of the Fourth National 
+              Conference on Artificial Intelligence",
+  pages   = "45-54",
+  address = "Menlo Park, Calif.",
+  publisher="AAAI Press",
+}
+@article{r:80,
+  author = {Robinson, Arthur L.},
+  title = {New Ways to Make Microcircuits Smaller},
+  volume = {208},
+  number = {4447},
+  pages = {1019--1022},
+  year = {1980},
+  doi = {10.1126/science.208.4447.1019},
+  publisher = {American Association for the Advancement of Science},
+  issn = {0036-8075},
+  URL = {https://science.sciencemag.org/content/208/4447/1019},
+  eprint = {https://science.sciencemag.org/content/208/4447/1019.full.pdf},
+  journal = {Science},
+}
+@article{r:80x,
+  author  = "Robinson, Arthur L.",
+  year    = 1980,
+  title   = "{New Ways to Make Microcircuits Smaller---Duplicate Entry}",
+  journal = "Science",
+  volume  =  208,
+  pages   = "1019-1026",
+}
+@article{hcr:83,
+title = {Strategic explanations for a diagnostic consultation system},
+journal = {International Journal of Man-Machine Studies},
+volume = {20},
+number = {1},
+pages = {3-19},
+year = {1984},
+issn = {0020-7373},
+doi = {https://doi.org/10.1016/S0020-7373(84)80003-6},
+url = {https://www.sciencedirect.com/science/article/pii/S0020737384800036},
+author = {Diane Warner Hasling and William J. Clancey and Glenn Rennels},
+abstract = {This article examines the problem of automatte explanation of reasoning, especially as it relates to expert systems. By explanation we mean the ability of a program to discuss what it is doing in some understandable way. We first present a general framework in which to view explanation and review some of the research done in this area. We then focus on the explanation system for NEOMYCIN, a medical consultation program. A consultation program interactively helps a user to solve a problem. Our goal is to have NEOMYCIN explain its problem-solving strategies. An explanation of strategy describes the plan the program is using to reach a solution. Such an explanation is usually concrete, referring to aspects of the current problem situation. Abstract explanations articulate a general principle, which can be applied in different situations; such explanations are useful in teaching and in explaining by analogy. We describe the aspects of NEOMYCIN that make abstract strategic explanations possible—the representation of strategic knowledge explicitly and separately from domain knowledge— and demonstrate how this representation can be used to generate explanations.}
+}
+@article{hcrt:83,
+  author  = "Hasling, Diane Warner and Clancey, William J. and Rennels, Glenn R. and Test, Thomas",
+  year    = 1983,
+  title   = "{Strategic Explanations in Consultation---Duplicate}",
+  journal = "The International Journal of Man-Machine Studies",
+  volume  = 20,
+  number  = 1,
+  pages   = "3-19",
+}
+@techreport{r:86,
+  author  = "Rice, James",
+  year    = 1986,
+  title   = "{Poligon: A System for Parallel Problem Solving}",
+  type    = "Technical Report", 
+  number  = "KSL-86-19", 
+  institution = "Dept.\ of Computer Science, Stanford Univ.",
+}
+@phdthesis{c:79,
+  author  = "Clancey, William J.",
+  year    = 1979,
+  title   = "{Transfer of Rule-Based Expertise
+through a Tutorial Dialogue}",
+  type    = "{Ph.D.} diss.",
+  school  = "Dept.\ of Computer Science, Stanford Univ.",
+  address = "Stanford, Calif.",
+}
+@unpublished{c:21,
+  author  = "Clancey, William J.",
+  title   = "{The Engineering of Qualitative Models}",
+  year    = 2021,
+  note    = "Forthcoming",
+}
+@misc{c:22,
+      title={Attention Is All You Need}, 
+      author={Ashish Vaswani and Noam Shazeer and Niki Parmar and Jakob Uszkoreit and Llion Jones and Aidan N. Gomez and Lukasz Kaiser and Illia Polosukhin},
+      year={2017},
+      eprint={1706.03762},
+      archivePrefix={arXiv},
+      primaryClass={cs.CL}
+}
+@misc{c:23,
+  title        = "Pluto: The 'Other' Red Planet",
+  author       = "{NASA}",
+  howpublished = "\url{https://www.nasa.gov/nh/pluto-the-other-red-planet}",
+  year         = 2015,
+  note         = "Accessed: 2018-12-06"
+}
+@article{r:80x,
+  author  = "Robinson, Arthur L.",
+  year    = 1980,
+  title   = "{New Ways to Make Microcircuits Smaller---Duplicate Entry}",
+  journal = "Science",
+  volume  =  208,
+  pages   = "1019-1026",
+}
+@article{hcrt:83,
+  author  = "Hasling, Diane Warner and Clancey, William J. and Rennels, Glenn R. and Test, Thomas",
+  year    = 1983,
+  title   = "{Strategic Explanations in Consultation---Duplicate}",
+  journal = "The International Journal of Man-Machine Studies",
+  volume  = 20,
+  number  = 1,
+  pages   = "3-19",
+}
+@article{xu2013online,
+  title={Online learning control using adaptive critic designs with sparse kernel machines},
+  author={Xu, Xin and Hou, Zhongsheng and Lian, Chuanqiang and He, Haibo},
+  journal={IEEE Trans. Neural Netw. Learn. Syst.},
+  volume={24},
+  number={5},
+  pages={762--775},
+  year={2013},
+  publisher={IEEE}
+}
+@article{bertsekas2017value,
+  title={Value and policy iterations in optimal control and adaptive dynamic programming},
+  author={Bertsekas, Dimitri P},
+  journal={IEEE Trans. Neural Netw. Learn. Syst.},
+  year={2017},
+  volume={28},
+  number={3},
+  pages={500 - 509},
+  publisher={IEEE}
+}
+@phdthesis{hackman2012faster,
+  title={Faster Gradient-TD Algorithms},
+  author={Hackman, Leah},
+  year={2012},
+  school={University of Alberta}
+}
+@inproceedings{harutyunyan2015multi,
+  title={Multi-scale reward shaping via an off-policy ensemble},
+  author={Harutyunyan, Anna and Brys, Tim and Vrancx, Peter and Now{\'e}, Ann},
+  booktitle={Proc. 2015 Int. Conf. Autonomous Agents and Multiagent Systems},
+  pages={1641--1642},
+  year={2015},
+  organization={International Foundation for Autonomous Agents and Multiagent Systems}
+}
+@inproceedings{harutyunyan2015expressing,
+  title={Expressing Arbitrary Reward Functions as Potential-Based Advice.},
+  author={Harutyunyan, Anna and Devlin, Sam and Vrancx, Peter and Now{\'e}, Ann},
+  booktitle={AAAI},
+  pages={2652--2658},
+  year={2015}
+}
+@article{wiewiora2003potential,
+  title={Potential-based shaping and Q-value initialization are equivalent},
+  author={Wiewiora, Eric},
+  journal={J. Artif. Intell. Res.},
+  volume={19},
+  pages={205--208},
+  year={2003}
+}
+@article{grzes2010online,
+  title={Online learning of shaping rewards in reinforcement learning},
+  author={Grze{\'s}, Marek and Kudenko, Daniel},
+  journal={Neural Netw.},
+  volume={23},
+  number={4},
+  pages={541--550},
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+  publisher={Elsevier}
+}
+@inproceedings{marthi2007automatic,
+  title={Automatic shaping and decomposition of reward functions},
+  author={Marthi, Bhaskara},
+  booktitle={Proc. 24th Int. Conf. Mach. Learn.},
+  pages={601--608},
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+}
+@inproceedings{laud2003influence,
+  title={The Influence of Reward on the Speed of Reinforcement Learning: An Analysis of Shaping},
+  author={Laud, Adam and Dejong, Gerald},
+  booktitle={Proc. 20th Int. Conf. Mach. Learn.},
+  pages={440--447},
+  year={2003}
+}
+@phdthesis{laud2004theory,
+  title={Theory and application of reward shaping in reinforcement learning},
+  author={Laud, Adam Daniel},
+  year={2004},
+  school={University of Illinois at Urbana-Champaign}
+}
+@article{geist2013algorithmic,
+  title={Algorithmic survey of parametric value function approximation},
+  author={Geist, Matthieu and Pietquin, Olivier},
+  journal={IEEE Trans. Neural Netw. Learn. Syst.},
+  volume={24},
+  number={6},
+  pages={845--867},
+  year={2013},
+  publisher={IEEE}
+}
+@article{furmston2016approximate,
+  title={Approximate Newton Methods for Policy Search in Markov Decision Processes},
+  author={Furmston, Thomas and Lever, Guy and Barber, David},
+  journal={J. Mach. Learn. Res.},
+  volume={17},
+  number={227},
+  pages={1--51},
+  year={2016}
+}
+@article{silver2016mastering,
+  title={Mastering the game of Go with deep neural networks and tree search},
+  author={Silver, David and Huang, Aja and Maddison, Chris J and Guez, Arthur and Sifre, Laurent and van den Driessche, George and Schrittwieser, Julian and Antonoglou, Ioannis and Panneershelvam, Veda and Lanctot, Marc and others},
+  journal={Nature},
+  volume={529},
+  number={7587},
+  pages={484--489},
+  year={2016},
+  publisher={Nature Publishing Group}
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+
+@article{mnih2015human,
+  title={Human-level control through deep reinforcement learning},
+  author={Mnih, Volodymyr and Kavukcuoglu, Koray and Silver, David and Rusu, Andrei A and Veness, Joel and Bellemare, Marc G and Graves, Alex and Riedmiller, Martin and Fidjeland, Andreas K and Ostrovski, Georg and others},
+  journal={Nature},
+  volume={518},
+  number={7540},
+  pages={529--533},
+  year={2015},
+  publisher={Nature Publishing Group}
+}
+@inproceedings{guo2014deep,
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+  AUTHOR ="Kurt Driessens",
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+  YEAR ="2004",
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+  organization={ACM}
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+  year={1999}
+}
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+  year={2005}
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+  year={2010}
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+  publisher={Cambridge, MA: MIT Press},
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+  year={2013}
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+  author =	 {Ron Breukelaar and Erik D. Demaine and Susan
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+  author =	 {Bertsekas, D. and Tsitsiklis, J. N.},
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+  year={2014}
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+  title={Proximal Gradient Temporal Difference Learning Algorithms.},
+  author={Liu, Bo and Liu, Ji and Ghavamzadeh, Mohammad and Mahadevan, Sridhar and Petrik, Marek},
+  booktitle={Proceedings of the International Joint Conference on Artificial Intelligence},
+  pages={4195--4199},
+  year={2016}
+}
+@article{liu2018proximal,
+  title={Proximal gradient temporal difference learning: Stable reinforcement learning with polynomial sample complexity},
+  author={Liu, Bo and Gemp, Ian and Ghavamzadeh, Mohammad and Liu, Ji and Mahadevan, Sridhar and Petrik, Marek},
+  journal={Journal of Artificial Intelligence Research},
+  volume={63},
+  pages={461--494},
+  year={2018}
+}
+@inproceedings{givchi2015quasi,
+  title={Quasi newton temporal difference learning},
+  author={Givchi, Arash and Palhang, Maziar},
+  booktitle={Asian Conference on Machine Learning},
+  pages={159--172},
+  year={2015}
+}
+@inproceedings{pan2017accelerated,
+  title={Accelerated gradient temporal difference learning},
+  author={Pan, Yangchen and White, Adam and White, Martha},
+  booktitle={Proceedings of the 21st AAAI Conference on Artificial Intelligence},
+  pages={2464--2470},
+  year={2017}
+}
+@inproceedings{hallak2016generalized,
+  title={Generalized emphatic temporal difference learning: bias-variance analysis},
+  author={Hallak, Assaf and Tamar, Aviv and Munos, Remi and Mannor, Shie},
+  booktitle={Proceedings of the 30th AAAI Conference on Artificial Intelligence},
+  pages={1631--1637},
+  year={2016}
+}
+@article{zhang2022truncated,
+  title={Truncated emphatic temporal difference methods for prediction and control},
+  author={Zhang, Shangtong and Whiteson, Shimon},
+  journal={The Journal of Machine Learning Research},
+  volume={23},
+  number={1},
+  pages={6859--6917},
+  year={2022},
+  publisher={JMLRORG}
+}
+@inproceedings{korda2015td,
+  title={On TD (0) with function approximation: Concentration bounds and a centered variant with exponential convergence},
+  author={Korda, Nathaniel and La, Prashanth},
+  booktitle={International conference on machine learning},
+  pages={626--634},
+  year={2015},
+  organization={PMLR}
+}
+@book{zhou2021machine,
+  title={Machine learning},
+  author={Zhou, Zhi-Hua},
+  year={2021},
+  publisher={Springer Nature}
+}
+@inproceedings{dalal2020tale,
+  title={A tale of two-timescale reinforcement learning with the tightest finite-time bound},
+  author={Dalal, Gal and Szorenyi, Balazs and Thoppe, Gugan},
+  booktitle={Proceedings of the AAAI Conference on Artificial Intelligence},
+  volume={34},
+  number={04},
+  pages={3701-3708},
+  year={2020}
+}
+@inproceedings{feng2019kernel,
+  title={A kernel loss for solving the Bellman equation},
+  author={Feng, Yihao and Li, Lihong and Liu, Qiang},
+  booktitle={Advances in Neural Information Processing Systems},
+  pages={15430--15441},
+  year={2019}
+}
+@inproceedings{basserrano2021logistic,
+  title={Logistic Q-Learning},
+  author={Bas-Serrano, Joan and Curi, Sebastian and Krause, Andreas and Neu, Gergely},
+  booktitle={International Conference on Artificial Intelligence and Statistics},
+  pages={3610--3618},
+  year={2021}
+}
+
+
+
+
+
+
+
+
+
diff --git a/NEW_aaai/aaai25.bst b/NEW_aaai/aaai25.bst
new file mode 100644
index 0000000..05b1d4e
--- /dev/null
+++ b/NEW_aaai/aaai25.bst
@@ -0,0 +1,1493 @@
+%%
+%% This is file `aaai22.bst',
+%% generated with the docstrip utility.
+%%
+%% The original source files were:
+%%
+%% merlin.mbs  (with options: `head,ay,nat,ed-au,nm-rev,ed-rev,jnrlst,aunm-semi,mcite,mct-1,mct-x3,keyxyr,dt-beg,yr-per,yrp-per,note-yr,atit-u,volp-sp,num-xser,bkpg-x,add-pub,isbn,ppx,ed,xedn,and-com,and-com-ed,etal-xc,nfss,,{}')
+%% merlin.mbs  (with options: `tail,ay,nat,ed-au,nm-rev,ed-rev,jnrlst,aunm-semi,mcite,mct-1,mct-x3,keyxyr,dt-beg,yr-per,yrp-per,note-yr,atit-u,volp-sp,num-xser,bkpg-x,add-pub,isbn,ppx,ed,xedn,and-com,and-com-ed,etal-xc,nfss,,{}')
+%% ----------------------------------------
+%% *** Natbib-compatible implementation of 'aaai' bib style ***
+%% 
+ % ===============================================================
+ % IMPORTANT NOTICE:
+ % This bibliographic style (bst) file has been generated from one or
+ % more master bibliographic style (mbs) files, listed above.
+ %
+ % This generated file can be redistributed and/or modified under the terms
+ % of the LaTeX Project Public License Distributed from CTAN
+ % archives in directory macros/latex/base/lppl.txt; either
+ % version 1 of the License, or any later version.
+ % ===============================================================
+ % Name and version information of the main mbs file:
+ % \ProvidesFile{merlin.mbs}[2011/11/18 4.33 (PWD, AO, DPC)]
+ %   For use with BibTeX version 0.99a or later
+ %-------------------------------------------------------------------
+ % This bibliography style file is intended for texts in ENGLISH
+ % This is an author-year citation style bibliography. As such, it is
+ % non-standard LaTeX, and requires a special package file to function properly.
+ % Such a package is    natbib.sty   by Patrick W. Daly
+ % The form of the \bibitem entries is
+ %   \bibitem[Jones et al.(1990)]{key}...
+ %   \bibitem[Jones et al.(1990)Jones, Baker, and Smith]{key}...
+ % The essential feature is that the label (the part in brackets) consists
+ % of the author names, as they should appear in the citation, with the year
+ % in parentheses following. There must be no space before the opening
+ % parenthesis!
+ % With natbib v5.3, a full list of authors may also follow the year.
+ % In natbib.sty, it is possible to define the type of enclosures that is
+ % really wanted (brackets or parentheses), but in either case, there must
+ % be parentheses in the label.
+ % The \cite command functions as follows:
+ %   \citet{key} ==>>                Jones et al. (1990)
+ %   \citet*{key} ==>>               Jones, Baker, and Smith (1990)
+ %   \citep{key} ==>>                (Jones et al., 1990)
+ %   \citep*{key} ==>>               (Jones, Baker, and Smith, 1990)
+ %   \citep[chap. 2]{key} ==>>       (Jones et al., 1990, chap. 2)
+ %   \citep[e.g.][]{key} ==>>        (e.g. Jones et al., 1990)
+ %   \citep[e.g.][p. 32]{key} ==>>   (e.g. Jones et al., 1990, p. 32)
+ %   \citeauthor{key} ==>>           Jones et al.
+ %   \citeauthor*{key} ==>>          Jones, Baker, and Smith
+ %   \citeyear{key} ==>>             1990
+ %---------------------------------------------------------------------
+
+ENTRY
+  { address
+    archivePrefix
+    author
+    booktitle
+    chapter
+    edition
+    editor
+    eid
+    eprint
+    howpublished
+    institution
+    isbn
+    journal
+    key
+    month
+    note
+    number
+    organization
+    pages
+    publisher
+    school
+    series
+    title
+    type
+    volume
+    year
+  }
+  {}
+  { label extra.label sort.label short.list }
+INTEGERS { output.state before.all mid.sentence after.sentence after.block }
+FUNCTION {init.state.consts}
+{ #0 'before.all :=
+  #1 'mid.sentence :=
+  #2 'after.sentence :=
+  #3 'after.block :=
+}
+STRINGS { s t}
+FUNCTION {output.nonnull}
+{ 's :=
+  output.state mid.sentence =
+    { ", " * write$ }
+    { output.state after.block =
+        { add.period$ write$
+          newline$
+          "\newblock " write$
+        }
+        { output.state before.all =
+            'write$
+            { add.period$ " " * write$ }
+          if$
+        }
+      if$
+      mid.sentence 'output.state :=
+    }
+  if$
+  s
+}
+FUNCTION {output}
+{ duplicate$ empty$
+    'pop$
+    'output.nonnull
+  if$
+}
+FUNCTION {output.check}
+{ 't :=
+  duplicate$ empty$
+    { pop$ "empty " t * " in " * cite$ * warning$ }
+    'output.nonnull
+  if$
+}
+FUNCTION {fin.entry}
+{ add.period$
+  write$
+  newline$
+}
+
+FUNCTION {new.block}
+{ output.state before.all =
+    'skip$
+    { after.block 'output.state := }
+  if$
+}
+FUNCTION {new.sentence}
+{ output.state after.block =
+    'skip$
+    { output.state before.all =
+        'skip$
+        { after.sentence 'output.state := }
+      if$
+    }
+  if$
+}
+FUNCTION {add.blank}
+{  " " * before.all 'output.state :=
+}
+
+FUNCTION {date.block}
+{
+  new.block
+}
+
+FUNCTION {not}
+{   { #0 }
+    { #1 }
+  if$
+}
+FUNCTION {and}
+{   'skip$
+    { pop$ #0 }
+  if$
+}
+FUNCTION {or}
+{   { pop$ #1 }
+    'skip$
+  if$
+}
+FUNCTION {new.block.checkb}
+{ empty$
+  swap$ empty$
+  and
+    'skip$
+    'new.block
+  if$
+}
+FUNCTION {field.or.null}
+{ duplicate$ empty$
+    { pop$ "" }
+    'skip$
+  if$
+}
+FUNCTION {emphasize}
+{ duplicate$ empty$
+    { pop$ "" }
+    { "\emph{" swap$ * "}" * }
+  if$
+}
+FUNCTION {tie.or.space.prefix}
+{ duplicate$ text.length$ #3 <
+    { "~" }
+    { " " }
+  if$
+  swap$
+}
+
+FUNCTION {capitalize}
+{ "u" change.case$ "t" change.case$ }
+
+FUNCTION {space.word}
+{ " " swap$ * " " * }
+ % Here are the language-specific definitions for explicit words.
+ % Each function has a name bbl.xxx where xxx is the English word.
+ % The language selected here is ENGLISH
+FUNCTION {bbl.and}
+{ "and"}
+
+FUNCTION {bbl.etal}
+{ "et~al." }
+
+FUNCTION {bbl.editors}
+{ "eds." }
+
+FUNCTION {bbl.editor}
+{ "ed." }
+
+FUNCTION {bbl.edby}
+{ "edited by" }
+
+FUNCTION {bbl.edition}
+{ "edition" }
+
+FUNCTION {bbl.volume}
+{ "volume" }
+
+FUNCTION {bbl.of}
+{ "of" }
+
+FUNCTION {bbl.number}
+{ "number" }
+
+FUNCTION {bbl.nr}
+{ "no." }
+
+FUNCTION {bbl.in}
+{ "in" }
+
+FUNCTION {bbl.pages}
+{ "" }
+
+FUNCTION {bbl.page}
+{ "" }
+
+FUNCTION {bbl.chapter}
+{ "chapter" }
+
+FUNCTION {bbl.techrep}
+{ "Technical Report" }
+
+FUNCTION {bbl.mthesis}
+{ "Master's thesis" }
+
+FUNCTION {bbl.phdthesis}
+{ "Ph.D. thesis" }
+
+MACRO {jan} {"January"}
+
+MACRO {feb} {"February"}
+
+MACRO {mar} {"March"}
+
+MACRO {apr} {"April"}
+
+MACRO {may} {"May"}
+
+MACRO {jun} {"June"}
+
+MACRO {jul} {"July"}
+
+MACRO {aug} {"August"}
+
+MACRO {sep} {"September"}
+
+MACRO {oct} {"October"}
+
+MACRO {nov} {"November"}
+
+MACRO {dec} {"December"}
+
+MACRO {acmcs} {"ACM Computing Surveys"}
+
+MACRO {acta} {"Acta Informatica"}
+
+MACRO {cacm} {"Communications of the ACM"}
+
+MACRO {ibmjrd} {"IBM Journal of Research and Development"}
+
+MACRO {ibmsj} {"IBM Systems Journal"}
+
+MACRO {ieeese} {"IEEE Transactions on Software Engineering"}
+
+MACRO {ieeetc} {"IEEE Transactions on Computers"}
+
+MACRO {ieeetcad}
+ {"IEEE Transactions on Computer-Aided Design of Integrated Circuits"}
+
+MACRO {ipl} {"Information Processing Letters"}
+
+MACRO {jacm} {"Journal of the ACM"}
+
+MACRO {jcss} {"Journal of Computer and System Sciences"}
+
+MACRO {scp} {"Science of Computer Programming"}
+
+MACRO {sicomp} {"SIAM Journal on Computing"}
+
+MACRO {tocs} {"ACM Transactions on Computer Systems"}
+
+MACRO {tods} {"ACM Transactions on Database Systems"}
+
+MACRO {tog} {"ACM Transactions on Graphics"}
+
+MACRO {toms} {"ACM Transactions on Mathematical Software"}
+
+MACRO {toois} {"ACM Transactions on Office Information Systems"}
+
+MACRO {toplas} {"ACM Transactions on Programming Languages and Systems"}
+
+MACRO {tcs} {"Theoretical Computer Science"}
+FUNCTION {bibinfo.check}
+{ swap$
+  duplicate$ missing$
+    {
+      pop$ pop$
+      ""
+    }
+    { duplicate$ empty$
+        {
+          swap$ pop$
+        }
+        { swap$
+          pop$
+        }
+      if$
+    }
+  if$
+}
+FUNCTION {bibinfo.warn}
+{ swap$
+  duplicate$ missing$
+    {
+      swap$ "missing " swap$ * " in " * cite$ * warning$ pop$
+      ""
+    }
+    { duplicate$ empty$
+        {
+          swap$ "empty " swap$ * " in " * cite$ * warning$
+        }
+        { swap$
+          pop$
+        }
+      if$
+    }
+  if$
+}
+FUNCTION {format.eprint}
+{ eprint duplicate$ empty$
+    'skip$
+    { archivePrefix duplicate$ empty$
+        'skip$
+        { ":" * swap$ }
+      if$
+      * "." *
+    }
+  if$
+}
+INTEGERS { nameptr namesleft numnames }
+
+
+STRINGS  { bibinfo}
+
+FUNCTION {format.names}
+{ 'bibinfo :=
+  duplicate$ empty$ 'skip$ {
+  's :=
+  "" 't :=
+  #1 'nameptr :=
+  s num.names$ 'numnames :=
+  numnames 'namesleft :=
+    { namesleft #0 > }
+    { s nameptr
+      "{vv~}{ll}{, f.}{, jj}"
+      format.name$
+      bibinfo bibinfo.check
+      't :=
+      nameptr #1 >
+        {
+          namesleft #1 >
+            { "; " * t * }
+            {
+              s nameptr "{ll}" format.name$ duplicate$ "others" =
+                { 't := }
+                { pop$ }
+              if$
+              ";" *
+              t "others" =
+                {
+                  " " * bbl.etal *
+                }
+                {
+                  bbl.and
+                  space.word * t *
+                }
+              if$
+            }
+          if$
+        }
+        't
+      if$
+      nameptr #1 + 'nameptr :=
+      namesleft #1 - 'namesleft :=
+    }
+  while$
+  } if$
+}
+FUNCTION {format.names.ed}
+{
+  format.names
+}
+FUNCTION {format.key}
+{ empty$
+    { key field.or.null }
+    { "" }
+  if$
+}
+
+FUNCTION {format.authors}
+{ author "author" format.names
+}
+FUNCTION {get.bbl.editor}
+{ editor num.names$ #1 > 'bbl.editors 'bbl.editor if$ }
+
+FUNCTION {format.editors}
+{ editor "editor" format.names duplicate$ empty$ 'skip$
+    {
+      "," *
+      " " *
+      get.bbl.editor
+      *
+    }
+  if$
+}
+FUNCTION {format.isbn}
+{ isbn "isbn" bibinfo.check
+  duplicate$ empty$ 'skip$
+    {
+      new.block
+      "ISBN " swap$ *
+    }
+  if$
+}
+
+FUNCTION {format.note}
+{
+ note empty$
+    { "" }
+    { note #1 #1 substring$
+      duplicate$ "{" =
+        'skip$
+        { output.state mid.sentence =
+          { "l" }
+          { "u" }
+        if$
+        change.case$
+        }
+      if$
+      note #2 global.max$ substring$ * "note" bibinfo.check
+    }
+  if$
+}
+
+FUNCTION {format.title}
+{ title
+  "title" bibinfo.check
+}
+FUNCTION {format.full.names}
+{'s :=
+ "" 't :=
+  #1 'nameptr :=
+  s num.names$ 'numnames :=
+  numnames 'namesleft :=
+    { namesleft #0 > }
+    { s nameptr
+      "{vv~}{ll}" format.name$
+      't :=
+      nameptr #1 >
+        {
+          namesleft #1 >
+            { ", " * t * }
+            {
+              s nameptr "{ll}" format.name$ duplicate$ "others" =
+                { 't := }
+                { pop$ }
+              if$
+              t "others" =
+                {
+                  " " * bbl.etal *
+                }
+                {
+                  numnames #2 >
+                    { "," * }
+                    'skip$
+                  if$
+                  bbl.and
+                  space.word * t *
+                }
+              if$
+            }
+          if$
+        }
+        't
+      if$
+      nameptr #1 + 'nameptr :=
+      namesleft #1 - 'namesleft :=
+    }
+  while$
+}
+
+FUNCTION {author.editor.key.full}
+{ author empty$
+    { editor empty$
+        { key empty$
+            { cite$ #1 #3 substring$ }
+            'key
+          if$
+        }
+        { editor format.full.names }
+      if$
+    }
+    { author format.full.names }
+  if$
+}
+
+FUNCTION {author.key.full}
+{ author empty$
+    { key empty$
+         { cite$ #1 #3 substring$ }
+          'key
+      if$
+    }
+    { author format.full.names }
+  if$
+}
+
+FUNCTION {editor.key.full}
+{ editor empty$
+    { key empty$
+         { cite$ #1 #3 substring$ }
+          'key
+      if$
+    }
+    { editor format.full.names }
+  if$
+}
+
+FUNCTION {make.full.names}
+{ type$ "book" =
+  type$ "inbook" =
+  or
+    'author.editor.key.full
+    { type$ "proceedings" =
+        'editor.key.full
+        'author.key.full
+      if$
+    }
+  if$
+}
+
+FUNCTION {output.bibitem}
+{ newline$
+  "\bibitem[{" write$
+  label write$
+  ")" make.full.names duplicate$ short.list =
+     { pop$ }
+     { * }
+   if$
+  "}]{" * write$
+  cite$ write$
+  "}" write$
+  newline$
+  ""
+  before.all 'output.state :=
+}
+
+FUNCTION {n.dashify}
+{
+  't :=
+  ""
+    { t empty$ not }
+    { t #1 #1 substring$ "-" =
+        { t #1 #2 substring$ "--" = not
+            { "--" *
+              t #2 global.max$ substring$ 't :=
+            }
+            {   { t #1 #1 substring$ "-" = }
+                { "-" *
+                  t #2 global.max$ substring$ 't :=
+                }
+              while$
+            }
+          if$
+        }
+        { t #1 #1 substring$ *
+          t #2 global.max$ substring$ 't :=
+        }
+      if$
+    }
+  while$
+}
+
+FUNCTION {word.in}
+{ bbl.in capitalize
+  " " * }
+
+FUNCTION {format.date}
+{ year "year" bibinfo.check duplicate$ empty$
+    {
+      "empty year in " cite$ * "; set to ????" * warning$
+       pop$ "????"
+    }
+    'skip$
+  if$
+  extra.label *
+  before.all 'output.state :=
+  after.sentence 'output.state :=
+}
+FUNCTION {format.btitle}
+{ title "title" bibinfo.check
+  duplicate$ empty$ 'skip$
+    {
+      emphasize
+    }
+  if$
+}
+FUNCTION {either.or.check}
+{ empty$
+    'pop$
+    { "can't use both " swap$ * " fields in " * cite$ * warning$ }
+  if$
+}
+FUNCTION {format.bvolume}
+{ volume empty$
+    { "" }
+    { bbl.volume volume tie.or.space.prefix
+      "volume" bibinfo.check * *
+      series "series" bibinfo.check
+      duplicate$ empty$ 'pop$
+        { swap$ bbl.of space.word * swap$
+          emphasize * }
+      if$
+      "volume and number" number either.or.check
+    }
+  if$
+}
+FUNCTION {format.number.series}
+{ volume empty$
+    { number empty$
+        { series field.or.null }
+        { series empty$
+            { number "number" bibinfo.check }
+            { output.state mid.sentence =
+                { bbl.number }
+                { bbl.number capitalize }
+              if$
+              number tie.or.space.prefix "number" bibinfo.check * *
+              bbl.in space.word *
+              series "series" bibinfo.check *
+            }
+          if$
+        }
+      if$
+    }
+    { "" }
+  if$
+}
+
+FUNCTION {format.edition}
+{ edition duplicate$ empty$ 'skip$
+    {
+      output.state mid.sentence =
+        { "l" }
+        { "t" }
+      if$ change.case$
+      "edition" bibinfo.check
+      " " * bbl.edition *
+    }
+  if$
+}
+INTEGERS { multiresult }
+FUNCTION {multi.page.check}
+{ 't :=
+  #0 'multiresult :=
+    { multiresult not
+      t empty$ not
+      and
+    }
+    { t #1 #1 substring$
+      duplicate$ "-" =
+      swap$ duplicate$ "," =
+      swap$ "+" =
+      or or
+        { #1 'multiresult := }
+        { t #2 global.max$ substring$ 't := }
+      if$
+    }
+  while$
+  multiresult
+}
+FUNCTION {format.pages}
+{ pages duplicate$ empty$ 'skip$
+    { duplicate$ multi.page.check
+        {
+          n.dashify
+        }
+        {
+        }
+      if$
+      "pages" bibinfo.check
+    }
+  if$
+}
+FUNCTION {format.journal.pages}
+{ pages duplicate$ empty$ 'pop$
+    { swap$ duplicate$ empty$
+        { pop$ pop$ format.pages }
+        {
+          ": " *
+          swap$
+          n.dashify
+          "pages" bibinfo.check
+          *
+        }
+      if$
+    }
+  if$
+}
+FUNCTION {format.journal.eid}
+{ eid "eid" bibinfo.check
+  duplicate$ empty$ 'pop$
+    { swap$ duplicate$ empty$ 'skip$
+      {
+          ": " *
+      }
+      if$
+      swap$ *
+    }
+  if$
+}
+FUNCTION {format.vol.num.pages}
+{ volume field.or.null
+  duplicate$ empty$ 'skip$
+    {
+      "volume" bibinfo.check
+    }
+  if$
+  number "number" bibinfo.check duplicate$ empty$ 'skip$
+    {
+      swap$ duplicate$ empty$
+        { "there's a number but no volume in " cite$ * warning$ }
+        'skip$
+      if$
+      swap$
+      "(" swap$ * ")" *
+    }
+  if$ *
+  eid empty$
+    { format.journal.pages }
+    { format.journal.eid }
+  if$
+}
+
+FUNCTION {format.chapter.pages}
+{ chapter empty$
+    'format.pages
+    { type empty$
+        { bbl.chapter }
+        { type "l" change.case$
+          "type" bibinfo.check
+        }
+      if$
+      chapter tie.or.space.prefix
+      "chapter" bibinfo.check
+      * *
+      pages empty$
+        'skip$
+        { ", " * format.pages * }
+      if$
+    }
+  if$
+}
+
+FUNCTION {format.booktitle}
+{
+  booktitle "booktitle" bibinfo.check
+  emphasize
+}
+FUNCTION {format.in.ed.booktitle}
+{ format.booktitle duplicate$ empty$ 'skip$
+    {
+      editor "editor" format.names.ed duplicate$ empty$ 'pop$
+        {
+          "," *
+          " " *
+          get.bbl.editor
+          ", " *
+          * swap$
+          * }
+      if$
+      word.in swap$ *
+    }
+  if$
+}
+FUNCTION {format.thesis.type}
+{ type duplicate$ empty$
+    'pop$
+    { swap$ pop$
+      "t" change.case$ "type" bibinfo.check
+    }
+  if$
+}
+FUNCTION {format.tr.number}
+{ number "number" bibinfo.check
+  type duplicate$ empty$
+    { pop$ bbl.techrep }
+    'skip$
+  if$
+  "type" bibinfo.check
+  swap$ duplicate$ empty$
+    { pop$ "t" change.case$ }
+    { tie.or.space.prefix * * }
+  if$
+}
+FUNCTION {format.article.crossref}
+{
+  word.in
+  " \cite{" * crossref * "}" *
+}
+FUNCTION {format.book.crossref}
+{ volume duplicate$ empty$
+    { "empty volume in " cite$ * "'s crossref of " * crossref * warning$
+      pop$ word.in
+    }
+    { bbl.volume
+      capitalize
+      swap$ tie.or.space.prefix "volume" bibinfo.check * * bbl.of space.word *
+    }
+  if$
+  " \cite{" * crossref * "}" *
+}
+FUNCTION {format.incoll.inproc.crossref}
+{
+  word.in
+  " \cite{" * crossref * "}" *
+}
+FUNCTION {format.org.or.pub}
+{ 't :=
+  ""
+  address empty$ t empty$ and
+    'skip$
+    {
+      address "address" bibinfo.check *
+      t empty$
+        'skip$
+        { address empty$
+            'skip$
+            { ": " * }
+          if$
+          t *
+        }
+      if$
+    }
+  if$
+}
+FUNCTION {format.publisher.address}
+{ publisher "publisher" bibinfo.warn format.org.or.pub
+}
+
+FUNCTION {format.organization.address}
+{ organization "organization" bibinfo.check format.org.or.pub
+}
+
+FUNCTION {article}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  format.date "year" output.check
+  date.block
+  format.title "title" output.check
+  new.block
+  crossref missing$
+    {
+      journal
+      "journal" bibinfo.check
+      emphasize
+      "journal" output.check
+      format.vol.num.pages output
+    }
+    { format.article.crossref output.nonnull
+      format.pages output
+    }
+  if$
+  new.block
+  format.note output
+  fin.entry
+}
+FUNCTION {book}
+{ output.bibitem
+  author empty$
+    { format.editors "author and editor" output.check
+      editor format.key output
+    }
+    { format.authors output.nonnull
+      crossref missing$
+        { "author and editor" editor either.or.check }
+        'skip$
+      if$
+    }
+  if$
+  format.date "year" output.check
+  date.block
+  format.btitle "title" output.check
+  crossref missing$
+    { format.bvolume output
+      new.block
+      format.number.series output
+      new.sentence
+      format.publisher.address output
+    }
+    {
+      new.block
+      format.book.crossref output.nonnull
+    }
+  if$
+  format.edition output
+  format.isbn output
+  new.block
+  format.note output
+  fin.entry
+}
+FUNCTION {booklet}
+{ output.bibitem
+  format.authors output
+  author format.key output
+  format.date "year" output.check
+  date.block
+  format.title "title" output.check
+  new.block
+  howpublished "howpublished" bibinfo.check output
+  address "address" bibinfo.check output
+  format.isbn output
+  new.block
+  format.note output
+  fin.entry
+}
+
+FUNCTION {inbook}
+{ output.bibitem
+  author empty$
+    { format.editors "author and editor" output.check
+      editor format.key output
+    }
+    { format.authors output.nonnull
+      crossref missing$
+        { "author and editor" editor either.or.check }
+        'skip$
+      if$
+    }
+  if$
+  format.date "year" output.check
+  date.block
+  format.btitle "title" output.check
+  crossref missing$
+    {
+      format.bvolume output
+      format.chapter.pages "chapter and pages" output.check
+      new.block
+      format.number.series output
+      new.sentence
+      format.publisher.address output
+    }
+    {
+      format.chapter.pages "chapter and pages" output.check
+      new.block
+      format.book.crossref output.nonnull
+    }
+  if$
+  format.edition output
+  crossref missing$
+    { format.isbn output }
+    'skip$
+  if$
+  new.block
+  format.note output
+  fin.entry
+}
+
+FUNCTION {incollection}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  format.date "year" output.check
+  date.block
+  format.title "title" output.check
+  new.block
+  crossref missing$
+    { format.in.ed.booktitle "booktitle" output.check
+      format.bvolume output
+      format.number.series output
+      format.chapter.pages output
+      new.sentence
+      format.publisher.address output
+      format.edition output
+      format.isbn output
+    }
+    { format.incoll.inproc.crossref output.nonnull
+      format.chapter.pages output
+    }
+  if$
+  new.block
+  format.note output
+  fin.entry
+}
+FUNCTION {inproceedings}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  format.date "year" output.check
+  date.block
+  format.title "title" output.check
+  new.block
+  crossref missing$
+    { format.in.ed.booktitle "booktitle" output.check
+      format.bvolume output
+      format.number.series output
+      format.pages output
+      new.sentence
+      publisher empty$
+        { format.organization.address output }
+        { organization "organization" bibinfo.check output
+          format.publisher.address output
+        }
+      if$
+      format.isbn output
+    }
+    { format.incoll.inproc.crossref output.nonnull
+      format.pages output
+    }
+  if$
+  new.block
+  format.note output
+  fin.entry
+}
+FUNCTION {conference} { inproceedings }
+FUNCTION {manual}
+{ output.bibitem
+  format.authors output
+  author format.key output
+  format.date "year" output.check
+  date.block
+  format.btitle "title" output.check
+  organization address new.block.checkb
+  organization "organization" bibinfo.check output
+  address "address" bibinfo.check output
+  format.edition output
+  new.block
+  format.note output
+  fin.entry
+}
+
+FUNCTION {mastersthesis}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  format.date "year" output.check
+  date.block
+  format.btitle
+  "title" output.check
+  new.block
+  bbl.mthesis format.thesis.type output.nonnull
+  school "school" bibinfo.warn output
+  address "address" bibinfo.check output
+  new.block
+  format.note output
+  fin.entry
+}
+
+FUNCTION {misc}
+{ output.bibitem
+  format.authors output
+  author format.key output
+  format.date "year" output.check
+  date.block
+  format.title output
+  new.block
+  howpublished "howpublished" bibinfo.check output
+  new.block
+  format.note output
+  format.eprint output
+  fin.entry
+}
+FUNCTION {phdthesis}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  format.date "year" output.check
+  date.block
+  format.btitle
+  "title" output.check
+  new.block
+  bbl.phdthesis format.thesis.type output.nonnull
+  school "school" bibinfo.warn output
+  address "address" bibinfo.check output
+  new.block
+  format.note output
+  fin.entry
+}
+
+FUNCTION {proceedings}
+{ output.bibitem
+  format.editors output
+  editor format.key output
+  format.date "year" output.check
+  date.block
+  format.btitle "title" output.check
+  format.bvolume output
+  format.number.series output
+  new.sentence
+  publisher empty$
+    { format.organization.address output }
+    { organization "organization" bibinfo.check output
+      format.publisher.address output
+    }
+  if$
+  format.isbn output
+  new.block
+  format.note output
+  fin.entry
+}
+
+FUNCTION {techreport}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  format.date "year" output.check
+  date.block
+  format.title
+  "title" output.check
+  new.block
+  format.tr.number output.nonnull
+  institution "institution" bibinfo.warn output
+  address "address" bibinfo.check output
+  new.block
+  format.note output
+  fin.entry
+}
+
+FUNCTION {unpublished}
+{ output.bibitem
+  format.authors "author" output.check
+  author format.key output
+  format.date "year" output.check
+  date.block
+  format.title "title" output.check
+  new.block
+  format.note "note" output.check
+  fin.entry
+}
+
+FUNCTION {default.type} { misc }
+READ
+FUNCTION {sortify}
+{ purify$
+  "l" change.case$
+}
+INTEGERS { len }
+FUNCTION {chop.word}
+{ 's :=
+  'len :=
+  s #1 len substring$ =
+    { s len #1 + global.max$ substring$ }
+    's
+  if$
+}
+FUNCTION {format.lab.names}
+{'s :=
+ "" 't :=
+  #1 'nameptr :=
+  s num.names$ 'numnames :=
+  numnames 'namesleft :=
+    { namesleft #0 > }
+    { s nameptr
+      "{vv~}{ll}" format.name$
+      't :=
+      nameptr #1 >
+        {
+          nameptr #2 =
+          numnames #3 > and
+            { "others" 't :=
+              #1 'namesleft := }
+            'skip$
+          if$
+          namesleft #1 >
+            { ", " * t * }
+            {
+              s nameptr "{ll}" format.name$ duplicate$ "others" =
+                { 't := }
+                { pop$ }
+              if$
+              t "others" =
+                {
+                  " " * bbl.etal *
+                }
+                {
+                  numnames #2 >
+                    { "," * }
+                    'skip$
+                  if$
+                  bbl.and
+                  space.word * t *
+                }
+              if$
+            }
+          if$
+        }
+        't
+      if$
+      nameptr #1 + 'nameptr :=
+      namesleft #1 - 'namesleft :=
+    }
+  while$
+}
+
+FUNCTION {author.key.label}
+{ author empty$
+    { key empty$
+        { cite$ #1 #3 substring$ }
+        'key
+      if$
+    }
+    { author format.lab.names }
+  if$
+}
+
+FUNCTION {author.editor.key.label}
+{ author empty$
+    { editor empty$
+        { key empty$
+            { cite$ #1 #3 substring$ }
+            'key
+          if$
+        }
+        { editor format.lab.names }
+      if$
+    }
+    { author format.lab.names }
+  if$
+}
+
+FUNCTION {editor.key.label}
+{ editor empty$
+    { key empty$
+        { cite$ #1 #3 substring$ }
+        'key
+      if$
+    }
+    { editor format.lab.names }
+  if$
+}
+
+FUNCTION {calc.short.authors}
+{ type$ "book" =
+  type$ "inbook" =
+  or
+    'author.editor.key.label
+    { type$ "proceedings" =
+        'editor.key.label
+        'author.key.label
+      if$
+    }
+  if$
+  'short.list :=
+}
+
+FUNCTION {calc.label}
+{ calc.short.authors
+  short.list
+  "("
+  *
+  year duplicate$ empty$
+  short.list key field.or.null = or
+     { pop$ "" }
+     'skip$
+  if$
+  *
+  'label :=
+}
+
+FUNCTION {sort.format.names}
+{ 's :=
+  #1 'nameptr :=
+  ""
+  s num.names$ 'numnames :=
+  numnames 'namesleft :=
+    { namesleft #0 > }
+    { s nameptr
+      "{vv{ } }{ll{ }}{  f{ }}{  jj{ }}"
+      format.name$ 't :=
+      nameptr #1 >
+        {
+          "   "  *
+          namesleft #1 = t "others" = and
+            { "zzzzz" 't := }
+            'skip$
+          if$
+          t sortify *
+        }
+        { t sortify * }
+      if$
+      nameptr #1 + 'nameptr :=
+      namesleft #1 - 'namesleft :=
+    }
+  while$
+}
+
+FUNCTION {sort.format.title}
+{ 't :=
+  "A " #2
+    "An " #3
+      "The " #4 t chop.word
+    chop.word
+  chop.word
+  sortify
+  #1 global.max$ substring$
+}
+FUNCTION {author.sort}
+{ author empty$
+    { key empty$
+        { "to sort, need author or key in " cite$ * warning$
+          ""
+        }
+        { key sortify }
+      if$
+    }
+    { author sort.format.names }
+  if$
+}
+FUNCTION {author.editor.sort}
+{ author empty$
+    { editor empty$
+        { key empty$
+            { "to sort, need author, editor, or key in " cite$ * warning$
+              ""
+            }
+            { key sortify }
+          if$
+        }
+        { editor sort.format.names }
+      if$
+    }
+    { author sort.format.names }
+  if$
+}
+FUNCTION {editor.sort}
+{ editor empty$
+    { key empty$
+        { "to sort, need editor or key in " cite$ * warning$
+          ""
+        }
+        { key sortify }
+      if$
+    }
+    { editor sort.format.names }
+  if$
+}
+FUNCTION {presort}
+{ calc.label
+  label sortify
+  "    "
+  *
+  type$ "book" =
+  type$ "inbook" =
+  or
+    'author.editor.sort
+    { type$ "proceedings" =
+        'editor.sort
+        'author.sort
+      if$
+    }
+  if$
+  #1 entry.max$ substring$
+  'sort.label :=
+  sort.label
+  *
+  "    "
+  *
+  title field.or.null
+  sort.format.title
+  *
+  #1 entry.max$ substring$
+  'sort.key$ :=
+}
+
+ITERATE {presort}
+SORT
+STRINGS { last.label next.extra }
+INTEGERS { last.extra.num last.extra.num.extended last.extra.num.blank number.label }
+FUNCTION {initialize.extra.label.stuff}
+{ #0 int.to.chr$ 'last.label :=
+  "" 'next.extra :=
+  #0 'last.extra.num :=
+  "a" chr.to.int$ #1 - 'last.extra.num.blank :=
+  last.extra.num.blank 'last.extra.num.extended :=
+  #0 'number.label :=
+}
+FUNCTION {forward.pass}
+{ last.label label =
+    { last.extra.num #1 + 'last.extra.num :=
+      last.extra.num "z" chr.to.int$ >
+       { "a" chr.to.int$ 'last.extra.num :=
+         last.extra.num.extended #1 + 'last.extra.num.extended :=
+       }
+       'skip$
+      if$
+      last.extra.num.extended last.extra.num.blank >
+        { last.extra.num.extended int.to.chr$
+          last.extra.num int.to.chr$
+          * 'extra.label := }
+        { last.extra.num int.to.chr$ 'extra.label := }
+      if$
+    }
+    { "a" chr.to.int$ 'last.extra.num :=
+      "" 'extra.label :=
+      label 'last.label :=
+    }
+  if$
+  number.label #1 + 'number.label :=
+}
+FUNCTION {reverse.pass}
+{ next.extra "b" =
+    { "a" 'extra.label := }
+    'skip$
+  if$
+  extra.label 'next.extra :=
+  extra.label
+  duplicate$ empty$
+    'skip$
+    { "{\natexlab{" swap$ * "}}" * }
+  if$
+  'extra.label :=
+  label extra.label * 'label :=
+}
+EXECUTE {initialize.extra.label.stuff}
+ITERATE {forward.pass}
+REVERSE {reverse.pass}
+FUNCTION {bib.sort.order}
+{ sort.label
+  "    "
+  *
+  year field.or.null sortify
+  *
+  "    "
+  *
+  title field.or.null
+  sort.format.title
+  *
+  #1 entry.max$ substring$
+  'sort.key$ :=
+}
+ITERATE {bib.sort.order}
+SORT
+FUNCTION {begin.bib}
+{ preamble$ empty$
+    'skip$
+    { preamble$ write$ newline$ }
+  if$
+  "\begin{thebibliography}{" number.label int.to.str$ * "}" *
+  write$ newline$
+  "\providecommand{\natexlab}[1]{#1}"
+  write$ newline$
+}
+EXECUTE {begin.bib}
+EXECUTE {init.state.consts}
+ITERATE {call.type$}
+FUNCTION {end.bib}
+{ newline$
+  "\end{thebibliography}" write$ newline$
+}
+EXECUTE {end.bib}
+%% End of customized bst file
+%%
+%% End of file `aaai22.bst'.
diff --git a/NEW_aaai/aaai25.sty b/NEW_aaai/aaai25.sty
new file mode 100644
index 0000000..4a20e41
--- /dev/null
+++ b/NEW_aaai/aaai25.sty
@@ -0,0 +1,315 @@
+\NeedsTeXFormat{LaTeX2e}%
+\ProvidesPackage{aaai25}[2025/05/08 AAAI 2025 Submission format]%
+\def\year{2025}%
+\typeout{Conference Style for AAAI for LaTeX 2e -- version for submission}%
+%
+\def\copyright@on{T}
+\def\showauthors@on{T}
+\def\nocopyright{\gdef\copyright@on{}} % Copyright notice is required for camera-ready only.
+\DeclareOption{submission}{%
+  \gdef\copyright@on{}%
+  \gdef\showauthors@on{}%
+  \long\gdef\pdfinfo #1{\relax}%
+}%
+\DeclareOption{draft}{%
+  \gdef\copyright@on{}%
+}%
+\ProcessOptions\relax%
+% WARNING: IF YOU ARE USING THIS STYLE SHEET FOR AN AAAI PUBLICATION, YOU
+% MAY NOT MODIFY IT FOR ANY REASON. MODIFICATIONS (IN YOUR SOURCE
+% OR IN THIS STYLE SHEET WILL RESULT IN REJECTION OF YOUR PAPER).
+%
+% WARNING: This style is NOT guaranteed to work. It is provided in the
+% hope that it might make the preparation of papers easier, but this style
+% file is provided "as is" without warranty of any kind, either express or
+% implied, including but not limited to the implied warranties of
+% merchantability, fitness for a particular purpose, or noninfringement.
+% You use this style file at your own risk. Standard disclaimers apply.
+% There are undoubtably bugs in this style. If you would like to submit
+% bug fixes, improvements, etc. please let us know. Please use the contact form
+% at www.aaai.org.
+%
+% Do not use this file unless you are an experienced LaTeX user.
+%
+% PHYSICAL PAGE LAYOUT
+\setlength\topmargin{-0.25in} \setlength\oddsidemargin{-0.25in}
+\setlength\textheight{9.0in} \setlength\textwidth{7.0in}
+\setlength\columnsep{0.375in} \newlength\titlebox \setlength\titlebox{2.25in}
+\setlength\headheight{0pt}  \setlength\headsep{0pt}
+%\setlength\footheight{0pt}  \setlength\footskip{0pt}
+\thispagestyle{empty} \pagestyle{empty}
+\flushbottom \twocolumn \sloppy
+% We're never going to need a table of contents, so just flush it to
+% save space --- suggested by drstrip@sandia-2
+\def\addcontentsline#1#2#3{}
+% gf: PRINT COPYRIGHT NOTICE
+\def\copyright@year{\number\year}
+\def\copyright@text{Copyright \copyright\space \copyright@year,
+Association for the Advancement of Artificial Intelligence (www.aaai.org).
+All rights reserved.}
+\def\copyrighttext#1{\gdef\copyright@on{T}\gdef\copyright@text{#1}}
+\def\copyrightyear#1{\gdef\copyright@on{T}\gdef\copyright@year{#1}}
+% gf: End changes for copyright notice (used in \maketitle, below)
+% Title stuff, taken from deproc.
+%
+\def\maketitle{%
+  \par%
+  \begingroup % to make the footnote style local to the title
+    \def\thefootnote{\fnsymbol{footnote}}
+    \twocolumn[\@maketitle] \@thanks%
+  \endgroup%
+  % Insert copyright slug unless turned off
+  \if T\copyright@on\insert\footins{\noindent\footnotesize\copyright@text}\fi%
+  %
+  \setcounter{footnote}{0}%
+  \let\maketitle\relax%
+  \let\@maketitle\relax%
+  \gdef\@thanks{}%
+  \gdef\@author{}%
+  \gdef\@title{}%
+  \let\thanks\relax%
+}%
+\long\gdef\affiliations #1{ \def \affiliations_{\if T\showauthors@on#1\fi}}%
+%
+\def\@maketitle{%
+  \def\theauthors{\if T\showauthors@on\@author\else Anonymous submission\fi}
+  \newcounter{eqfn}\setcounter{eqfn}{0}%
+  \newsavebox{\titlearea}
+  \sbox{\titlearea}{
+    \let\footnote\relax\let\thanks\relax%
+    \setcounter{footnote}{0}%
+    \def\equalcontrib{%
+      \ifnum\value{eqfn}=0%
+        \footnote{These authors contributed equally.}%
+        \setcounter{eqfn}{\value{footnote}}%
+      \else%
+        \footnotemark[\value{eqfn}]%
+      \fi%
+    }%
+    \vbox{%
+      \hsize\textwidth%
+      \linewidth\hsize%
+      \vskip 0.625in minus 0.125in%
+      \centering%
+      {\LARGE\bf \@title \par}%
+      \vskip 0.1in plus 0.5fil minus 0.05in%
+      {\Large{\textbf{\theauthors\ifhmode\\\fi}}}%
+      \vskip .2em plus 0.25fil%
+      {\normalsize \affiliations_\ifhmode\\\fi}%
+      \vskip .5em plus 2fil%
+    }%
+  }%
+%
+  \newlength\actualheight%
+  \settoheight{\actualheight}{\usebox{\titlearea}}%
+  \ifdim\actualheight>\titlebox%
+    \setlength{\titlebox}{\actualheight}%
+  \fi%
+%
+  \vbox to \titlebox {%
+    \let\footnote\thanks\relax%
+    \setcounter{footnote}{0}%
+    \def\equalcontrib{%
+      \ifnum\value{eqfn}=0%
+        \footnote{These authors contributed equally.}%
+        \setcounter{eqfn}{\value{footnote}}%
+      \else%
+        \footnotemark[\value{eqfn}]%
+      \fi%
+    }%
+    \hsize\textwidth%
+    \linewidth\hsize%
+    \vskip 0.625in minus 0.125in%
+    \centering%
+    {\LARGE\bf \@title \par}%
+    \vskip 0.1in plus 0.5fil minus 0.05in%
+    {\Large{\textbf{\theauthors\ifhmode\\\fi}}}%
+    \vskip .2em plus 0.25fil%
+    {\normalsize \affiliations_\ifhmode\\\fi}%
+    \vskip .5em plus 2fil%
+  }%
+}%
+%
+\renewenvironment{abstract}{%
+  \centerline{\bf Abstract}%
+  \vspace{0.5ex}%
+  \setlength{\leftmargini}{10pt}%
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diff --git a/NEW_aaai/anonymous-submission-latex-2025.bbl b/NEW_aaai/anonymous-submission-latex-2025.bbl
new file mode 100644
index 0000000..d058220
--- /dev/null
+++ b/NEW_aaai/anonymous-submission-latex-2025.bbl
@@ -0,0 +1,134 @@
+\begin{thebibliography}{26}
+\providecommand{\natexlab}[1]{#1}
+
+\bibitem[{Baird et~al.(1995)}]{baird1995residual}
+Baird, L.; et~al. 1995.
+\newblock Residual algorithms: Reinforcement learning with function approximation.
+\newblock In \emph{Proc. 12th Int. Conf. Mach. Learn.}, 30--37.
+
+\bibitem[{Bas-Serrano et~al.(2021)Bas-Serrano, Curi, Krause, and Neu}]{basserrano2021logistic}
+Bas-Serrano, J.; Curi, S.; Krause, A.; and Neu, G. 2021.
+\newblock Logistic Q-Learning.
+\newblock In \emph{International Conference on Artificial Intelligence and Statistics}, 3610--3618.
+
+\bibitem[{Borkar(1997)}]{borkar1997stochastic}
+Borkar, V.~S. 1997.
+\newblock Stochastic approximation with two time scales.
+\newblock \emph{Syst. \& Control Letters}, 29(5): 291--294.
+
+\bibitem[{Borkar and Meyn(2000)}]{borkar2000ode}
+Borkar, V.~S.; and Meyn, S.~P. 2000.
+\newblock The ODE method for convergence of stochastic approximation and reinforcement learning.
+\newblock \emph{SIAM J. Control Optim.}, 38(2): 447--469.
+
+\bibitem[{Chen et~al.(2023)Chen, Ma, Li, Yang, Yang, and Gao}]{chen2023modified}
+Chen, X.; Ma, X.; Li, Y.; Yang, G.; Yang, S.; and Gao, Y. 2023.
+\newblock Modified Retrace for Off-Policy Temporal Difference Learning.
+\newblock In \emph{Uncertainty in Artificial Intelligence}, 303--312. PMLR.
+
+\bibitem[{Devlin and Kudenko(2012)}]{devlin2012dynamic}
+Devlin, S.; and Kudenko, D. 2012.
+\newblock Dynamic potential-based reward shaping.
+\newblock In \emph{Proc. 11th Int. Conf. Autonomous Agents and Multiagent Systems}, 433--440.
+
+\bibitem[{Feng, Li, and Liu(2019)}]{feng2019kernel}
+Feng, Y.; Li, L.; and Liu, Q. 2019.
+\newblock A kernel loss for solving the Bellman equation.
+\newblock In \emph{Advances in Neural Information Processing Systems}, 15430--15441.
+
+\bibitem[{Givchi and Palhang(2015)}]{givchi2015quasi}
+Givchi, A.; and Palhang, M. 2015.
+\newblock Quasi newton temporal difference learning.
+\newblock In \emph{Asian Conference on Machine Learning}, 159--172.
+
+\bibitem[{Hackman(2012)}]{hackman2012faster}
+Hackman, L. 2012.
+\newblock \emph{Faster Gradient-TD Algorithms}.
+\newblock Ph.D. thesis, University of Alberta.
+
+\bibitem[{Hallak et~al.(2016)Hallak, Tamar, Munos, and Mannor}]{hallak2016generalized}
+Hallak, A.; Tamar, A.; Munos, R.; and Mannor, S. 2016.
+\newblock Generalized emphatic temporal difference learning: bias-variance analysis.
+\newblock In \emph{Proceedings of the 30th AAAI Conference on Artificial Intelligence}, 1631--1637.
+
+\bibitem[{Hirsch(1989)}]{hirsch1989convergent}
+Hirsch, M.~W. 1989.
+\newblock Convergent activation dynamics in continuous time networks.
+\newblock \emph{Neural Netw.}, 2(5): 331--349.
+
+\bibitem[{Johnson and Zhang(2013)}]{johnson2013accelerating}
+Johnson, R.; and Zhang, T. 2013.
+\newblock Accelerating stochastic gradient descent using predictive variance reduction.
+\newblock In \emph{Advances in Neural Information Processing Systems}, 315--323.
+
+\bibitem[{Korda and La(2015)}]{korda2015td}
+Korda, N.; and La, P. 2015.
+\newblock On TD (0) with function approximation: Concentration bounds and a centered variant with exponential convergence.
+\newblock In \emph{International conference on machine learning}, 626--634. PMLR.
+
+\bibitem[{Liu et~al.(2018)Liu, Gemp, Ghavamzadeh, Liu, Mahadevan, and Petrik}]{liu2018proximal}
+Liu, B.; Gemp, I.; Ghavamzadeh, M.; Liu, J.; Mahadevan, S.; and Petrik, M. 2018.
+\newblock Proximal gradient temporal difference learning: Stable reinforcement learning with polynomial sample complexity.
+\newblock \emph{Journal of Artificial Intelligence Research}, 63: 461--494.
+
+\bibitem[{Liu et~al.(2015)Liu, Liu, Ghavamzadeh, Mahadevan, and Petrik}]{liu2015finite}
+Liu, B.; Liu, J.; Ghavamzadeh, M.; Mahadevan, S.; and Petrik, M. 2015.
+\newblock Finite-sample analysis of proximal gradient TD algorithms.
+\newblock In \emph{Proceedings of the 21st Conference on Uncertainty in Artificial Intelligence}, 504--513.
+
+\bibitem[{Liu et~al.(2016)Liu, Liu, Ghavamzadeh, Mahadevan, and Petrik}]{liu2016proximal}
+Liu, B.; Liu, J.; Ghavamzadeh, M.; Mahadevan, S.; and Petrik, M. 2016.
+\newblock Proximal Gradient Temporal Difference Learning Algorithms.
+\newblock In \emph{Proceedings of the International Joint Conference on Artificial Intelligence}, 4195--4199.
+
+\bibitem[{Ng, Harada, and Russell(1999)}]{ng1999policy}
+Ng, A.~Y.; Harada, D.; and Russell, S. 1999.
+\newblock Policy invariance under reward transformations: Theory and application to reward shaping.
+\newblock In \emph{Proc. 16th Int. Conf. Mach. Learn.}, 278--287.
+
+\bibitem[{Pan, White, and White(2017)}]{pan2017accelerated}
+Pan, Y.; White, A.; and White, M. 2017.
+\newblock Accelerated gradient temporal difference learning.
+\newblock In \emph{Proceedings of the 21st AAAI Conference on Artificial Intelligence}, 2464--2470.
+
+\bibitem[{Sutton et~al.(2009)Sutton, Maei, Precup, Bhatnagar, Silver, Szepesv{\'a}ri, and Wiewiora}]{sutton2009fast}
+Sutton, R.; Maei, H.; Precup, D.; Bhatnagar, S.; Silver, D.; Szepesv{\'a}ri, C.; and Wiewiora, E. 2009.
+\newblock Fast gradient-descent methods for temporal-difference learning with linear function approximation.
+\newblock In \emph{Proc. 26th Int. Conf. Mach. Learn.}, 993--1000.
+
+\bibitem[{Sutton(1988)}]{sutton1988learning}
+Sutton, R.~S. 1988.
+\newblock Learning to predict by the methods of temporal differences.
+\newblock \emph{Machine learning}, 3(1): 9--44.
+
+\bibitem[{Sutton and Barto(2018)}]{Sutton2018book}
+Sutton, R.~S.; and Barto, A.~G. 2018.
+\newblock \emph{Reinforcement Learning: An Introduction}.
+\newblock The MIT Press, second edition.
+
+\bibitem[{Sutton, Maei, and Szepesv{\'a}ri(2008)}]{sutton2008convergent}
+Sutton, R.~S.; Maei, H.~R.; and Szepesv{\'a}ri, C. 2008.
+\newblock A Convergent $ O (n) $ Temporal-difference Algorithm for Off-policy Learning with Linear Function Approximation.
+\newblock In \emph{Advances in Neural Information Processing Systems}, 1609--1616. Cambridge, MA: MIT Press.
+
+\bibitem[{Sutton, Mahmood, and White(2016)}]{sutton2016emphatic}
+Sutton, R.~S.; Mahmood, A.~R.; and White, M. 2016.
+\newblock An emphatic approach to the problem of off-policy temporal-difference learning.
+\newblock \emph{The Journal of Machine Learning Research}, 17(1): 2603--2631.
+
+\bibitem[{Tsitsiklis and Van~Roy(1997)}]{tsitsiklis1997analysis}
+Tsitsiklis, J.~N.; and Van~Roy, B. 1997.
+\newblock Analysis of temporal-diffference learning with function approximation.
+\newblock In \emph{Advances in Neural Information Processing Systems}, 1075--1081.
+
+\bibitem[{Xu et~al.(2019)Xu, Wang, Zhou, and Liang}]{xu2019reanalysis}
+Xu, T.; Wang, Z.; Zhou, Y.; and Liang, Y. 2019.
+\newblock Reanalysis of Variance Reduced Temporal Difference Learning.
+\newblock In \emph{International Conference on Learning Representations}.
+
+\bibitem[{Zhang and Whiteson(2022)}]{zhang2022truncated}
+Zhang, S.; and Whiteson, S. 2022.
+\newblock Truncated emphatic temporal difference methods for prediction and control.
+\newblock \emph{The Journal of Machine Learning Research}, 23(1): 6859--6917.
+
+\end{thebibliography}
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+% Title
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+% Your title must be in mixed case, not sentence case.
+% That means all verbs (including short verbs like be, is, using,and go),
+% nouns, adverbs, adjectives should be capitalized, including both words in hyphenated terms, while
+% articles, conjunctions, and prepositions are lower case unless they
+% directly follow a colon or long dash
+\title{A Variance Minimization Approach to  Off-policy Temporal-Difference Learning}
+\author{
+    %Authors
+    % All authors must be in the same font size and format.
+    Written by AAAI Press Staff\textsuperscript{\rm 1}\thanks{With help from the AAAI Publications Committee.}\\
+    AAAI Style Contributions by Pater Patel Schneider,
+    Sunil Issar,\\
+    J. Scott Penberthy,
+    George Ferguson,
+    Hans Guesgen,
+    Francisco Cruz\equalcontrib,
+    Marc Pujol-Gonzalez\equalcontrib
+}
+\affiliations{
+    %Afiliations
+    \textsuperscript{\rm 1}Association for the Advancement of Artificial Intelligence\\
+    % If you have multiple authors and multiple affiliations
+    % use superscripts in text and roman font to identify them.
+    % For example,
+
+    % Sunil Issar\textsuperscript{\rm 2},
+    % J. Scott Penberthy\textsuperscript{\rm 3},
+    % George Ferguson\textsuperscript{\rm 4},
+    % Hans Guesgen\textsuperscript{\rm 5}
+    % Note that the comma should be placed after the superscript
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+    1101 Pennsylvania Ave, NW Suite 300\\
+    Washington, DC 20004 USA\\
+    % email address must be in roman text type, not monospace or sans serif
+    proceedings-questions@aaai.org
+%
+% See more examples next
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+
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+\author {
+    Author Name
+}
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+    Affiliation\\
+    Affiliation Line 2\\
+    name@example.com
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+\fi
+
+\iffalse
+%Example, Multiple Authors, ->> remove \iffalse,\fi and place them surrounding AAAI title to use it
+\title{My Publication Title --- Multiple Authors}
+\author {
+    % Authors
+    First Author Name\textsuperscript{\rm 1},
+    Second Author Name\textsuperscript{\rm 2},
+    Third Author Name\textsuperscript{\rm 1}
+}
+\affiliations {
+    % Affiliations
+    \textsuperscript{\rm 1}Affiliation 1\\
+    \textsuperscript{\rm 2}Affiliation 2\\
+    firstAuthor@affiliation1.com, secondAuthor@affilation2.com, thirdAuthor@affiliation1.com
+}
+\fi
+
+
+% REMOVE THIS: bibentry
+% This is only needed to show inline citations in the guidelines document. You should not need it and can safely delete it.
+\usepackage{bibentry}
+% END REMOVE bibentry
+
+\begin{document}
+\setcounter{theorem}{0}
+\maketitle
+% \setcounter{theorem}{0}
+\begin{abstract}
+    In this paper, we introduce the concept of improving the performance of parametric 
+    Temporal-Difference (TD) learning algorithms by the Variance Minimization (VM) parameter, $\omega$, 
+    which is dynamically updated at each time step. Specifically, we incorporate the VM parameter into off-policy linear algorithms such as TDC and ETD, resulting in the 
+    Variance Minimization TDC (VMTDC) algorithm and the Variance Minimization ETD (VMETD) algorithm. In the two-state counterexample, 
+    we analyze 
+    the convergence speed of these algorithms by calculating the minimum eigenvalue of the key 
+    matrices and find that the VMTDC algorithm converges faster than TDC, while VMETD is more stable in convergence than ETD
+     through the 
+    experiment.In controlled experiments, the VM algorithms demonstrate 
+    superior performance.
+\end{abstract}
+
+% Uncomment the following to link to your code, datasets, an extended version or similar.
+%
+% \begin{links}
+%     \link{Code}{https://aaai.org/example/code}
+%     \link{Datasets}{https://aaai.org/example/datasets}
+%     \link{Extended version}{https://aaai.org/example/extended-version}
+% \end{links}
+
+\input{main/introduction.tex}
+\input{main/preliminaries.tex}
+\input{main/motivation.tex}
+\input{main/theory.tex}
+\input{main/experiment.tex}
+% \input{main/relatedwork.tex}
+\input{main/conclusion.tex}
+
+\bibliography{aaai25}
+
+\end{document}
diff --git a/NEW_aaai/figure1.pdf b/NEW_aaai/figure1.pdf
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diff --git a/NEW_aaai/figure2.pdf b/NEW_aaai/figure2.pdf
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diff --git a/NEW_aaai/main/conclusion.tex b/NEW_aaai/main/conclusion.tex
new file mode 100644
index 0000000..a5ff5d1
--- /dev/null
+++ b/NEW_aaai/main/conclusion.tex
@@ -0,0 +1,12 @@
+\section{Conclusion and Future Work}
+Value-based reinforcement learning typically aims 
+to minimize error as an optimization objective. 
+As an alternation, this study proposes new objective 
+functions: VBE and VPBE, and derives many variance minimization algorithms, including VMTD, 
+VMTDC and VMETD. 
+All algorithms demonstrated superior performance in policy 
+evaluation and control experiments.
+Future work may include, but are not limited
+to, (1) analysis of the convergence rate of VMTDC and VMETD. 
+(2) extensions of VBE and VPBE to multi-step returns. 
+(3) extensions to nonlinear approximations, such as neural networks. 
\ No newline at end of file
diff --git a/NEW_aaai/main/experiment.tex b/NEW_aaai/main/experiment.tex
new file mode 100644
index 0000000..10bff8a
--- /dev/null
+++ b/NEW_aaai/main/experiment.tex
@@ -0,0 +1,35 @@
+\section{Experimental Studies}
+This section assesses algorithm performance through experiments, 
+which are divided into policy evaluation experiments and control experiments.
+The control algorithms for TDC, ETD, VMTDC, and VMETD are named GQ, EQ, VMGQ, and VMEQ, respectively.
+The evaluation experimental environments are the 2-state and 7-state counterexample. 
+The control experimental environments are Maze, CliffWalking-v0, MountainCar-v0, and Acrobot-v1.
+For specific experimental parameters, please refer to the appendix.
+
+For the evaluation experiment, the experimental results 
+align with our previous analysis. In the 2-state counterexample 
+environment, the TDC algorithm has the smallest minimum 
+eigenvalue of the key matrix, resulting in the slowest 
+convergence speed. In contrast, the minimum eigenvalue 
+of VMTDC is larger, leading to faster convergence. 
+Although VMETD's minimum eigenvalue is larger than ETD's, 
+causing VMETD to converge more slowly than ETD in the 
+2-state counterexample, the standard deviation (shaded area) 
+of VMETD is smaller than that of ETD, indicating that VMETD 
+converges more smoothly. In the 7-state counterexample 
+environment, VMTDC converges faster than TDC and both VMETD and ETD are diverge.
+
+For the control experiments, the results for the maze and 
+cliff walking environments are similar: VMGQ 
+outperforms GQ, EQ outperforms VMGQ, and VMEQ performs 
+the best. In the mountain car and Acrobot experiments, 
+VMGQ and VMEQ show comparable performance, both outperforming 
+GQ and EQ. In summary, for control experiments, VM algorithms 
+outperform non-VM algorithms.
+
+In summary, the performance of VMSarsa, 
+VMQ, and VMGQ(0) is better than that of other algorithms. 
+In the Cliff Walking environment, 
+the performance of VMGQ(0) is slightly better than that of 
+VMSarsa and VMQ. In the other three experimental environments, 
+the performances of VMSarsa, VMQ, and VMGQ(0) are close.
\ No newline at end of file
diff --git a/NEW_aaai/main/introduction.tex b/NEW_aaai/main/introduction.tex
new file mode 100644
index 0000000..d70882b
--- /dev/null
+++ b/NEW_aaai/main/introduction.tex
@@ -0,0 +1,93 @@
+\section{Introduction}
+\label{introduction}
+Reinforcement learning can be mainly divided into two
+categories: value-based reinforcement learning
+and policy gradient-based reinforcement learning. This
+paper focuses on temporal difference learning based on
+linear approximated valued functions. Its research is
+usually divided into two steps: the first step is to establish the convergence of the algorithm, and the second
+step is to accelerate the algorithm.
+
+
+In terms of stability, \citet{sutton1988learning} established the
+ convergence of on-policy TD(0), and \citet{tsitsiklis1997analysis}
+ established the convergence of on-policy TD($\lambda$).
+ However, ``The deadly triad'' consisting of off-policy learning, 
+ bootstrapping, and function approximation makes 
+ the stability  a difficult problem \citep{Sutton2018book}.
+ To solve this problem, convergent off-policy temporal difference
+  learning algorithms are proposed, e.g., BR \cite{baird1995residual},
+ GTD \cite{sutton2008convergent},  GTD2 and TDC \cite{sutton2009fast},
+  ETD \cite{sutton2016emphatic}, and MRetrace \cite{chen2023modified}.
+
+In terms of acceleration, \citet{hackman2012faster} 
+proposed Hybrid TD algorithm with on-policy matrix.
+\citet{liu2015finite,liu2016proximal,liu2018proximal} proposed
+true stochastic algorithms, i.e., GTD-MP and GTD2-MP, from 
+a convex-concave saddle-point formulation.
+Second-order  methods are used to accelerate TD learning,
+e.g.,  Quasi Newton TD \cite{givchi2015quasi} and 
+accelerated TD (ATD)  \citep{pan2017accelerated}.
+\citet{hallak2016generalized} introduced an new parameter 
+to reduce variance for ETD.
+\citet{zhang2022truncated} proposed truncated ETD with a lower variance.
+Variance Reduced TD with direct variance reduction technique \citep{johnson2013accelerating} is proposed by \cite{korda2015td} 
+and analysed by  \cite{xu2019reanalysis}.
+How to further improve the convergence rates of reinforcement learning 
+algorithms is currently still an open problem.
+
+Algorithm stability is prominently reflected in the changes 
+to the objective function, transitioning from mean squared 
+errors (MSE) \citep{Sutton2018book} to mean squared bellman errors (MSBE) \cite{baird1995residual}, then to 
+norm of the expected TD update \cite{sutton2009fast}, and further to 
+mean squared projected Bellman errors (MSPBE) \cite{sutton2009fast}. On the other hand, algorithm 
+acceleration is more centered around optimizing the iterative 
+update formula of the algorithm itself without altering the 
+objective function, thereby speeding up the convergence rate 
+of the algorithm. The emergence of new optimization objective 
+functions often leads to the development of novel algorithms. 
+The introduction of new algorithms, in turn, tends to inspire 
+researchers to explore methods for accelerating algorithms, 
+leading to the iterative creation of increasingly superior algorithms.
+
+The kernel loss function can be optimized using standard 
+gradient-based methods, addressing the issue of double 
+sampling in residual gradient algorithm \cite{feng2019kernel}. It ensures convergence 
+in both on-policy and off-policy scenarios. The logistic bellman 
+error is convex and smooth in the action-value function parameters, 
+with bounded gradients \cite{basserrano2021logistic}. In contrast, the squared Bellman error is 
+not convex in the action-value function parameters, and RL algorithms 
+based on recursive optimization using it are known to be unstable.
+
+
+% The value-based algorithms mentioned above aim to
+% minimize some errors, e.g., mean squared errors \citep{Sutton2018book},
+% mean squared Bellman errors \cite{baird1995residual}, norm
+% of the expected TD update \cite{sutton2009fast}, 
+% mean squared projected Bellman errors (MSPBE) \cite{sutton2009fast}, etc.
+It is necessary to propose a new objective function, but the mentioned objective functions above are all some form of error.
+Is minimizing error the only option for value-based reinforcement learning?
+
+For policy evaluation experiments, 
+differences in objective functions may result 
+in inconsistent fixed points. This inconsistency 
+makes it difficult to uniformly compare the superiority 
+of algorithms derived from different objective functions. 
+However, for control experiments, since the choice of actions 
+depends on the relative values of the Q values rather than their
+ absolute values, the presence of solution bias is acceptable.
+
+Based on this observation, we propose  alternate objective functions 
+instead of minimizing errors. We minimize 
+Variance of Projected Bellman Error (VPBE)
+and derive Variance Minimization (VM) algorithms.
+These algorithms preserve the invariance of the optimal policy in the control environments,
+but significantly reduce the variance of gradient estimation,
+and thus hastening convergence.
+
+The contributions of this paper are as follows:
+(1) Introduction of  novel objective functions based on
+the invariance of the optimal policy.
+(2) Propose two off-policy variance minimization algorithms.
+(3) Proof of their convergence.
+(5) Experiments demonstrating the faster convergence speed of the proposed algorithms.
diff --git a/NEW_aaai/main/motivation.tex b/NEW_aaai/main/motivation.tex
new file mode 100644
index 0000000..3223cb5
--- /dev/null
+++ b/NEW_aaai/main/motivation.tex
@@ -0,0 +1,319 @@
+\section{Variance Minimization Algorithms}
+To derive an algorithm with a larger minimum eigenvalue for matrix 
+$\textbf{A}$, it is necessary to propose new objective functions. 
+The mentioned objective functions in the Introduction 
+are all forms of error. Is minimizing error the only option 
+for value-based reinforcement learning? Based on this observation, 
+we propose alternative objective functions instead of minimizing errors. 
+We minimize the Variance of Projected Bellman Error (VPBE) and derive the 
+VMTDC algorithm. This idea is then innovatively applied to ETD, resulting 
+in the VMETD algorithm.
+% \subsection{Motivation}
+% gagagga
+\begin{algorithm}[t]
+    \caption{VMTDC algorithm with linear function approximation in the off-policy setting}
+    \label{alg:algorithm 2}
+\begin{algorithmic}
+    \STATE {\bfseries Input:} $\bm{\theta}_{0}$, $\bm{u}_0$, $\omega_{0}$, $\gamma
+    $, learning rate $\alpha_t$, $\zeta_t$ and $\beta_t$, behavior policy $\mu$ and target policy $\pi$
+    \REPEAT
+    \STATE For any episode, initialize $\bm{\theta}_{0}$ arbitrarily, $\bm{u}_{0}$ and $\omega_{0}$ to $0$, $\gamma \in (0,1]$, and $\alpha_t$, $\zeta_t$ and $\beta_t$ are constant.\\
+    % \textbf{Output}: $\bm{\theta}^*$.\\
+    \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 $\delta_t = R_{t+1}+\gamma\bm{\theta}_t^{\top}\bm{\phi}_{t+1}-\bm{\theta}_t^{\top}\bm{\phi}_t$
+    \STATE $\rho_{t} \leftarrow \frac{\pi(A_t | S_t)}{\mu(A_t | S_t)}$
+    \STATE $\bm{\theta}_{t+1}\leftarrow \bm{\theta}_{t}+\alpha_t [ (\rho_t\delta_t-\omega_t)\bm{\phi}_t - \gamma \rho_t\bm{\phi}_{t+1}(\bm{\phi}^{\top}_{t} \bm{u}_{t})]$
+    \STATE $\bm{u}_{t+1}\leftarrow \bm{u}_{t}+\zeta_t[(\rho_t\delta_t-\omega_t) - \bm{\phi}^{\top}_{t} \bm{u}_{t}] \bm{\phi}_t$
+    \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]
+    \caption{VMETD algorithm with linear function approximation in the off-policy setting}
+    \label{alg:algorithm 5}
+\begin{algorithmic}
+    \STATE {\bfseries Input:} $\bm{\theta}_{0}$, $F_0$, $\omega_{0}$, $\gamma
+    $, learning rate $\alpha_t$, $\zeta_t$ and $\beta_t$, behavior policy $\mu$ and target policy $\pi$
+    \REPEAT
+    \STATE For any episode, initialize $\bm{\theta}_{0}$ arbitrarily, $F_{0}$ to $1$ 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 $\delta_t = R_{t+1}+\gamma\bm{\theta}_t^{\top}\bm{\phi}_{t+1}-\bm{\theta}_t^{\top}\bm{\phi}_t$
+    \STATE $\rho_{t} \leftarrow \frac{\pi(A_t | S_t)}{\mu(A_t | S_t)}$
+    \STATE $F_{t}\leftarrow \gamma \rho_t F_{t-1} +1$
+    \STATE $\bm{\theta}_{t+1}\leftarrow \bm{\theta}_{t}+\alpha_t (F_t \rho_t\delta_t-\omega_t)\bm{\phi}_t$
+    \STATE $\omega_{t+1}\leftarrow \omega_{t}+\beta_t (F_t \rho_t\delta_t-\omega_t)$
+    \STATE $S_t=S_{t+1}$
+    \ENDFOR
+    \UNTIL{terminal episode}
+\end{algorithmic}
+\end{algorithm}
+
+\subsection{Variance Minimization TDC Learning: VMTDC}
+For off-policy learning, we propose a new objective function, 
+called Variance of Projected Bellman error (VPBE), 
+and the corresponding algorithm is called VMTDC.
+\begin{align}
+    \text{VPBE}(\bm{\theta}) &= \mathbb{E}[(\delta-\mathbb{E}[\delta]) \bm{\phi}]^{\top} \mathbb{E}[\bm{\phi} \bm{\phi}^{\top}]^{-1} \\ 
+    & \mathbb{E}[(\delta -\mathbb{E}[\delta ])\bm{\phi}] \notag \\
+    &= (\bm{\Phi}^{\top}\textbf{D}(\textbf{W}_{\bm{\theta}} + \textbf{T}\textbf{V}_{\bm{\theta}} -\textbf{V}_{\bm{\theta}}))^{\top}(\bm{\Phi}^{\top}\textbf{D}\bm{\Phi})^{-1} \notag \\
+    & \bm{\Phi}^{\top}\textbf{D}(\textbf{W}_{\bm{\theta}} + \textbf{T}\textbf{V}_{\bm{\theta}} -\textbf{V}_{\bm{\theta}}) \notag \\
+    &= (\textbf{W}_{\bm{\theta}} + \textbf{T}\textbf{V}_{\bm{\theta}} -\textbf{V}_{\bm{\theta}})^{\top}\textbf{D}^{\top}\bm{\Phi}(\bm{\Phi}^{\top}\textbf{D}\bm{\Phi})^{-1} \notag \\
+    & \bm{\Phi}^{\top}\textbf{D}(\textbf{W}_{\bm{\theta}} + \textbf{T}\textbf{V}_{\bm{\theta}} -\textbf{V}_{\bm{\theta}}) \notag \\
+    &= (\textbf{W}_{\bm{\theta}} + \textbf{T}\textbf{V}_{\bm{\theta}} -\textbf{V}_{\bm{\theta}})^{\top}\Pi^{\top}\textbf{D}\Pi \notag \\
+    & (\textbf{W}_{\bm{\theta}} + \textbf{T}\textbf{V}_{\bm{\theta}} -\textbf{V}_{\bm{\theta}}) \notag \\
+    &= (\Pi(\textbf{V}_{\bm{\theta}} - \textbf{T}\textbf{V}_{\bm{\theta}}-\textbf{W}_{\bm{\theta}}))^{\top}\textbf{D} \notag \\
+    & (\Pi(\textbf{V}_{\bm{\theta}} - \textbf{T}\textbf{V}_{\bm{\theta}}-\textbf{W}_{\bm{\theta}})) \notag \\
+    &= ||\Pi(\textbf{V}_{\bm{\theta}} - \textbf{T}\textbf{V}_{\bm{\theta}} - \textbf{W}_{\bm{\theta}})||^{2}_{\mu} \notag \\
+    &= ||\Pi(\textbf{V}_{\bm{\theta}} - \textbf{T}\textbf{V}_{\bm{\theta}}) - \Pi\textbf{W}_{\bm{\theta}}||^{2}_{\mu} \notag \\
+    &= \mathbb{E}[(\delta-\omega) \bm{\phi}]^{\top} \mathbb{E}[\bm{\phi} \bm{\phi}^{\top}]^{-1}\mathbb{E}[(\delta -\omega)\bm{\phi}] 
+\end{align}
+where $\textbf{W}_{\bm{\theta}}$ is viewed as vectors with every element being equal to $||\textbf{V}_{\bm{\theta}} - \textbf{T}\textbf{V}_{\bm{\theta}}||^{2}_{\mu}$ and $\omega$ is used to approximate $\mathbb{E}[\delta]$, i.e., $\omega \doteq\mathbb{E}[\delta] $.
+
+The gradient of the (3) with respect to $\theta$ is
+\begin{equation*}
+    \begin{array}{ccl}
+    -\frac{1}{2}\nabla \text{VPBE}(\bm{\theta}) &=& -\mathbb{E}\Big[\Big( (\gamma \bm{\phi}' - \bm{\phi}) - \mathbb{E}[ (\gamma \bm{\phi}' - \bm{\phi})]\Big)\bm{\phi}^{\top} \Big] \\
+    & & \mathbb{E}[\bm{\phi} \bm{\phi}^{\top}]^{-1} \mathbb{E}[( \delta -\mathbb{E}[ \delta])\bm{\phi}]\\
+    &=& \mathbb{E}\Big[\Big( (\bm{\phi} - \gamma \bm{\phi}')- \mathbb{E}[ (\bm{\phi} - \gamma \bm{\phi}')]\Big)\bm{\phi}^{\top} \Big]  \\
+    & & \mathbb{E}[\bm{\phi} \bm{\phi}^{\top}]^{-1}\\
+    & & \mathbb{E}\Big[\Big( r + \gamma {\bm{\phi}'}^{\top} \bm{\theta} -\bm{\phi}^{\top} \bm{\theta}\\
+    & & \hspace{2em} -\mathbb{E}[ r + \gamma {\bm{\phi}'}^{\top} \bm{\theta} -\bm{\phi}^{\top} \bm{\theta}]\Big)\bm{\phi} \Big]\\
+    &=& \textbf{A}^{\top} \textbf{C}^{-1}(-\textbf{A}\bm{\theta} + \textbf{b}) 
+    \end{array}
+\end{equation*}
+where
+\begin{equation*}
+    \begin{array}{ccl}
+        \textbf{A} &=& \mathbb{E}\Big[\Big( (\bm{\phi} - \gamma \bm{\phi}')- \mathbb{E}[ (\bm{\phi} - \gamma \bm{\phi}')]\Big)\bm{\phi}^{\top} \Big] \\
+                   &=& \mathbb{E}[(\bm{\phi} - \gamma \bm{\phi}')\bm{\phi}^{\top}] - \mathbb{E}[\bm{\phi} - \gamma \bm{\phi}']\mathbb{E}[\bm{\phi}^{\top}]\\
+                    &=& \mathrm{Cov}(\bm{\bm{\phi}},\bm{\bm{\phi}}-\gamma\bm{\bm{\phi}}'),
+    \end{array}
+\end{equation*}
+\begin{equation*}
+    \begin{array}{ccl}
+        \textbf{C} &=& \mathbb{E}[\bm{\bm{\phi}}\bm{\bm{\phi}}^{\top}],
+    \end{array}
+\end{equation*}
+\begin{equation*}
+    \begin{array}{ccl}
+        \textbf{b} &=& \mathbb{E}( r- \mathbb{E}[r])\bm{\phi}  \\
+        &=& \mathbb{E}[r\bm{\phi}] - \mathbb{E}[r]\mathbb{E}[\bm{\phi}]\\
+         &=& \mathrm{Cov}(r,\bm{\bm{\phi}}),
+    \end{array}
+\end{equation*}
+where $\mathrm{Cov}(\cdot,\cdot )$ is a covariance operator. 
+
+In the process of computing the gradient of the (4) with respect to $\theta$, 
+$\omega$ is treated as a constant.
+So, the derivation process of the VMTDC algorithm is the same 
+as that of the TDC algorithm, the only difference is that the original $\delta$ is replaced by $\delta-\omega$.
+Therefore, we can easily get the updated formula of VMTDC, as follows:
+% \begin{equation*}
+%     \begin{array}{ccl}
+%     \text{VPBE}(\bm{\theta})&=&\mathbb{E}[(\rho_k \delta_k-\mathbb{E}[\rho_k \delta_k]) \bm{\phi}_k]^{\top} \mathbb{E}[\bm{\phi}_k \bm{\phi}^{\top}_k]^{-1}\\
+%     & &\mathbb{E}[(\rho_k \delta_k -\mathbb{E}[\rho_k \delta_k ])\bm{\phi}_k],
+%     % &=&\mathbb{E}[(\delta-\omega) \bm{\phi}]^{\top} \mathbb{E}[\bm{\phi} \bm{\phi}^{\top}]^{-1}\mathbb{E}[(\delta-\omega)\bm{\phi}],
+%     \end{array}
+% \end{equation*}
+\begin{equation}
+    \bm{\theta}_{k+1}\leftarrow\bm{\theta}_{k}+\alpha_{k}[(\delta_{k}- \omega_k) \bm{\phi}_k\\
+     - \gamma\bm{\phi}_{k+1}(\bm{\phi}^{\top}_k \bm{u}_{k})],
+\label{thetavmtdc}
+\end{equation}
+\begin{equation}
+    \bm{u}_{k+1}\leftarrow \bm{u}_{k}+\zeta_{k}[\delta_{k}-\omega_k - \bm{\phi}^{\top}_k \bm{u}_{k}]\bm{\phi}_k,
+\label{uvmtdc}
+\end{equation}
+and
+\begin{equation}
+    \omega_{k+1}\leftarrow \omega_{k}+\beta_k (\delta_k- \omega_k),
+    \label{omegavmtdc}
+\end{equation}
+
+The VMTDC algorithm (\ref{thetavmtdc}) is derived to work 
+with a given set of sub-samples—in the form of 
+triples $(S_k, R_k, S'_k)$ that match transitions 
+from both the behavior and target policies. What if 
+we wanted to use all the data? The data 
+is generated according to the behavior policy $\pi_b$, 
+while our objective is to learn about the target 
+policy $\pi$. We should use  importance-sampling. 
+The VPBE with importance sampling is:
+\begin{equation}
+    \label{rho_VPBE}
+    \begin{array}{ccl}
+    \text{VPBE}(\bm{\theta})&=&\mathbb{E}[(\rho\delta-\mathbb{E}[\rho\delta]) \bm{\phi}]^{\top} \mathbb{E}[\bm{\phi} \bm{\phi}^{\top}]^{-1}\\
+    & &\mathbb{E}[(\rho\delta -\mathbb{E}[\rho\delta ])\bm{\phi}],
+\end{array}
+\end{equation}
+Following the linear VMTDC derivation, we get the following algorithm (linear VMTDC algorithm
+based on importance weighting scenario):
+\begin{equation}
+    \bm{\theta}_{k+1}\leftarrow\bm{\theta}_{k}+\alpha_{k}[(\rho_k\delta_{k}- \omega_k) \bm{\phi}_k\\
+     - \gamma\rho_k\bm{\phi}_{k+1}(\bm{\phi}^{\top}_k \bm{u}_{k})],
+\end{equation}
+\begin{equation}
+    \bm{u}_{k+1}\leftarrow \bm{u}_{k}+\zeta_{k}[(\rho_k\delta_{k}-\omega_k) - \bm{\phi}^{\top}_k \bm{u}_{k}]\bm{\phi}_k,
+\end{equation}
+and
+\begin{equation}
+    \omega_{k+1}\leftarrow \omega_{k}+\beta_k (\rho_k\delta_k- \omega_k),
+\end{equation}
+
+The gradient of the (\ref{rho_VPBE}) with respect to $\theta$ is
+\begin{equation*}
+    \begin{array}{ccl}
+    -\frac{1}{2}\nabla \text{VPBE}(\bm{\theta}) &=& \mathbb{E}\Big[\Big( \rho(\bm{\phi} - \gamma \bm{\phi}')- \mathbb{E}[ \rho(\bm{\phi} - \gamma \bm{\phi}')]\Big)\bm{\phi}^{\top} \Big]  \\
+    & & \mathbb{E}[\bm{\phi} \bm{\phi}^{\top}]^{-1}\\
+    & & \mathbb{E}\Big[\Big( \rho(r + \gamma {\bm{\phi}'}^{\top} \bm{\theta} -\bm{\phi}^{\top} \bm{\theta})\\
+    & & \hspace{2em} -\mathbb{E}[ \rho(r + \gamma {\bm{\phi}'}^{\top} \bm{\theta} -\bm{\phi}^{\top} \bm{\theta})]\Big)\bm{\phi} \Big]\\
+    &=& \mathbb{E}[ \rho(\bm{\phi} - \gamma \bm{\phi}')\bm{\phi}^{\top}]- \mathbb{E}[ \rho(\bm{\phi} - \gamma \bm{\phi}')]\mathbb{E}[\bm{\phi}^{\top}]   \\
+    & & \mathbb{E}[\bm{\phi} \bm{\phi}^{\top}]^{-1}\\
+    & & \mathbb{E}\Big[\Big( \rho(r + \gamma {\bm{\phi}'}^{\top} \bm{\theta} -\bm{\phi}^{\top} \bm{\theta})\\
+    & & \hspace{2em} -\mathbb{E}[ \rho(r + \gamma {\bm{\phi}'}^{\top} \bm{\theta} -\bm{\phi}^{\top} \bm{\theta})]\Big)\bm{\phi} \Big]\\
+    % &=&\bm{\Phi}^{\top}(\textbf{D}_{\mu}- \textbf{d}_{\mu}\textbf{d}_{\mu}^{\top})(\textbf{I}-\gamma \textbf{P}_{\pi})\bm{\Phi}\\
+    &=& \textbf{A}^{\top} \textbf{C}^{-1}(-\textbf{A}\bm{\theta} + \textbf{b}), 
+    \end{array}
+\end{equation*}
+where $\textbf{A}=\bm{\Phi}^{\top}(\textbf{D}_{\mu}- \textbf{d}_{\mu}\textbf{d}_{\mu}^{\top})(\textbf{I}-\gamma \textbf{P}_{\pi})\bm{\Phi}$, 
+$\textbf{b}=\bm{\Phi}^{\top}(\textbf{D}_{\mu}- \textbf{d}_{\mu}\textbf{d}_{\mu}^{\top})\textbf{r}_{\pi}$ and 
+$\textbf{r}_{\pi}$ is viewed as vectors.
+In the 2-state counterexample, 
+$\textbf{A}_{\text{VMTDC}}=0.025$, meaning that VMTDC can stably converge and converges faster than TDC.
+
+
+
+
+% The control algorithm corresponding to TDC is called GQ. 
+% Now, we will introduce the improved version of the GQ algorithm, named VMGQ:
+% \begin{equation}
+%     \begin{array}{ccl}
+%         \bm{\theta}_{k+1}\leftarrow\bm{\theta}_{k}&+&\alpha_{k}[(\delta_{k}- \omega_k) \bm{\phi}(s_k,a_k)\\
+%      &-& \gamma\bm{\phi}(s_{k+1},A^{*}_{k+1})(\bm{\phi}^{\top} (s_k,a_k) \bm{u}_{k})],
+%     \end{array}
+% \end{equation}
+% \begin{equation}
+%     \bm{u}_{k+1}\leftarrow \bm{u}_{k}+\zeta_{k}[(\delta_{k}-\bm{u}_{k}) - \bm{\phi}^{\top} (s_k,a_k) \bm{u}_{k}]\bm{\phi}(s_k,a_k),
+% \end{equation}
+% and
+% \begin{equation}
+%     \omega_{k+1}\leftarrow \omega_{k}+\beta_k(\delta_k- \omega_k),
+% \end{equation}
+% where 
+% $\delta_k$ in VMGQ is:
+% \begin{equation}
+%     \delta_{k}=r_{k+1}+\gamma \max_{a\in A}\bm{\theta}_{k}^{\top}\bm{\phi}(s_{k+1},a) - \bm{\theta}_{k}^{\top}\bm{\phi}(s_{k},a_{k}),
+%     \label{deltaQ}
+% \end{equation}
+% and $A^{*}_{k+1}={\arg \max}_{a}(\bm{\theta}_{k}^{\top}\bm{\phi}(s_{k+1},a))$.
+\begin{table*}[t]
+    \caption{Minimum eigenvalues of various algorithms in the 2-state counterexample.}
+    \vskip 0.15in
+    \begin{center}
+    \begin{small}
+    \begin{sc}
+    \begin{tabular}{lcccccr}
+    \toprule
+    algorithm & off-policy TD & TDC & ETD & VMTDC & VMETD \\
+    \midrule
+    Minimum eigenvalues&$-0.2$ & $0.016$ & $3.4$ & $0.025$ & $1.15$ \\
+    \bottomrule
+    \end{tabular}
+    \end{sc}
+    \end{small}
+    \end{center}
+    \vskip -0.1in
+\end{table*}
+\subsection{Variance Minimization ETD Learning: VMETD}
+Based on the off-policy TD algorithm, a scalar, $F$,  
+is introduced to obtain the ETD algorithm, 
+which ensures convergence under off-policy 
+conditions. This paper further introduces a scalar, 
+$\omega$, based on the ETD algorithm to obtain 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}
+    \label{thetavmetd}
+    \bm{\theta}_{k+1}\leftarrow \bm{\theta}_k+\alpha_k (F_k \rho_k\delta_k - \omega_{k})\bm{\phi}_k,
+\end{equation}
+\begin{equation}
+    \label{omegavmetd}
+    \omega_{k+1} \leftarrow \omega_k+\beta_k(F_k  \rho_k \delta_k - \omega_k),
+\end{equation}
+where $\omega$  is used to estimate $\mathbb{E}[F \rho\delta]$, i.e., $\omega \doteq \mathbb{E}[F \rho\delta]$.
+
+(\ref{thetavmetd}) can be rewritten as
+\begin{equation*}
+    \begin{array}{ccl}
+        \bm{\theta}_{k+1}&\leftarrow& \bm{\theta}_k+\alpha_k (F_k \rho_k\delta_k - \omega_k)\bm{\phi}_k -\alpha_k \omega_{k+1}\bm{\phi}_k\\
+        &=&\bm{\theta}_{k}+\alpha_k(F_k\rho_k\delta_k-\mathbb{E}_{\mu}[F_k\rho_k\delta_k|\bm{\theta}_k])\bm{\phi}_k\\
+    &=&\bm{\theta}_k+\alpha_k F_k \rho_k (R_{k+1}+\gamma \bm{\theta}_k^{\top}\bm{\phi}_{k+1}-\bm{\theta}_k^{\top}\bm{\phi}_k)\bm{\phi}_k\\
+    & & \hspace{2em} -\alpha_k \mathbb{E}_{\mu}[F_k \rho_k \delta_k]\bm{\phi}_k\\
+    &=& \bm{\theta}_k+\alpha_k \{\underbrace{(F_k\rho_kR_{k+1}-\mathbb{E}_{\mu}[F_k\rho_k R_{k+1}])\bm{\phi}_k}_{\textbf{b}_{\text{VMETD},k}}\\
+    &&\hspace{-5em}- \underbrace{(F_k\rho_k\bm{\phi}_k(\bm{\phi}_k-\gamma\bm{\phi}_{k+1})^{\top}-\bm{\phi}_k\mathbb{E}_{\mu}[F_k\rho_k (\bm{\phi}_k-\gamma\bm{\phi}_{k+1})]^{\top})}_{\textbf{A}_{\text{VMETD},k}}\bm{\theta}_k\}.
+    \end{array}
+\end{equation*}
+Therefore, 
+\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 \bm{\phi}_k (\bm{\phi}_k - \gamma \bm{\phi}_{k+1})^{\top}]\\  
+        &&\hspace{-1em}- \lim_{k\rightarrow \infty} \mathbb{E}_{\mu}[  \bm{\phi}_k]\mathbb{E}_{\mu}[F_k \rho_k (\bm{\phi}_k - \gamma \bm{\phi}_{k+1})]^{\top}\\
+        &=& \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[\bm{\phi}_kF_k \rho_k  (\bm{\phi}_k - \gamma \bm{\phi}_{k+1})^{\top}]\\ 
+        &&\hspace{-1em}- \lim_{k\rightarrow \infty} \mathbb{E}_{\mu}[  \bm{\phi}_k]\mathbb{E}_{\mu}[F_k \rho_k (\bm{\phi}_k - \gamma \bm{\phi}_{k+1})]^{\top}\\
+        &=& \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[\bm{\phi}_kF_k \rho_k (\bm{\phi}_k - \gamma \bm{\phi}_{k+1})^{\top}]\\   
+        &&\hspace{-2em}-\lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[ \bm{\phi}_k]\lim_{k \rightarrow \infty}\mathbb{E}_{\mu}[F_k \rho_k (\bm{\phi}_k - \gamma \bm{\phi}_{k+1})]^{\top}\\
+        && \hspace{-9em}=\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+1})^{\top}|S_k = s]\\   
+        &&\hspace{-2em}-\sum_{s} d_{\mu}(s)\phi(s)\sum_{s} d_{\mu}(s)\lim_{k \rightarrow \infty}\mathbb{E}_{\mu}[F_k|S_k = s]\\
+        &&\hspace{2em}\mathbb{E}_{\mu}[\rho_k(\phi_k - \gamma \phi_{k+1})^{\top}|S_k = s]\\
+        &=& \sum_{s} f(s)\mathbb{E}_{\pi}[\phi_k(\phi_k - \gamma \phi_{k+1})^{\top}|S_k = s]\\   
+        &&\hspace{-3em}-\sum_{s} d_{\mu}(s)\phi(s)\sum_{s} f(s)\mathbb{E}_{\pi}[(\phi_k - \gamma \phi_{k+1})^{\top}|S_k = s]\\
+        &=&\sum_{s} f(s) \bm{\phi}(s)(\bm{\phi}(s) - \gamma \sum_{s'}[\textbf{P}_{\pi}]_{ss'}\bm{\phi}(s'))^{\top}  \\
+        &&\hspace{-4em} -\sum_{s} d_{\mu}(s) \bm{\phi}(s) * \sum_{s} f(s)(\bm{\phi}(s) - \gamma \sum_{s'}[\textbf{P}_{\pi}]_{ss'}\bm{\phi}(s'))^{\top}\\
+        &=&{\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}  \\
+        &=&{\bm{\Phi}}^{\top} (\textbf{F} - \textbf{d}_{\mu} \textbf{f}^{\top}) (\textbf{I} - \gamma \textbf{P}_{\pi}){\bm{\Phi}} \\
+        &=&{\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}} \\
+        &=&{\bm{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} ){\bm{\Phi}},
+    \end{array}
+\end{equation*}
+\begin{equation*}
+    \begin{array}{ccl}
+        \textbf{b}_{\text{VMETD}}&=&\lim_{k \rightarrow \infty} \mathbb{E}[\textbf{b}_{\text{VMETD},k}]\\
+        &=& \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[F_k\rho_kR_{k+1}\bm{\phi}_k]\\
+        &&- \lim_{k\rightarrow \infty} \mathbb{E}_{\mu}[\bm{\phi}_k]\mathbb{E}_{\mu}[F_k\rho_kR_{k+1}]\\  
+        &=& \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[\bm{\phi}_kF_k\rho_kR_{k+1}]\\
+        &&- \lim_{k\rightarrow \infty} \mathbb{E}_{\mu}[  \bm{\phi}_k]\mathbb{E}_{\mu}[\bm{\phi}_k]\mathbb{E}_{\mu}[F_k\rho_kR_{k+1}]\\ 
+        &=& \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[\bm{\phi}_kF_k\rho_kR_{k+1}]\\
+        &&- \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[ \bm{\phi}_k]\lim_{k \rightarrow \infty}\mathbb{E}_{\mu}[F_k\rho_kR_{k+1}]\\  
+        &=&\sum_{s} f(s) \bm{\phi}(s)r_{\pi} - \sum_{s} d_{\mu}(s) \bm{\phi}(s) * \sum_{s} f(s)r_{\pi}  \\
+        &=&\bm{\bm{\Phi}}^{\top}(\textbf{F}-\textbf{d}_{\mu} \textbf{f}^{\top})\textbf{r}_{\pi}.
+    \end{array}
+\end{equation*}
+Therefore, in the 2-state counterexample, 
+$\textbf{A}_{\text{VMETD}}=1.15$, meaning that VMETD can stably converge and converges slower than ETD.
+However, subsequent experiments showed that the VMETD algorithm converges more smoothly and performs better in controlled experiments.
+% In this paper, we refer to the control algorithm for ETD as EQ. 
+% Now, we will introduce the improved version of the EQ algorithm, named VMEQ:
+% \begin{equation}
+%     \begin{array}{ccl}
+%         \bm{\theta}_{k+1}\leftarrow\bm{\theta}_{k}&+&\alpha_{k}(F_k  \rho_k \delta_k- \omega_k) \bm{\phi}(s_k,a_k),
+%     \end{array}
+% \end{equation}
+% \begin{equation}
+%     \omega_{k+1}\leftarrow \omega_{k}+\beta_k(F_k  \rho_k \delta_k- \omega_k),
+% \end{equation}
+% where 
+% $\delta_k$ is (\ref{deltaQ}) 
+% and $A^{*}_{k+1}={\arg \max}_{a}(\bm{\theta}_{k}^{\top}\bm{\phi}(s_{k+1},a))$.
+
+
+
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+}
+
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diff --git a/NEW_aaai/main/preliminaries.tex b/NEW_aaai/main/preliminaries.tex
new file mode 100644
index 0000000..1a14e0f
--- /dev/null
+++ b/NEW_aaai/main/preliminaries.tex
@@ -0,0 +1,213 @@
+\section{Background}
+\label{preliminaries}
+\subsection{Markov Decision Process}
+Reinforcement learning agent interacts with environment, observes state,
+ takes sequential decision makings to influence environment, and obtains
+ rewards.
+ Consider an infinite-horizon discounted 
+ Markov Decision Process (MDP), defined by a tuple $\langle S,A,R,P,\gamma
+ \rangle$, where $S=\{1,2,\ldots,N\}$ is a finite set of states of the environment;  $A$
+ is a finite set of actions of the agent; 
+ $R:S\times A \times S \rightarrow \mathbb{R}$ is a bounded deterministic reward
+ function; $P:S\times A\times S \rightarrow [0,1]$ is the transition
+ probability distribution;  and $\gamma\in (0,1)$
+  is the discount factor \cite{Sutton2018book}.
+  Due to the requirements of  online learning, value iteration based on sampling
+  is considered in this paper. 
+  In each sampling, an experience (or transition) $\langle s, a, s', r\rangle$ is
+  obtained.
+
+  A policy is a mapping $\pi:S\times A \rightarrow [0,1]$. The goal of the
+  agent is to find an optimal policy $\pi^*$ to maximize the expectation of a
+  discounted cumulative rewards in a long period.
+  State value function $V^{\pi}(s)$  for a stationary policy $\pi$ is 
+  defined as:
+  \begin{equation*}
+  V^{\pi}(s)=\mathbb{E}_{\pi}[\sum_{k=0}^{\infty} \gamma^k R_{k}|s_0=s].
+  \label{valuefunction}
+  \end{equation*}
+  Linear value function for state $s\in S$ is defined as:
+   \begin{equation}
+   V_{{\theta}}(s):= {\bm{\theta}}^{\top}{\bm{\phi}}(s) = \sum_{i=1}^{m}
+   \theta_i \phi_i(s),
+   \label{linearvaluefunction}
+   \end{equation}
+  where ${\bm{\theta}}:=(\theta_1,\theta_2,\ldots,\theta_m)^{\top}\in
+  \mathbb{R}^m$ is a parameter vector, 
+  ${\bm{\phi}}:=(\phi_1,\phi_2,\ldots,\phi_m)^{\top}\in \mathbb{R}^m$ is a feature
+  function  defined on state space $S$, and $m$ is the feature size. 
+
+  Tabular temporal difference (TD) learning \cite{Sutton2018book} has been successfully applied to small-scale problems.
+  To deal with the well-known curse of dimensionality of large scale MDPs, value
+  function is usually approximated by a linear model (the focus of this paper), kernel methods, decision
+   trees, or neural networks, etc. 
+   % This paper focuses on the linear model.
+% TD learning can also be used to find optimal strategies. The problem of finding an optimal policy is 
+% often called the control problem. Two popular TD methods are Sarsa and Q-leaning. The former is an on-policy 
+% TD control, while the latter is an off-policy control. 
+
+% It is well known that TDC algorithm \cite{sutton2009fast} guarantees 
+% convergence under off-policy conditions while the off-policy TD algorithm may diverge. The 
+% objective function of TDC is MSPBE. 
+% TDC is essentially an adjustment or correction of the TD update so that it
+% follows the gradient of the MSPBE objective function. In the context of the TDC algorithm, the control algorithm 
+% is known as Greedy-GQ($\lambda$) \cite{sutton2009fast}. When $\lambda$ is set to 0, it is denoted 
+% as GQ(0).
+
+
+\subsection{On-policy and Off-policy}
+On-policy and off-policy algorithms are currently hot topics in research. 
+Off-policy algorithms, in particular, present greater challenges due to the 
+difficulty in ensuring their convergence, making them more complex to study. 
+The main difference between the two lies in the fact that in on-policy algorithms, 
+the behavior policy $\mu$ and the target policy $\pi$ are the same during the learning process. 
+The algorithm directly generates data from the current policy and optimizes it. 
+In off-policy algorithms, however, the behavior policy and the target policy are different. 
+The algorithm uses data generated from the behavior policy to optimize the 
+target policy, which leads to higher sample efficiency and complex stability issues.
+
+Taking the TD(0) algorithm as an example can help understand the different 
+performances of on-policy and off-policy:
+
+In the on-policy TD(0) algorithm, the behavior policy and the target policy 
+are the same. The algorithm uses the data generated by the current policy to 
+update its value estimates. Since the behavior policy and the target policy 
+are consistent, the convergence of TD(0) is more assured. In each step of the 
+update, the algorithm is based on the actual behavior of the current policy, 
+which gradually leads the value function estimate to converge to the true 
+value of the target policy.
+
+The on-policy TD(0) update formula is
+\begin{equation*}
+   \label{thetatd_onpolicy}
+   \begin{array}{ccl}
+      \bm{\theta}_{k+1}&\leftarrow&\bm{\theta}_k+\alpha_k \delta_k\bm{\phi}_k,
+   \end{array}
+\end{equation*}
+where $\delta_k = r_{k+1}+\gamma \bm{\theta}_k^{\top}\bm{\phi}_{k+1}-\bm{\theta}_k^{\top}\bm{\phi}_k$ and 
+the key matrix $\textbf{A}_{\text{on}}$ of on-policy TD(0) is
+\begin{equation*}
+   \textbf{A}_{\text{on}} = \bm{\Phi}^{\top}\textbf{D}_{\pi}(\textbf{I}-\gamma \textbf{P}_{\pi})\bm{\Phi},
+\end{equation*}
+where $\bm{\Phi}$ is the $N \times n$ matrix with the $\phi(s)$ as its rows, and $\textbf{D}_{\pi}$ is the $N \times N$ diagonal
+matrix with $\textbf{d}_{\pi}$ on its diagonal. $\textbf{d}_{\pi}$ is a vector, each component representing the steady-state
+distribution under $\pi$. $\textbf{P}_{\pi}$ denote the $N \times N$ matrix of transition probabilities under $\pi$. And $\textbf{P}_{\pi}^{\top}\textbf{d}_{\pi}=\textbf{d}_{\pi}$.
+
+An $\bm{\Phi}^{\top}\bm{\text{X}}\bm{\Phi}$ matrix of this
+        form will be positive definite whenever the matrix $\bm{\text{X}}$ is positive definite.
+        Any matrix $\bm{\text{X}}$ is positive definite if and only if
+        the symmetric matrix $\bm{\text{S}}=\bm{\text{X}}+\bm{\text{X}}^{\top}$ 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\cite{sutton2016emphatic}.
+
+All components of the matrix $\textbf{D}_{\pi}(\textbf{I}-\gamma \textbf{P}_{\pi})$ are positive.
+The row sums of $\textbf{D}_{\pi}(\textbf{I}-\gamma \textbf{P}_{\pi})$ are positive. And The row sums of 
+$\textbf{D}_{\pi}(\textbf{I}-\gamma \textbf{P}_{\pi})$ are
+\begin{equation*}
+   \begin{array}{ccl}
+      \textbf{1}^{\top}\textbf{D}_{\pi}(\textbf{I}-\gamma \textbf{P}_{\pi})&=&\textbf{d}_{\pi}^{\top}(\textbf{I}-\gamma \textbf{P}_{\pi})\\
+      &=& \textbf{d}_{\pi}^{\top} - \gamma \textbf{d}_{\pi}^{\top}\textbf{P}_{\pi}\\
+      &=& \textbf{d}_{\pi}^{\top}  - \gamma \textbf{d}_{\pi}^{\top}\\
+      &=& (1-\gamma)\textbf{d}_{\pi}^{\top},
+   \end{array}
+\end{equation*}
+all components of which are positive. Thus, the key matrix and its $\textbf{A}_{\text{on}}$ matrix are positive
+definite, and on-policy TD(0) is stable
+
+The off-policy TD(0) update formula is
+\begin{equation*}
+   \label{thetatd_offpolicy}
+   \begin{array}{ccl}
+      \bm{\theta}_{k+1}&\leftarrow&\bm{\theta}_k+\alpha_k \rho_k \delta_k\bm{\phi}_k,
+   \end{array}
+\end{equation*}
+where $\rho_k =\frac{\pi(A_k | S_k)}{\mu(A_k | S_k)}$, called importance sampling ratio, 
+ and the key matrix $\textbf{A}_{\text{off}}$ of off-policy TD(0) is
+\begin{equation*}
+   \textbf{A}_{\text{off}} = \bm{\Phi}^{\top}\textbf{D}_{\mu}(\textbf{I}-\gamma \textbf{P}_{\pi})\bm{\Phi}.
+\end{equation*}
+where $\textbf{D}_{\mu}$ is the $N \times N$ diagonal
+matrix with $\textbf{d}_{\mu}$ on its diagonal. $\textbf{d}_{\mu}$ is a vector, each component representing the steady-state
+distribution under $\mu$
+
+If the key matrix 
+$\textbf{A}$ in the algorithm is positive definite, then the 
+algorithm is stable and converges. However, in the off-policy TD(0)
+ algorithm, it cannot be guaranteed that 
+$\textbf{A}$ is a positive definite matrix. In the 2-state counterexample, 
+$\textbf{A}_{\text{off}}=-0.2$, which means that off-policy TD(0) cannot stably converge.
+
+TDC and ETD are two well-known off-policy algorithms. 
+The former is an off-policy algorithm derived from the 
+objective function Mean Squared Projected Bellman error (MSPBE), while the latter employs a technique 
+to transform the key matrix 
+$\textbf{A}$ in the original off-policy TD(0) from non-positive 
+definite to positive definite, thereby ensuring the algorithm's 
+convergence under off-policy conditions.
+
+The MSPBE with importance sampling is
+\begin{equation*}
+   \begin{array}{ccl}
+   \text{MSPBE}(\bm{\theta})&=&||\textbf{V}_{\bm{\theta}} - \Pi \textbf{T}^{\pi}\textbf{V}_{\bm{\theta}}||^{2}_{\mu}\\
+   &=&||\Pi(\textbf{V}_{\bm{\theta}} - \textbf{T}^{\pi}\textbf{V}_{\bm{\theta}})||^{2}_{\mu}\\
+   &=&\mathbb{E}[\rho \delta \bm{\phi}]^{\top} \mathbb{E}[\bm{\phi} \bm{\phi}^{\top}]^{-1}\mathbb{E}[\rho \delta \bm{\phi}],
+   \end{array}
+\end{equation*}
+where $\textbf{V}_{\bm{\theta}}$ is viewed as vectors with one element for each state,
+ the norm $||\bm{v}||^{2}_{\mu}=\sum_{s}^{}\mu(s)\bm{v}^{2}(s)$, $\textbf{T}^{\pi}$, simplified to 
+ $\textbf{T}$ in the following text, is Bellman operator and $\bm{\Pi}=\bm{\Phi}(\bm{\Phi}^{\top}\textbf{D}\bm{\Phi})^{-1}\bm{\Phi}^{\top}\textbf{D}$.
+The TDC update formula with importance sampling is
+\begin{equation*}
+   \bm{\theta}_{k+1}\leftarrow\bm{\theta}_{k}+\alpha_{k} \rho_{k}[\delta_{k} \bm{\phi}_k- \gamma\bm{\phi}_{k+1}(\bm{\phi}^{\top}_k \bm{u}_{k})],
+\label{thetatdc}
+\end{equation*}
+\begin{equation*}
+   \bm{u}_{k+1}\leftarrow \bm{u}_{k}+\zeta_{k}[\rho_k \delta_{k} - \bm{\phi}^{\top}_k \bm{u}_{k}]\bm{\phi}_k.
+\label{utdc}
+\end{equation*}
+The key matrix $\textbf{A}_{\text{TDC}}= \textbf{A}^{\top}_{\text{off}}\textbf{C}^{-1}\textbf{A}_{\text{off}}$, 
+where $\textbf{C}=\mathbb{E}[\bm{\bm{\phi}}\bm{\bm{\phi}}^{\top}]$. 
+In the 2-state counterexample, 
+$\textbf{A}_{\text{TDC}}=0.016$, which means that TDC can stably converge.
+
+The ETD update formula is
+\begin{equation}
+   \label{fvmetd}
+   F_k \leftarrow \gamma \rho_{k-1}F_{k-1}+1,
+\end{equation}
+\begin{equation*}
+   \label{thetaetd}
+   \bm{\theta}_{k+1}\leftarrow \bm{\theta}_k+\alpha_k F_k \rho_k\delta_k\bm{\phi}_k,
+\end{equation*}
+where $F_t$ is a scalar variable and $F_0=1$.
+The key matrix $\textbf{A}_{\text{ETD}}= \bm{\Phi}^{\top}\textbf{F}(\textbf{I}-\gamma \textbf{P}_{\pi})\bm{\Phi}$, 
+where 
+$\textbf{F}$ 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*}. 
+The row sums of 
+$\textbf{F}(\textbf{I}-\gamma \textbf{P}_{\pi})$ are
+\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}_{\mu}^{\top}(\textbf{I}-\gamma \textbf{P}_{\pi})^{-1}(\textbf{I}-\gamma \textbf{P}_{\pi})\\
+      &=& \textbf{d}_{\mu}^{\top},
+   \end{array}
+\end{equation*}
+and in the 2-state counterexample, 
+$\textbf{A}_{\text{ETD}}=3.4$, which means that ETD can stably converge.
+
+The convergence rate of the algorithm is related to the matrix 
+$\textbf{A}$. The larger the minimum eigenvalue of 
+$\textbf{A}$, the faster the convergence rate. In the 2-state case, the minimum eigenvalue of the matrix 
+$\textbf{A}$ in ETD is the largest, so it converges the fastest. 
+Based on this theorem, can we derive an algorithm with a larger minimum eigenvalue for matrix $\textbf{A}$.
diff --git a/NEW_aaai/main/relatedwork.tex b/NEW_aaai/main/relatedwork.tex
new file mode 100644
index 0000000..99bfcff
--- /dev/null
+++ b/NEW_aaai/main/relatedwork.tex
@@ -0,0 +1,52 @@
+\section{Related Work}
+\subsection{Difference between VMQ and R-learning}
+Tabular VMQ's update formula bears some resemblance 
+to R-learning's update formula. As shown in Table \ref{differenceRandVMQ}, the update formulas of the two algorithms have the following differences:
+\\(1) The goal of the R-learning algorithm \cite{schwartz1993reinforcement} is to maximize the average 
+reward, rather than the cumulative reward, by learning an estimate 
+of the average reward. This estimate $m$ is then used to update the Q-values.
+On the contrary, the $\omega$ in the tabular VMQ update formula eventually converges to $\mathbb{E}[\delta]$.
+\\(2) When $\gamma=1$ in the tabular VMQ update formula, the 
+R-learning update formula is formally 
+the same as the tabular VMQ update formula. 
+Therefore, R-learning algorithm can be 
+considered as a special case of VMQ algorithm in form.
+
+\subsection{Variance Reduction for TD Learning}
+ The TD with centering algorithm (CTD) \cite{korda2015td} 
+was proposed, which directly applies variance reduction techniques to 
+the TD algorithm. The CTD algorithm updates its parameters using the 
+average gradient of a batch of Markovian samples and a projection operator. 
+Unfortunately, the authors’ analysis of the CTD algorithm contains technical 
+errors. The VRTD algorithm \cite{xu2020reanalysis} is also a variance-reduced algorithm that updates 
+its parameters using the average gradient of a batch of i.i.d. samples. The 
+authors of VRTD provide a technically sound analysis to demonstrate the 
+advantages of variance reduction. 
+
+\subsection{Variance Reduction for Policy Gradient Algorithms}
+Policy gradient algorithms are a class of reinforcement 
+learning algorithms that directly optimize cumulative rewards. 
+REINFORCE  is a Monte Carlo algorithm that estimates 
+gradients through sampling, but may have a high variance. 
+Baselines are introduced to reduce variance and to
+accelerate learning \cite{Sutton2018book}. In  Actor-Critic, 
+value function as a baseline and bootstrapping 
+ are used to reduce variance, also accelerating convergence \cite{Sutton2018book}.
+ TRPO \cite{schulman2015trust} and PPO \cite{schulman2017proximal}
+  use generalized advantage 
+estimation, which combines multi-step bootstrapping and Monte Carlo 
+estimation to reduce variance, making gradient estimation more stable and 
+accelerating convergence. 
+
+In Variance Minimization, 
+the incorporation of $\omega \doteq \mathbb{E}[\delta]$ 
+bears a striking resemblance to the use of a baseline 
+in policy gradient methods. The introduction of a baseline 
+in policy gradient techniques does not alter 
+the expected value of the update; 
+rather, it significantly impacts the variance of gradient estimation. 
+The addition of $\omega \doteq \mathbb{E}[\delta]$ in Variance Minimization 
+ preserves the invariance of the optimal 
+policy while stabilizing gradient estimation, 
+reducing the variance of gradient estimation, 
+and hastening convergence.
\ No newline at end of file
diff --git a/NEW_aaai/main/theory.tex b/NEW_aaai/main/theory.tex
new file mode 100644
index 0000000..604b341
--- /dev/null
+++ b/NEW_aaai/main/theory.tex
@@ -0,0 +1,581 @@
+\section{Theoretical Analysis}
+The purpose of this section is to establish the stabilities of the VMTDC algorithm
+and the VMETD algorithm.
+\begin{theorem}
+    \label{theorem2}(Convergence of VMTDC).
+    In the case of off-policy learning, consider the iterations (\ref{omegavmtdc}), (\ref{uvmtdc}) and (\ref{thetavmtdc})   of VMTDC.
+     Let the step-size sequences $\alpha_k$, $\zeta_k$ and $\beta_k$, $k\geq 0$ satisfy in this case $\alpha_k,\zeta_k,\beta_k>0$, for all $k$,
+    $
+    \sum_{k=0}^{\infty}\alpha_k=\sum_{k=0}^{\infty}\beta_k=\sum_{k=0}^{\infty}\zeta_k=\infty,
+    $
+    $
+    \sum_{k=0}^{\infty}\alpha_k^2<\infty,
+    $
+    $
+    \sum_{k=0}^{\infty}\zeta_k^2<\infty,
+    $
+    $
+    \sum_{k=0}^{\infty}\beta_k^2<\infty,
+    $
+    and  
+    $
+    \alpha_k = o(\zeta_k),
+    $
+    $
+    \zeta_k = o(\beta_k).
+    $
+    Assume that $(\bm{\bm{\phi}}_k,r_k,\bm{\bm{\phi}}_k')$ is an i.i.d. sequence with
+    uniformly bounded second moments.
+    Let $\textbf{A} = \mathrm{Cov}(\bm{\bm{\phi}},\bm{\bm{\phi}}-\gamma\bm{\bm{\phi}}')$,
+    $\bm{b}=\mathrm{Cov}(r,\bm{\bm{\phi}})$, and $\textbf{C}=\mathbb{E}[\bm{\bm{\phi}}\bm{\bm{\phi}}^{\top}]$.
+    Assume that  $\textbf{A}$ and $\textbf{C}$ are  non-singular matrices. 
+    Then the parameter vector $\bm{\theta}_k$ converges with probability one 
+    to $\textbf{A}^{-1}\bm{b}$.
+\end{theorem}
+\begin{proof}
+    The proof is similar to that given by \cite{sutton2009fast} for TDC, but it is based on multi-time-scale stochastic approximation.
+    % For the VMTDC algorithm, a new one-step linear TD solution is defined as:
+    % \begin{equation*}
+    %     0=\mathbb{E}[(\bm{\phi} - \gamma \bm{\phi}' - \mathbb{E}[\bm{\phi} - \gamma \bm{\phi}'])\bm{\phi}^\top]\mathbb{E}[\bm{\phi} \bm{\phi}^{\top}]^{-1}\mathbb{E}[(\delta -\mathbb{E}[\delta])\bm{\phi}]=\textbf{A}^{\top}\textbf{C}^{-1}(-\textbf{A}\bm{\theta}+\bm{b}).
+    % \end{equation*}
+    % The matrix $\textbf{A}^{\top}\textbf{C}^{-1}\textbf{A}$ is positive definite. Thus, the  VMTD's solution is
+    % $\bm{\theta}_{\text{VMTDC}}=\bm{\theta}_{\text{VMTD}}=\textbf{A}^{-1}\bm{b}$.
+    
+    First, note that recursion (\ref{thetavmtdc}) and (\ref{uvmtdc}) can be rewritten as, respectively, 
+\begin{equation*}
+    \theta_{k+1}\leftarrow \theta_k+\zeta_k x(k),
+\end{equation*}
+\begin{equation*}
+    u_{k+1}\leftarrow u_k+\beta_k y(k),
+\end{equation*}
+where 
+\begin{equation*}
+    x(k)=\frac{\alpha_k}{\zeta_k}[(\delta_{k}- \omega_k) \phi_k - \gamma\phi'_{k}(\phi^{\top}_k u_k)],
+\end{equation*}
+\begin{equation*}
+    y(k)=\frac{\zeta_k}{\beta_k}[\delta_{k}-\omega_k - \phi^{\top}_k u_k]\phi_k.
+\end{equation*}
+
+Recursion (\ref{thetavmtdc}) can also be rewritten as
+\begin{equation*}
+    \theta_{k+1}\leftarrow \theta_k+\beta_k z(k),
+\end{equation*}
+where
+\begin{equation*}
+    z(k)=\frac{\alpha_k}{\beta_k}[(\delta_{k}- \omega_k) \phi_k - \gamma\phi'_{k}(\phi^{\top}_k u_k)],
+\end{equation*}
+
+Due to the settings of step-size schedule 
+$\alpha_k = o(\zeta_k)$, $\zeta_k = o(\beta_k)$, $x(k)\rightarrow 0$, $y(k)\rightarrow 0$, $z(k)\rightarrow 0$ almost surely as $k\rightarrow 0$.
+That is that the increments in iteration (\ref{omegavmtdc}) are uniformly larger than
+those in (\ref{uvmtdc}) and  the increments in iteration (\ref{uvmtdc}) are uniformly larger than
+those in (\ref{thetavmtdc}), thus (\ref{omegavmtdc}) is the fastest recursion, (\ref{uvmtdc}) is the second fast recursion and (\ref{thetavmtdc}) is the slower recursion.
+Along the fastest time scale, iterations of (\ref{thetavmtdc}), (\ref{uvmtdc}) and (\ref{omegavmtdc})
+are associated to ODEs system as follows:
+\begin{equation}
+    \dot{\theta}(t) = 0,
+    \label{thetavmtdcFastest}
+\end{equation}
+\begin{equation}
+    \dot{u}(t) = 0,
+    \label{uvmtdcFastest}
+\end{equation}
+\begin{equation}
+    \dot{\omega}(t)=\mathbb{E}[\delta_t|u(t),\theta(t)]-\omega(t).
+    \label{omegavmtdcFastest}
+\end{equation}
+
+Based on the ODE (\ref{thetavmtdcFastest}) and (\ref{uvmtdcFastest}), both $\theta(t)\equiv \theta$
+and $u(t)\equiv u$ when viewed from the fastest timescale.
+By the Hirsch lemma \cite{hirsch1989convergent}, it follows that
+$||\theta_k-\theta||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
+$\theta$ that depends on the initial condition $\theta_0$ of recursion
+(\ref{thetavmtdc}) and $||u_k-u||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
+$u$ that depends on the initial condition $u_0$ of recursion
+(\ref{uvmtdc}). Thus, the ODE pair (\ref{thetavmtdcFastest})-(ref{omegavmtdcFastest})
+can be written as 
+\begin{equation}
+    \dot{\omega}(t)=\mathbb{E}[\delta_t|u,\theta]-\omega(t).
+    \label{omegavmtdcFastestFinal}
+\end{equation}
+
+Consider the function $h(\omega)=\mathbb{E}[\delta|\theta,u]-\omega$,
+i.e., the driving vector field of the ODE (\ref{omegavmtdcFastestFinal}).
+It is easy to find that the function $h$ is Lipschitz with coefficient
+$-1$.
+Let $h_{\infty}(\cdot)$ be the function defined by
+ $h_{\infty}(\omega)=\lim_{r\rightarrow \infty}\frac{h(r\omega)}{r}$.
+ Then  $h_{\infty}(\omega)= -\omega$,  is well-defined. 
+ For (\ref{omegavmtdcFastestFinal}), $\omega^*=\mathbb{E}[\delta|\theta,u]$
+is the unique globally asymptotically stable equilibrium.
+For the ODE
+\begin{equation}
+ \dot{\omega}(t) = h_{\infty}(\omega(t))= -\omega(t),
+ \label{omegavmtdcInfty}
+\end{equation}
+apply $\vec{V}(\omega)=(-\omega)^{\top}(-\omega)/2$ as its
+associated strict Liapunov function. Then,
+the origin of (\ref{omegavmtdcInfty}) is a globally asymptotically stable
+equilibrium.
+
+Consider now the recursion (\ref{omegavmtdc}).
+Let
+$M_{k+1}=(\delta_k-\omega_k)
+-\mathbb{E}[(\delta_k-\omega_k)|\mathcal{F}(k)]$,
+where $\mathcal{F}(k)=\sigma(\omega_l,u_l,\theta_l,l\leq k;\phi_s,\phi_s',r_s,s<k)$, 
+$k\geq 1$ are the sigma fields
+generated by $\omega_0,u_0,\theta_0,\omega_{l+1},u_{l+1},\theta_{l+1},\phi_l,\phi_l'$,
+$0\leq l<k$.
+It is easy to verify that $M_{k+1},k\geq0$ are integrable random variables that 
+satisfy $\mathbb{E}[M_{k+1}|\mathcal{F}(k)]=0$, $\forall k\geq0$.
+Because $\phi_k$, $r_k$, and $\phi_k'$   have
+uniformly bounded second moments, it can be seen that for some constant
+$c_1>0$, $\forall k\geq0$,
+\begin{equation*}
+\mathbb{E}[||M_{k+1}||^2|\mathcal{F}(k)]\leq
+c_1(1+||\omega_k||^2+||u_k||^2+||\theta_k||^2).
+\end{equation*}
+
+
+Now Assumptions (A1) and (A2) of \cite{borkar2000ode} are verified.
+Furthermore, Assumptions (TS) of \cite{borkar2000ode} is satisfied by our
+conditions on the step-size sequences $\alpha_k$,$\zeta_k$, $\beta_k$. Thus,
+by Theorem 2.2 of \cite{borkar2000ode} we obtain that
+$||\omega_k-\omega^*||\rightarrow 0$ almost surely as $k\rightarrow \infty$.
+
+Recursion (\ref{uvmtdc}) is considered the second timescale.
+Recursion (\ref{thetavmtdc}) is considered the slower timescale.
+For the convergence properties of $u$ and $\theta$, please refer to the appendix.
+\end{proof}
+
+% \begin{proof}
+%     The proof is similar to that given by \cite{sutton2009fast} for TDC, but it is based on multi-time-scale stochastic approximation.
+    
+%     For the VMTDC algorithm, a new one-step linear TD solution is defined as:
+%     \begin{equation*}
+%         \begin{array}{ccl}
+%         0&=&\mathbb{E}[(\bm{\phi} - \gamma \bm{\phi}' - \mathbb{E}[\bm{\phi} - \gamma \bm{\phi}'])\bm{\phi}^\top]\\ 
+%          & &\mathbb{E}[\bm{\phi} \bm{\phi}^{\top}]^{-1}\mathbb{E}[(\delta -\mathbb{E}[\delta])\bm{\phi}]\\
+%          &=&\textbf{A}^{\top}\textbf{C}^{-1}(-\textbf{A}\bm{\theta}+\bm{b})\\
+%          &=&-\textbf{A}_{\text{VMTDC}}\bm{\theta}+\bm{b}_{\text{VMTDC}}.
+%         \end{array}
+%     \end{equation*}
+%     The matrix $\textbf{A}^{\top}\textbf{C}^{-1}\textbf{A}$ is positive definite. Thus, the  VMTD's solution is
+%     $\bm{\theta}_{\text{VMTDC}}=\textbf{A}^{-1}\bm{b}$.
+    
+%     First, note that recursion (\ref{thetavmtdc}) and (\ref{uvmtdc}) can be rewritten as, respectively, 
+%     \begin{equation*}
+%         \bm{\theta}_{k+1}\leftarrow \bm{\theta}_k+\zeta_k \bm{x}(k),
+%     \end{equation*}
+%     \begin{equation*}
+%         \bm{u}_{k+1}\leftarrow \bm{u}_k+\beta_k \bm{y}(k),
+%     \end{equation*}
+%     where 
+%     \begin{equation*}
+%         \bm{x}(k)=\frac{\alpha_k}{\zeta_k}[(\delta_{k}- \omega_k) \bm{\phi}_k - \gamma\bm{\phi}'_{k}(\bm{\phi}^{\top}_k \bm{u}_k)],
+%     \end{equation*}
+%     \begin{equation*}
+%         \bm{y}(k)=\frac{\zeta_k}{\beta_k}[\delta_{k}-\omega_k - \bm{\phi}^{\top}_k \bm{u}_k]\bm{\phi}_k.
+%     \end{equation*}
+    
+%     Recursion (\ref{thetavmtdc}) can also be rewritten as
+%     \begin{equation*}
+%         \bm{\theta}_{k+1}\leftarrow \bm{\theta}_k+\beta_k z(k),
+%     \end{equation*}
+%     where
+%     \begin{equation*}
+%         z(k)=\frac{\alpha_k}{\beta_k}[(\delta_{k}- \omega_k) \bm{\phi}_k - \gamma\bm{\phi}'_{k}(\bm{\phi}^{\top}_k \bm{u}_k)],
+%     \end{equation*}
+    
+%     Due to the settings of step-size schedule 
+%     $\alpha_k = o(\zeta_k)$, $\zeta_k = o(\beta_k)$, $x(k)\rightarrow 0$, $y(k)\rightarrow 0$, $z(k)\rightarrow 0$ almost surely as $k\rightarrow 0$.
+%     That is that the increments in iteration (\ref{omegavmtdc}) are uniformly larger than
+%     those in (\ref{uvmtdc}) and  the increments in iteration (\ref{uvmtdc}) are uniformly larger than
+%     those in (\ref{thetavmtdc}), thus (\ref{omegavmtdc}) is the fastest recursion, (\ref{uvmtdc}) is the second fast recursion and (\ref{thetavmtdc}) is the slower recursion.
+%     Along the fastest time scale, iterations of (\ref{thetavmtdc}), (\ref{uvmtdc}) and (\ref{omegavmtdc})
+%     are associated to ODEs system as follows:
+%     \begin{equation}
+%         \dot{\bm{\theta}}(t) = 0,
+%         \label{thetavmtdcFastest}
+%     \end{equation}
+%     \begin{equation}
+%         \dot{\bm{u}}(t) = 0,
+%         \label{uvmtdcFastest}
+%     \end{equation}
+%     \begin{equation}
+%         \dot{\omega}(t)=\mathbb{E}[\delta_t|\bm{u}(t),\bm{\theta}(t)]-\omega(t).
+%         \label{omegavmtdcFastest}
+%     \end{equation}
+    
+%     Based on the ODE (\ref{thetavmtdcFastest}) and (\ref{uvmtdcFastest}), both $\theta(t)\equiv \theta$
+%     and $u(t)\equiv u$ when viewed from the fastest timescale.
+%     By the Hirsch lemma \cite{hirsch1989convergent}, it follows that
+%     $||\theta_k-\theta||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
+%     $\theta$ that depends on the initial condition $\theta_0$ of recursion
+%     (\ref{thetavmtdc}) and $||u_k-u||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
+%     $u$ that depends on the initial condition $u_0$ of recursion
+%     (\ref{uvmtdc}). Thus, the ODE pair (\ref{thetavmtdcFastest})-(ref{omegavmtdcFastest})
+%     can be written as 
+%     \begin{equation}
+%         \dot{\omega}(t)=\mathbb{E}[\delta_t|\bm{u},\bm{\theta}]-\omega(t).
+%         \label{omegavmtdcFastestFinal}
+%     \end{equation}
+    
+%     Consider the function $h(\omega)=\mathbb{E}[\delta|\theta,u]-\omega$,
+%     i.e., the driving vector field of the ODE (\ref{omegavmtdcFastestFinal}).
+%     It is easy to find that the function $h$ is Lipschitz with coefficient
+%     $-1$.
+%     Let $h_{\infty}(\cdot)$ be the function defined by
+%      $h_{\infty}(\omega)=\lim_{r\rightarrow \infty}\frac{h(r\omega)}{r}$.
+%      Then  $h_{\infty}(\omega)= -\omega$,  is well-defined. 
+%      For (\ref{omegavmtdcFastestFinal}), $\omega^*=\mathbb{E}[\delta|\bm{\theta},\bm{u}]$
+%     is the unique globally asymptotically stable equilibrium.
+%     For the ODE
+%     \begin{equation}
+%      \dot{\omega}(t) = h_{\infty}(\omega(t))= -\omega(t),
+%      \label{omegavmtdcInfty}
+%     \end{equation}
+%     apply $\vec{V}(\omega)=(-\omega)^{\top}(-\omega)/2$ as its
+%     associated strict Liapunov function. Then,
+%     the origin of (\ref{omegavmtdcInfty}) is a globally asymptotically stable
+%     equilibrium.
+    
+%     Consider now the recursion (\ref{omegavmtdc}).
+%     Let
+%     $M_{k+1}=(\delta_k-\omega_k)
+%     -\mathbb{E}[(\delta_k-\omega_k)|\mathcal{F}(k)]$,
+%     where $\mathcal{F}(k)=\sigma(\omega_l,\bm{u}_l,\bm{\theta}_l,l\leq k;\bm{\phi}_s,\bm{\phi}_s',r_s,s<k)$, 
+%     $k\geq 1$ are the sigma fields
+%     generated by $\omega_0,u_0,\bm{\theta}_0,\omega_{l+1},\bm{u}_{l+1},\bm{\theta}_{l+1},\bm{\phi}_l,\bm{\phi}_l'$,
+%     $0\leq l<k$.
+%     It is easy to verify that $M_{k+1},k\geq0$ are integrable random variables that 
+%     satisfy $\mathbb{E}[M_{k+1}|\mathcal{F}(k)]=0$, $\forall k\geq0$.
+%     Because $\bm{\phi}_k$, $r_k$, and $\bm{\phi}_k'$   have
+%     uniformly bounded second moments, it can be seen that for some constant
+%     $c_1>0$, $\forall k\geq0$,
+%     \begin{equation*}
+%     \mathbb{E}[||M_{k+1}||^2|\mathcal{F}(k)]\leq
+%     c_1(1+||\omega_k||^2+||\bm{u}_k||^2+||\bm{\theta}_k||^2).
+%     \end{equation*}
+    
+    
+%     Now Assumptions (A1) and (A2) of \cite{borkar2000ode} are verified.
+%     Furthermore, Assumptions (TS) of \cite{borkar2000ode} is satisfied by our
+%     conditions on the step-size sequences $\alpha_k$,$\zeta_k$, $\beta_k$. Thus,
+%     by Theorem 2.2 of \cite{borkar2000ode} we obtain that
+%     $||\omega_k-\omega^*||\rightarrow 0$ almost surely as $k\rightarrow \infty$.
+    
+%     Consider now the second time scale recursion (\ref{uvmtdc}).
+%     Based on the above analysis, (\ref{uvmtdc}) can be rewritten as
+%     % \begin{equation*}
+%     %     \bm{u}_{k+1}\leftarrow u_{k}+\zeta_{k}[\delta_{k}-\mathbb{E}[\delta_k|\bm{u}_k,\bm{\theta}_k] - \bm{\phi}^{\top} (s_k) \bm{u}_k]\bm{\phi}(s_k).
+%     % \end{equation*}
+%     \begin{equation}
+%         \dot{\bm{\theta}}(t) = 0,
+%         \label{thetavmtdcFaster}
+%     \end{equation}
+%     \begin{equation}
+%         \dot{u}(t) = \mathbb{E}[(\delta_t-\mathbb{E}[\delta_t|\bm{u}(t),\bm{\theta}(t)])\bm{\phi}_t|\bm{\theta}(t)] - \textbf{C}\bm{u}(t).
+%         \label{uvmtdcFaster}
+%     \end{equation}
+%     The ODE (\ref{thetavmtdcFaster}) suggests that $\theta(t)\equiv \theta$ (i.e., a time invariant parameter)
+%     when viewed from the second fast timescale.
+%     By the Hirsch lemma \cite{hirsch1989convergent}, it follows that
+%     $||\theta_k-\theta||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
+%     $\theta$ that depends on the initial condition $\theta_0$ of recursion
+%     (\ref{thetavmtdc}). 
+    
+%     Consider now the recursion (\ref{uvmtdc}).
+%     Let
+%     $N_{k+1}=((\delta_k-\mathbb{E}[\delta_k]) - \bm{\phi}_k \bm{\phi}^{\top}_k \bm{u}_k) -\mathbb{E}[((\delta_k-\mathbb{E}[\delta_k]) - \bm{\phi}_k \bm{\phi}^{\top}_k \bm{u}_k)|\mathcal{I} (k)]$,
+%     where $\mathcal{I}(k)=\sigma(\bm{u}_l,\bm{\theta}_l,l\leq k;\bm{\phi}_s,\bm{\phi}_s',r_s,s<k)$, 
+%     $k\geq 1$ are the sigma fields
+%     generated by $\bm{u}_0,\bm{\theta}_0,\bm{u}_{l+1},\bm{\theta}_{l+1},\bm{\phi}_l,\bm{\phi}_l'$,
+%     $0\leq l<k$.
+%     It is easy to verify that $N_{k+1},k\geq0$ are integrable random variables that 
+%     satisfy $\mathbb{E}[N_{k+1}|\mathcal{I}(k)]=0$, $\forall k\geq0$.
+%     Because $\bm{\phi}_k$, $r_k$, and $\bm{\phi}_k'$   have
+%     uniformly bounded second moments, it can be seen that for some constant
+%     $c_2>0$, $\forall k\geq0$,
+%     \begin{equation*}
+%     \mathbb{E}[||N_{k+1}||^2|\mathcal{I}(k)]\leq
+%     c_2(1+||\bm{u}_k||^2+||\bm{\theta}_k||^2).
+%     \end{equation*}
+    
+%     Because $\bm{\theta}(t)\equiv \bm{\theta}$ from (\ref{thetavmtdcFaster}), the ODE pair (\ref{thetavmtdcFaster})-(\ref{uvmtdcFaster})
+%     can be written as 
+%     \begin{equation}
+%         \dot{\bm{u}}(t) = \mathbb{E}[(\delta_t-\mathbb{E}[\delta_t|\bm{\theta}])\bm{\phi}_t|\bm{\theta}] - \textbf{C}\bm{u}(t).
+%         \label{uvmtdcFasterFinal}
+%     \end{equation}
+%     Now consider the function $h(\bm{u})=\mathbb{E}[\delta_t-\mathbb{E}[\delta_t|\bm{\theta}]|\bm{\theta}] -\textbf{C}\bm{u}$, i.e., the
+%     driving vector field of the ODE (\ref{uvmtdcFasterFinal}). For (\ref{uvmtdcFasterFinal}),
+%     $\bm{u}^* = \textbf{C}^{-1}\mathbb{E}[(\delta-\mathbb{E}[\delta|\bm{\theta}])\bm{\phi}|\bm{\theta}]$ is the unique globally asymptotically
+%     stable equilibrium. Let $h_{\infty}(\bm{u})=-\textbf{C}\bm{u}$.
+%     For the ODE
+%     \begin{equation}
+%         \dot{\bm{u}}(t) = h_{\infty}(\bm{u}(t))= -\textbf{C}\bm{u}(t),
+%         \label{uvmtdcInfty}
+%     \end{equation}
+%     the origin of (\ref{uvmtdcInfty}) is a globally asymptotically stable
+%     equilibrium because $\textbf{C}$ is a positive definite matrix (because it is nonnegative definite and nonsingular).
+%     Now Assumptions (A1) and (A2) of \cite{borkar2000ode} are verified.
+%     Furthermore, Assumptions (TS) of \cite{borkar2000ode} is satisfied by our
+%     conditions on the step-size sequences $\alpha_k$,$\zeta_k$, $\beta_k$. Thus,
+%     by Theorem 2.2 of \cite{borkar2000ode} we obtain that
+%     $||\bm{u}_k-\bm{u}^*||\rightarrow 0$ almost surely as $k\rightarrow \infty$.
+    
+%     Consider now the slower timescale recursion (\ref{thetavmtdc}). In the light of the above,
+%     (\ref{thetavmtdc}) can be rewritten as 
+%     % \begin{equation}
+%     %     \bm{\theta}_{k+1} \leftarrow \bm{\theta}_{k} + \alpha_k (\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k]) \bm{\phi}_k\\
+%     %     - \alpha_k \gamma\bm{\phi}'_{k}(\bm{\phi}^{\top}_k \textbf{C}^{-1}\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k])\bm{\phi}|\bm{\theta}_k]).
+%     % \end{equation}
+%     \begin{equation}
+%         \begin{array}{ccl}
+%         \bm{\theta}_{k+1}&\leftarrow&\bm{\theta}_{k} + \alpha_k (\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k]) \bm{\phi}_k\\ 
+%          & &- \alpha_k \gamma\bm{\phi}'_{k}(\bm{\phi}^{\top}_k \textbf{C}^{-1}\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k])\bm{\phi}|\bm{\theta}_k]).
+%         \end{array}
+%     \end{equation}
+%     Let $\mathcal{G}(k)=\sigma(\bm{\theta}_l,l\leq k;\bm{\phi}_s,\bm{\phi}_s',r_s,s<k)$, 
+%     $k\geq 1$ be the sigma fields
+%     generated by $\bm{\theta}_0,\bm{\theta}_{l+1},\bm{\phi}_l,\bm{\phi}_l'$,
+%     $0\leq l<k$. Let
+%     \begin{equation*}
+%         \begin{array}{ccl}
+%         Z_{k+1}&=&\left(\right.(\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k]) \bm{\phi}_k\\
+%         & &- \gamma \bm{\phi}'_{k}\bm{\phi}^{\top}_k \textbf{C}^{-1}\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k])\bm{\phi}|\bm{\theta}_k]\left.\right)\\
+%          & &-\mathbb{E}\left[\right.((\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k]) \bm{\phi}_k\\
+%          & &- \gamma \bm{\phi}'_{k}\bm{\phi}^{\top}_k \textbf{C}^{-1}\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k])\bm{\phi}|\bm{\theta}_k])|\mathcal{G}(k)\left.\right]\\
+%          &=&\left(\right.(\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k]) \bm{\phi}_k\\
+%          & &- \gamma \bm{\phi}'_{k}\bm{\phi}^{\top}_k \textbf{C}^{-1}\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k])\bm{\phi}|\bm{\theta}_k]\left.\right)\\
+%         & &-\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k]) \bm{\phi}_k|\bm{\theta}_k]\\
+%         & &- \gamma\mathbb{E}[\bm{\phi}' \bm{\phi}^{\top}]\textbf{C}^{-1}\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|\bm{\theta}_k]) \bm{\phi}_k|\bm{\theta}_k].
+%         \end{array}
+%     \end{equation*}
+%     It is easy to see that $Z_k$, $k\geq 0$ are integrable random variables and $\mathbb{E}[Z_{k+1}|\mathcal{G}(k)]=0$, $\forall k\geq0$. Further,
+%     \begin{equation*}
+%     \mathbb{E}[||Z_{k+1}||^2|\mathcal{G}(k)]\leq
+%     c_3(1+||\bm{\theta}_k||^2), k\geq 0
+%     \end{equation*}
+%     for some constant $c_3 \geq 0$, again beacuse $\bm{\phi}_k$, $r_k$, and $\bm{\phi}_k'$   have
+%     uniformly bounded second moments, it can be seen that for some constant.
+    
+%     Consider now the following ODE associated with (\ref{thetavmtdc}):
+%     \begin{equation}
+%         \dot{\bm{\theta}}(t) = (\textbf{I} - \mathbb{E}[\gamma \bm{\phi}' \bm{\phi}^{\top}]\textbf{C}^{-1})\mathbb{E}[(\delta -\mathbb{E}[\delta|\bm{\theta}(t)]) \bm{\phi}|\bm{\theta}(t)].
+%         \label{thetavmtdcSlowerFinal}
+%     \end{equation}
+%     Let 
+%     \begin{equation*}
+%     \begin{array}{ccl}
+%         \vec{h}(\bm{\theta}(t))&=&(\textbf{I} - \mathbb{E}[\gamma \bm{\phi}' \bm{\phi}^{\top}]\textbf{C}^{-1})\mathbb{E}[(\delta -\mathbb{E}[\delta|\bm{\theta}(t)]) \bm{\phi}|\bm{\theta}(t)]\\
+%         &=&(\textbf{C} - \mathbb{E}[\gamma \bm{\phi}' \bm{\phi}^{\top}])\textbf{C}^{-1}\mathbb{E}[(\delta -\mathbb{E}[\delta|\bm{\theta}(t)]) \bm{\phi}|\bm{\theta}(t)]\\
+%         &=& (\mathbb{E}[\bm{\phi} \bm{\phi}^{\top}] - \mathbb{E}[\gamma \bm{\phi}' \bm{\phi}^{\top}])\textbf{C}^{-1}\\
+%         & &       \mathbb{E}[(\delta -\mathbb{E}[\delta|\bm{\theta}(t)]) \bm{\phi}|\bm{\theta}(t)]\\
+%         &=& \textbf{A}^{\top}\textbf{C}^{-1}(-\textbf{A}\bm{\theta}(t)+\bm{b}),
+%     \end{array}
+%     \end{equation*}
+%     because $\mathbb{E}[(\delta -\mathbb{E}[\delta|\bm{\theta}(t)]) \bm{\phi}|\bm{\theta}(t)]=-\textbf{A}\bm{\theta}(t)+\bm{b}$, where 
+%     $\textbf{A} = \mathrm{Cov}(\bm{\phi},\bm{\phi}-\gamma\bm{\phi}')$, $\bm{b}=\mathrm{Cov}(r,\bm{\phi})$, and $\textbf{C}=\mathbb{E}[\bm{\phi}\bm{\phi}^{\top}]$
+    
+%     Therefore,
+%     $\bm{\theta}^*=\textbf{A}^{-1}\bm{b}$ can be seen to be the unique globally asymptotically
+%     stable equilibrium for ODE (\ref{thetavmtdcSlowerFinal}).
+%     Let $\vec{h}_{\infty}(\bm{\theta})=\lim_{r\rightarrow
+%     \infty}\frac{\vec{h}(r\bm{\theta})}{r}$. Then
+%     $\vec{h}_{\infty}(\bm{\theta})=-\textbf{A}^{\top}\textbf{C}^{-1}\textbf{A}\bm{\theta}$ is well-defined. 
+%     Consider now the ODE
+%     \begin{equation}
+%     \dot{\bm{\theta}}(t)=-\textbf{A}^{\top}\textbf{C}^{-1}\textbf{A}\bm{\theta}(t).
+%     \label{odethetavmtdcfinal}
+%     \end{equation}
+    
+%     Because $\textbf{C}^{-1}$ is positive definite and $\textbf{A}$ has full rank (as it
+%     is nonsingular by assumption), the matrix $\textbf{A}^{\top} \textbf{C}^{-1}\textbf{A}$ is also
+%     positive definite. 
+%     The ODE (\ref{odethetavmtdcfinal}) has the origin as its unique globally asymptotically stable equilibrium.
+%     Thus, the assumption (A1) and (A2) are verified.
+    
+%     The proof is given above.
+%     In the fastest time scale, the parameter $w$ converges to
+%     $\mathbb{E}[\delta|\bm{u}_k,\bm{\theta}_k]$.
+%     In the second fast time scale,
+%     the parameter $u$ converges to $\textbf{C}^{-1}\mathbb{E}[(\delta-\mathbb{E}[\delta|\bm{\theta}_k])\bm{\phi}|\bm{\theta}_k]$.
+%     In the slower time scale,
+%     the parameter $\bm{\theta}$ converges to $\textbf{A}^{-1}\bm{b}$.
+%     \end{proof}
+% 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{fvmetd}), (\ref{omegavmetd}) and (\ref{thetavmetd}) 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 $(\bm{\bm{\phi}}_k,r_k,\bm{\bm{\phi}}_k')$ is an i.i.d. sequence with
+    uniformly bounded second moments, where $\bm{\bm{\phi}}_k$ and $\bm{\bm{\phi}}'_{k}$ are sampled from the same Markov chain.
+    Let $\textbf{A}_{\textbf{VMETD}} ={\bm{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} ){\bm{\Phi}}$,
+    $\bm{b}_{\textbf{VMETD}}=\bm{\Phi}^{\top}(\textbf{F}-\textbf{d}_{\mu} \textbf{f}^{\top})\textbf{r}_{\pi}$.
+    Assume that matrix $A$ is  non-singular. 
+    Then the parameter vector $\bm{\theta}_k$ converges with probability one 
+    to $\textbf{A}_{\textbf{VMETD}}^{-1}\bm{b}_{\textbf{VMETD}}$.
+\end{theorem}
+\begin{proof}
+    The proof of VMETD's convergence is also based on Borkar's Theorem   for
+    general stochastic approximation recursions with two time scales
+    \cite{borkar1997stochastic}. 
+
+    Recursion (\ref{omegavmetd}) is considered the faster timescale. For the convergence properties of $\omega$, please refer to the appendix.
+Recursion (\ref{thetavmetd}) is considered the slower timescale. 
+If the key matrix 
+$\textbf{A}_{\text{VMETD}}$ is positive definite, then 
+$\theta$ converges.
+\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. 
+So $\textbf{A}_{\text{VMETD}}$ is positive definite.
+
+
+\end{proof}
+
+\begin{figure*}[htb]
+    \vskip 0.2in
+    \begin{center}
+    \subfigure[2-state counterexample]{
+        \includegraphics[width=0.55\columnwidth, height=0.475\columnwidth]{main/pic/2-state.pdf}
+        \label{2-state}
+    }
+    \subfigure[7-state counterexample]{
+        \includegraphics[width=0.55\columnwidth, height=0.475\columnwidth]{main/pic/7-state.pdf}
+        \label{7-state}
+    }
+    \subfigure[Maze]{
+        \includegraphics[width=0.55\columnwidth, height=0.475\columnwidth]{main/pic/maze.pdf}
+        \label{MazeFull}
+    }\\
+    \subfigure[Cliff Walking]{
+        \includegraphics[width=0.55\columnwidth, height=0.475\columnwidth]{main/pic/cl.pdf}
+        \label{CliffWalkingFull}
+    }
+    \subfigure[Mountain Car]{
+        \includegraphics[width=0.55\columnwidth, height=0.475\columnwidth]{main/pic/mt.pdf}
+        \label{MountainCarFull}
+    }
+    \subfigure[Acrobot]{
+        \includegraphics[width=0.55\columnwidth, height=0.475\columnwidth]{main/pic/acrobot.pdf}
+        \label{AcrobotFull}
+    }
+        \caption{Learning curses of two evaluation environments and four contral environments.}
+        \label{Complete_full}
+    \end{center}
+    \vskip -0.2in
+  \end{figure*}
+\subsection{Optimal Policy Invariance}
+This section prove 
+the optimal policy invariance of 
+VMTDC and VMETD in control experiments, 
+laying the groundwork for subsequent experiments.
+
+ As shown 
+in Table \ref{example_bias}, although there is a bias between the 
+true value and the predicted value, action $a_3$ is 
+still chosen under the greedy-policy.
+On the contrary, supervised learning is usually used to predict temperature, humidity, morbidity, etc. If the bias is too large, the consequences could be serious. 
+
+\begin{table}[t]
+    \caption{Comparison of action selection with and without
+    constant bias in $Q$ values.}
+    \label{example_bias}
+    \vskip 0.15in
+    \begin{center}
+    \begin{small}
+    \begin{sc}
+    \begin{tabular}{lcccr}
+    \toprule
+    action & $Q$ value & $Q$ value with bias \\
+    \midrule
+    $Q(s, a_0)$ & 1& 5 \\
+    $Q(s, a_1)$ & 2& 6 \\
+    $Q(s, a_2)$ & 3& 7 \\
+    $Q(s, a_3)$ & 4& 8 \\
+    $\arg \min_{a}Q(s,a)$ & $a_3$& $a_3$\\
+    \bottomrule
+    \end{tabular}
+    \end{sc}
+    \end{small}
+    \end{center}
+    \vskip -0.1in
+\end{table}
+
+In addition, reward shaping can significantly speed up the learning  by adding a shaping
+reward $F(s,s')$ to the original reward  $r$, 
+where $F(s,s')$ is the general form of any state-based shaping reward.
+Static potential-based reward shaping (Static PBRS) maintains the policy invariance if the
+shaping reward follows from $F(s,s')=\gamma
+f(s')-f(s)$ \cite{ng1999policy}.
+
+This means that we can make changes to the TD error $\delta = r+\gamma \bm{\theta}^{\top}\bm{\phi}'-\bm{\theta}^{\top}\bm{\phi} $ while still ensuring the invariance of the optimal policy,
+\begin{equation*}
+    \delta - \omega= r+\gamma \bm{\theta}^{\top}\bm{\phi}'-\bm{\theta}^{\top}\bm{\phi} - \omega,
+\end{equation*}
+where $\omega$ is a constant, acting as a static PBRS. 
+This also means that algorithms with the optimization goal 
+of minimizing errors, after introducing reward shaping, 
+may result in larger or smaller bias. Fortunately, 
+as discussed above, bias is acceptable in reinforcement 
+learning. 
+However, the problem is that selecting an appropriate 
+$\omega$ requires expert knowledge. This forces us to learn 
+$\omega$ dynamically, i.e., $\omega=\omega_t $ and dynamic PBRS can also maintain the policy 
+invariance if the shaping reward is $F(s,t,s',t')=\gamma f(s',t')-f(s,t)$,
+where $t$ is the time-step the agent reaches in  state $s$
+\cite{devlin2012dynamic}.
+However, this result requires the convergence guarantee of the dynamic potential
+function $f(s,t)$. If $f(s,t)$ does not converge as the time-step
+$t\rightarrow\infty$, the Q-values of  dynamic PBRS are not 
+guaranteed to converge.
+
+Let  $f_{\omega_t}(s)=\frac{\omega_t}{\gamma-1}$.
+Thus, $F_{\omega_t}(s,s')=\gamma f_{\omega_t}(s')-f_{\omega_t}(s)= \omega_t$
+is a dynamic PBRS. And if $\omega$ converges finally, the dynamic potential
+function $f(s,t)$ will converge.
+Bias is the expected difference between the predicted value 
+and the true value. Therefore, under the premise of bootstrapping, we first think of 
+letting $\omega \doteq\mathbb{E}[\delta]$ or 
+$\omega \doteq \mathbb{E}[F \rho\delta]$. 
diff --git "a/\350\256\272\346\226\207\350\215\211\347\250\277.txt" "b/\350\256\272\346\226\207\350\215\211\347\250\277.txt"
new file mode 100644
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@@ -0,0 +1,106 @@
+题目：A Variance Minimization Approach to  Off-policy Temporal-Difference Learning
+摘要：
+在本文中，我们介绍了通过引入一个新的参数来提高参数化时序差分学习算法的性能的思想，文中称这个新的参数为方差最小化参数，并且该参数在每一时间步上动态更新。特别的，我们将方差最小化参数引入到非策略算法，如TDC和ETD，中去，得到variance minimization TDC算法和Variance minimization ETD算法。本文在特定评估实验环境中计算算法关键矩阵的最小特征值来分析算法的收敛速度，并通过实验得到验证。在控制实验中，variance minimization算法拥有更好的性能。
+
+In this paper, we introduce the concept of improving the performance of parametric Temporal-Difference (TD) learning algorithms by the variance minimization parameter, which is dynamically updated at each time step. Specifically, we incorporate the variance minimization parameter into off-policy algorithms such as TDC and ETD, resulting in the Variance Minimization TDC algorithm and the Variance Minimization ETD algorithm.We analyze the convergence speed of these algorithms by calculating the minimum eigenvalue of the key matrices in a specific evaluation experimental environment and validate the results through experiments.In controlled experiments, the variance minimization algorithms demonstrate superior performance.
+
+background：
+1）简单介绍怎么从onpolicy变为offpolicy
+计算两状态的最小特征值
+2）介绍TDC和ETD
+offpolicy TD发散
+TDC从MSPBE角度出发解决发散
+ETD从构造正定矩阵解决发散
+计算两状态的最小特征值
+3）引用，矩阵的最小特征值
+说明ETD最快
+如何找出收敛速度更快的算法
+结束
+
+On-policy和off-policy算法是当前研究的热门话题。尤其是off-policy算法，由于其收敛性更难以保证，使得研究更具挑战性。两者的主要区别在于，on-policy算法在学习过程中，行为策略与目标策略相同，算法直接从当前策略生成数据并进行优化。而在off-policy算法中，行为策略与目标策略不同，算法利用从行为策略生成的数据来优化目标策略，这带来了更高的样本效率和复杂的稳定性问题。
+以TD(0)算法为例，可以帮助理解on-policy和off-policy的不同表现：
+在on-policy TD(0)算法中，行为策略和目标策略是相同的。算法使用当前策略生成的数据来更新自身的价值估计。由于行为策略与目标策略一致，TD(0)的收敛性比较有保障。算法在每一步更新时，都是基于当前策略的实际行为，这使得价值函数的估计逐渐收敛于目标策略的真实价值。
+on policy TD(0)更新公式如下：
+on policy TD(0)的公式
+给出on policy TD(0)的A的公式推导
+从数学的角度分析，行和＋列和大于0，则A正定，即on policy TD(0)稳定收敛。
+
+在off-policy TD(0)算法中，行为策略与目标策略不同。算法利用从行为策略生成的数据来估计目标策略的价值。由于行为策略和目标策略之间存在差异，TD(0)算法面临着额外的挑战：
+数据分布不匹配：行为策略生成的数据与目标策略的数据分布不同，这会导致估计偏差和方差增大，从而影响收敛性。
+重要性采样的高方差：off-policy TD(0)通常需要使用重要性采样来调整数据分布。如果重要性采样的权重过大，可能会导致高方差，进而使得算法难以收敛。
+off policy TD(0)更新公式如下：
+off policy TD(0)的公式
+给出off policy TD(0)的A的公式推导
+从数学的角度分析，行和＋列和小于0，则A非正定，即off policy TD(0)收敛不稳定。
+在2-state counterexample中，A=[-0.2]
+
+TDC和ETD是两个著名的offpolicy算法，前者由目标函数MSPBE推导得出的offpolicy算法，后者则通过技巧将原本offpolicy TD(0)的A由非正定转变为正定，从而使得算法在offpolicy下收敛。
+带有重要性采样的MSPBE公式如下：
+带有重要性采样的MSPBE的公式（看那个博士论文）
+带有重要性采样的TDC的更新公式如下：
+带有重要性采样的TDC的更新公式（看那个博士论文）
+给出TDC的A的公式推导
+在2-state counterexample中，A=[0.016]
+
+ETD更新公式如下：
+ETD的公式
+给出ETD的A的公式推导
+从数学的角度分析，行和＋列和大于0，则A正定，即ETD稳定收敛。
+在2-state counterexample中，A=[??]
+
+陈老师arxiv那篇论文中的定理描述，得到算法的收敛速度跟矩阵A有关系，A的最小特征值越大，收敛速度越快。
+由于在2-state中，ETD的矩阵A的最小特征值最大，因此收敛速度最快。基于这个定理，我们是否可以推导出矩阵A最小特征值越大的算法？
+
+
+
+
+
+Variance Minimization Algorithms
+其他都是误差最小化
+我们提出方差最小化
+控制算法去掉
+可以在实验部分提一下
+算法的伪代码可能也需要去掉
+需要A的计算过程，并且带入到2-state具体计算
+需要一个表格，总结一下四个算法的最小特征值
+
+为了推导出矩阵A最小特征值越大的算法，我们有必要提出新的目标函数。
+The mentioned objective functions in Introduction are all some form
+of error. Is minimizing error the only option for value-based
+reinforcement learning?
+Based on this observation, we propose alternate objec-
+tive functions instead of minimizing errors. We minimize
+Variance of Projected Bellman Error (VPBE)  and derive the VMTDC algorithm.
+并将此想法对ETD进行创新，得出VMETD算法。
+
+表格总结：表1展示了四种算法关键矩阵A的最小特征值，VMTDC的最小特征是大于TDC，VMTDC的收敛速度比TDC快在后文的实验中得到验证。ETD的最小特征值大于VMETD，但后文实验表明VMETD的稳定性由于ETD。
+
+
+
+Theoretical Analysis
+1）VMTDC收敛性，证明简要说，具体放到附录
+2）VMETD收敛性，证明简要说，具体放到附录
+3）策略不变性
+
+实验：
+评估：2-state 7-state
+控制：maze cliffwalking mt acrobot
+实验环境具体介绍：放到附录
+
+Reproducibility Checklist
+把nips模板复制过来
+
+总结：
+本文主要研究线性近似下的offpolicy算法的研究。基于陈的定理，算法的关键矩阵的最小特征值越大算法收敛速度越快的结论，我们提出了一个新的目标函数——VPBE，并推导出VMTDC。VMTDC比TDC多一个动态更新的标量参数，我们将该想法引入到ETD中得到VMETD。通过数值分析和实验发现VM算法有着更优越的性能或者收敛更稳定。
+Future work may include, but are not limited
+to, (1) 将标量参数引入到更多的TD算法中. 
+(2) extensions of VMTDC and VMETD to multi-step returns. 
+(3) extensions to nonlinear approximations, such as neural networks. 
+
+
+对于评估实验，实验结果与我们上分分析的结果一样，在2-state counterexample环境中，TDC的关键矩阵的最小特征最小，因此收敛速度最慢，相比之下，VMTDC的最小特征值更大，因此收敛速度更快。虽然VMETD的最小特征值比ETD要大使得在2-state counterexample中收敛速度慢于ETD，但VMETD的阴影部分（std）小于ETD，意味着VMETD收敛更加平稳。
+
+
+对于控制实验，迷宫和cliff walking的实验结果相似，VMGQ表现优于GQ，EQ表现优于VMGQ，而VMEQ的性能最优。
+mountain car和Acrobot的实验结果相似，VMGQ和VMEQ的性能接近都优于GQ和EQ。总之对于控制实验，VM算法优于非VM算法
+
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