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Apendix/anonymous-submission-latex-2024.aux
\relax
\bibstyle{aaai24}
\citation{borkar1997stochastic}
\citation{hirsch1989convergent}
\citation{borkar2000ode}
\citation{borkar2000ode}
\citation{borkar2000ode}
\newlabel{proofth1}{{A.1}{1}}
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\newlabel{omegaInfty}{{A-4}{1}}
\citation{sutton2009fast}
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\newlabel{covariance}{{A-6}{2}}
\newlabel{odethetafinal}{{A-7}{2}}
\newlabel{proofth2}{{A.2}{2}}
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\citation{hirsch1989convergent}
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\citation{sutton2016emphatic}
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\citation{baird1995residual,sutton2009fast}
\citation{baird1995residual,sutton2009fast,maei2011gradient}
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
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\bibdata{aaai24}
\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}}{{}}}
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\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}}{{}}}
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Apendix/anonymous-submission-latex-2024.bbl
\begin{thebibliography}{7}
\begin{thebibliography}{5}
\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.
......@@ -21,11 +16,6 @@ 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.
......
Apendix/anonymous-submission-latex-2024.blg
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Apendix/anonymous-submission-latex-2024.tex
......@@ -187,122 +187,287 @@
\onecolumn
\appendix
\section{Relevant proofs}
% \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{proofth1}
\begin{proof}
\label{th1proof}
The proof is based on Borkar's Theorem for
general stochastic approximation recursions with two time scales
\cite{borkar1997stochastic}.
% The new TD error for the linear setting is
% \begin{equation*}
% \delta_{\text{new}}=r+\gamma
% \theta^{\top}\phi'-\theta^{\top}\phi-\mathbb{E}[\delta].
% \end{equation*}
A new one-step
linear TD solution is defined
as:
\begin{equation*}
0=\mathbb{E}[(\delta-\mathbb{E}[\delta]) \phi]=-A\theta+b.
\end{equation*}
Thus, the VMTD's solution is
$\theta_{\text{VMTD}}=A^{-1}b$.
First, note that recursion (5) can be rewritten as
\begin{equation*}
\theta_{k+1}\leftarrow \theta_k+\beta_k\xi(k),
\end{equation*}
where
\begin{equation*}
\xi(k)=\frac{\alpha_k}{\beta_k}(\delta_k-\omega_k)\phi_k
\end{equation*}
Due to the settings of step-size schedule $\alpha_k = o(\beta_k)$,
$\xi(k)\rightarrow 0$ almost surely as $k\rightarrow\infty$.
That is the increments in iteration (4) are uniformly larger than
those in (5), thus (4) is the faster recursion.
Along the faster time scale, iterations of (4) and (5)
are associated to ODEs system as follows:
\begin{equation}
\dot{\theta}(t) = 0,
\label{thetaFast}
\end{equation}
\begin{equation}
\dot{\omega}(t)=\mathbb{E}[\delta_t|\theta(t)]-\omega(t).
\label{omegaFast}
\end{equation}
Based on the ODE (\ref{thetaFast}), $\theta(t)\equiv \theta$ when
viewed from the faster 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
(5).
Thus, the ODE pair (\ref{thetaFast})-(\ref{omegaFast}) can be written as
\begin{equation}
\dot{\omega}(t)=\mathbb{E}[\delta_t|\theta]-\omega(t).
\label{omegaFastFinal}
\end{equation}
Consider the function $h(\omega)=\mathbb{E}[\delta|\theta]-\omega$,
i.e., the driving vector field of the ODE (\ref{omegaFastFinal}).
It is easy to find that the function $h$ is Lipschitz with coefficient
$-1$.
Let $h_{\infty}(\cdot)$ be the function defined by
$h_{\infty}(\omega)=\lim_{x\rightarrow \infty}\frac{h(x\omega)}{x}$.
Then $h_{\infty}(\omega)= -\omega$, is well-defined.
For (\ref{omegaFastFinal}), $\omega^*=\mathbb{E}[\delta|\theta]$
is the unique globally asymptotically stable equilibrium.
For the ODE
\begin{equation}
\dot{\omega}(t) = h_{\infty}(\omega(t))= -\omega(t),
\label{omegaInfty}
\end{equation}
apply $\vec{V}(\omega)=(-\omega)^{\top}(-\omega)/2$ as its
associated strict Liapunov function. Then,
the origin of (\ref{omegaInfty}) is a globally asymptotically stable
equilibrium.
\subsection{Proof of Theorem 1}
Consider now the recursion (\ref{omega}).
Let
$M_{k+1}=(\delta_k-\omega_k)
-\mathbb{E}[(\delta_k-\omega_k)|\mathcal{F}(k)]$,
where $\mathcal{F}(k)=\sigma(\omega_l,\theta_l,l\leq k;\phi_s,\phi_s',r_s,s<k)$,
$k\geq 1$ are the sigma fields
generated by $\omega_0,\theta_0,\omega_{l+1},\theta_{l+1},\phi_l,\phi_l'$,
$0\leq l<k$.
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+||\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 $\alpha_k$, $\beta_k$. 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 slower time scale recursion (5).
Based on the above analysis, (5) can be rewritten as
\begin{equation*}
\theta_{k+1}\leftarrow
\theta_{k}+\alpha_k(\delta_k-\mathbb{E}[\delta_k|\theta_k])\phi_k.
\end{equation*}
Let $\mathcal{G}(k)=\sigma(\theta_l,l\leq k;\phi_s,\phi_s',r_s,s<k)$,
$k\geq 1$ be the sigma fields
generated by $\theta_0,\theta_{l+1},\phi_l,\phi_l'$,
$0\leq l<k$.
Let
$
Z_{k+1} = Y_{t}-\mathbb{E}[Y_{t}|\mathcal{G}(k)],
$
where
\begin{equation*}
Y_{k}=(\delta_k-\mathbb{E}[\delta_k|\theta_k])\phi_k.
\end{equation*}
Consequently,
\begin{equation*}
\begin{array}{ccl}
\mathbb{E}[Y_t|\mathcal{G}(k)]&=&\mathbb{E}[(\delta_k-\mathbb{E}[\delta_k|\theta_k])\phi_k|\mathcal{G}(k)]\\
&=&\mathbb{E}[\delta_k\phi_k|\theta_k]
-\mathbb{E}[\mathbb{E}[\delta_k|\theta_k]\phi_k]\\
&=&\mathbb{E}[\delta_k\phi_k|\theta_k]
-\mathbb{E}[\delta_k|\theta_k]\mathbb{E}[\phi_k]\\
&=&\mathrm{Cov}(\delta_k|\theta_k,\phi_k),
\end{array}
\end{equation*}
where $\mathrm{Cov}(\cdot,\cdot)$ is a covariance operator.
Thus,
\begin{equation*}
\begin{array}{ccl}
Z_{k+1}&=&(\delta_k-\mathbb{E}[\delta_k|\theta_k])\phi_k-\mathrm{Cov}(\delta_k|\theta_k,\phi_k).
\end{array}
\end{equation*}
It is easy to verify that $Z_{k+1},k\geq0$ are integrable random variables that
satisfy $\mathbb{E}[Z_{k+1}|\mathcal{G}(k)]=0$, $\forall k\geq0$.
Also, because $\phi_k$, $r_k$, and $\phi_k'$ have
uniformly bounded second moments, it can be seen that for some constant
$c_2>0$, $\forall k\geq0$,
\begin{equation*}
\mathbb{E}[||Z_{k+1}||^2|\mathcal{G}(k)]\leq
c_2(1+||\theta_k||^2).
\end{equation*}
Consider now the following ODE associated with (5):
\begin{equation}
\begin{array}{ccl}
\dot{\theta}(t)&=&\mathrm{Cov}(\delta|\theta(t),\phi)\\
&=&\mathrm{Cov}(r+(\gamma\phi'-\phi)^{\top}\theta(t),\phi)\\
&=&\mathrm{Cov}(r,\phi)-\mathrm{Cov}(\theta(t)^{\top}(\phi-\gamma\phi'),\phi)\\
&=&\mathrm{Cov}(r,\phi)-\theta(t)^{\top}\mathrm{Cov}(\phi-\gamma\phi',\phi)\\
&=&\mathrm{Cov}(r,\phi)-\mathrm{Cov}(\phi-\gamma\phi',\phi)^{\top}\theta(t)\\
&=&\mathrm{Cov}(r,\phi)-\mathrm{Cov}(\phi,\phi-\gamma\phi')\theta(t)\\
&=&-A\theta(t)+b.
\end{array}
\label{odetheta}
\end{equation}
Let $\vec{h}(\theta(t))$ be the driving vector field of the ODE
(\ref{odetheta}).
\begin{equation*}
\vec{h}(\theta(t))=-A\theta(t)+b.
\end{equation*}
Consider the cross-covariance matrix,
\begin{equation}
\begin{array}{ccl}
A &=& \mathrm{Cov}(\phi,\phi-\gamma\phi')\\
&=&\frac{\mathrm{Cov}(\phi,\phi)+\mathrm{Cov}(\phi-\gamma\phi',\phi-\gamma\phi')-\mathrm{Cov}(\gamma\phi',\gamma\phi')}{2}\\
&=&\frac{\mathrm{Cov}(\phi,\phi)+\mathrm{Cov}(\phi-\gamma\phi',\phi-\gamma\phi')-\gamma^2\mathrm{Cov}(\phi',\phi')}{2}\\
&=&\frac{(1-\gamma^2)\mathrm{Cov}(\phi,\phi)+\mathrm{Cov}(\phi-\gamma\phi',\phi-\gamma\phi')}{2},\\
\end{array}
\label{covariance}
\end{equation}
where we eventually used $\mathrm{Cov}(\phi',\phi')=\mathrm{Cov}(\phi,\phi)$
\footnote{The covariance matrix $\mathrm{Cov}(\phi',\phi')$ is equal to
the covariance matrix $\mathrm{Cov}(\phi,\phi)$ if the initial state is re-reachable or
initialized randomly in a Markov chain for on-policy update.}.
Note that the covariance matrix $\mathrm{Cov}(\phi,\phi)$ and
$\mathrm{Cov}(\phi-\gamma\phi',\phi-\gamma\phi')$ are semi-positive
definite. Then, the matrix $A$ is semi-positive definite because $A$ is
linearly combined by two positive-weighted semi-positive definite matrice
(\ref{covariance}).
Furthermore, $A$ is nonsingular due to the assumption.
Hence, the cross-covariance matrix $A$ is positive definite.
Therefore,
$\theta^*=A^{-1}b$ can be seen to be the unique globally asymptotically
stable equilibrium for ODE (\ref{odetheta}).
Let $\vec{h}_{\infty}(\theta)=\lim_{r\rightarrow
\infty}\frac{\vec{h}(r\theta)}{r}$. Then
$\vec{h}_{\infty}(\theta)=-A\theta$ is well-defined.
Consider now the ODE
\begin{equation}
\dot{\theta}(t)=-A\theta(t).
\label{odethetafinal}
\end{equation}
The ODE (\ref{odethetafinal}) has the origin as its unique globally asymptotically stable equilibrium.
Thus, the assumption (A1) and (A2) are verified.
\end{proof}
\subsection{Proof of Theorem 2}
\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.
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}).
0=\mathbb{E}[({\phi} - \gamma {\phi}' - \mathbb{E}[{\phi} - \gamma {\phi}']){\phi}^\top]\mathbb{E}[{\phi} {\phi}^{\top}]^{-1}\mathbb{E}[(\delta -\mathbb{E}[\delta]){\phi}]=\textbf{A}^{\top}\textbf{C}^{-1}(-\textbf{A}{\theta}+{b}).
\end{equation*}
The matrix $\textbf{A}^{\top}\textbf{C}^{-1}\textbf{A}$ is positive definite. Thus, the VMTD's solution is
$\bm{\theta}_{\text{VMTDC}}=\textbf{A}^{-1}\bm{b}$.
${\theta}_{\text{VMTDC}}=\textbf{A}^{-1}{b}$.
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),
{\theta}_{k+1}\leftarrow {\theta}_k+\zeta_k {x}(k),
\end{equation*}
\begin{equation*}
\bm{u}_{k+1}\leftarrow \bm{u}_k+\beta_k \bm{y}(k),
{u}_{k+1}\leftarrow {u}_k+\beta_k {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)],
{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*}
\bm{y}(k)=\frac{\zeta_k}{\beta_k}[\delta_{k}-\omega_k - \bm{\phi}^{\top}_k \bm{u}_k]\bm{\phi}_k.
{y}(k)=\frac{\zeta_k}{\beta_k}[\delta_{k}-\omega_k - {\phi}^{\top}_k {u}_k]{\phi}_k.
\end{equation*}
Recursion (5) can also be rewritten as
\begin{equation*}
\bm{\theta}_{k+1}\leftarrow \bm{\theta}_k+\beta_k z(k),
{\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) \bm{\phi}_k - \gamma\bm{\phi}'_{k}(\bm{\phi}^{\top}_k \bm{u}_k)],
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
$\alpha_k = o(\zeta_k)$, $\zeta_k = o(\beta_k)$, $\bm{x}(k)\rightarrow 0$, $\bm{y}(k)\rightarrow 0$, $z(k)\rightarrow 0$ almost surely as $k\rightarrow 0$.
$\alpha_k = o(\zeta_k)$, $\zeta_k = o(\beta_k)$, ${x}(k)\rightarrow 0$, ${y}(k)\rightarrow 0$, $z(k)\rightarrow 0$ almost surely as $k\rightarrow 0$.
That is that the increments in iteration (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,
\dot{{\theta}}(t) = 0,
\label{thetavmtdcFastest}
\end{equation}
\begin{equation}
\dot{\bm{u}}(t) = 0,
\dot{{u}}(t) = 0,
\label{uvmtdcFastest}
\end{equation}
\begin{equation}
\dot{\omega}(t)=\mathbb{E}[\delta_t|\bm{u}(t),\bm{\theta}(t)]-\omega(t).
\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 $\bm{\theta}(t)\equiv \bm{\theta}$
and $\bm{u}(t)\equiv \bm{u}$ when viewed from the fastest timescale.
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
$||\bm{\theta}_k-\bm{\theta}||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
$\bm{\theta}$ that depends on the initial condition $\bm{\theta}_0$ of recursion
(5) and $||\bm{u}_k-\bm{u}||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
$||{\theta}_k-{\theta}||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
${\theta}$ that depends on the initial condition ${\theta}_0$ of recursion
(5) 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
(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).
\dot{\omega}(t)=\mathbb{E}[\delta_t|{u},{\theta}]-\omega(t).
\label{omegavmtdcFastestFinal}
\end{equation}
Consider the function $h(\omega)=\mathbb{E}[\delta|\bm{\theta},\bm{u}]-\omega$,
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}), $\omega^*=\mathbb{E}[\delta|\bm{\theta},\bm{u}]$
For (\ref{omegavmtdcFastestFinal}), $\omega^*=\mathbb{E}[\delta|{\theta},{u}]$
is the unique globally asymptotically stable equilibrium.
For the ODE
\begin{equation}
......@@ -318,18 +483,18 @@ Consider now the recursion (7).
Let
$M_{k+1}=(\delta_k-\omega_k)
-\mathbb{E}[(\delta_k-\omega_k)|\mathcal{F}(k)]$,
where $\mathcal{F}(k)=\sigma(\omega_l,\bm{u}_l,\bm{\theta}_l,l\leq k;\bm{\phi}_s,\bm{\phi}_s',r_s,s<k)$,
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 $\omega_0,u_0,\bm{\theta}_0,\omega_{l+1},\bm{u}_{l+1},\bm{\theta}_{l+1},\bm{\phi}_l,\bm{\phi}_l'$,
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 $\bm{\phi}_k$, $r_k$, and $\bm{\phi}_k'$ have
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+||\bm{u}_k||^2+||\bm{\theta}_k||^2).
c_1(1+||\omega_k||^2+||{u}_k||^2+||{\theta}_k||^2).
\end{equation*}
......@@ -342,53 +507,53 @@ $||\omega_k-\omega^*||\rightarrow 0$ almost surely as $k\rightarrow \infty$.
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).
% {u}_{k+1}\leftarrow u_{k}+\zeta_{k}[\delta_{k}-\mathbb{E}[\delta_k|{u}_k,{\theta}_k] - {\phi}^{\top} (s_k) {u}_k]{\phi}(s_k).
% \end{equation*}
\begin{equation}
\dot{\bm{\theta}}(t) = 0,
\dot{{\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).
\dot{u}(t) = \mathbb{E}[(\delta_t-\mathbb{E}[\delta_t|{u}(t),{\theta}(t)]){\phi}_t|{\theta}(t)] - \textbf{C}{u}(t).
\label{uvmtdcFaster}
\end{equation}
The ODE (\ref{thetavmtdcFaster}) suggests that $\bm{\theta}(t)\equiv \bm{\theta}$ (i.e., a time invariant parameter)
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
$||\bm{\theta}_k-\bm{\theta}||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
$\bm{\theta}$ that depends on the initial condition $\bm{\theta}_0$ of recursion
$||{\theta}_k-{\theta}||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
${\theta}$ that depends on the initial condition ${\theta}_0$ of recursion
(5).
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 $\mathcal{I}(k)=\sigma(\bm{u}_l,\bm{\theta}_l,l\leq k;\bm{\phi}_s,\bm{\phi}_s',r_s,s<k)$,
$N_{k+1}=((\delta_k-\mathbb{E}[\delta_k]) - {\phi}_k {\phi}^{\top}_k {u}_k) -\mathbb{E}[((\delta_k-\mathbb{E}[\delta_k]) - {\phi}_k {\phi}^{\top}_k {u}_k)|\mathcal{I} (k)]$,
where $\mathcal{I}(k)=\sigma({u}_l,{\theta}_l,l\leq k;{\phi}_s,{\phi}_s',r_s,s<k)$,
$k\geq 1$ are the sigma fields
generated by $\bm{u}_0,\bm{\theta}_0,\bm{u}_{l+1},\bm{\theta}_{l+1},\bm{\phi}_l,\bm{\phi}_l'$,
generated by ${u}_0,{\theta}_0,{u}_{l+1},{\theta}_{l+1},{\phi}_l,{\phi}_l'$,
$0\leq l<k$.
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
Because ${\phi}_k$, $r_k$, and ${\phi}_k'$ 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).
c_2(1+||{u}_k||^2+||{\theta}_k||^2).
\end{equation*}
Because $\bm{\theta}(t)\equiv \bm{\theta}$ from (\ref{thetavmtdcFaster}), the ODE pair (\ref{thetavmtdcFaster})-(\ref{uvmtdcFaster})
Because ${\theta}(t)\equiv {\theta}$ 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).
\dot{{u}}(t) = \mathbb{E}[(\delta_t-\mathbb{E}[\delta_t|{\theta}]){\phi}_t|{\theta}] - \textbf{C}{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
Now consider the function $h({u})=\mathbb{E}[\delta_t-\mathbb{E}[\delta_t|{\theta}]|{\theta}] -\textbf{C}{u}$, 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 $h_{\infty}(\bm{u})=-\textbf{C}\bm{u}$.
${u}^* = \textbf{C}^{-1}\mathbb{E}[(\delta-\mathbb{E}[\delta|{\theta}]){\phi}|{\theta}]$ is the unique globally asymptotically
stable equilibrium. Let $h_{\infty}({u})=-\textbf{C}{u}$.
For the ODE
\begin{equation}
\dot{\bm{u}}(t) = h_{\infty}(\bm{u}(t))= -\textbf{C}\bm{u}(t),
\dot{{u}}(t) = h_{\infty}({u}(t))= -\textbf{C}{u}(t),
\label{uvmtdcInfty}
\end{equation}
the origin of (\ref{uvmtdcInfty}) is a globally asymptotically stable
......@@ -397,60 +562,60 @@ 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 $k\rightarrow \infty$.
$||{u}_k-{u}^*||\rightarrow 0$ almost surely as $k\rightarrow \infty$.
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]).
{\theta}_{k+1} \leftarrow {\theta}_{k} + \alpha_k (\delta_k -\mathbb{E}[\delta_k|{\theta}_k]) {\phi}_k\\
- \alpha_k \gamma{\phi}'_{k}({\phi}^{\top}_k \textbf{C}^{-1}\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|{\theta}_k]){\phi}|{\theta}_k]).
\end{equation}
Let $\mathcal{G}(k)=\sigma(\bm{\theta}_l,l\leq k;\bm{\phi}_s,\bm{\phi}_s',r_s,s<k)$,
Let $\mathcal{G}(k)=\sigma({\theta}_l,l\leq k;{\phi}_s,{\phi}_s',r_s,s<k)$,
$k\geq 1$ be the sigma fields
generated by $\bm{\theta}_0,\bm{\theta}_{l+1},\bm{\phi}_l,\bm{\phi}_l'$,
generated by ${\theta}_0,{\theta}_{l+1},{\phi}_l,{\phi}_l'$,
$0\leq l<k$. Let
\begin{equation*}
\begin{array}{ccl}
Z_{k+1}&=&((\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])\\
& &-\mathbb{E}[((\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)]\\
&=&((\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])\\
& &-\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].
Z_{k+1}&=&((\delta_k -\mathbb{E}[\delta_k|{\theta}_k]) {\phi}_k - \gamma {\phi}'_{k}{\phi}^{\top}_k \textbf{C}^{-1}\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|{\theta}_k]){\phi}|{\theta}_k])\\
& &-\mathbb{E}[((\delta_k -\mathbb{E}[\delta_k|{\theta}_k]) {\phi}_k - \gamma {\phi}'_{k}{\phi}^{\top}_k \textbf{C}^{-1}\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|{\theta}_k]){\phi}|{\theta}_k])|\mathcal{G}(k)]\\
&=&((\delta_k -\mathbb{E}[\delta_k|{\theta}_k]) {\phi}_k - \gamma {\phi}'_{k}{\phi}^{\top}_k \textbf{C}^{-1}\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|{\theta}_k]){\phi}|{\theta}_k])\\
& &-\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|{\theta}_k]) {\phi}_k|{\theta}_k] - \gamma\mathbb{E}[{\phi}' {\phi}^{\top}]\textbf{C}^{-1}\mathbb{E}[(\delta_k -\mathbb{E}[\delta_k|{\theta}_k]) {\phi}_k|{\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
c_3(1+||{\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
for some constant $c_3 \geq 0$, again beacuse ${\phi}_k$, $r_k$, and ${\phi}_k'$ have
uniformly bounded second moments, it can be seen that for some constant.
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)].
\dot{{\theta}}(t) = (\textbf{I} - \mathbb{E}[\gamma {\phi}' {\phi}^{\top}]\textbf{C}^{-1})\mathbb{E}[(\delta -\mathbb{E}[\delta|{\theta}(t)]) {\phi}|{\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}),
\vec{h}({\theta}(t))&=&(\textbf{I} - \mathbb{E}[\gamma {\phi}' {\phi}^{\top}]\textbf{C}^{-1})\mathbb{E}[(\delta -\mathbb{E}[\delta|{\theta}(t)]) {\phi}|{\theta}(t)]\\
&=&(\textbf{C} - \mathbb{E}[\gamma {\phi}' {\phi}^{\top}])\textbf{C}^{-1}\mathbb{E}[(\delta -\mathbb{E}[\delta|{\theta}(t)]) {\phi}|{\theta}(t)]\\
&=& (\mathbb{E}[{\phi} {\phi}^{\top}] - \mathbb{E}[\gamma {\phi}' {\phi}^{\top}])\textbf{C}^{-1}\mathbb{E}[(\delta -\mathbb{E}[\delta|{\theta}(t)]) {\phi}|{\theta}(t)]\\
&=& \textbf{A}^{\top}\textbf{C}^{-1}(-\textbf{A}{\theta}(t)+{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 $\textbf{C}=\mathbb{E}[\bm{\phi}\bm{\phi}^{\top}]$
because $\mathbb{E}[(\delta -\mathbb{E}[\delta|{\theta}(t)]) {\phi}|{\theta}(t)]=-\textbf{A}{\theta}(t)+{b}$, where
$\textbf{A} = \mathrm{Cov}({\phi},{\phi}-\gamma{\phi}')$, ${b}=\mathrm{Cov}(r,{\phi})$, and $\textbf{C}=\mathbb{E}[{\phi}{\phi}^{\top}]$
Therefore,
$\bm{\theta}^*=\textbf{A}^{-1}\bm{b}$ can be seen to be the unique globally asymptotically
${\theta}^*=\textbf{A}^{-1}{b}$ 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.
Let $\vec{h}_{\infty}({\theta})=\lim_{r\rightarrow
\infty}\frac{\vec{h}(r{\theta})}{r}$. Then
$\vec{h}_{\infty}({\theta})=-\textbf{A}^{\top}\textbf{C}^{-1}\textbf{A}{\theta}$ is well-defined.
Consider now the ODE
\begin{equation}
\dot{\bm{\theta}}(t)=-\textbf{A}^{\top}\textbf{C}^{-1}\textbf{A}\bm{\theta}(t).
\dot{{\theta}}(t)=-\textbf{A}^{\top}\textbf{C}^{-1}\textbf{A}{\theta}(t).
\label{odethetavmtdcfinal}
\end{equation}
......@@ -462,11 +627,11 @@ Thus, the assumption (A1) and (A2) are verified.
The proof is given above.
In the fastest time scale, the parameter $w$ converges to
$\mathbb{E}[\delta|\bm{u}_k,\bm{\theta}_k]$.
$\mathbb{E}[\delta|{u}_k,{\theta}_k]$.
In the second fast time scale,
the parameter $u$ converges to $\textbf{C}^{-1}\mathbb{E}[(\delta-\mathbb{E}[\delta|\bm{\theta}_k])\bm{\phi}|\bm{\theta}_k]$.
the parameter $u$ converges to $\textbf{C}^{-1}\mathbb{E}[(\delta-\mathbb{E}[\delta|{\theta}_k]){\phi}|{\theta}_k]$.
In the slower time scale,
the parameter $\bm{\theta}$ converges to $\textbf{A}^{-1}\bm{b}$.
the parameter ${\theta}$ converges to $\textbf{A}^{-1}{b}$.
\end{proof}
\subsection{Proof of Theorem 2}
......@@ -478,49 +643,49 @@ the parameter $\bm{\theta}$ converges to $\textbf{A}^{-1}\bm{b}$.
\cite{borkar1997stochastic}.
The VMTD's solution is
$\bm{\theta}_{\text{VMETD}}=\textbf{A}_{\text{VMETD}}^{-1}\bm{b}_{\text{VMETD}}$.
${\theta}_{\text{VMETD}}=\textbf{A}_{\text{VMETD}}^{-1}{b}_{\text{VMETD}}$.
First, note that recursion (19) can be rewritten as
\begin{equation*}
\bm{\theta}_{k+1}\leftarrow \bm{\theta}_k+\beta_k\bm{\xi}(k),
{\theta}_{k+1}\leftarrow {\theta}_k+\beta_k{\xi}(k),
\end{equation*}
where
\begin{equation*}
\bm{\xi}(k)=\frac{\alpha_k}{\beta_k} (F_k \rho_k\delta_k - \omega_{k+1})\bm{\phi}_k
{\xi}(k)=\frac{\alpha_k}{\beta_k} (F_k \rho_k\delta_k - \omega_{k+1}){\phi}_k
\end{equation*}
Due to the settings of step-size schedule $\alpha_k = o(\beta_k)$,
$\bm{\xi}(k)\rightarrow 0$ almost surely as $k\rightarrow\infty$.
${\xi}(k)\rightarrow 0$ almost surely as $k\rightarrow\infty$.
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,
\dot{{\theta}}(t) = 0,
\label{thetaFast}
\end{equation}
\begin{equation}
\dot{\omega}(t)=\mathbb{E}_{\mu}[F_t\rho_t\delta_t|\bm{\theta}(t)]-\omega(t).
\dot{\omega}(t)=\mathbb{E}_{\mu}[F_t\rho_t\delta_t|{\theta}(t)]-\omega(t).
\label{omegaFast}
\end{equation}
Based on the ODE (\ref{thetaFast}), $\bm{\theta}(t)\equiv \bm{\theta}$ when
Based on the ODE (\ref{thetaFast}), ${\theta}(t)\equiv {\theta}$ when
viewed from the faster 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
$||{\theta}_k-{\theta}||\rightarrow 0$ a.s. as $k\rightarrow \infty$ for some
${\theta}$ that depends on the initial condition ${\theta}_0$ of recursion
(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).
\dot{\omega}(t)=\mathbb{E}_{\mu}[F_t\rho_t\delta_t|{\theta}]-\omega(t).
\label{omegaFastFinal}
\end{equation}
Consider the function $h(\omega)=\mathbb{E}_{\mu}[F\rho\delta|\bm{\theta}]-\omega$,
Consider the function $h(\omega)=\mathbb{E}_{\mu}[F\rho\delta|{\theta}]-\omega$,
i.e., the driving vector field of the ODE (\ref{omegaFastFinal}).
It is easy to find that the function $h$ is Lipschitz with coefficient
$-1$.
Let $h_{\infty}(\cdot)$ be the function defined by
$h_{\infty}(\omega)=\lim_{x\rightarrow \infty}\frac{h(x\omega)}{x}$.
Then $h_{\infty}(\omega)= -\omega$, is well-defined.
For (\ref{omegaFastFinal}), $\omega^*=\mathbb{E}_{\mu}[F\rho\delta|\bm{\theta}]$
For (\ref{omegaFastFinal}), $\omega^*=\mathbb{E}_{\mu}[F\rho\delta|{\theta}]$
is the unique globally asymptotically stable equilibrium.
For the ODE
\begin{equation}
......@@ -537,18 +702,18 @@ The VMTD's solution is
Let
$M_{k+1}=(F_k\rho_k\delta_k-\omega_k)
-\mathbb{E}_{\mu}[(F_k\rho_k\delta_k-\omega_k)|\mathcal{F}(k)]$,
where $\mathcal{F}(k)=\sigma(\omega_l,\bm{\theta}_l,l\leq k;\bm{\phi}_s,\bm{\phi}_s',r_s,s<k)$,
where $\mathcal{F}(k)=\sigma(\omega_l,{\theta}_l,l\leq k;{\phi}_s,{\phi}_s',r_s,s<k)$,
$k\geq 1$ are the sigma fields
generated by $\omega_0,\bm{\theta}_0,\omega_{l+1},\bm{\theta}_{l+1},\bm{\phi}_l,\bm{\phi}_l'$,
generated by $\omega_0,{\theta}_0,\omega_{l+1},{\theta}_{l+1},{\phi}_l,{\phi}_l'$,
$0\leq l<k$.
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 $\bm{\phi}_k$, $r_k$, and $\bm{\phi}_k'$ have
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+||\bm{\theta}_k||^2).
c_1(1+||\omega_k||^2+||{\theta}_k||^2).
\end{equation*}
......@@ -561,23 +726,23 @@ The VMTD's solution is
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.
% {\theta}_{k+1}\leftarrow
% {\theta}_{k}+\alpha_k(F_k\rho_k\delta_k-\mathbb{E}_{\mu}[F_k\rho_k\delta_k|{\theta}_k]){\phi}_k.
% \end{equation*}
\begin{equation*}
\begin{split}
\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 -\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}_{\bm{b}_{\text{VMETD},k}}
-\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\}
{\theta}_{k+1}&\leftarrow {\theta}_k+\alpha_k (F_k \rho_k\delta_k - \omega_k){\phi}_k -\alpha_k \omega_{k+1}{\phi}_k\\
&={\theta}_{k}+\alpha_k(F_k\rho_k\delta_k-\mathbb{E}_{\mu}[F_k\rho_k\delta_k|{\theta}_k]){\phi}_k\\
&={\theta}_k+\alpha_k F_k \rho_k (R_{k+1}+\gamma {\theta}_k^{\top}{\phi}_{k+1}-{\theta}_k^{\top}{\phi}_k){\phi}_k -\alpha_k \mathbb{E}_{\mu}[F_k \rho_k \delta_k]{\phi}_k\\
&= {\theta}_k+\alpha_k \{\underbrace{(F_k\rho_kR_{k+1}-\mathbb{E}_{\mu}[F_k\rho_k R_{k+1}]){\phi}_k}_{{b}_{\text{VMETD},k}}
-\underbrace{(F_k\rho_k{\phi}_k({\phi}_k-\gamma{\phi}_{k+1})^{\top}-{\phi}_k\mathbb{E}_{\mu}[F_k\rho_k ({\phi}_k-\gamma{\phi}_{k+1})]^{\top})}_{\textbf{A}_{\text{VMETD},k}}{\theta}_k\}
\end{split}
\end{equation*}
Let $\mathcal{G}(k)=\sigma(\bm{\theta}_l,l\leq k;\bm{\phi}_s,\bm{\phi}_s',r_s,s<k)$,
Let $\mathcal{G}(k)=\sigma({\theta}_l,l\leq k;{\phi}_s,{\phi}_s',r_s,s<k)$,
$k\geq 1$ be the sigma fields
generated by $\bm{\theta}_0,\bm{\theta}_{l+1},\bm{\phi}_l,\bm{\phi}_l'$,
generated by ${\theta}_0,{\theta}_{l+1},{\phi}_l,{\phi}_l'$,
$0\leq l<k$.
Let
$
......@@ -585,17 +750,17 @@ The VMTD's solution is
$
where
\begin{equation*}
Y_{k}=(F_k\rho_k\delta_k-\mathbb{E}_{\mu}[F_k\rho_k\delta_k|\bm{\theta}_k])\bm{\phi}_k.
Y_{k}=(F_k\rho_k\delta_k-\mathbb{E}_{\mu}[F_k\rho_k\delta_k|{\theta}_k]){\phi}_k.
\end{equation*}
Consequently,
\begin{equation*}
\begin{array}{ccl}
\mathbb{E}_{\mu}[Y_k|\mathcal{G}(k)]&=&\mathbb{E}_{\mu}[(F_k\rho_k\delta_k-\mathbb{E}_{\mu}[F_k\rho_k\delta_k|\bm{\theta}_k])\bm{\phi}_k|\mathcal{G}(k)]\\
&=&\mathbb{E}_{\mu}[F_k\rho_k\delta_k\bm{\phi}_k|\bm{\theta}_k]
-\mathbb{E}_{\mu}[\mathbb{E}_{\mu}[F_k\rho_k\delta_k|\bm{\theta}_k]\bm{\phi}_k]\\
&=&\mathbb{E}_{\mu}[F_k\rho_k\delta_k\bm{\phi}_k|\bm{\theta}_k]
-\mathbb{E}_{\mu}[F_k\rho_k\delta_k|\bm{\theta}_k]\mathbb{E}_{\mu}[\bm{\phi}_k]\\
&=&\mathrm{Cov}(F_k\rho_k\delta_k|\bm{\theta}_k,\bm{\phi}_k),
\mathbb{E}_{\mu}[Y_k|\mathcal{G}(k)]&=&\mathbb{E}_{\mu}[(F_k\rho_k\delta_k-\mathbb{E}_{\mu}[F_k\rho_k\delta_k|{\theta}_k]){\phi}_k|\mathcal{G}(k)]\\
&=&\mathbb{E}_{\mu}[F_k\rho_k\delta_k{\phi}_k|{\theta}_k]
-\mathbb{E}_{\mu}[\mathbb{E}_{\mu}[F_k\rho_k\delta_k|{\theta}_k]{\phi}_k]\\
&=&\mathbb{E}_{\mu}[F_k\rho_k\delta_k{\phi}_k|{\theta}_k]
-\mathbb{E}_{\mu}[F_k\rho_k\delta_k|{\theta}_k]\mathbb{E}_{\mu}[{\phi}_k]\\
&=&\mathrm{Cov}(F_k\rho_k\delta_k|{\theta}_k,{\phi}_k),
\end{array}
\end{equation*}
where $\mathrm{Cov}(\cdot,\cdot)$ is a covariance operator.
......@@ -603,61 +768,61 @@ The VMTD's solution is
Thus,
\begin{equation*}
\begin{array}{ccl}
Z_{k+1}&=&(F_k\rho_k\delta_k-\mathbb{E}[\delta_k|\bm{\theta}_k])\bm{\phi}_k-\mathrm{Cov}(F_k\rho_k\delta_k|\bm{\theta}_k,\bm{\phi}_k).
Z_{k+1}&=&(F_k\rho_k\delta_k-\mathbb{E}[\delta_k|{\theta}_k]){\phi}_k-\mathrm{Cov}(F_k\rho_k\delta_k|{\theta}_k,{\phi}_k).
\end{array}
\end{equation*}
It is easy to verify that $Z_{k+1},k\geq0$ are integrable random variables that
satisfy $\mathbb{E}[Z_{k+1}|\mathcal{G}(k)]=0$, $\forall k\geq0$.
Also, because $\bm{\phi}_k$, $r_k$, and $\bm{\phi}_k'$ have
Also, because ${\phi}_k$, $r_k$, and ${\phi}_k'$ have
uniformly bounded second moments, it can be seen that for some constant
$c_2>0$, $\forall k\geq0$,
\begin{equation*}
\mathbb{E}[||Z_{k+1}||^2|\mathcal{G}(k)]\leq
c_2(1+||\bm{\theta}_k||^2).
c_2(1+||{\theta}_k||^2).
\end{equation*}
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}}.
\dot{{\theta}}(t)&=&-\textbf{A}_{\text{VMETD}}{\theta}(t)+{b}_{\text{VMETD}}.
\end{array}
\label{odetheta}
\end{equation}
\begin{equation}
\begin{split}
\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}]- \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}[\underbrace{\bm{\phi}_k}_{X}\underbrace{F_k \rho_k (\bm{\phi}_k - \gamma \bm{\phi}_{k+1})^{\top}}_{Y}]- \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}]- \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}]- \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}\\
&=\sum_{s} f(s) \bm{\phi}(s)(\bm{\phi}(s) - \gamma \sum_{s'}[\textbf{P}_{\pi}]_{ss'}\bm{\phi}(s'))^{\top} - \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}} \\
&= \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[F_k \rho_k {\phi}_k ({\phi}_k - \gamma {\phi}_{k+1})^{\top}]- \lim_{k\rightarrow \infty} \mathbb{E}_{\mu}[ {\phi}_k]\mathbb{E}_{\mu}[F_k \rho_k ({\phi}_k - \gamma {\phi}_{k+1})]^{\top}\\
% &= \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[\underbrace{{\phi}_k}_{X}\underbrace{F_k \rho_k ({\phi}_k - \gamma {\phi}_{k+1})^{\top}}_{Y}]- \lim_{k\rightarrow \infty} \mathbb{E}_{\mu}[ {\phi}_k]\mathbb{E}_{\mu}[F_k \rho_k ({\phi}_k - \gamma {\phi}_{k+1})]^{\top}\\
&= \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[{\phi}_kF_k \rho_k ({\phi}_k - \gamma {\phi}_{k+1})^{\top}]- \lim_{k\rightarrow \infty} \mathbb{E}_{\mu}[ {\phi}_k]\mathbb{E}_{\mu}[F_k \rho_k ({\phi}_k - \gamma {\phi}_{k+1})]^{\top}\\
&= \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[{\phi}_kF_k \rho_k ({\phi}_k - \gamma {\phi}_{k+1})^{\top}]- \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[ {\phi}_k]\lim_{k \rightarrow \infty}\mathbb{E}_{\mu}[F_k \rho_k ({\phi}_k - \gamma {\phi}_{k+1})]^{\top}\\
&=\sum_{s} f(s) {\phi}(s)({\phi}(s) - \gamma \sum_{s'}[\textbf{P}_{\pi}]_{ss'}{\phi}(s'))^{\top} - \sum_{s} d_{\mu}(s) {\phi}(s) * \sum_{s} f(s)({\phi}(s) - \gamma \sum_{s'}[\textbf{P}_{\pi}]_{ss'}{\phi}(s'))^{\top} \\
&={{\Phi}}^{\top} \textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi}) {\Phi} - {{\Phi}}^{\top} \textbf{d}_{\mu} \textbf{f}^{\top} (\textbf{I} - \gamma \textbf{P}_{\mu}) {\Phi} \\
&={{\Phi}}^{\top} (\textbf{F} - \textbf{d}_{\mu} \textbf{f}^{\top}) (\textbf{I} - \gamma \textbf{P}_{\pi}){{\Phi}} \\
&={{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{f}^{\top} (\textbf{I} - \gamma \textbf{P}_{\pi})){{\Phi}} \\
&={{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} ){{\Phi}} \\
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\bm{b}_{\text{VMETD}}&=\lim_{k \rightarrow \infty} \mathbb{E}[\bm{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} \\
{b}_{\text{VMETD}}&=\lim_{k \rightarrow \infty} \mathbb{E}[{b}_{\text{VMETD},k}]\\
&= \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[F_k\rho_kR_{k+1}{\phi}_k]- \lim_{k\rightarrow \infty} \mathbb{E}_{\mu}[{\phi}_k]\mathbb{E}_{\mu}[F_k\rho_kR_{k+1}]\\
&= \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[{\phi}_kF_k\rho_kR_{k+1}]- \lim_{k\rightarrow \infty} \mathbb{E}_{\mu}[ {\phi}_k]\mathbb{E}_{\mu}[{\phi}_k]\mathbb{E}_{\mu}[F_k\rho_kR_{k+1}]\\
&= \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[{\phi}_kF_k\rho_kR_{k+1}]- \lim_{k \rightarrow \infty} \mathbb{E}_{\mu}[ {\phi}_k]\lim_{k \rightarrow \infty}\mathbb{E}_{\mu}[F_k\rho_kR_{k+1}]\\
&=\sum_{s} f(s) {\phi}(s)r_{\pi} - \sum_{s} d_{\mu}(s) {\phi}(s) * \sum_{s} f(s)r_{\pi} \\
&={{\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
Let $\vec{h}({\theta}(t))$ be the driving vector field of the ODE
(\ref{odetheta}).
\begin{equation*}
\vec{h}(\bm{\theta}(t))=-\textbf{A}_{\text{VMETD}}\bm{\theta}(t)+\bm{b}_{\text{VMETD}}.
\vec{h}({\theta}(t))=-\textbf{A}_{\text{VMETD}}{\theta}(t)+{b}_{\text{VMETD}}.
\end{equation*}
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
An ${\Phi}^{\top}{\text{X}}{\Phi}$ matrix of this
form will be positive definite whenever the matrix ${\text{X}}$ is positive definite.
Any matrix ${\text{X}}$ is positive definite if and only if
the symmetric matrix ${\text{S}}={\text{X}}+{\text{X}}^{\top}$ is positive definite.
Any symmetric real matrix ${\text{S}}$ 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}.
......@@ -693,14 +858,14 @@ The VMTD's solution is
Therefore,
$\bm{\theta}^*=\textbf{A}_{\text{VMETD}}^{-1}\bm{b}_{\text{VMETD}}$ can be seen to be the unique globally asymptotically
${\theta}^*=\textbf{A}_{\text{VMETD}}^{-1}{b}_{\text{VMETD}}$ can be seen to be the unique globally asymptotically
stable equilibrium for ODE (\ref{odetheta}).
Let $\vec{h}_{\infty}(\bm{\theta})=\lim_{r\rightarrow
\infty}\frac{\vec{h}(r\bm{\theta})}{r}$. Then
$\vec{h}_{\infty}(\bm{\theta})=-\textbf{A}_{\text{VMETD}}\bm{\theta}$ is well-defined.
Let $\vec{h}_{\infty}({\theta})=\lim_{r\rightarrow
\infty}\frac{\vec{h}(r{\theta})}{r}$. Then
$\vec{h}_{\infty}({\theta})=-\textbf{A}_{\text{VMETD}}{\theta}$ is well-defined.
Consider now the ODE
\begin{equation}
\dot{\bm{\theta}}(t)=-\textbf{A}_{\text{VMETD}}\bm{\theta}(t).
\dot{{\theta}}(t)=-\textbf{A}_{\text{VMETD}}{\theta}(t).
\label{odethetafinal}
\end{equation}
The ODE (\ref{odethetafinal}) has the origin as its unique globally asymptotically stable equilibrium.
......@@ -719,25 +884,25 @@ The VMTD's solution is
% {
% \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&${\bm{\Phi}}^{\top}(\textbf{D}_{\pi}-\textbf{d}_{\pi} \textbf{d}_{\pi}^{\top} )(\textbf{I} - \gamma\textbf{P}_{\pi}){\bm{\Phi}}$
% &$\checkmark$&$\bm{\Phi}^{\top}(\textbf{D}_{\pi}-\textbf{d}_{\pi} \textbf{d}_{\pi}^{\top})\textbf{r}_{\pi}$\\
% Algorithm&Key matrix $\textbf{A}$&{Positive definite}&{${b}$}\\\midrule
% On-policy TD&${\Phi}^{\top}\textbf{D}_{\pi}(\textbf{I}-\gamma
% \textbf{P}_{\pi}){\Phi}$&$\checkmark$&${b}_{\text{on}}={\Phi}^{\top}\textbf{D}_{\pi}\textbf{r}_{\pi}$\\
% On-policy VMTD&${{\Phi}}^{\top}(\textbf{D}_{\pi}-\textbf{d}_{\pi} \textbf{d}_{\pi}^{\top} )(\textbf{I} - \gamma\textbf{P}_{\pi}){{\Phi}}$
% &$\checkmark$&${\Phi}^{\top}(\textbf{D}_{\pi}-\textbf{d}_{\pi} \textbf{d}_{\pi}^{\top})\textbf{r}_{\pi}$\\
% \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& $\textbf{A}_{\text{off}}^{\top}\textbf{C}^{-1}\textbf{A}_{\text{off}}$&$\checkmark$&$\textbf{A}_{\text{off}}^{\top}\textbf{C}^{-1}\bm{b}_{\text{off}}$
% Off-policy TD&$\textbf{A}_{\text{off}}={{\Phi}}^{\top}\textbf{D}_{\mu}(\textbf{I}-\gamma
% \textbf{P}_{\pi}){{\Phi}}$&$\times$&${b}_{\text{off}}={\Phi}^{\top}\textbf{D}_{\mu}\textbf{r}_{\pi}$\\
% TDC& $\textbf{A}_{\text{off}}^{\top}\textbf{C}^{-1}\textbf{A}_{\text{off}}$&$\checkmark$&$\textbf{A}_{\text{off}}^{\top}\textbf{C}^{-1}{b}_{\text{off}}$
% \\
% ETD& ${\bm{\Phi}}^{\top}\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi}){\bm{\Phi}}$
% &$\checkmark$&$\bm{\Phi}^{\top}\textbf{F}\textbf{r}_{\pi}$\\
% ETD& ${{\Phi}}^{\top}\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi}){{\Phi}}$
% &$\checkmark$&${\Phi}^{\top}\textbf{F}\textbf{r}_{\pi}$\\
% \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& $\textbf{A}_{\text{VMTD}}^{\top}\textbf{C}^{-1}\textbf{A}_{\text{VMTD}}$&$\checkmark$&$\textbf{A}_{\text{VMTD}}^{\top}\textbf{C}^{-1}\bm{b}_{\text{VMTD}}$
% Off-policy VMTD&$\textbf{A}_{\text{VMTD}}={{\Phi}}^{\top} (\textbf{D}_{\mu}-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} )(\textbf{I} - \gamma\textbf{P}_{\pi}){{\Phi}}$
% &$\times$&${b}_{\text{VMTD}}={\Phi}^{\top}(\textbf{D}_{\mu}-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top})\textbf{r}_{\pi}$\\
% VMTDC& $\textbf{A}_{\text{VMTD}}^{\top}\textbf{C}^{-1}\textbf{A}_{\text{VMTD}}$&$\checkmark$&$\textbf{A}_{\text{VMTD}}^{\top}\textbf{C}^{-1}{b}_{\text{VMTD}}$
% \\
% VMETD& ${\bm{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} ){\bm{\Phi}}$
% &$\checkmark$&$\bm{\Phi}^{\top}(\textbf{F}-\textbf{d}_{\mu} \textbf{f}^{\top})\textbf{r}_{\pi}$\\
% VMETD& ${{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} ){{\Phi}}$
% &$\checkmark$&${\Phi}^{\top}(\textbf{F}-\textbf{d}_{\mu} \textbf{f}^{\top})\textbf{r}_{\pi}$\\
% \bottomrule
% \end{tabular}
% }
......@@ -835,50 +1000,57 @@ The VMTD's solution is
\section{Experimental details}
\label{experimentaldetails}
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:
\begin{equation*}
\Phi_{Counter}=\left[
\begin{array}{cccccccc}
1 & 2& 0& 0& 0& 0& 0& 0\\
1 & 0& 2& 0& 0& 0& 0& 0\\
1 & 0& 0& 2& 0& 0& 0& 0\\
1 & 0& 0& 0& 2& 0& 0& 0\\
1 & 0& 0& 0& 0& 2& 0& 0\\
1 & 0& 0& 0& 0& 0& 2& 0\\
2 & 0& 0& 0& 0& 0& 0& 1
\end{array}\right]
\end{equation*}
% 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:
% \begin{equation*}
% \Phi_{Counter}=\left[
% \begin{array}{cccccccc}
% 1 & 2& 0& 0& 0& 0& 0& 0\\
% 1 & 0& 2& 0& 0& 0& 0& 0\\
% 1 & 0& 0& 2& 0& 0& 0& 0\\
% 1 & 0& 0& 0& 2& 0& 0& 0\\
% 1 & 0& 0& 0& 0& 2& 0& 0\\
% 1 & 0& 0& 0& 0& 0& 2& 0\\
% 2 & 0& 0& 0& 0& 0& 0& 1
% \end{array}\right]
% \end{equation*}
2-state version of Baird's off-policy counterexample: All learning rates follow linear learning rate decay.
For TD algorithm, $\frac{\alpha_k}{\omega_k}=4$ and $\alpha_0 = 0.1$.
For TDC algorithm, $\frac{\alpha_k}{\zeta_k}=5$ and $\alpha_0 = 0.1$.
For VMTDC algorithm, $\frac{\alpha_k}{\zeta_k}=5$, $\frac{\alpha_k}{\omega_k}=4$,and $\alpha_0 = 0.1$.
For VMTD algorithm, $\frac{\alpha_k}{\omega_k}=4$ and $\alpha_0 = 0.1$.
2-state version of Baird's off-policy counterexample: All learning rates follow linear learning rate decay.
For TD algorithm, $\frac{\alpha_k}{\omega_k}=4$ and $\alpha_0 = 0.1$.
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 $\alpha_0 = 0.1$.
For VMTDC algorithm, $\frac{\alpha_k}{\zeta_k}=5$, $\frac{\alpha_k}{\omega_k}=4$,and $\alpha_0 = 0.1$.For VMETD algorithm, $\frac{\alpha_k}{\omega_k}=4$ and $\alpha_0 = 0.1$.
For VMTD 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 $\alpha_0 = 0.1$.
% 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.
......@@ -887,28 +1059,28 @@ 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.
% \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
......@@ -918,15 +1090,14 @@ while the default settings are used for the other two environments. In Mountain
\hline
\multicolumn{1}{c|}{\diagbox{algorithms($lr$)}{envs}} &Maze &Cliff walking &Mountain Car &Acrobot \\
\hline
% Sarsa($\alpha$)&$0.1$ &$0.1$ &$0.1$ &$0.1$ \\
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($\alpha,\beta$)&$0.1,0.001$ &$0.1,\text{1e-4}$ &$0.1,\text{1e-4}$ &$0.1,\text{1e-4}$ \\
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($\alpha,\beta$)&$0.1,0.001$ &$0.1,\text{1e-4}$ &$0.1,\text{1e-4}$ &$0.1,\text{1e-4}$ \\
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}
......
NEW_aaai/anonymous-submission-latex-2025.aux
......@@ -26,81 +26,69 @@
\newlabel{introduction}{{}{1}}
\citation{Sutton2018book}
\citation{Sutton2018book}
\citation{sutton2016emphatic}
\newlabel{preliminaries}{{}{2}}
\newlabel{valuefunction}{{}{2}}
\newlabel{linearvaluefunction}{{1}{2}}
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\newlabel{fvmetd}{{2}{3}}
\newlabel{thetaetd}{{}{3}}
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{alg:algorithm 2}{{1}{3}}
\newlabel{alg:algorithm 5}{{2}{4}}
\newlabel{thetavmtdc}{{5}{4}}
\newlabel{uvmtdc}{{6}{4}}
\newlabel{omegavmtdc}{{7}{4}}
\newlabel{rho_VPBE}{{8}{4}}
\newlabel{tab:min_eigenvalues}{{1}{3}}
\newlabel{delta}{{3}{3}}
\newlabel{omega}{{4}{3}}
\newlabel{theta}{{5}{3}}
\newlabel{thetavmtdc}{{8}{4}}
\newlabel{uvmtdc}{{9}{4}}
\newlabel{omegavmtdc}{{10}{4}}
\newlabel{fvmetd}{{11}{4}}
\newlabel{thetavmetd}{{12}{4}}
\newlabel{omegavmetd}{{13}{4}}
\citation{borkar1997stochastic}
\citation{sutton2009fast}
\citation{hirsch1989convergent}
\newlabel{theorem2}{{1}{5}}
\newlabel{thetavmtdcFastest}{{14}{5}}
\newlabel{uvmtdcFastest}{{15}{5}}
\newlabel{omegavmtdcFastest}{{16}{5}}
\newlabel{omegavmtdcFastestFinal}{{17}{5}}
\newlabel{omegavmtdcInfty}{{18}{5}}
\citation{borkar2000ode}
\citation{borkar2000ode}
\citation{borkar2000ode}
\citation{borkar1997stochastic}
\newlabel{theorem1}{{1}{5}}
\newlabel{th1proof}{{}{5}}
\newlabel{covariance}{{14}{5}}
\newlabel{theorem2}{{2}{5}}
\newlabel{theorem3}{{3}{5}}
\newlabel{rowsum}{{15}{5}}
\newlabel{columnsum}{{16}{5}}
\citation{ng1999policy}
\citation{devlin2012dynamic}
\newlabel{theorem3}{{2}{6}}
\newlabel{rowsum}{{19}{6}}
\newlabel{example_bias}{{2}{6}}
\newlabel{columnsum}{{20}{6}}
\bibdata{aaai25}
\bibcite{baird1995residual}{{1}{1995}{{Baird et~al.}}{{}}}
\newlabel{2-state}{{1(a)}{7}}
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\newlabel{sub@2-state}{{(a)}{7}}
\newlabel{7-state}{{1(b)}{7}}
\newlabel{7-state}{{3(b)}{7}}
\newlabel{sub@7-state}{{(b)}{7}}
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\newlabel{CliffWalkingFull}{{3(d)}{7}}
\newlabel{sub@CliffWalkingFull}{{(d)}{7}}
\newlabel{MountainCarFull}{{1(e)}{7}}
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\newlabel{sub@MountainCarFull}{{(e)}{7}}
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\bibcite{basserrano2021logistic}{{2}{2021}{{Bas-Serrano et~al.}}{{Bas-Serrano, Curi, Krause, and Neu}}}
\bibcite{borkar1997stochastic}{{3}{1997}{{Borkar}}{{}}}
\bibcite{borkar2000ode}{{4}{2000}{{Borkar and Meyn}}{{}}}
\bibcite{chen2023modified}{{5}{2023}{{Chen et~al.}}{{Chen, Ma, Li, Yang, Yang, and Gao}}}
\bibcite{devlin2012dynamic}{{6}{2012}{{Devlin and Kudenko}}{{}}}
\bibcite{feng2019kernel}{{7}{2019}{{Feng, Li, and Liu}}{{}}}
\bibcite{givchi2015quasi}{{8}{2015}{{Givchi and Palhang}}{{}}}
\bibcite{hackman2012faster}{{9}{2012}{{Hackman}}{{}}}
\bibcite{hallak2016generalized}{{10}{2016}{{Hallak et~al.}}{{Hallak, Tamar, Munos, and Mannor}}}
\bibcite{hirsch1989convergent}{{11}{1989}{{Hirsch}}{{}}}
\bibcite{johnson2013accelerating}{{12}{2013}{{Johnson and Zhang}}{{}}}
\bibcite{korda2015td}{{13}{2015}{{Korda and La}}{{}}}
\bibcite{liu2018proximal}{{14}{2018}{{Liu et~al.}}{{Liu, Gemp, Ghavamzadeh, Liu, Mahadevan, and Petrik}}}
\bibcite{liu2015finite}{{15}{2015}{{Liu et~al.}}{{Liu, Liu, Ghavamzadeh, Mahadevan, and Petrik}}}
\bibcite{liu2016proximal}{{16}{2016}{{Liu et~al.}}{{Liu, Liu, Ghavamzadeh, Mahadevan, and Petrik}}}
\bibcite{ng1999policy}{{17}{1999}{{Ng, Harada, and Russell}}{{}}}
\bibcite{pan2017accelerated}{{18}{2017}{{Pan, White, and White}}{{}}}
\bibcite{sutton2009fast}{{19}{2009}{{Sutton et~al.}}{{Sutton, Maei, Precup, Bhatnagar, Silver, Szepesv{\'a}ri, and Wiewiora}}}
\bibcite{sutton1988learning}{{20}{1988}{{Sutton}}{{}}}
\bibcite{Sutton2018book}{{21}{2018}{{Sutton and Barto}}{{}}}
\bibcite{sutton2008convergent}{{22}{2008}{{Sutton, Maei, and Szepesv{\'a}ri}}{{}}}
\bibcite{sutton2016emphatic}{{23}{2016}{{Sutton, Mahmood, and White}}{{}}}
\bibcite{tsitsiklis1997analysis}{{24}{1997}{{Tsitsiklis and Van~Roy}}{{}}}
\bibcite{xu2019reanalysis}{{25}{2019}{{Xu et~al.}}{{Xu, Wang, Zhou, and Liang}}}
\bibcite{zhang2022truncated}{{26}{2022}{{Zhang and Whiteson}}{{}}}
\bibcite{chen2023modified}{{4}{2023}{{Chen et~al.}}{{Chen, Ma, Li, Yang, Yang, and Gao}}}
\bibcite{devlin2012dynamic}{{5}{2012}{{Devlin and Kudenko}}{{}}}
\bibcite{feng2019kernel}{{6}{2019}{{Feng, Li, and Liu}}{{}}}
\bibcite{givchi2015quasi}{{7}{2015}{{Givchi and Palhang}}{{}}}
\bibcite{hackman2012faster}{{8}{2012}{{Hackman}}{{}}}
\bibcite{hallak2016generalized}{{9}{2016}{{Hallak et~al.}}{{Hallak, Tamar, Munos, and Mannor}}}
\bibcite{johnson2013accelerating}{{10}{2013}{{Johnson and Zhang}}{{}}}
\bibcite{korda2015td}{{11}{2015}{{Korda and La}}{{}}}
\bibcite{liu2018proximal}{{12}{2018}{{Liu et~al.}}{{Liu, Gemp, Ghavamzadeh, Liu, Mahadevan, and Petrik}}}
\bibcite{liu2015finite}{{13}{2015}{{Liu et~al.}}{{Liu, Liu, Ghavamzadeh, Mahadevan, and Petrik}}}
\bibcite{liu2016proximal}{{14}{2016}{{Liu et~al.}}{{Liu, Liu, Ghavamzadeh, Mahadevan, and Petrik}}}
\bibcite{ng1999policy}{{15}{1999}{{Ng, Harada, and Russell}}{{}}}
\bibcite{pan2017accelerated}{{16}{2017}{{Pan, White, and White}}{{}}}
\bibcite{sutton2009fast}{{17}{2009}{{Sutton et~al.}}{{Sutton, Maei, Precup, Bhatnagar, Silver, Szepesv{\'a}ri, and Wiewiora}}}
\bibcite{sutton1988learning}{{18}{1988}{{Sutton}}{{}}}
\bibcite{Sutton2018book}{{19}{2018}{{Sutton and Barto}}{{}}}
\bibcite{sutton2008convergent}{{20}{2008}{{Sutton, Maei, and Szepesv{\'a}ri}}{{}}}
\bibcite{sutton2016emphatic}{{21}{2016}{{Sutton, Mahmood, and White}}{{}}}
\bibcite{tsitsiklis1997analysis}{{22}{1997}{{Tsitsiklis and Van~Roy}}{{}}}
\bibcite{xu2019reanalysis}{{23}{2019}{{Xu et~al.}}{{Xu, Wang, Zhou, and Liang}}}
\bibcite{zhang2022truncated}{{24}{2022}{{Zhang and Whiteson}}{{}}}
\gdef \@abspage@last{8}
NEW_aaai/anonymous-submission-latex-2025.bbl
\begin{thebibliography}{26}
\begin{thebibliography}{24}
\providecommand{\natexlab}[1]{#1}
\bibitem[{Baird et~al.(1995)}]{baird1995residual}
......@@ -16,11 +16,6 @@ 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.
......@@ -51,11 +46,6 @@ 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.
......
NEW_aaai/anonymous-submission-latex-2025.blg
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NEW_aaai/anonymous-submission-latex-2025.tex
......@@ -119,7 +119,7 @@
% 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}
\title{A Variance Minimization Approach to Temporal-Difference Learning}
\author{
%Authors
% All authors must be in the same font size and format.
......@@ -194,16 +194,31 @@
\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,
% 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.
Under certain conditions, the larger the smallest
eigenvalue of the key matrix of an algorithm, the
faster the algorithm converges. By observation, most
current objective functions aim to minimize error.
Therefore, in this paper, we propose two new objective
functions and derive three Variance Minimization (VM) algorithms, including VMTD, VMTDC and VMETD.
A scalar parameter, $\omega$, is introduced, to improve the performance of parametric
Temporal-Difference (TD) learning algorithms.
In the policy evaluation experiment, two-state,
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
matrices both on-policy and off-policy.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.
......
NEW_aaai/main/conclusion.tex
\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.
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.
As an alternation, this study proposes two new objective
functions: VBE and VPBE, and derives an on-policy algorithm:
VMTD and two off-policy algorithms: VMTDC and VMETD.
% The VMTD algorithm
% is essentially an adjustment or correction to the traditional
% TD update.
% Both
% algorithms are capable of stabilizing gradient estimation, reducing
% the variance of gradient estimation and accelerating convergence.
All algorithms demonstrated superior performance in policy
evaluation and control experiments.
Both 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
to,
\begin{itemize}
\item analysis of the convergence rate of VMTDC and VMETD.
\item extensions of VBE and VPBE to multi-step returns.
\item extensions to nonlinear approximations, such as neural networks.
\end{itemize}
\ No newline at end of file
NEW_aaai/main/experiment.tex
% \subsection{Testing Tasks}
\begin{figure}[h]
\centering
\includegraphics[scale=0.2]{main/pic/maze_13_13.pdf}
\caption{Maze.}
\end{figure}
\begin{figure*}[tb]
\vskip 0.2in
\begin{center}
\subfigure[on-policy 2-state]{
\includegraphics[width=0.65\columnwidth, height=0.58\columnwidth]{main/pic/2-state-onpolicy.pdf}
\label{2-state}
}
\subfigure[off-policy 2-state]{
\includegraphics[width=0.65\columnwidth, height=0.58\columnwidth]{main/pic/2-state-offpolicy.pdf}
\label{7-state}
}
\subfigure[Maze]{
\includegraphics[width=0.65\columnwidth, height=0.58\columnwidth]{main/pic/maze.pdf}
\label{MazeFull}
}\\
\subfigure[Cliff Walking]{
\includegraphics[width=0.65\columnwidth, height=0.58\columnwidth]{main/pic/cl.pdf}
\label{CliffWalkingFull}
}
\subfigure[Mountain Car]{
\includegraphics[width=0.65\columnwidth, height=0.58\columnwidth]{main/pic/mt.pdf}
\label{MountainCarFull}
}
\subfigure[Acrobot]{
\includegraphics[width=0.65\columnwidth, height=0.58\columnwidth]{main/pic/acrobot.pdf}
\label{AcrobotFull}
}
\caption{Learning curses of one evaluation environment and four contral environments.}
\label{Complete_full}
\end{center}
\vskip -0.2in
\end{figure*}
\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 evaluation experimental environments is the 2-state.
In a 2-state environment, we conducted two types of experiments—on-policy
and off-policy—to verify the relationship between the convergence speed of
the algorithm and the smallest eigenvalue of the key matrix $\textbf{A}$.
Control experiments, by allowing the algorithm to interact
with the environment to optimize the policy, can evaluate its
performance in learning the optimal policy. This provides a more
comprehensive assessment of the algorithm's overall capabilities.
The control experimental environments are Maze, CliffWalking-v0, MountainCar-v0, and Acrobot-v1.
The control algorithms for TDC, ETD, VMTDC, and VMETD are named GQ, EQ, VMGQ, and VMEQ, respectively.
For TD and VMTD control algorithms, there are two variants each: Sarsa and Q-learning for TD, and VMSarsa and VMQ for VMTD.
% For specific experimental parameters, please refer to the appendix.
% \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{main/pic/BairdExample.tex}
% \caption{7-state.}
% \label{bairdexample}
% \end{center}
% \end{figure}
% The feature matrix of 7-state version of Baird's off-policy counterexample is
% defined as follow:
% \begin{equation*}
% \Phi_{Counter}=\left[
% \begin{array}{cccccccc}
% 1 & 2& 0& 0& 0& 0& 0& 0\\
% 1 & 0& 2& 0& 0& 0& 0& 0\\
% 1 & 0& 0& 2& 0& 0& 0& 0\\
% 1 & 0& 0& 0& 2& 0& 0& 0\\
% 1 & 0& 0& 0& 0& 2& 0& 0\\
% 1 & 0& 0& 0& 0& 0& 2& 0\\
% 2 & 0& 0& 0& 0& 0& 0& 1
% \end{array}\right]
% \end{equation*}
\subsection{Testing Tasks}
% \begin{figure}[h]
% \centering
% \includegraphics[scale=0.2]{main/pic/maze_13_13.pdf}
% \caption{Maze.}
% \end{figure}
\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.
\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.
For all policy evaluation experiments, each experiment
is independently run 100 times.
For all control experiments, each experiment is independently run 50 times.
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
\subsection{Experimental Results and Analysis}
Figure \ref{2-state} shows the learning curves for the on-policy
2-state policy evaluation experiment. In this setup,
the convergence speed of TD, VMTD, TDC, and VMTDC decreases
sequentially. Table \ref{tab:min_eigenvalues} indicates that the smallest eigenvalue
of the key matrix for these four algorithms is greater than 0
and decreases sequentially, which is consistent with the
experimental curves and table values.
Figure B displays the learning curves for the off-policy
2-state policy evaluation experiment. In this setup,
the convergence speed of ETD, VMETD, VMTD, VMTDC, and
TDC decreases sequentially, while TD diverges. Table \ref{tab:min_eigenvalues}
shows that the smallest eigenvalue of the key matrix for
ETD, VMETD, VMTD, VMTDC, and TDC is greater than 0 and
decreases sequentially, while the smallest eigenvalue
for TD is less than 0. This is consistent with the
experimental curves and table values. Remarkably,
although VMTD is guaranteed to converge under
on-policy conditions, it still converges in the
off-policy 2-state scenario. The update formula
of VMTD indicates that it is essentially an
adjustment and correction of the TD update,
with the introduction of the parameter $\omega$
making the variance of the gradient estimate
more stable, thereby making the update of theta more stable.
Figures \ref{MazeFull}, \ref{CliffWalkingFull}, \ref{MountainCarFull} and \ref{AcrobotFull} show the learning curves
for four control experiments. A common feature
observed across these experiments is that VMEQ
outperforms EQ, VMGQ outperforms GQ, VMQ outperforms
Q-learning, and VMSarsa outperforms Sarsa. For the
Maze and Cliffwalking experiments, VMEQ demonstrated
the best performance with the fastest convergence speed.
In the Mountain Car and Acrobot experiments, the performance
of the four VM algorithms was nearly identical and all
outperformed the other algorithms.
Overall, whether in policy evaluation experiments or
control experiments, the VM algorithms have
demonstrated superior performance,
especially excelling in the control experiments.
\ No newline at end of file
NEW_aaai/main/introduction.tex
......@@ -68,26 +68,28 @@ based on recursive optimization using it are known to be unstable.
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.
% 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
instead of minimizing errors. We minimize Variance of Bellman Error (VBE) and
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 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.
\begin{itemize}
\item Introduction of novel objective functions, VBE and VPBE.
\item Propose a on-policy VM algorithm and two off-policy VM algorithms.
\item Proof of their convergence.
\item The experiments demonstrate the superiority of the VM algorithms.
\end{itemize}
NEW_aaai/main/motivation.tex
\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.
This section will introduce two new objective functions and
three new algorithms, including one on-policy algorithm and two off-policy algorithms, and calculate the minimum eigenvalue
of $\textbf{A}$ for each of the three algorithms under on-policy and
off-policy in a 2-state environment, thereby comparing the
convergence speed of the three 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}
% \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 TD Learning: VMTD}
For on-policy learning,
a novel objective function, Variance of Bellman Error (VBE), is proposed as follows:
% \begin{equation}
% \begin{array}{ccl}
% \arg \min_{\theta}\text{VBE}(\theta)&=&\arg \min_{\theta}\mathbb{E}[(\mathbb{E}[\delta|s]-\mathbb{E}[\mathbb{E}[\delta|s]])^2]\\
% &=&\arg \min_{\theta,\omega} \mathbb{E}[(\mathbb{E}[\delta|s]-\omega)^2].
% \end{array}
% \end{equation}
\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}]
\arg \min_{\theta}\text{VBE}(\theta) &= \arg \min_{\theta}\mathbb{E}[(\mathbb{E}[\delta_t|S_t]-\mathbb{E}[\mathbb{E}[\delta_t|S_t]])^2] \\
&= \arg \min_{\theta,\omega} \mathbb{E}[(\mathbb{E}[\delta_t|S_t]-\omega)^2]\notag
\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] $.
where $\delta_t$ is the TD error as follows:
\begin{equation}
\delta_t = r_{t+1}+\gamma
\theta_t^{\top}\phi_{t+1}-\theta_t^{\top}\phi_t.
\label{delta}
\end{equation}
Clearly, it is no longer to minimize Bellman errors.
First, the parameter $\omega$ is derived directly based on
stochastic gradient descent:
\begin{equation}
\omega_{t+1}\leftarrow \omega_{t}+\beta_t(\delta_t-\omega_t),
\label{omega}
\end{equation}
The gradient of the (3) with respect to $\theta$ is
Then, based on stochastic semi-gradient descent, the update of
the parameter $\theta$ is as follows:
\begin{equation}
\theta_{t+1}\leftarrow
\theta_{t}+\alpha_t(\delta_t-\omega_t)\phi_t.
\label{theta}
\end{equation}
The semi-gradient of the (2) 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}
&&-\frac{1}{2}\nabla \text{VBE}({\theta}) \\
&=& \mathbb{E}[(\mathbb{E}[\delta_t|S_t]-\mathbb{E}[\mathbb{E}[\delta_t|S_t]])(\phi_t -\mathbb{E}[\phi_t])] \\
&=& \mathbb{E}[\delta_t \phi_t] -\mathbb{E}[\delta_t] \mathbb{E}[\phi_t] ,
% &=&-\mathbb{E}\Big[\Big( (\phi_t - \gamma\phi_t')- \mathbb{E}[ (\phi_t- \gamma {\phi_t}')]\Big)\phi_t^{\top} \Big]\theta + \mathbb{E}( r_{t+1}- \mathbb{E}[r_{t+1}])\bm{\phi_t}
\end{array}
\end{equation*}
where
The key matrix $\textbf{A}_{\text{VMTD}}$ and $b_{\text{VMTD}}$ of on-policy VMTD is
\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}}'),
&&\textbf{A}_{\text{VMTD}} \\
&=& \mathbb{E}[(\phi - \gamma \phi')\phi^{\top}] - \mathbb{E}[\phi - \gamma \phi']\mathbb{E}[\phi^{\top}]\\
&=&\sum_{s}d_{\pi}(s)\phi(s)\Big(\phi(s) -\gamma \sum_{s'}[\textbf{P}_{\pi}]_{ss'}\phi(s') \Big)^{\top} \\
&& -\sum_{s}d_{\pi}(s)\phi(s) \cdot \sum_{s}d_{\pi}(s)\Big(\phi(s) -\gamma \sum_{s'}[\textbf{P}_{\pi}]_{ss'}\phi(s') \Big)^{\top}\\
&=& \bm{\Phi}^{\top}\textbf{D}_{\mu}(\textbf{I}-\gamma \textbf{P}_{\pi})\bm{\Phi} -\bm{\Phi}^{\top}d_{\pi}d_{\pi}^{\top}(\textbf{I}-\gamma \textbf{P}_{\pi})\bm{\Phi}\\
&=& \bm{\Phi}^{\top}(\textbf{D}_{\pi}-d_{\pi}d_{\pi}^{\top})(\textbf{I}-\gamma \textbf{P}_{\pi})\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{C} &=& \mathbb{E}[\bm{\bm{\phi}}\bm{\bm{\phi}}^{\top}],
&&b_{\text{VMTD}}\\
&=& \mathbb{E}( r- \mathbb{E}[r])\phi \\
&=& \mathbb{E}[r\phi] - \mathbb{E}[r]\mathbb{E}[\phi]\\
&=& \bm{\Phi}^{\top}(\textbf{D}_{\pi}-d_{\pi}d_{\pi}^{\top})r_\pi.
\end{array}
\end{equation*}
It can be easily obtained that The key matrix $\textbf{A}_{\text{VMTD}}$ and $b_{\text{VMTD}}$ of off-policy VMTD are, respectively,
\begin{equation*}
\textbf{A}_{\text{VMTD}} = \bm{\Phi}^{\top}(\textbf{D}_{\mu}-d_{\mu}d_{\mu}^{\top})(\textbf{I}-\gamma \textbf{P}_{\pi})\bm{\Phi},
\end{equation*}
\begin{equation*}
b_{\text{VMTD}}=\bm{\Phi}^{\top}(\textbf{D}_{\mu}-d_{\mu}d_{\mu}^{\top})r_\pi,
\end{equation*}
In the on-policy 2-state environment, the minimum eigenvalue
of the key matrix for VMTD is greater than that of on-policy TDC and smaller than that of on-policy TD(0),
indicating that VMTD converges faster than TDC and slower than TD(0) in this
environment. In the off-policy 2-state environment, the
minimum eigenvalue of the key matrix for VMTD is greater than 0,
suggesting that VMTD can converge stably.
\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}) \notag\\
&= \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}] \\
% &= (\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 (6) with respect to $\theta$ is
\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}}),
-\frac{1}{2}\nabla \text{VPBE}({\theta}) &=& -\mathbb{E}\Big[\Big( (\gamma {\phi}' - {\phi}) - \mathbb{E}[ (\gamma {\phi}' - {\phi})]\Big){\phi}^{\top} \Big] \\
& & \mathbb{E}[{\phi} {\phi}^{\top}]^{-1} \mathbb{E}[( \delta -\mathbb{E}[ \delta]){\phi}]\\
&=& \mathbb{E}\Big[\Big( ({\phi} - \gamma {\phi}')- \mathbb{E}[ ({\phi} - \gamma {\phi}')]\Big){\phi}^{\top} \Big] \\
& & \mathbb{E}[{\phi} {\phi}^{\top}]^{-1}\\
& & \mathbb{E}\Big[\Big( r + \gamma {{\phi}'}^{\top} {\theta} -{\phi}^{\top} {\theta}\\
& & \hspace{2em} -\mathbb{E}[ r + \gamma {{\phi}'}^{\top} {\theta} -{\phi}^{\top} {\theta}]\Big){\phi} \Big].
% &=& \textbf{A}^{\top} \textbf{C}^{-1}(-\textbf{A}\bm{\theta} + \textbf{b})
\end{array}
\end{equation*}
where $\mathrm{Cov}(\cdot,\cdot )$ is a covariance operator.
It can be easily obtained that The key matrix $\textbf{A}_{\text{VMTDC}}$ and $b_{\text{VMTDC}}$ of VMTDC are, respectively,
\begin{equation*}
\textbf{A}_{\text{VMTDC}} = \textbf{A}_{\text{VMTD}}^{\top} \textbf{C}^{-1}\textbf{A}_{\text{VMTD}},
\end{equation*}
\begin{equation*}
b_{\text{VMTDC}}=\textbf{A}_{\text{VMTD}}^{\top} \textbf{C}^{-1}b_{\text{VMTD}},
\end{equation*}
where, for on-policy, $\textbf{A}_{\text{VMTD}}=\bm{\Phi}^{\top}(\textbf{D}_{\pi}-d_{\pi}d_{\pi}^{\top})(\textbf{I}-\gamma \textbf{P}_{\pi})\bm{\Phi}$
and $b_{\text{VMTD}}=\bm{\Phi}^{\top}(\textbf{D}_{\pi}-d_{\pi}d_{\pi}^{\top})r_\pi$ and, for off-policy,
$\textbf{A}_{\text{VMTD}}=\bm{\Phi}^{\top}(\textbf{D}_{\mu}-d_{\mu}d_{\mu}^{\top})(\textbf{I}-\gamma \textbf{P}_{\pi})\bm{\Phi}$
and $b_{\text{VMTD}}=\bm{\Phi}^{\top}(\textbf{D}_{\mu}-d_{\mu}d_{\mu}^{\top})r_\pi$.
In the process of computing the gradient of the (4) with respect to $\theta$,
In the process of computing the gradient of the (7) 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 $\delta-\omega$.
......@@ -133,60 +215,67 @@ Therefore, we can easily get the updated formula of VMTDC, as follows:
\end{equation}
and
\begin{equation}
\omega_{k+1}\leftarrow \omega_{k}+\beta_k (\delta_k- \omega_k),
\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}
from both the behavior and target policies.
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.
% 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 on-policy 2-state environment, the minimum eigenvalue
of the key matrix for VMTDC is smaller than that of TD(0), TDC and VMTD
indicating that VMTDC converges slower than them in this
on-policy. In the off-policy 2-state environment, the
minimum eigenvalue of the key matrix for VMTD is greater than TDC,
suggesting that VMTDC converges faster than them in off-policy
environment.
......@@ -213,24 +302,24 @@ $\textbf{A}_{\text{VMTDC}}=0.025$, meaning that VMTDC can stably converge and co
% \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*}
% \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,
......@@ -242,65 +331,73 @@ VMETD by the following update:
% \delta_{t}= R_{t+1}+\gamma \theta_t^{\top}\phi_{t+1}-\theta_t^{\top}\phi_t.
% \end{equation}
\begin{equation}
\label{fvmetd}
F_t \leftarrow \gamma \rho_{t-1}F_{t-1}+1,
\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,
{\theta}_{t+1}\leftarrow {\theta}_t+\alpha_t (F_t \rho_t\delta_t - \omega_{t}){\phi}_t,
\end{equation}
\begin{equation}
\label{omegavmetd}
\omega_{k+1} \leftarrow \omega_k+\beta_k(F_k \rho_k \delta_k - \omega_k),
\omega_{t+1} \leftarrow \omega_t+\beta_t(F_t \rho_t \delta_t - \omega_t),
\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\}.
{\theta}_{t+1}&\leftarrow& {\theta}_t+\alpha_t (F_t \rho_t\delta_t - \omega_t){\phi}_t -\alpha_t \omega_{t+1}{\phi}_t\\
&=&{\theta}_{t}+\alpha_t(F_t\rho_t\delta_t-\mathbb{E}_{\mu}[F_t\rho_t\delta_t|{\theta}_t]){\phi}_t\\
&=&{\theta}_t+\alpha_t F_t \rho_t (r_{t+1}+\gamma {\theta}_t^{\top}{\phi}_{t+1}-{\theta}_t^{\top}{\phi}_t){\phi}_t\\
& & \hspace{2em} -\alpha_t \mathbb{E}_{\mu}[F_t \rho_t \delta_t]{\phi}_t\\
&=& {\theta}_t+\alpha_t \{\underbrace{(F_t\rho_tr_{t+1}-\mathbb{E}_{\mu}[F_t\rho_t r_{t+1}]){\phi}_t}_{{b}_{\text{VMETD},t}}\\
&&\hspace{-7em}- \underbrace{(F_t\rho_t{\phi}_t({\phi}_t-\gamma{\phi}_{t+1})^{\top}-{\phi}_t\mathbb{E}_{\mu}[F_t\rho_t ({\phi}_t-\gamma{\phi}_{t+1})]^{\top})}_{\textbf{A}_{\text{VMETD},t}}{\theta}_t\}.
\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]\\
&&\textbf{A}_{\text{VMETD}}\\
&=&\lim_{t \rightarrow \infty} \mathbb{E}[\textbf{A}_{\text{VMETD},t}]\\
&=& \lim_{t \rightarrow \infty} \mathbb{E}_{\mu}[F_t \rho_t {\phi}_t ({\phi}_t - \gamma {\phi}_{t+1})^{\top}]\\
&&\hspace{1em}- \lim_{t\rightarrow \infty} \mathbb{E}_{\mu}[ {\phi}_t]\mathbb{E}_{\mu}[F_t \rho_t ({\phi}_t - \gamma {\phi}_{t+1})]^{\top}\\
% &=& \lim_{t \rightarrow \infty} \mathbb{E}_{\mu}[{\phi}_tF_t \rho_t ({\phi}_t - \gamma {\phi}_{t+1})^{\top}]\\
% &&\hspace{1em}- \lim_{t\rightarrow \infty} \mathbb{E}_{\mu}[ {\phi}_t]\mathbb{E}_{\mu}[F_t \rho_t ({\phi}_t - \gamma {\phi}_{t+1})]^{\top}\\
&=& \lim_{t \rightarrow \infty} \mathbb{E}_{\mu}[{\phi}_tF_t \rho_t ({\phi}_t - \gamma {\phi}_{t+1})^{\top}]\\
&&\hspace{1em}-\lim_{t \rightarrow \infty} \mathbb{E}_{\mu}[ {\phi}_t]\lim_{t \rightarrow \infty}\mathbb{E}_{\mu}[F_t \rho_t ({\phi}_t - \gamma {\phi}_{t+1})]^{\top}\\
&& \hspace{-2em}=\sum_{s} d_{\mu}(s)\lim_{t \rightarrow \infty}\mathbb{E}_{\mu}[F_t|S_t = s]\mathbb{E}_{\mu}[\rho_t\phi_t(\phi_t - \gamma \phi_{t+1})^{\top}|S_t= s]\\
&&\hspace{1em}-\sum_{s} d_{\mu}(s)\phi(s)\sum_{s} d_{\mu}(s)\lim_{t \rightarrow \infty}\mathbb{E}_{\mu}[F_t|S_t = s]\\
&&\hspace{7em}\mathbb{E}_{\mu}[\rho_t(\phi_t - \gamma \phi_{t+1})^{\top}|S_t = s]\\
&=& \sum_{s} f(s)\mathbb{E}_{\pi}[\phi_t(\phi_t- \gamma \phi_{t+1})^{\top}|S_t = s]\\
&&\hspace{1em}-\sum_{s} d_{\mu}(s)\phi(s)\sum_{s} f(s)\mathbb{E}_{\pi}[(\phi_t- \gamma \phi_{t+1})^{\top}|S_t = 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}},
&&\hspace{1em} -\sum_{s} d_{\mu}(s) {\phi}(s) * \sum_{s} f(s)({\phi}(s) - \gamma \sum_{s'}[\textbf{P}_{\pi}]_{ss'}{\phi}(s'))^{\top}\\
&=&{\bm{\Phi}}^{\top} \textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi}) \bm{\Phi} - {\bm{\Phi}}^{\top} {d}_{\mu} {f}^{\top} (\textbf{I} - \gamma \textbf{P}_{\mu}) \bm{\Phi} \\
&=&{\bm{\Phi}}^{\top} (\textbf{F} - {d}_{\mu} {f}^{\top}) (\textbf{I} - \gamma \textbf{P}_{\pi}){\bm{\Phi}} \\
&=&{\bm{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-{d}_{\mu} {f}^{\top} (\textbf{I} - \gamma \textbf{P}_{\pi})){\bm{\Phi}} \\
&=&{\bm{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-{d}_{\mu} {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}.
&&{b}_{\text{VMETD}}\\
&=&\lim_{t \rightarrow \infty} \mathbb{E}[{b}_{\text{VMETD},t}]\\
&=& \lim_{t \rightarrow \infty} \mathbb{E}_{\mu}[F_t\rho_tR_{t+1}{\phi}_t]\\
&&\hspace{2em} - \lim_{t\rightarrow \infty} \mathbb{E}_{\mu}[{\phi}_t]\mathbb{E}_{\mu}[F_t\rho_kR_{k+1}]\\
&=& \lim_{t \rightarrow \infty} \mathbb{E}_{\mu}[{\phi}_tF_t\rho_tr_{t+1}]\\
&&\hspace{2em} - \lim_{t\rightarrow \infty} \mathbb{E}_{\mu}[ {\phi}_t]\mathbb{E}_{\mu}[{\phi}_t]\mathbb{E}_{\mu}[F_t\rho_tr_{t+1}]\\
&=& \lim_{t \rightarrow \infty} \mathbb{E}_{\mu}[{\phi}_tF_t\rho_tr_{t+1}]\\
&&\hspace{2em} - \lim_{t \rightarrow \infty} \mathbb{E}_{\mu}[ {\phi}_t]\lim_{t \rightarrow \infty}\mathbb{E}_{\mu}[F_t\rho_tr_{t+1}]\\
&=&\sum_{s} f(s) {\phi}(s)r_{\pi} - \sum_{s} d_{\mu}(s) {\phi}(s) * \sum_{s} f(s)r_{\pi} \\
&=&\bm{\bm{\Phi}}^{\top}(\textbf{F}-{d}_{\mu} {f}^{\top}){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 the off-policy 2-state environment, the minimum eigenvalue
of the key matrix for VMETD is greater than that of TD(0), TDC and VMTD and smaller than that of ETD,
indicating that VMTDC converges faster than TD(0), TDC and VMTD and slower than ETD in this
off-policy.
However, subsequent experiments showed that the VMETD algorithm converges more smoothly and performs best 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}
......
NEW_aaai/main/pic/BairdExample copy 2.tex 0 → 100644
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NEW_aaai/main/preliminaries.tex
......@@ -19,172 +19,243 @@ Reinforcement learning agent interacts with environment, observes state,
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.
discounted cumulative rewards in a long period. For each discrete time step
$t=0,1,2,3,…$,
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].
V^{\pi}(s)=\mathbb{E}_{\pi}[\sum_{k=0}^{\infty} \gamma^k R_{t+k+1}|S_t=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}
V_{{\theta}}(s):= {{\theta}}^{\top}{{\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
where ${{\theta}}:=(\theta_1,\theta_2,\ldots,\theta_m)^{\top}\in
\mathbb{R}^m$ is a parameter vector,
${\bm{\phi}}:=(\phi_1,\phi_2,\ldots,\phi_m)^{\top}\in \mathbb{R}^m$ is a feature
${{\phi}}:=(\phi_1,\phi_2,\ldots,\phi_m)^{\top}\in \mathbb{R}^m$ 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}
\begin{table*}[t]
\caption{Minimum eigenvalues of various algorithms in the 2-state counterexample.}
\label{tab:min_eigenvalues} % 添加标签
\vskip 0.15in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lccccccr}
\toprule
algorithm &TD & TDC & ETD & VMTD & VMTDC & VMETD \\
\midrule
on-policy 2-state&$0.475$ & $0.09025$ & \text{\textbackslash}& $0.25$ & $0.025$ & \text{\textbackslash} \\
off-policy 2-state&$-0.2$ & $0.016$ & $3.4$ & $0.25$ & $0.025$ & $1.15$\\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.1in
\end{table*}
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, since the behavior policy and the target policy
% are consistent, the convergence of TD(0) is more assured. In each time step $t$ 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.
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
From the theory of stochastic methods, the
convergence point of linear TD algorithms, is a parameter vector, say $\bm{\theta}$, that satisfies
\begin{equation*}
\label{thetatd_onpolicy}
\begin{array}{ccl}
\bm{\theta}_{k+1}&\leftarrow&\bm{\theta}_k+\alpha_k \delta_k\bm{\phi}_k,
b - \textbf{A}{\theta}&=&0,
\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
where $\textbf{A}\in \mathbb{R}^{|S| \times m}$ and $b\in \mathbb{R}^{m}$.
If the matrix
$\textbf{A}$ is positive definite, then the algorithm converges.
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.
Next, we will compute the minimum eigenvalue of
$\textbf{A}$ for TD(0), TDC, and ETD in both on-policy and off-policy settings in a 2-state environment.
First, we will introduce the environment setup for the 2-state case in both on-policy and off-policy settings.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.3\columnwidth, height=0.15\columnwidth]{main/pic/2StateExample.pdf}
\caption{2-state}
\end{center}
\end{figure}
The "1"$\rightarrow$"2" problem has only two states. From each
state, there are two actions, left and right, which take
the agent to the left or right state. All rewards are zero.
The feature $\bm{\Phi}=(1,2)^{\top}$
are assigned to the left and
the right state. The first policy takes the equal
probability to left or right
in both states, i.e.,
$
\textbf{P}_{1}=
\begin{bmatrix}
0.5 & 0.5 \\
0.5 & 0.5
\end{bmatrix}
$.
The second policy only selects action right in both states, i.e.,
$
\textbf{P}_{2}=
\begin{bmatrix}
0 & 1 \\
0 & 1
\end{bmatrix}
$.
The state distribution of
the first policy is $d_1 =(0.5,0.5)^{\top}$.
The state distribution of
the second policy is $d_1 =(0,1)^{\top}$.
The discount factor is $\gamma=0.9$.
In the on-policy setting, the behavior policy
and the target policy are the same, so
let $\textbf{P}_{\mu}=\textbf{P}_{\pi}=\textbf{P}_{1}$.
In the off-policy setting,
let $\textbf{P}_{\mu}=\textbf{P}_{1}$ and $\textbf{P}_{\pi}=\textbf{P}_{2}$.
% The on-policy TD(0) update formula is
% \begin{equation*}
% \label{thetatd_onpolicy}
% \begin{array}{ccl}
% \bm{\theta}_{t+1}&\leftarrow&\bm{\theta}_t+\alpha_t \delta_t\bm{\phi}_t,
% \end{array}
% \end{equation*}
% where $\delta_t = r_{t+1}+\gamma \bm{\theta}_t^{\top}\bm{\phi}_{t+1}-\bm{\theta}_t^{\top}\bm{\phi}_t$ is one-step TD error 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
where $\bm{\Phi}$ is the $|S| \times m$ matrix with the $\phi(s)$ as its rows, and $\textbf{D}_{\pi}$ is the $|S| \times |S|$ diagonal
matrix with $d_{\pi}$ on its diagonal. $d_{\pi}$ is a vector, each component representing the steady-state
distribution under policy $\pi$. $\textbf{P}_{\pi}$ denote the $|S| \times |S|$ matrix of transition probabilities under $\pi$. And $\textbf{P}_{\pi}^{\top}d_{\pi}=d_{\pi}$.
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*}
% 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*}
\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}
\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 $|S| \times |S|$ diagonal
matrix with $d_{\mu}$ on its diagonal. $d_{\mu}$ is a vector, each component representing the steady-state
distribution under behavior policy $\mu$.
% 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 off-policy 2-state,
$\textbf{A}_{\text{off}}=-0.2$, which means that off-policy TD(0) cannot stably converge,
while , in the on-policy 2-state, $\textbf{A}_{\text{on}}=0.475$, which means that on-policy TD(0) can 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}
The key matrix $\textbf{A}_{\text{TDC}}$ of on-policy TDC is
\begin{equation*}
\label{thetaetd}
\bm{\theta}_{k+1}\leftarrow \bm{\theta}_k+\alpha_k F_k \rho_k\delta_k\bm{\phi}_k,
\textbf{A}_{\text{TDC}} = \textbf{A}^{\top}_{\text{on}}\textbf{C}^{-1}\textbf{A}_{\text{on}}.
\end{equation*}
The key matrix $\textbf{A}_{\text{TDC}}$ of off-policy TDC is
\begin{equation*}
\textbf{A}_{\text{TDC}} = \textbf{A}^{\top}_{\text{off}}\textbf{C}^{-1}\textbf{A}_{\text{off}}.
\end{equation*}
$\textbf{A}_{\text{TDC}}=0.016$ in the off-policy 2-state and $\textbf{A}_{\text{TDC}}=0.09025$
in the on-policy 2-state, which means that TDC can stably converge in two settings.
To address the issue of the key matrix $\textbf{A}_{\text{off}}$
in off-policy TD(0) being non-positive definite,
a scalar variable, $F_t$,
is introduced to obtain the off-policy TD(0) algorithm,
which ensures convergence under off-policy
conditions.
The key matrix $\textbf{A}_{\text{ETD}}$ is
\begin{equation*}
\textbf{A}_{\text{ETD}} = \bm{\Phi}^{\top}\textbf{F}(\textbf{I}-\gamma \textbf{P}_{\pi})\bm{\Phi},
\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
$f(s)\dot{=}d_{\mu}(s)\lim_{t\rightarrow \infty}\mathbb{E}_{\mu}[F_k|S_k=s]$,
$f(s)\dot{=}d_{\mu}(s)\lim_{t\rightarrow \infty}\mathbb{E}_{\mu}[F_t|S_t=s]$,
which we assume exists.
The vector $\textbf{f}\in \mathbb{R}^N$ with components
$[\textbf{f}]_s\dot{=}f(s)$ can be written as
......@@ -193,21 +264,71 @@ $[\textbf{f}]_s\dot{=}f(s)$ can be written as
\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,
In the off-policy 2-state,
$\textbf{A}_{\text{ETD}}=3.4$, 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 $\textbf{A}$.
Table \ref{tab:min_eigenvalues} shows Minimum eigenvalues
of various algorithms in the 2-state counterexample.
In the on-policy 2-state environment, the minimum eigenvalue
of the key matrix for TDC is greater than that of TD(0),
indicating that TDC converges faster than TD(0) in this
environment. In the off-policy 2-state environment, the
minimum eigenvalue of the key matrix for ETD is the largest,
suggesting that ETD has the fastest convergence rate.
Minimum eigenvalue larger, algorithm's convergence faster.
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.
% 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.
% 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 $\textbf{A}$.
NEW_aaai/main/theory.tex
\section{Theoretical Analysis}
The purpose of this section is to establish the stabilities of the VMTDC algorithm
and the VMETD algorithm.
This section primarily focuses on proving the convergence of VMTD, VMTDC, and VMETD.
\begin{theorem}
\label{theorem1}(Convergence of VMTD).
In the case of on-policy learning, consider the iterations (\ref{omega}) and (\ref{theta}) with (\ref{delta}) of VMTD.
Let the step-size sequences $\alpha_k$ and $\beta_k$, $k\geq 0$ satisfy in this case $\alpha_k,\beta_k>0$, for all $k$,
$
\sum_{k=0}^{\infty}\alpha_k=\sum_{k=0}^{\infty}\beta_k=\infty,
$
$
\sum_{k=0}^{\infty}\alpha_k^2<\infty,
$
$
\sum_{k=0}^{\infty}\beta_k^2<\infty,
$
and
$
\alpha_k = o(\beta_k).
$
Assume that $(\phi_k,r_k,\phi_k')$ is an i.i.d. sequence with
uniformly bounded second moments, where $\phi_k$ and $\phi'_{k}$ are sampled from the same Markov chain.
Let $\textbf{A} = \mathrm{Cov}(\phi,\phi-\gamma\phi')$,
$b=\mathrm{Cov}(r,\phi)$.
Assume that matrix $\textbf{A}$ is non-singular.
Then the parameter vector $\theta_k$ converges with probability one
to $\textbf{A}^{-1}b$.
\end{theorem}
\begin{proof}
\label{th1proof}
The proof is based on Borkar's Theorem for
general stochastic approximation recursions with two time scales
\cite{borkar1997stochastic}.
A sketch proof is given as follows.
In the fast time scale, the parameter $w$ converges to
$\mathbb{E}[\delta|\theta_k]$.
In the slow time scale,
the associated ODE is
\begin{equation*}
\vec{h}(\theta(t))=-\textbf{A}\theta(t)+b.
\end{equation*}
\begin{equation}
\begin{array}{ccl}
A &=& \mathrm{Cov}(\phi,\phi-\gamma\phi')\\
&=&\frac{\mathrm{Cov}(\phi,\phi)+\mathrm{Cov}(\phi-\gamma\phi',\phi-\gamma\phi')-\mathrm{Cov}(\gamma\phi',\gamma\phi')}{2}\\
&=&\frac{\mathrm{Cov}(\phi,\phi)+\mathrm{Cov}(\phi-\gamma\phi',\phi-\gamma\phi')-\gamma^2\mathrm{Cov}(\phi',\phi')}{2}\\
&=&\frac{(1-\gamma^2)\mathrm{Cov}(\phi,\phi)+\mathrm{Cov}(\phi-\gamma\phi',\phi-\gamma\phi')}{2},\\
\end{array}
\label{covariance}
\end{equation}
where we eventually used $\mathrm{Cov}(\phi',\phi')=\mathrm{Cov}(\phi,\phi)$
\footnote{The covariance matrix $\mathrm{Cov}(\phi',\phi')$ is equal to
the covariance matrix $\mathrm{Cov}(\phi,\phi)$ if the initial state is re-reachable or
initialized randomly in a Markov chain for on-policy update.}.
Note that the covariance matrix $\mathrm{Cov}(\phi,\phi)$ and
$\mathrm{Cov}(\phi-\gamma\phi',\phi-\gamma\phi')$ are semi-positive
definite. Then, the matrix $\textbf{A}$ is semi-positive definite because $\textbf{A}$ is
linearly combined by two positive-weighted semi-positive definite matrice
(\ref{covariance}).
Furthermore, $\textbf{A}$ is nonsingular due to the assumption.
Hence, the matrix $\textbf{A}$ is positive definite. And,
the parameter $\theta$ converges to $\textbf{A}^{-1}b$.
\end{proof}
Please refer to the appendix for VMTD's detailed proof process.
\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.
......@@ -24,129 +87,147 @@ and the VMETD algorithm.
$
\zeta_k = o(\beta_k).
$
Assume that $(\bm{\bm{\phi}}_k,r_k,\bm{\bm{\phi}}_k')$ is an i.i.d. sequence with
Assume that $(\phi_k,r_k,\phi_k')$ 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 $\textbf{C}=\mathbb{E}[\bm{\bm{\phi}}\bm{\bm{\phi}}^{\top}]$.
Let $\textbf{A} = \mathrm{Cov}(\phi,\phi-\gamma\phi')$,
$b=\mathrm{Cov}(r,\phi)$, and $\textbf{C}=\mathbb{E}[\phi\phi^{\top}]$.
Assume that $\textbf{A}$ and $\textbf{C}$ are non-singular matrices.
Then the parameter vector $\bm{\theta}_k$ converges with probability one
to $\textbf{A}^{-1}\bm{b}$.
Then the parameter vector $\theta_k$ converges with probability one
to $\textbf{A}^{-1}b$.
\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*}
The proof is similar to that given by \cite{sutton2009fast} for TDC,
but it is based on multi-time-scale stochastic approximation.
A sketch proof is given as follows.
In the fastest time scale, the parameter $w$ converges to
$\mathbb{E}[\delta|u_k,\theta_k]$.
In the second fast time scale,
the parameter $u$ converges to $\textbf{C}^{-1}\mathbb{E}[(\delta-\mathbb{E}[\delta|\theta_k])\phi|\theta_k]$.
In the slower time scale,
the associated ODE is
\begin{equation*}
\vec{h}(\theta(t))=\textbf{A}^{\top}\textbf{C}^{-1}(-\textbf{A}\theta(t)+b).
\end{equation*}
The matrix $\textbf{A}^{\top}\textbf{C}^{-1}\textbf{A}$ is positive definite. Thus,
the parameter $\theta$ converges to $\textbf{A}^{-1}b$.
\end{proof}
Please refer to the appendix for VMTDC's detailed proof process.
% \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
% % $\bm{\theta}_{\text{VMTDC}}=\bm{\theta}_{\text{VMTD}}=\textbf{A}^{-1}\bm{b}$.
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*}
% 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*}
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}
% 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*}
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}
% 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}
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*}
% 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}
% 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.
......@@ -426,18 +507,20 @@ For the convergence properties of $u$ and $\theta$, please refer to the appendix
$
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 $\textbf{A}_{\textbf{VMETD}}^{-1}\bm{b}_{\textbf{VMETD}}$.
Let $\textbf{A}_{\textbf{VMETD}} ={\bm{\Phi}}^{\top} (\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-d_{\mu} d_{\mu}^{\top} ){\bm{\Phi}}$,
$b_{\text{VMETD}}=\bm{\Phi}^{\top}(\textbf{F}-d_{\mu} f^{\top})r_{\pi}$.
Assume that matrix $\textbf{A}$ is non-singular.
Then the parameter vector $\theta_k$ converges with probability one
to $\textbf{A}_{\textbf{VMETD}}^{-1}b_{\textbf{VMETD}}$.
\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.
A sketch proof is given as follows.
In the fast time scale, the parameter $\omega$ converges to
$\mathbb{E}_{\mu}[F\rho\delta|\theta_k]$.
Recursion (\ref{thetavmetd}) is considered the slower timescale.
If the key matrix
$\textbf{A}_{\text{VMETD}}$ is positive definite, then
......@@ -445,70 +528,37 @@ $\theta$ 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 \\
&(\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-{d}_{\mu} {d}_{\mu}^{\top} )e\\
&=\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})e-{d}_{\mu} {d}_{\mu}^{\top} e\\
% &=\textbf{F}(\textbfe-\gamma \textbf{P}_{\pi} \textbfe)-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} \textbfe\\
&=(1-\gamma)\textbf{F}e-{d}_{\mu} {d}_{\mu}^{\top} e\\
% &=(1-\gamma)\textbf{f}-\textbf{d}_{\mu} \textbf{d}_{\mu}^{\top} \textbfe\\
&=(1-\gamma){f}-{d}_{\mu} \\
&=(1-\gamma)(\textbf{I}-\gamma\textbf{P}_{\pi}^{\top})^{-1}{d}_{\mu}-{d}_{\mu} \\
&=(1-\gamma)[(\textbf{I}-\gamma\textbf{P}_{\pi}^{\top})^{-1}-\textbf{I}]{d}_{\mu} \\
&=(1-\gamma)[\sum_{t=0}^{\infty}(\gamma\textbf{P}_{\pi}^{\top})^{t}-\textbf{I}]{d}_{\mu} \\
&=(1-\gamma)[\sum_{t=1}^{\infty}(\gamma\textbf{P}_{\pi}^{\top})^{t}]{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
&e^{\top}(\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-\textbf{d}_{\mu} {d}_{\mu}^{\top} )\\
&=e^{\top}\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-e^{\top}{d}_{\mu} {d}_{\mu}^{\top} \\
&={d}_{\mu}^{\top}-e^{\top}{d}_{\mu} {d}_{\mu}^{\top} \\
&={d}_{\mu}^{\top}- {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
where $e$ is the all-ones vector.
(\ref{rowsum}) and (\ref{columnsum}) show that the matrix $\textbf{F} (\textbf{I} - \gamma \textbf{P}_{\pi})-d_{\mu} d_{\mu}^{\top}$ 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
the optimal policy invariance of VMTD,
VMTDC and VMETD in control experiments,
laying the groundwork for subsequent experiments.
......@@ -518,7 +568,7 @@ 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]
\begin{table}[ht]
\caption{Comparison of action selection with and without
constant bias in $Q$ values.}
\label{example_bias}
......
画图.pptx 0 → 100644
File added
论文草稿.txt
题目：A Variance Minimization Approach to Off-policy Temporal-Difference Learning
题目：A Variance Minimization Approach to Off-policy Temporal-Difference Learning
......@@ -104,3 +104,29 @@ to, (1) 将标量参数引入到更多的TD算法中.
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
接下来，我们将在2-state环境中计算TD(0)、TDC、ETD的分别在on-policy和off-policy下的各自A的最小特征值。
如果矩阵A正定，则算法收敛。
首先，我们将介绍2-state分别在on-policy和off-policy下的环境设定。
在on-policy设定下，行为策略与目标策略一样，令A=B。
为了解决off-policy TD(0)的关键矩阵A_off非正定问题，
为了方便
在2-state环境中，我们进行了两种实验——on-policy实验和off-policy实验，来验证算法的收敛速度与关键矩阵的最小特征值的关系。
图A是on-policy 2-state的策略评估实验的曲线图。在该实验设定下，TD、VMTD、TDC以及VMTDC的收敛速度在依次递减，而表1可以得到这四个算法的关键矩阵的最小特征值都大于0，并且依次递减。实验曲线和表格数值相照应。
图B是off-policy 2-state的策略评估实验的曲线图。在该实验设定下，ETD、VMETD、VMTD、VMTDC以及TDC的收敛速度在依次递减，TD则发散。而表1可以得到ETD、VMETD、VMTD、VMTDC以及TDC这五个算法的关键矩阵的最小特征值都大于0，并且依次递减，TD算法的关键矩阵的最小特征值小于0。实验曲线和表格数值相照应。令人惊喜的是，尽管VMTD是on-policy下保证收敛的算法，但在off-policy 2-state下依旧可以收敛。由VMTD的更新公式可以看出，VMTD的更新公式相当于是对TD更新的调整与修正，参数omega的引入使得梯度估计的方差更加稳定，从而让theta的更新更加稳定。
图1，2，3，4分别是四个控制实验的曲线图。四个控制实验都表现出了一个共性特征：VMEQ的表现优于EQ，VMGQ优于GQ，VMQ优于Q-learning，VMSarsa优于Sarsa。对于Maze和Cliffwalking实验，VMEQ都表现出了最佳的性能，收敛速度最快。对于Mountain car和 Acrobot实验，四个VM算法的表现近乎一样，并且都优于其他算法。
总的来说，不管是策略评估实验还是控制实验，VM算法都表现较为优秀，尤其在控制实验中特别突出。
在本论文中，
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