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Apendix/anonymous-submission-latex-2024.aux
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Apendix/anonymous-submission-latex-2024.bbl
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\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.
......
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\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/pic/BairdExample copy 2.tex 0 → 100644
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NEW_aaai/main/pic/mt.pdf
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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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