\begin{document}


\twocolumn[

\aistatstitle{$Q$-learning with Logarithmic Regret}

\aistatsauthor{ Kunhe Yang \And Lin F. Yang \And  Simon S. Du}

\aistatsaddress{ \texttt{ykh17@mails.tsinghua.edu.cn}\\
Tsinghua University \And  
\texttt{linyang@ee.ucla.edu}\\
University of California, Los Angeles \And 
\texttt{ssdu@cs.washington.edu}\\
University of Washington }]

\begin{abstract}
\input{abs}
\end{abstract}


\section{Introduction}
\input{intro.tex}
\label{sec:intro}
\subsection{Related Work}
\label{sec:rel}
\input{rel.tex}


\section{Preliminaries}
\label{sec:pre}
\input{prelim.tex}


\section{Main Theoretical Results}
\label{sec:main_results}
\input{main_results.tex}

\section{Proof of Theorem~\ref{thm:episodic}}
\label{sec:proof_sketch}
\input{proof_sketch.tex}

\section{Conclusion and Future Directions}
\label{sec:conclusion}
\input{conclusion.tex}

\bibliographystyle{plainnat}
\bibliography{ref}


\onecolumn


\input{discounted_proof}

\section{Difficulty in Applying Optimistic Surplus}
\label{sec:opt_surplus}
\input{optsurplus.tex}

\end{document}
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