\documentclass{article} \usepackage{graphicx} \newcommand{\di}{{d}} \newcommand{\nexp}{{n}} \newcommand{\nf}{{p}} \newcommand{\vcd}{{\textbf{D}}} \usepackage{nccmath} \usepackage{mathtools} \usepackage{graphicx,caption} \usepackage{enumitem} \usepackage{epstopdf,subcaption} \usepackage{psfrag} \usepackage{amsmath,amssymb,epsf} \usepackage{verbatim} \usepackage[hyphens]{url} \usepackage{color} \usepackage{bbm} \usepackage{listings} \usepackage{setspace} \usepackage{float} \usepackage{natbib} \definecolor{Code}{rgb}{0,0,0} \definecolor{Decorators}{rgb}{0.5,0.5,0.5} \definecolor{Numbers}{rgb}{0.5,0,0} \definecolor{MatchingBrackets}{rgb}{0.25,0.5,0.5} \definecolor{Keywords}{rgb}{0,0,1} \definecolor{self}{rgb}{0,0,0} \definecolor{Strings}{rgb}{0,0.63,0} \definecolor{Comments}{rgb}{0,0.63,1} \definecolor{Backquotes}{rgb}{0,0,0} \definecolor{Classname}{rgb}{0,0,0} \definecolor{FunctionName}{rgb}{0,0,0} \definecolor{Operators}{rgb}{0,0,0} \definecolor{Background}{rgb}{0.98,0.98,0.98} \lstdefinelanguage{Python}{ numbers=left, numberstyle=\footnotesize, numbersep=1em, xleftmargin=1em, framextopmargin=2em, framexbottommargin=2em, showspaces=false, showtabs=false, showstringspaces=false, frame=l, tabsize=4, % Basic basicstyle=\ttfamily\footnotesize\setstretch{1}, backgroundcolor=\color{Background}, % Comments commentstyle=\color{Comments}\slshape, % Strings stringstyle=\color{Strings}, morecomment=[s][\color{Strings}]{"""}{"""}, morecomment=[s][\color{Strings}]{'''}{'''}, % keywords morekeywords={import,from,class,def,for,while,if,is,in,elif,else,not,and,or ,print,break,continue,return,True,False,None,access,as,,del,except,exec ,finally,global,import,lambda,pass,print,raise,try,assert}, keywordstyle={\color{Keywords}\bfseries}, % additional keywords morekeywords={[2]@invariant}, keywordstyle={[2]\color{Decorators}\slshape}, emph={self}, emphstyle={\color{self}\slshape}, % } \pagestyle{empty} \addtolength{\textwidth}{1.0in} \addtolength{\textheight}{0.5in} \addtolength{\oddsidemargin}{-0.5in} \addtolength{\evensidemargin}{-0.5in} \newcommand{\ruleskip}{\bigskip\hrule\bigskip} \newcommand{\nodify}[1]{{\sc #1}} \newcommand{\points}[1]{{\textbf{[#1 points]}}} \newcommand{\subquestionpoints}[1]{{[#1 points]}} \newenvironment{answer}{{\bf Answer:} \sf \begingroup\color{red}}{\endgroup}% \newcommand{\bitem}{\begin{list}{$\bullet$}% {\setlength{\itemsep}{0pt}\setlength{\topsep}{0pt}% \setlength{\rightmargin}{0pt}}} \newcommand{\eitem}{\end{list}} \setlength{\parindent}{0pt} \setlength{\parskip}{0.5ex} \setlength{\unitlength}{1cm} \renewcommand{\Re}{{\mathbb R}} \newcommand{\R}{\mathbb{R}} \newcommand{\what}[1]{\widehat{#1}} \renewcommand{\comment}[1]{} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\half}{\frac{1}{2}} \def\KL{D_{KL}} \def\xsi{x^{(i)}} \def\ysi{y^{(i)}} \def\zsi{z^{(i)}} \def\E{\mathbb{E}} \def\calN{\mathcal{N}} \def\calD{\mathcal{D}} \usepackage{tikz} \usepackage{bbding} \usepackage{pifont} \usepackage{wasysym} \usepackage{amssymb} \usepackage{booktabs} \usepackage{verbatim}