% packages 

\usepackage{amsmath,amssymb,amsthm}
\usepackage{bbm}
\usepackage{paralist}
\usepackage[linesnumbered,ruled,vlined]{algorithm2e}
\usepackage{authblk}
	\renewcommand{\Authsep}{\qquad}
	\renewcommand{\Authand}{\qquad}
	\renewcommand{\Authands}{\qquad}
	\renewcommand\Affilfont{\itshape\small}
\usepackage[left=1.1in,right=1.1in,top=1.1in,bottom=1.1in]{geometry}
\usepackage[bookmarks=false]{hyperref}
    \hypersetup{
        linktocpage=true,
        colorlinks=true,				
        linkcolor=DarkBlue,				
        citecolor=DarkBlue,				
        urlcolor=DarkBlue,			
    }
\usepackage[tt=false]{libertine}
    \usepackage[libertine]{newtxmath}
    \usepackage[T1]{fontenc}
    \renewcommand{\baselinestretch}{1.00}
\usepackage{lipsum}
\usepackage{microtype}
\usepackage{multirow}
\usepackage{enumitem}
\usepackage{nicefrac}
\usepackage{tikz}
	\usetikzlibrary{positioning}
	\definecolor{DarkGreen}{rgb}{0.2,0.6,0.2}
	\definecolor{DarkRed}{rgb}{0.6,0.2,0.2}
	\definecolor{DarkBlue}{rgb}{0.2,0.2,0.6}
	\definecolor{DarkPurple}{rgb}{0.4,0.2,0.4}   
\usepackage{url}
\usepackage{verbatim}
\usepackage{wrapfig}
\usepackage{framed}
% \usepackage{ulem}
% \normalem

   
% comments
\setlength\marginparwidth{62pt}
\setlength\marginparsep{5pt}
\ifnum\comments=1
    \newcommand{\mynote}[2]{{\marginpar{\color{#1}\sf \tiny #2}}}
    \newcommand{\mynoteinline}[2]{{\color{#1} \sf \small #2}}
\else
\newcommand{\mynote}[2]{}
    \newcommand{\mynoteinline}[2]{}
\fi
\newcommand{\jnote}[1]{\mynote{blue}{JU: #1}}
\newcommand{\as}[1]{\mynote{red}{ADS: #1}}
\newcommand{\anote}{\as}
\newcommand{\asinline}[1]{\mynoteinline{red}{ADS: #1}}

\renewcommand{\epsilon}{\eps}

% fixing left-right spacing
%\let\originalleft\left
%\let\originalright\right
%\renewcommand{\left}{\mathopen{}\mathclose\bgroup\originalleft}
%\renewcommand{\right}{\aftergroup\egroup\originalright}

% math macros
\newcommand{\ex}[2]{{\ifx&#1& \mathbb{E} \else \underset{#1}{\mathbb{E}} \fi \left(#2\right)}}
% \newcommand{\pr}[2]{{\ifx&#1& \mathbb{P} \else
%     \underset{#1}{\mathbb{P}} \fi \left(#2\right)}}
\newcommand{\pr}[2]{{\operatorname*{\mathbb{P}}_{#1} \paren{#2}}}
\newcommand{\var}[2]{{\ifx&#1& \mathrm{Var} \else \underset{#1}{\mathrm{Var}} \fi \left(#2\right)}}

\newcommand{\mypar}[1]{\medskip\textbf{#1}}

\newcommand{\from}{:}
\newcommand{\poly}{\mathrm{poly}}
\newcommand{\polylog}{\mathrm{polylog}}
\newcommand{\eps}{\varepsilon}

\newcommand{\ind}{\mathbb{I}}
\newcommand{\pmo}{\{\pm 1\}}
\newcommand{\zo}{\{0,1\}}
\newcommand{\bit}[1]{{\zo^{#1}}}
\newcommand{\cond}[1]{\vert_{#1}}
\newcommand{\norm}[2]{\|#1\|_{#2}}
\newcommand{\abs}[1]{{\left | {#1} \right|}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\nicehalf}{\nicefrac{1}{2}}
\newcommand{\set}[1]{\left\{ #1 \right\}}
\newcommand{\cardset}[1]{\left| \set{#1} \right|}
\newcommand{\paren}[1]{{\left ( {#1} \right)}}
\newcommand{\bparen}[1]{{\big( {#1} \big)}}
\newcommand{\Bparen}[1]{{\Big( {#1} \Big)}}
\newcommand{\bracket}[1]{{\left [ {#1} \right]}}
\newcommand{\ip}[1]{{\left \langle {#1} \right\rangle }}

\newcommand{\distance}{\mathrm{d}}
\newcommand{\dtv}{\distance_{\mathrm{TV}}}
\newcommand{\dkl}{\distance_{\mathrm{KL}}}
\newcommand{\dhe}{\distance_{\mathrm{H^2}}}
\newcommand{\dcs}{\distance_{\mathrm{\chi^2}}}

\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argmax}{arg\,max}

\newcommand{\N}{\mathbb{N}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}

\newcommand{\cA}{\mathcal{A}}
\newcommand{\cB}{\mathcal{B}}
\newcommand{\cC}{\mathcal{C}}
\newcommand{\cD}{\mathcal{D}}
\newcommand{\cE}{\mathcal{E}}
\newcommand{\cF}{\mathcal{F}}
\newcommand{\cG}{\mathcal{G}}
\newcommand{\cH}{\mathcal{H}}
\newcommand{\cI}{\mathcal{I}}
\newcommand{\cJ}{\mathcal{J}}
\newcommand{\cK}{\mathcal{K}}
\newcommand{\cL}{\mathcal{L}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\cN}{\mathcal{N}}
\newcommand{\cO}{\mathcal{O}}
\newcommand{\cP}{\mathcal{P}}
\newcommand{\cQ}{\mathcal{Q}}
\newcommand{\cR}{\mathcal{R}}
\newcommand{\cS}{\mathcal{S}}
\newcommand{\cT}{\mathcal{T}}
\newcommand{\cU}{\mathcal{U}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\cW}{\mathcal{W}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cY}{\mathcal{Y}}
\newcommand{\cZ}{\mathcal{Z}}


\newcommand{\bc}{\mathbf{c}}
\newcommand{\bp}{\mathbf{p}}
\newcommand{\bw}{\mathbf{w}}
\newcommand{\by}{\mathbf{y}}



\newcommand{\defeq}{\stackrel{{\mbox{\tiny def}}}{=}}


%%%% notes-wide macros %%%%
% Old
% \newcommand{\dataset}{X}
% \newcommand{\data}{x}
% \newcommand{\univ}{\cX}
% \newcommand{\query}{f}
% \newcommand{\querypred}{\varphi}
% \newcommand{\ans}{a}
% \newcommand{\querymat}{F}

\newcommand{\dataset}{{\mathbf{x}}}
\newcommand{\ds}{\dataset}
\newcommand{\data}{x}
\newcommand{\Dataset}{{\mathbf{X}}}
\newcommand{\DS}{\Dataset}
\newcommand{\Data}{X}
\newcommand{\univ}{\cU}
\newcommand{\query}{f}
\newcommand{\querypred}{\varphi}
\newcommand{\ans}{a}
\newcommand{\querymat}{F}
\newcommand{\GS}[1]{{{GS}_{#1}}}
\newcommand{\GSone}[1]{{{GS}_{{#1},\ell_1}}}

\newcommand{\hist}{\mathbf{h}}
\newcommand{\qmat}{\mathbf{F}}
\newcommand{\mmat}{\mathbf{M}}
\newcommand{\rmat}{\mathbf{R}}
\newcommand{\emat}{\mathbf{E}}
\newcommand{\noise}{\mathbf{Z}}
\newcommand{\mnorm}[1]{\| #1 \|_{1 \to 2}}
\newcommand{\rnorm}[1]{\| #1 \|_{\mathit{F}}}
\newcommand{\id}{\mathbb{I}}
\newcommand{\const}{c_{\eps,\delta}}

\newcommand{\vol}{{\text{\rm Vol}}}

\newcommand{\RRbasic}{{RR_{\text{basic}}}}
\newcommand{\RR}{RR}

\newcommand{\Lap}{\mathsf{Lap}}
\newcommand{\Exp}{\mathsf{Exp}}

\newcommand{\score}{q}
\newcommand{\smax}{{\score_{\max}}}
\newcommand{\mech}{A}


\newcommand{\PLoss}[3]{{I_{{#1},{#2}}\paren{{#3}}}}
\newcommand{\ploss}[1]{{\PLoss{\dataset}{\dataset'}{#1}}}

\newcommand{\trans}{y}


\newcommand{\dpl}{\cD} %decision player
\newcommand{\adv}{\cA} % adversary
\newcommand{\acts}{A}





\newcommand{\Time}{{\text{\sf Time}}}

\newtheorem{thm}{Theorem}[section]
    \newtheorem{clm}[thm]{Claim}
    \newtheorem{lem}[thm]{Lemma}
    \newtheorem{prop}[thm]{Proposition}
    \newtheorem{cor}[thm]{Corollary}
    \newtheorem{fact}[thm]{Fact}
    \newtheorem{claim}[thm]{Claim}
\theoremstyle{definition}
    \newtheorem{defn}[thm]{Definition}
    \newtheorem{example}[thm]{Example}
    \newtheorem{rem}[thm]{Remark}
    \newtheorem{exer}[thm]{Exercise}

\newcommand{\printtitle}[4]{\begin{figure}[t!]{\bfseries \Large
      \color{DarkBlue}  \noindent Privacy in Statistics and Machine
      Learning \hfill Spring 2023} \\[0.2em] {\bfseries \Large
      \color{DarkBlue} Lecture #1: #2} \\[1em] {\bfseries \large Adam
      Smith (based on materials developed with Jonathan Ullman)} \\[1em]\end{figure}}

\newcommand{\HWtitle}[2]{\begin{figure}[t!]{\bfseries \Large \color{DarkBlue}  \noindent Privacy in Statistics and Machine Learning \hfill Spring 2023} \\[0.2em] {\bfseries \Large \color{DarkBlue} Homework #1: Due {#2}} \\[1em] {\bfseries \large Adam Smith (based on materials developed with Jonathan Ullman)} \\[1em]\end{figure}}

% For this macro: #1 is lecture number, #2 is topic, #3 is the date.
\newcommand{\inclasstitle}[3]{\begin{figure}[t!]{\bfseries \Large
      \color{DarkBlue}  \noindent Privacy in Statistics and Machine
      Learning \hfill Spring 2023} \\[0.2em] {\bfseries \Large
      \color{DarkBlue} In-class Exercises for Lecture {#1}  ({#2}) \\[0.2em] {#3}} \\[1em] {\bfseries \large Adam Smith (based on materials developed with Jonathan Ullman)} \\[1em]\end{figure}}

\newcommand{\misctitle}[1]{\begin{figure}[t!]{\bfseries \Large
      \color{DarkBlue}  \noindent Privacy in Statistics and Machine
      Learning \hfill Spring 2023} \\[0.2em] {\bfseries \Large
      \color{DarkBlue} {#1} } \\[1em] {\bfseries \large Adam Smith (based on materials developed with Jonathan Ullman)} \\[1em]\end{figure}}
