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\lecture{28}{5/6/2013}{Pseudorandomness for Half-spaces}{Nick Pappas}

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\section{Preliminaries}
Here are the notions that we need:
\begin{definition}[Half-Space] A half-space is defined as a function
$H_{w,\theta}:\{-1,1\}^n\rightarrow\{-1,1\}$
for some vector $w\in \R^n$ and
some constant threshold value $\theta\in\R$. For any vector
$x\in\R^n$, we define
$H_{w,\theta}(x)=\text{sign}(\langle w,x\rangle-\theta)$.
\end{definition}

I.e, we 
report $1$ if the projection of $x$ onto $v$ is greater than $\theta$, and we
report $-1$ if the projection is less than $\theta$.We assume that
the length of $w$ will be 1 ($\|w\|_2=1$).


\begin{definition}[Regular Half-Space] 
(We remind the definition of the supreme norm denoted as ${\left\| w \right\|_\infty } $):\\
${\left\| w \right\|_\infty } \buildrel\textstyle.\over= \max |{w_i}|$\\

A half-space is regular if ${\left\| w \right\|_\infty } \le \epsilon$
\end{definition}

\begin{definition}[Supreme distance] We define the supreme distance denoted as $d_\infty (A,B)$:
${d_\infty }(A,B) \buildrel\textstyle.\over= \mathop {\sup }\limits_{t \in R} |\Pr (A < t) - \Pr (B < t)|$
\end{definition}

\begin{theorem} {Barry-Esseen Theorem}\\
Let $Y_1,\ldots,Y_n$ be independent random
variables with the following properties:
\begin{enumerate}
\item $\E[Y_i]=0$
\item $\sum\limits_i {E[{Y_i}^2] = 1} $
\item $\sum\limits_i {E[{Y_i}^4] \le {\varepsilon ^2}} $
\end{enumerate}
Let $S_n = {Y_1+\ldots+Y_n}$ and let $N(01)$ denote the normal distribution with
mean 0 and variance 1.
Then the supreme distance ${d_\infty }({S_n},N(01)) \le \varepsilon $
\end{theorem}

\begin{corollary} {Corollary of the Barry-Esseen Theorem}\\
${d_\infty }( < w,{U_n} > ,N(01)) \le \varepsilon $
\end{corollary} 


\section{PRG construction attempt}
Let ${Y_i} = w{X_i}$ where ${X_i}$ is uniform in {-1,1}.
We pick ${X_i}$s from a 4-wise uniform distribution and we end up with :\\
${X_1},{X_2},...,{X_n}$\\
However picking ${X_i}$s like that gives us a problem; the ${Y_i}$s will fail the third condition of the Barry -Esseen Theorem.
Therefore we split them in $t$ groups:\\
\[\underbrace {\left( {{X_1},...,{X_{\frac{n}{t}}}} \right)}_{{D^1}}\underbrace {\left( {{X_{\frac{n}{t} + 1}},...} \right)}_{{D^2}}......\underbrace {\left( {..,{X_{\frac{n}{t}}}} \right)}_{{D^t}}\]

The seed length is $\approx t\log \frac{n}{t}$

\begin{theorem} 
${d_\infty }({S_n},N(01)) \le \varepsilon $
\end{theorem}
\begin{proof}
We use the B-E theorem:\\
\[w = \underbrace {\left( {{w_1}.....{w_{\frac{n}{t}}}} \right)}_{{w^1}}\underbrace {\left( {{w_{\frac{n}{t}}}.....} \right)}_{{w^2}}......\underbrace {\left( {.....{w_n}} \right)}_{{w^t}}\]\\

\noindent Now let :\\
${Y_i} \buildrel\textstyle.\over=  < {D^i},{w^i} > $ then ${S_t} =  < w,D > $
What we have now is the following:

\begin{enumerate}
\item $E[{Y_i}] = \underbrace {\sum\limits_j {{w^i}_jE[{D^i}_j]} }_0 = 0$
\item $\sum\limits_i {E[{Y_i}^2]}  = \sum\limits_i {\sum\limits_{{j_1} + {j_2}} {\underbrace {E[{w^i}_{{j_1}}{D^i}_{{j_1}}]E[{w^i}_{{j_2}}{D^i}_{{j_2}}]}_{{j_1} \ne {j_2} \to 0}} }  + \sum\limits_i {\sum\limits_j {{{\left( {{w^i}_{{j_1}}} \right)}^2}\underbrace {E[{{\left( {{D^i}_j} \right)}^2}]}_1} }  = \sum\limits_k {{w_k}^2 = ||w|| = 1} $
\item $\sum\limits_i {E[{Y_i}^4]}  = \sum\limits_{{j_1} + {j_2} + {j_3} + {j_4}} {{w_{{j_1}}}^i...{w_{{j_4}}}^iE[D{j_1}...D{j_4}]}  \le 3\sum\limits_{j,k} {{{\left( {w_j^i} \right)}^2}{{\left( {w_k^i} \right)}^2}}  = 3{\left( {\sum\limits_j {{{\left( {w_j^i} \right)}^2}} } \right)^2} = 3||{w^i}|{|_2}^4$
\end{enumerate}

For the last part we require that $\sum\limits_t {||{w^i}|{|_2}^4}  \le {\varepsilon ^2}$ for which it is sufficient that 
$||{w^i}|{|_2}^4 \le \frac{{{\varepsilon ^2}}}{t}$\\

For example:
${w^1} = (\varepsilon .....\varepsilon )$ will give us :
\[||{w^i}|{|_2}^4 = {\left( {\frac{n}{t}} \right)^2}{\varepsilon ^4} \le \frac{{{\varepsilon ^2}}}{t} \Leftrightarrow t \ge {n^2}{\varepsilon ^2}\]
\end{proof}

Idea: We pick the t partitions of the ${w^i}$s at random using a hash function $h[n] \to [t]$.
Now the last part becomes :
\[........ \le 3\sum\limits_{j,k} {\mathop E\limits_h {{\left( {w_j^i} \right)}^2}{{\left( {w_k^i} \right)}^2}}  = 3{\left( {\sum\limits_j {\mathop E\limits_h {{\left( {w_j^i} \right)}^2}} } \right)^2} = 3\mathop E\limits_h ||{w^i}|{|_2}^4\]

\begin{theorem} 
$\mathop E\limits_h \left[ {||{w^i}|{|_2}^4} \right] \le \Theta \left( {\frac{{{\varepsilon ^2}}}{t}} \right)$
\end{theorem}
\begin{proof}
Let ${H_j} = \left\{ {\begin{array}{*{20}{c}}
1&{h(j) = i}\\
0&{otherwise}
\end{array}} \right.$\\
Then \[\mathop E\limits_h \left[ {||{w^i}|{|_2}^4} \right] = \sum\limits_j^n {{H_j}^4{w_j}^4}  + \sum {\underbrace {E\left[ {{H_j}{H_k}} \right]{w_j}^2{w_k}^2}_{\frac{1}{{{t^2}}}}}  \le \frac{{{\varepsilon ^2}}}{t} + \frac{1}{{{t^2}}}\]
and so finally:
\[||{w^i}|{|_2}^4 \le \frac{{{\varepsilon ^2}}}{t} + \frac{1}{{{t^2}}}\underbrace  = _{t = \frac{1}{{{\varepsilon ^2}}}}\Theta \left( {\frac{{{\varepsilon ^2}}}{t}} \right)\]
\end{proof}

Now PRG ${D'} = (D,h)$ satisfies 
\[{d_\infty }\left( { < w,D' > ,N(01)} \right) \le \varepsilon  \Rightarrow d\left( {{H_{w,\Theta }}(U),{H_{w,\Theta }}(D')} \right) \le O(\varepsilon )\]
and the seed length of D' is \\
\[t\log \frac{n}{t} + \log (nt) \sim {\varepsilon ^2}\log \frac{1}{\varepsilon }\log n\]
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