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Started the 1 \pm \epsilon approx alg.

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Aaron Huber 2020-08-05 16:08:40 -04:00
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28
approx_alg.tex
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%root: main.tex
\section{$1 \pm \epsilon$ Approximation Algorithm}
\AH{There are things in the poly write up for this section which \textit{I am not exactly sure on how to verify as of yet}, and with Atri being gone this week, I may not be able to verify everything, so my approach will be to \LaTeX this section up, and {\Large \bf leave notes for things that need to be verified.}}
\begin{Lemma}\label{lem:approx-alg}
For any query polynomial $\poly(X_1,\ldots, X_n)$, an approximation of $\rpoly(\prob_1,\ldots, \prob_n)$ can be computed in $O\left(|\poly|\cdot k \frac{\log\frac{1}{\conf}}{\error^2}\right)$, within $1 \pm \error$ multiplicative error with probability $\geq 1 - \conf$, where $k$ denotes the product width of $\poly$.
\end{Lemma}
\begin{proof}[Proof of Lemma \ref{lem:approx-alg}]
Assume a set of $\samplesize$ random variables $\vct{\randvar}$, where $\randvar_i \sim \unidist{\coeffset}$, such that $\coeffset = \{\coeffitem{1},\ldots, \coeffitem{\setsize}\}$, where $\coeffitem{i}$ is the value of the $i^{th}$ monomial term in $\rpoly(\prob_1,\ldots, \prob_n)$, with $c_i$ being the coefficient, and $\distinctvars$ being the number of distinct variables appearing in the $i^{th}$ monomial of $\poly$.
Given random variable $\randvar_1$, it is the case that $\expct\pbox{\randvar_1} = \sum_{i = 1}^{\setsize}\frac{\coeffitem{i}}{\setsize} = \ave{\coeffset}$. Let $\hoeffest = \frac{1}{\samplesize}\sum_{i = 1}^{\samplesize}\randvar_i$, and then it is true that
\[\expct\pbox{\hoeffest} = \expct\pbox{ \frac{1}{\samplesize}\sum_{i = 1}^{\samplesize}\randvar_i} = \frac{1}{\samplesize}\sum_{i = 1}^{\samplesize}\expct\pbox{\randvar_i} = \frac{1}{\samplesize}\sum_{i = 1}^{\samplesize}\frac{1}{\setsize}\sum_{j = 1}^{\setsize}\coeffitem{j} = \ave{\coeffset}.\]
Denote $\hoeffestsum = \hoeffest \cdot \setsize$ and $\setsum = \ave{\coeffset} \cdot \setsize$.
Given the range $[a, b]$ for every $\randvar_i$ in $\vct{\randvar}$, by Hoeffding, it is true that $Pr\pbox{| \hoeffestsum - \setsum | \geq \error\setsize} \leq 2\exp{-\frac{2\samplesize^2\setsize^2\error^2}{\sum_{i = 1}^{\samplesize}\left(b_i - a_i\right)^2}} \leq \conf$.
Solving for the number of samples $\samplesize$ we get
\begin{align}
&\conf \geq 2\exp{-\frac{2\samplesize^2\setsize^2\error^2}{\sum_{i = 1}^{\samplesize}\left(b_i - a_i\right)^2}}\nonumber\\
&\frac{\conf}{2} \geq \exp{-\frac{2\samplesize^2\setsize^2\error^2}{\sum_{i = 1}^{\samplesize}\left(b_i - a_i\right)^2}}\nonumber\\
&\frac{2}{\conf} \leq \exp{\frac{2\samplesize^2\setsize^2\error^2}{\sum_{i = 1}^{\samplesize}\left(b_i - a_i\right)^2}}\nonumber\\
&\log{\frac{2}{\conf}} \leq \frac{2\samplesize^2\setsize^2\error^2}{\sum_{i = 1}^{\samplesize}\left(b_i - a_i\right)^2}\nonumber\\
&\log{\frac{2}{\conf}} \leq \frac{2\samplesize\setsize^2\error^2}{\left(b_i - a_i\right)^2}\nonumber\\
&\frac{\log{\frac{2}{\conf}}\left(b - a\right)^2}{2\setsize^2\error^2} \leq \samplesize.\nonumber
\end{align}
Let us now show a sampling scheme which can run in $O\left(|\poly|\cdot k\right)$ per sample.
\end{proof}

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macros.tex
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\newcommand{\mtrix}[1]{M_{#1}}
\newcommand{\dtrm}[1]{Det\left(#1\right)}
%Approx Alg
\newcommand{\randvar}{Y}
\newcommand{\coeffset}{S}
\newcommand{\distinctvars}{d}
\newcommand{\coeffitem}[1]{c_{#1}\cdot\prob^{\distinctvars_{#1}}}
\newcommand{\unidist}[1]{Uniform\left(#1\right)}
\newcommand{\samplesize}{N}
\newcommand{\setsize}{m}
\newcommand{\hoeffest}{\overline{\randvar}}
\newcommand{\setsum}{SUM}
\newcommand{\ave}[1]{AVE(#1)}
\newcommand{\hoeffestsum}{EST_{\setsum}}
\newcommand{\error}{\epsilon}
\newcommand{\conf}{\delta}
%PDBs
\newcommand{\ti}{TIDB}