60 lines
2.7 KiB
TeX
60 lines
2.7 KiB
TeX
% -*- root: main.tex -*-
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\section{Bounding the Estimates}
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\newcommand{\bMu}{\epsilon\mu_{\sketchCols_{sum}}}
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\newcommand{\bBnd}{\sketchCols_{sketch}}
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\newcommand{\mBnd}{\sketchRows_{sketch}}
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\newcommand{\sBnd}{m_{sketch}}
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For a $\sketchCols$ estimate, denoted $\sketchCols_{est}$, and given the following:
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\begin{align*}
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&\bMu \text{ is the expectation for the sum of estimates.}\\
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&X = \sum_{i = 1}^{\sketchRows}X_i \\
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&X_i\text{ is i.i.d. r.v.} \in [0, 1], i \in \sketchRows \\
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&X_i = \begin{cases}
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0 &\sketchCols_{est} > \bMu\\
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1 &\sketchCols_{est} \leq \bMu
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\end{cases}\\
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&p[X_i = 1] \geq \frac{2}{3}\\
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&p[X_i = 0] \leq \frac{1}{3}\\
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&\mu = \frac{2}{3}\sketchRows\\
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&\epsilon = 0.5
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\end{align*}
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Because Chebyshev bounds tell us that the probability of a bad row estimate is $\leq \frac{1}{3}$, we set epsilon to the value that, when multiplied to $\mu$, outputs $\frac{1}{3}$. We then derive bounds for $\sketchRows$.
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Note, because we are only concerned with the left side of the tail, we can use the generic Chernoff bounds for the left tail,
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\begin{equation*}
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Pr[|X - \mu| \leq (1 - \epsilon)\mu] \leq e^{-\frac{\epsilon^2}{2 + \epsilon}\mu}.
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\end{equation*}
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Solving for $\delta$,
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\begin{align*}
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\delta \geq e^{-\frac{(\frac{1}{2})^2}{2 + \frac{1}{2}}\frac{2}{3}\sketchRows}\\
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\delta \geq e^{-\frac{1}{15}\sketchRows}\\
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e^{\frac{1}{15}\sketchRows} \geq \frac{1}{\delta}\\
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\sketchRows \geq \frac{15}{1}ln(\frac{1}{\delta})
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\end{align*}
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We are now ready to combine the bounds we have derived for both $\sketchCols$ and $\sketchRows$ to which we will refer to as $\bBnd$ and $\mBnd$ respectively.
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\begin{align*}
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&\mBnd \cdot \bBnd \\
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= & \frac{15}{1}ln(\frac{1}{\delta}) \cdot \frac{3\left(\norm{\genV}_2^2\left(|\pw|\right) + \norm{\genV}_1^2\right)}{\epsilon^2 p^2}\\
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= & \frac{45\left(\norm{\genV}_2^2\left(|\pw|\right) + \norm{\genV}_1^2\right)}{\epsilon^2 p^2}ln(\frac{1}{\delta})
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\end{align*}
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Sampling bounds, $\sBnd$, are obtained via Chernoff Bounds. Given,
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\begin{align*}
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&X = \sum_{i = 1}^{m}X_i\\
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&X_i \text{is i.i.d. r.v.} \in [0, 1] \\
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&p = \frac{\norm{\genV}_1}{|W|}\\
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&\bar{X} = \frac{X}{m}\\
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&P[|\bar{X} - p| \geq \epsilon p] \leq 2e^{-\frac{\epsilon^2}{2 + \epsilon}pm} \rightarrow\\
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&\delta \geq 2e^{-\frac{\epsilon^2}{2 + \epsilon}pm} \\
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&e^{\frac{\epsilon^2}{2 + \epsilon}pm} \geq \frac{2}{\delta} \\
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&\frac{\epsilon^2}{2 + \epsilon}pm \geq ln(\frac{2}{\delta})\\
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&m \geq \frac{2 + \epsilon}{\epsilon^2 p}ln(\frac{2}{\delta})
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\end{align*}
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We are particularly interested when the former are a lower bound to the latter. We want to know when the following is true.
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\begin{equation*}
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\frac{2 + \epsilon}{\epsilon^2 p}ln(\frac{2}{\delta}) > \frac{45\left(\norm{\genV}_2^2\left(|\pw|\right) + \norm{\genV}_1^2\right)}{\epsilon^2 p^2}ln(\frac{1}{\delta})
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\end{equation*} |