34 lines
1.3 KiB
TeX
34 lines
1.3 KiB
TeX
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% -*- 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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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}{3})^2}{2 + \frac{1}{3}}\frac{2}{3}\sketchRows}\\
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\delta \geq e^{-\frac{63}{2}\sketchRows}\\
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e^{\frac{63}{2}\sketchRows} \geq \frac{1}{\delta}\\
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\sketchRows \geq \frac{63}{2}ln(\frac{1}{\delta})
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\end{align*}
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