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lecture 8 completed
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@ -40,7 +40,7 @@ $$
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We cannot have a sample of $n = \infty$ so we will introduce another assumption:
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the \red{Chernoff-Hoffding bound}
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\subsection{Chernoff-Hoffding bound}
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\section{Chernoff-Hoffding bound}
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$$
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Z_1,...,Z_n \quad \textit{iid random variable} \qquad \barra{E}\left[Z_t \right] = u
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$$
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@ -61,7 +61,7 @@ $
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\\
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We are using the bound of e to bound the deviation of this.
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\subsubsection{Union Bound}
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\section{Union Bound}
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Union bound: a collection of event not necessary disjoint, then i know
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that probability of the union of this event is the at most the sum of the
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probabilities of individual events
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@ -177,4 +177,124 @@ In risk we get opt in $h^*$ but in empirical one we could get another $h’$ bet
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\\\\
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In order to fix on a concrete algorithm we are going to take the empirical Islam
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minimiser (ERM) algorithm.
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\\
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$A$ is $ERM$ on $H$ \qquad $\left(A\right) = \hat{h} = (\in) \, argmin \, \hat{\ell}_S\left(h\right)
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$
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\\
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Once I piack $\hat{h}$ i can look at training error of ERM
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\\
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$$ \hat{\ell}_S\left(\hat{h}\right) of \hat{h} = A(S)$$
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where $\hat{\ell}_S$ is the training error
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\\\\
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Should $\hat{\ell}_S\left(\hat{h}\right)$ be close to $\ell_D\left(\hat{h}\right)$ ?
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\\
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I’m interested in empirical error minimiser and do a trivial decomposition.
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\\\\
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$$
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\ell_d\left(\hat{h}\right) = \quad \ell_D\left(\hat{h}\right) - \ell_d\left(h^*\right) + \qquad \longrightarrow \red{\textbf{ Variance error $\Rightarrow$ Overfitting}}
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$$
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$$
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\qquad \quad +\, \ell_d\left(h^+\right) - \ell_d\left(f^*\right) + \qquad \longrightarrow \red{\textbf{ Bias error $\Rightarrow$ Underfitting}}
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$$
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$$ \qquad \qquad \quad
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+ \, \ell_D\left(f^*\right)\qquad \qquad \quad \longrightarrow \red{\textbf{ Bayes risk $\Rightarrow$ Unavoidable}}
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$$\\
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Even the best predictor is going to suffer that\\
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$$ f^* \textit{ is \textbf{Bayes Optimal} for $(D,\ell)$ }
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$$
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$$\forall \, h \qquad \ell_D\left(h\right) \geq \ell_D\left(f^*\right)
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$$
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If $f^* \not\in H$ then $\ell_D\left(h^*\right) > \ell_D (f^*) $
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\\\\
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If i pick $h^*$ I will pick some error because we are not close enough to the risk.\\
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We called this component \red{\textbf{bias error}}.\\
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Bias error is responsible for underfitting (when training and test are close
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to each but they are both high :( )\\
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\red{\textbf{Variance error}} over fitting
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\\
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\begin{figure}[h]
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\centering
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\includegraphics[width=0.5\linewidth]{../img/lez7-img6.JPG}
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\caption{Draw of how $\hat{h}$, $h^*$ and $f^*$ are represented}
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%\label{fig:}
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\end{figure}\\
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Variance is a random quantity and we want to study this.
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We can always get risk from training error.
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\\\\
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\section{Studying overfitting of a ERM}
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We can bound it with probability.\\
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\bred{I add and subtract trivial traning error $\hat{\ell}_S\left(h\right)$}
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$$
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\ell_D \left(\hat{h}\right)
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-\ell_d \left(h^*\right)
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\quad
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=
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\quad
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\ell_D \left(\hat{h}\right) -
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\hat{\ell}_S\left( h \right)
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+
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\hat{\ell}_S \left( \hat{h} \right)
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- \ell_D\left( h^* \right) \leq
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$$
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$$
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\leq \, \ell_D \left(\hat{h}\right) -
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\hat{\ell}_S\left( \hat{h} \right)
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+
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\hat{\ell}_S \left( h^* \right)
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- \ell_D\left( h^* \right) \leq \,
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$$
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$$
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\leq \, | \, \ell_D\left(\hat{h}\right) - \hat{\ell}_S\left(h\right) \, | + | \, \hat{\ell}_S\left(h^+\right) - \ell_D\left(h^*\right) \, |\, \leq
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$$
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$$
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\leq \quad 2 \cdot max \, |\hat{\ell}_S\left(h\right) - \ell_D\left(h\right) |
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$$
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(no probability here)\\
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\textbf{Any given $\hat{h}$ minising $\hat{\ell}_S\left(h\right)$}
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\\\\
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Now assume we have a large deviation
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\\
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$$
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\textit{Assume \quad } \ell_D\left(\hat{h}\right) - \ell_D \left(h^* \right) > \varepsilon \qquad \Rightarrow \qquad max \, | \, \hat{\ell}_S\left(h\right) - \ell_D \left(h\right) \, | > \frac{\varepsilon}{2}
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$$
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\\
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We know $\ell_d\left(\hat{h}\right) - \ell_D\left(h^*\right)
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\quad \leq \quad 2
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\cdot max \,|\, \hat{\ell}_S \left(h\right) - \ell_D\left(h\right) \, |$ \quad $\Rightarrow$
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\\
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$$
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\Rightarrow \quad \exists h \in H \qquad | \, \hat{\ell}_S\left(h\right) - \ell_D\left(h \right) \, | \, > \frac{3}{2} \qquad \Rightarrow
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$$
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with $|H| < \infty$
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$$
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\Rightarrow U \left( \, | \, \hat{\ell}_S\left(h\right) - \ell_D \left(h\right) \, | \, \right) > \frac{3}{2}
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$$
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\\
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$$
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\barra{P} \left( \ell_D \left(\hat{h}\right) - \ell_D \left( h^* \right)
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>
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\varepsilon \right) \quad \leq \quad \barra{P} \left( U \left( \, | \, \hat{\ell}_S\left(h\right) - \ell_D \left(h\right) \, | \, \right) > \frac{3}{2} \right) \quad \leq
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$$
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$$
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\red{\leq} \quad \sum_{h \in H}{} \, \barra{P} \left( \, | \, \hat{\ell}_S\left(h\right) - \ell_D \left(h\right) \, | \, > \frac{3}{2} \right) \qquad \leq \qquad \sum_{h \in H}{} 2 \cdot e^{-2 \, \left(\frac{\varepsilon}{2}\right)^2 \, m} \qquad \leq
|
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$$
|
||||
\bred{Union Bound } \blue{Chernoff. Hoffding bound ($\barra{P} \left( ... \right) $)}
|
||||
$$
|
||||
\leq \quad 2 \cdot |H| e^{- \, \frac{\varepsilon^2}{2} \, m}
|
||||
$$
|
||||
\\
|
||||
Solve for $\varepsilon$ \qquad
|
||||
$
|
||||
2 \cdot |H| e^{- \, \frac{\varepsilon^2}{2} \, m} \quad = \quad \delta
|
||||
$
|
||||
$$
|
||||
\textit{ Solve for } \varepsilon \longrightarrow \quad \varepsilon = \sqrt[]{\frac{2}{m} \cdot \ln \cdot \frac{2|H|}{\delta}}
|
||||
$$
|
||||
$$
|
||||
\ell_D\left(\hat{h}\right) - \ell_D \left( h^* \right) \quad\leq \quad \sqrt[]{\frac{2}{m} \cdot \ln \cdot \frac{2|H|}{\delta}}
|
||||
$$
|
||||
\\
|
||||
With probability at least $1 - \delta$ with respect to random draw of $S$.\\
|
||||
We want $m >> ln |H|$ \quad $\longrightarrow$ in order to avoid overfitting
|
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\\
|
||||
\end{document}
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2.9/fonts/type1/public/amsfonts/cm/cmr12.pfb><C:/Program Files/MiKTeX 2.9/font
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s/type1/public/amsfonts/cm/cmr6.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/pu
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blic/amsfonts/cm/cmr8.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfo
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msy6.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmsy8.pfb>
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<C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts/symbols/msam10.pfb><C:
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/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts/symbols/msbm10.pfb>
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Output written on main.pdf (44 pages, 1382132 bytes).
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rm2074.pk> <C:\Users\AndreDany\AppData\Local\MiKTeX\2.9\fonts/pk/ljfour/jknappe
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n/ec/dpi600\ecbx2488.pk><C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfont
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s/cm/cmex10.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmm
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i12.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmmi6.pfb><
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C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmmi8.pfb><C:/Progra
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m Files/MiKTeX 2.9/fonts/type1/public/amsfonts/cm/cmr12.pfb><C:/Program Files/M
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iKTeX 2.9/fonts/type1/public/amsfonts/cm/cmr6.pfb><C:/Program Files/MiKTeX 2.9/
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fonts/type1/public/amsfonts/cm/cmr8.pfb><C:/Program Files/MiKTeX 2.9/fonts/type
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1/public/amsfonts/cm/cmsy10.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/public
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msam10.pfb><C:/Program Files/MiKTeX 2.9/fonts/type1/public/amsfonts/symbols/msb
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m10.pfb>
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Output written on main.pdf (47 pages, 1413965 bytes).
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PDF statistics:
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599 PDF objects out of 1000 (max. 8388607)
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628 PDF objects out of 1000 (max. 8388607)
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0 named destinations out of 1000 (max. 500000)
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41 words of extra memory for PDF output out of 10000 (max. 10000000)
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46 words of extra memory for PDF output out of 10000 (max. 10000000)
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Binary file not shown.
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@ -27,6 +27,8 @@
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\newcommand\nl{N_{\ell}}
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\newcommand\sll{S_{\ell}}
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\newcommand\red[1]{\textcolor{BrickRed}{#1}}
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\newcommand\bred[1]{\textcolor{BrickRed}{\textbf{#1}}}
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\newcommand\blue[1]{\textcolor{Blue}{\textbf{#1}}}
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\newcommand\gray[1]{\textcolor{mygray}{#1}}
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\newcommand\col[2]{\textcolor{#2}{#1}}
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@ -28,9 +28,10 @@
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\contentsline {subsection}{\numberline {6.1.2}Zero-one loss for binary classification}{31}%
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\contentsline {section}{\numberline {6.2}Bayes Risk}{33}%
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\contentsline {chapter}{\numberline {7}Lecture 7 - 07-04-2020}{34}%
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\contentsline {subsection}{\numberline {7.0.1}Chernoff-Hoffding bound}{34}%
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\contentsline {subsubsection}{Union Bound}{35}%
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\contentsline {chapter}{\numberline {8}Lecture 8 - 07-04-2020}{38}%
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\contentsline {chapter}{\numberline {9}Lecture 9 - 07-04-2020}{39}%
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\contentsline {chapter}{\numberline {10}Lecture 10 - 07-04-2020}{40}%
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\contentsline {section}{\numberline {10.1}TO BE DEFINE}{40}%
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\contentsline {section}{\numberline {7.1}Chernoff-Hoffding bound}{34}%
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\contentsline {section}{\numberline {7.2}Union Bound}{35}%
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\contentsline {section}{\numberline {7.3}Studying overfitting of a ERM}{39}%
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\contentsline {chapter}{\numberline {8}Lecture 8 - 07-04-2020}{41}%
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\contentsline {chapter}{\numberline {9}Lecture 9 - 07-04-2020}{42}%
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\contentsline {chapter}{\numberline {10}Lecture 10 - 07-04-2020}{43}%
|
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\contentsline {section}{\numberline {10.1}TO BE DEFINE}{43}%
|
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|
||||
Loading…
Reference in New Issue
Block a user