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Tian Cheng (Riccardo) Xia
2024-06-15 14:19:41 +02:00
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src/year1/languages-and-algorithms-for-ai/module3/sections/_computational_learning.tex
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@ -233,7 +233,7 @@ In other words, a point is misclassified if it is in $R$ but not in $T$ or vice
We want to prove that: We want to prove that:
\begin{equation} \label{eq:rect_points_to_error} \begin{equation} \label{eq:rect_points_to_error}
\begin{aligned} \begin{aligned}
\forall i: \left( \parbox{2.7cm}{Some red points in the training data is in $F_i$} \right) &\Rightarrow \forall i: \left( \parbox{2.7cm}{Some red points in the training data are in $F_i$} \right) &\Rightarrow
E_i \subseteq F_i \\ E_i \subseteq F_i \\
& \Rightarrow \mathcal{P}_{x \sim \mathcal{D}}[x \in E_i] \leq \mathcal{P}_{x \sim \mathcal{D}}[x \in F_i] \\ & \Rightarrow \mathcal{P}_{x \sim \mathcal{D}}[x \in E_i] \leq \mathcal{P}_{x \sim \mathcal{D}}[x \in F_i] \\
& \Rightarrow \mathcal{P}_{x \sim \mathcal{D}}[x \in E_i] \leq \frac{\varepsilon}{4} & \Rightarrow \mathcal{P}_{x \sim \mathcal{D}}[x \in E_i] \leq \frac{\varepsilon}{4}