minor changes to example slide for hypothesis spaces

slides/advriskmin/slides-advriskmin-risk-minimizer.tex

@@ -57,21 +57,21 @@
 \end{vbframe}
 
 \begin{vbframe}{Two short examples}
-\textbf{Linear Model with squared loss:}\\
+\textbf{Regression with linear model:}\\
 \begin{itemize}
     \item Model: $f(\xi) = \thetab^\top \xi + \theta_0$
-    \item Loss:  
-    $L(\yi, f(\xi)) = \left(\yi - \thetab^\top \xi - \theta_0\right)^2$
+    \item Squared loss:  
+    $L(\yi, f(\xi)) = \left(\yi - f(\xi)\right)^2$
     \item Hypothesis space: $$\Hspace_{\text{lin}} = \left\{ \xi \mapsto \thetab^\top \xi + \theta_0 : \thetab \in \mathbb{R}^d, \theta_0 \in \mathbb{R} \right\}$$
 \end{itemize}
 
 \vspace{0.3cm}
 
-\textbf{Shallow MLP with binary CE loss:}\\
+\textbf{Binary classification with shallow MLP:}\\
 \begin{itemize}
     \item Model: $f(\xi) = \bm{w}_2^{\top} \text{ReLU}(\bm{W}_1 \xi + \bm{b}_1) + b_2$
-    \item Loss: $L(\yi, f(\xi)) = -(\yi\log(p^{(i)})+(1-\yi)\log(1-p^{(i)}))$ where $p^{(i)} = \sigma(f(\xi))$
-    \item Hypothesis space: $$\Hspace_{\text{MLP}} = \left\{ \xi \mapsto \bm{w}_2^{\top} \text{ReLU}(\bm{W}_1 \xi + \bm{b}_1) + b_2: \mathbf{W}_1 \in \mathbb{R}^{h \times d}, \mathbf{b}_1 \in \mathbb{R}^h, \mathbf{w}_2 \in \mathbb{R}^h, b_2 \in \mathbb{R} \right\}$$
+    \item Binary cross-entropy loss: $L(\yi, f(\xi)) = -(\yi\log(p^{(i)})+(1-\yi)\log(1-p^{(i)}))$\\ where $p^{(i)} = \sigma(f(\xi))$ (logistic sigmoid)
+    \item Hypothesis space: {\small $$\Hspace_{\text{MLP}} = \left\{ \xi \mapsto \bm{w}_2^{\top} \text{ReLU}(\bm{W}_1 \xi + \bm{b}_1) + b_2: \mathbf{W}_1 \in \mathbb{R}^{h \times d}, \mathbf{b}_1 \in \mathbb{R}^h, \mathbf{w}_2 \in \mathbb{R}^h, b_2 \in \mathbb{R} \right\}$$}
 \end{itemize}
 
 \end{vbframe}