sl03 il

Aufgabe 1

Klassifikationsfunktion:

$\hat{y}=\{\begin{array}{l}1\text{ if }p>0.5\\ 0\text{ otherwise }\end{array}$

Mit:

• $p=f(x,\theta )$

• $f(x,\theta )=\frac{1}{1+e^{\left({\sum }_{i=1}^n{w}_i{x}_i\right)+b}}$

• $\theta =\left[\begin{array}{c}{w}_1\\ w_2\\ b\end{array}\right]$

Aufgabe 2

$L(p,y)=-(y\log (p)+(1-y)\log (1-p))$

• $\frac{\partial }{\partial w_1}L(p,y)$ = $(p-y)x_1$
• $\frac{\partial }{\partial w_2}L(p,y)=(p-y)x_2$
• $\frac{\partial }{\partial b}L(p,y)=p-y$

${\theta }^{t+1}={\theta }^t-\eta {\nabla }_{\theta }L\left(f\left(x,\theta \right),y\right)$

${\nabla }_{\theta }L\left(f\left(x,\theta \right),y\right)=\left[\begin{array}{c}(p-y)x_1\\ (p-y)x_2\\ p-y\end{array}\right]$