Ex. 10.7
Ex. 10.7
Derive expression (10.32).
Soln. 10.7
We are looking for the optimum of
\[\begin{eqnarray}
\hat \gamma_{jm} &=& \underset{\gamma_{jm}}{\operatorname{argmin}}\sum_{x_i\in R_{jm}}e^{-y_if_{m-1}(x_i)-y_i\gamma_{jm}}\non\\
&=&\underset{\gamma_{jm}}{\operatorname{argmin}}\sum_{x_i\in R_{jm}}w_i^{(m)}e^{-y_i\gamma_{jm}},\non
\end{eqnarray}\]
where \(w_i^{(m)} = e^{-y_if_{m-1}(x_i)}\).
Let
\[\begin{equation}
F(\gamma_{jm}) = \sum_{x_i\in R_{jm}}w_i^{(m)}e^{-y_i\gamma_{jm}}.\non
\end{equation}\]
We have
\[\begin{eqnarray}
\frac{\partial F}{\partial \gamma_{jm}} &=& \sum_{x_i\in R_{jm}}w_i^{(m)}e^{-y_i\gamma_{jm}}\cdot (-y_i)\non\\
&=&-\sum_{x_i\in R_{jm}, y_i = 1}w_i^{(m)}e^{-\gamma_{jm}} + \sum_{x_i\in R_{jm}, y_i = -1}w_i^{(m)}e^{\gamma_{jm}}.\non
\end{eqnarray}\]
Setting it to be zero and solve for \(\gamma_{jm}\) we obtain
\[\begin{equation}
\hat\gamma_{jm} = \frac{1}{2}\log \frac{\sum_{x_i\in R_{jm}}w_i^{(m)}\bb{1}(y_i=1)}{\sum_{x_i\in R_{jm}}w_i^{(m)}\bb{1}(y_i=-1)}.\non
\end{equation}\]