Ex. 10.7

Ex. 10.7

Derive expression (10.32).

Soln. 10.7

We are looking for the optimum of

$$\begin{array}{rcl}{\hat{\gamma }}_{jm}& =& \underset{{\gamma }_{jm}}{\mathrm{argmin}}\sum _{x_i\in R_{jm}}{e}^{-y_i{f}_{m-1}(x_i)-y_i{\gamma }_{jm}}\\ & =& \underset{{\gamma }_{jm}}{\mathrm{argmin}}\sum _{x_i\in R_{jm}}{w}_i^{(m)}{e}^{-y_i{\gamma }_{jm}},\end{array}$$

where $w_i^{(m)}=e^{-y_i{f}_{m-1}(x_i)}$.

Let

$$F({\gamma }_{jm})=\sum _{x_i\in R_{jm}}{w}_i^{(m)}{e}^{-y_i{\gamma }_{jm}}.$$

We have

$$\begin{array}{rcl}\frac{\partial F}{\partial {\gamma }_{jm}}& =& \sum _{x_i\in R_{jm}}{w}_i^{(m)}{e}^{-y_i{\gamma }_{jm}}\cdot (-y_i)\\ & =& -\sum _{x_i\in R_{jm},y_i=1}{w}_i^{(m)}{e}^{-{\gamma }_{jm}}+\sum _{x_i\in R_{jm},y_i=-1}{w}_i^{(m)}{e}^{{\gamma }_{jm}}.\end{array}$$

Setting it to be zero and solve for ${\gamma }_{jm}$ we obtain

$${\hat{\gamma }}_{jm}=\frac{1}{2}\log \frac{\sum _{x_i\in R_{jm}}{w}_i^{(m)}\mathbf{\text{1}}(y_i=1)}{\sum _{x_i\in R_{jm}}{w}_i^{(m)}\mathbf{\text{1}}(y_i=-1)}.$$