Algorithm 4 LogitBoost algorithm for multi-class classificaiton problem

$$\begin{array}{cc}\hfill & \text{Initialize:}{w}_{ij}=1∕n,i=1,\dots ,n,j=1,\dots ,J,\hfill \\ \hfill & F_j({\mathit{x}}_i)=0\text{and}{p}_j({\mathit{x}}_i)=1∕J,\forall j\hfill \\ \hfill & \text{Repeat}\text{for}m=1,\dots ,M:\hfill \\ \hfill & 1)\text{Repeat}\text{for}j=1,\dots ,J:\hfill \\ \hfill & \mathrm{a})\text{Compute the working responses and weights}\hfill \\ \hfill & z_{ij}=\frac{y_{ij}-p_j({\mathit{x}}_i)}{p({\mathit{x}}_i)[1-p_j({\mathit{x}}_i)]},\hfill \\ \hfill & w_{ij}=p_j({\mathit{x}}_i)[1-p_j({\mathit{x}}_i)];\hfill \\ \hfill & \mathrm{b})\text{Fit a weighted least square regression},f_{mj}(\mathit{x})\hfill \\ \hfill & \text{of training points}{x}_i\text{to response values}{z}_{ij}\text{with}\hfill \\ \hfill & \text{weights}{w}_{ij},\text{and denote the fitted values}by{\widehat{f}}_l;\hfill \\ \hfill & 2)\hfill \\ \hfill & f_{mj}\leftarrow \frac{J-1}{J}[f_{mj}(\mathit{x})-\frac{1}{J}\sum _{k=1}^J{f}_{mk}(\mathit{x})],\hfill \\ \hfill & F_j(\mathit{x})\leftarrow F_j(\mathit{x})+f_{mj}(\mathit{x});\hfill \\ \hfill & 3)\text{Update}{p}_j\text{by}\hfill \\ \hfill & p_j=\frac{e^{F_j(\mathit{x})}}{{\sum }_{k=1}^J{e}^{F_k(\mathit{x})}};\hfill \\ \hfill & {\text{Output the classifier argmax}}_j{F}_j(\mathit{x}).\hfill \end{array}$$