Algorithm 2. Gradient boosting decision tree (GBDT) algorithm.

Inputs: Training data set: $\begin{array}{cc}T=\left\{\right(x_1{,\mathrm{y}}_1),(x_2{,\mathrm{y}}_2),\cdots ,(x_n{,\mathrm{y}}_n\left)\right\}& \begin{array}{ccc},& x_i\in \chi \subseteq R^n,y_i\in \Upsilon \subseteq R& ;\end{array}\end{array}$      Loss function $L(y,f(x))=\frac{1}{2}{(y-f(x))}^2$ (1) Initialize:      ${{\displaystyle f}}_0(x)=argmi{n}_{\gamma }{\displaystyle {\sum }_{i=1}^{N}L(y_i,\gamma )}$ (2) For m = 1, 2, …, M (a) For i = 1, 2, …, N, compute      $r_{im}=-{\left[\frac{\partial L(y_i,f(x_i))}{\partial f(x_i)}\right]}_{f=f_{m-1}}$ (b) Fit a regression tree to the targets ${\mathrm{r}}_{\mathrm{im}}$ giving terminal regions Rjm, j = 1, 2,…, Jm. (c) For j = 1, 2, …, Jm compute      ${\gamma }_{\mathrm{jm}}=\arg \underset{\gamma }{\min }{\displaystyle \sum _{x_i\in R_{jm}}L(y_i,f_{m-1}(x_i)+\gamma )}$ (d) Update $f_m(x)=f_{m-1}(x)+{\displaystyle {\sum }_{j=1}^{j_m}{\gamma }_{jm}I(x\in R_{jm})}$ Outputs: $\hat{f}(\mathrm{x})=f_M(x)$