GBDT
GBDT （Gradient Boosting Decison Tree）=Gradient Boosting+cart回归树
注意是cart回归树，不是cart分类树
说白了就是gradient boosting基学习器为cart回归树
gradient boosting算法流程:
1.初始化：$f_0(x) = \mathop{\arg\min}\limits_\gamma \sum\limits_{i=1}^N L(y_i, \gamma)$
2.for m=1 to M:
(a). 计算负梯度： $\tilde{y}_i = -\frac{\partial L(y_i,f_{m-1}(x_i))}{\partial f_{m-1}(x_i)}, \quad i = 1,2 \cdots N$
(b). 通过最小化平方误差，用基学习器$h_m(x)$拟合$\tilde{y_i}$，$w_m = \mathop{\arg\min}\limits_w \sum\limits_{i=1}^{N} \left[\tilde{y}_i - h_m(x_i\,;\,w) \right]^2$
(c). 使用line search确定步长$ρ_m$，使$L$最小，$\rho_m = \mathop{\arg\min}\limits_{\rho} \sum\limits_{i=1}^{N} L(y_i,f_{m-1}(x_i) + \rho h_m(x_i\,;\,w_m))$
(d). $f_m(x) = f_{m-1}(x) + \rho_m h_m(x\,;\,w_m)$
3.输出$f_M(x)$
GBDT算法流程：
- 初始化： $f_0(x) = \mathop{\arg\min}\limits_\gamma \sum\limits_{i=1}^N L(y_i, \gamma)$
- for m=1 to M:
(a). 计算负梯度： $\tilde{y}_i = -\frac{\partial L(y_i,f_{m-1}(x_i))}{\partial f_{m-1}(x_i)}, \qquad i = 1,2 \cdots N$
(b). $\left \{ R_{jm} \right\}_1^J = \mathop{\arg\min}\limits_{\left \{ R_{jm} \right\}_1^J}\sum\limits_{i=1}^N \left [\tilde{y}_i - h_m(x_i\,;\,\left \{R_{jm},b_{jm} \right\}_1^J) \right]^2$
(c). $\gamma_{jm} = \mathop{\arg\min}\limits_\gamma \sum\limits_{x_i \in R_{jm}}L(y_i,f_{m-1}(x_i)+\gamma)$
(d). $f_m(x) = f_{m-1}(x) + \sum\limits_{j=1}^J \gamma_{jm}I(x \in R_{jm})$ - 输出$f_M(x)$