GBDT和XGBOOST算法原理

GBDT

以多分类问题为例介绍GBDT的算法，针对多分类问题，每次迭代都需要生成K个树（K为分类的个数），记为$F_{mk}(x)$，其中m为迭代次数，k为分类。

针对每个训练样本，使用的损失函数通常为\[L(y_i, F_{m1}(x_i), ..., F_{mK}(x_i))=-\sum_{k=1}^{K}I({y_i}=k)ln[p_{mk}(x_i)]=-\sum_{k=1}^{K}I({y_i}=k)ln(\frac{e^{F_{mk}(x_i)}}{\sum_{l=1}^{K}e^{F_{ml}(x_i)}})\]
此时损失函数的梯度可以表示为\[g_{mki}=-\frac{\partial{L(y_i, F_{m1}(x_i), ..., F_{mK}(x_i))}}{\partial{F_{mk}(x_i)}}=I({y_i}=k)-p_{mk}(x_i)=I({y_i}=k)-\frac{e^{F_{mk}(x_i)}}{\sum_{l=1}^{K}e^{F_{ml}(x_i)}}\]

GBDT算法的流程如下所示：

for k=1 to K: Initialize $F_{0k}(x)=0$
for m=1 to M:
- for k=1 to K: compute $g_{m-1,ki}$ for each sample $(x_i, y_i)$
- for k=1 to K: build up regression tree $R_{mkj}$(j=1 to J_mk refer to the leaf nodes) from training samples $(x_i, g_{m-1,ki})_{i=1,...,N}$
- for k=1 to K: compute leaf weights $w_{mkj}$ for j=1 to J_mk
- for k=1 to K: $F_{mk}(x)=F_{m-1,k}(x)+\eta*\sum_{j=1}^{J_{mk}}w_{mkj}I({x}\in{R_{mkj}})$, $\eta$为学习率

针对$w_{mkj}$的计算，有\[w_{mkj(j=1...J_{mk},k=1...K)}=argmin_{w_{kj(j=1...J_{mk},k=1...K)}}\sum_{i=1}^{N}L(y_i, ..., F_{m-1,k}(x_i)+\sum_{j=1}^{J_{mk}}w_{kj}I({x_i}\in{R_{mkj}}), ...)\]
为了求得w的值，使上述公式的一阶导数为0，利用Newton-Raphson公式（在这个问题中将初始值设为0，只进行一步迭代，并且Hessian矩阵只取对角线上的值），记$L_i=L(y_i, ..., F_{m-1,k}(x_i)+\sum_{j=1}^{J_{mk}}w_{kj}I({x_i}\in{R_{mkj}}), ...)$，有\[w_{mkj}=-\frac{\sum_{i=1}^N\partial{L_i}/\partial{w_{kj}}}{\sum_{i=1}^N\partial^2{L_i}/\partial{w_{kj}^2}}=\frac{\sum_{i=1}^{N}I({x_i}\in{R_{mkj}})[I({y_i}=k)-p_{m-1,k}(x_i)]}{\sum_{i=1}^{N}I^2({x_i}\in{R_{mkj}})p_{m-1,k}(x_i)[1-p_{m-1,k}(x_i)]}=\frac{\sum_{x_i\in{R_{mkj}}}g_{m-1,ki}}{\sum_{x_i\in{R_{mkj}}}\lvert{g_{m-1,ki}}\rvert(1-\lvert{g_{m-1,ki}}\rvert)}\]

文献[1]在$w_{mkj}$的前面乘以了$\frac{K-1}{K}$的系数，可能原因是在分类器的建立过程中，可以把任意一个$F_k(x)$始终设为0，只计算剩余的$F_k(x)$，因此$w_{mkj} \gets \frac{1}{K}*0+\frac{K-1}{K}w_{mkj}$

参考文献

[1] Friedman, Jerome H. Greedy function approximation: A gradient boosting machine. Ann. Statist. 29 (2001), 1189--1232.

XGBOOST

仍以多分类问题介绍XGBOOST的算法，相对于GBDT的一阶导数，XGBOOST还引入了二阶导数，并且加入了正则项。

区别于上文的符号，记\[g_{mki}=\frac{\partial{L(y_i, F_{m1}(x_i), ..., F_{mK}(x_i))}}{\partial{F_{mk}(x_i)}}=p_{mk}(x_i)-I({y_i}=k)=\frac{e^{F_{mk}(x_i)}}{\sum_{l=1}^{K}e^{F_{ml}(x_i)}}-I({y_i}=k)\]以及\[h_{mki}=\frac{\partial^2{L(y_i, F_{m1}(x_i), ..., F_{mK}(x_i))}}{\partial{F_{mk}(x_i)}^2}=p_{mk}(x_i)[1-p_{mk}(x_i)]\]

XGBOOST的计算流程如下：

for k=1 to K: Initialize $F_{0k}(x)=0$
for m=1 to M:
- for k=1 to K: compute $g_{m-1,ki}$ and $h_{m-1,ki}$ for each sample $(x_i, y_i)$
- for k=1 to K: compute regression tree $f_{mk}(x)=\sum_{j=1}^{J_{mk}}w_{mkj}I({x}\in{R_{mkj}})$, where $R_{mkj}$(j=1 to J_mk) refer to the leaf nodes, to minimize the function $\sum_{i=1}^{N}[g_{m-1,ki}f_{mk}(x_i)+\frac{1}{2}h_{m-1,ki}f_{mk}(x_i)^2]+\Omega(f_{mk})$, where $\Omega$ is the regularization function
- for k=1 to K: $F_{mk}(x)=F_{m-1,k}(x)+\eta*\sum_{j=1}^{J_{mk}}w_{mkj}I({x}\in{R_{mkj}})$, $\eta$为学习率

针对回归树$f_{mk}(x)=\sum_{j=1}^{J_{mk}}w_{mkj}I({x}\in{R_{mkj}})$的求解，记$\Omega(f_{mk})=\gamma J_{mk}+\frac \lambda {2}\sum_{j=1}^{J_{mk}}w_{mkj}^2$，最小化问题变为J_mk个独立的二次函数之和：\[\sum_{j=1}^{J_{mk}}[(\sum_{x_i \in R_{mkj}}g_{m-1,ki})w_{mkj}+\frac{1}{2}(\lambda+\sum_{x_i \in R_{mkj}}h_{m-1,ki})w_{mkj}^2]+\gamma J_{mk}\]

假设回归树的结构已经固定，记$T_k=J_{mk},G_{kj}=\sum_{x_i \in R_{mkj}}g_{m-1,ki}和H_{kj}=\sum_{x_i \in R_{mkj}}h_{m-1,ki}$，此时在$w_{mkj}=-\frac{G_{kj}}{H_{kj}+\lambda}$时目标函数取得最小值，最小值为$-\frac{1}{2}\sum_{j=1}^{T_k}\frac{G_{kj}^2}{H_{kj}+\lambda}+\gamma T_k$

使用贪心算法确定回归树的结构，从一个节点开始不断进行分裂，分裂的规则是挑选使分裂增益最大的特征和分裂点进行分裂，直到达到某个停止条件为止。假设待分裂的节点为$I$，分裂后的两个节点分别为$I_L$和$I_R$，那么分裂增益可以表示为：\[-\frac{1}{2}\frac{(\sum_{x_i \in I}g_{m-1,ki})^2}{\lambda+\sum_{x_i \in I}h_{m-1,ki}}+\frac{1}{2}[\frac{(\sum_{x_i \in I_L}g_{m-1,ki})^2}{\lambda+\sum_{x_i \in I_L}h_{m-1,ki}}+\frac{(\sum_{x_i \in I_R}g_{m-1,ki})^2}{\lambda+\sum_{x_i \in I_R}h_{m-1,ki}}]-\gamma\]

XGBOOST的Python包中的重要参数

eta(alias:learning_rate): learning rate or shrinkage，同上文的$\eta$，常用值0.01~0.2
gamma(alias: min_split_loss): min loss reduction to create new tree split，同上文的$\gamma$
lambda(alias: reg_lambda): L2 regularization on leaf weights，同上文的$\lambda$
alpha(alias: reg_alpha): L1 regularization on leaf weights
max_depth: max depth per tree，常用值3~10
min_child_weight: minimum sum of instance weight(hessian) of all observations required in a child，即在一个节点中的样本的损失函数二阶导数之和，以多分类为例，$\sum_{x_i \in I}h_{mki}=\sum_{x_i \in I}p_{mk}(x_i)[1-p_{mk}(x_i)]$
subsample: % samples used per tree，常用值0.5~1
colsample_bytree: % features used per tree，常用值0.5~1
针对样本类别不均衡问题，若是二分类问题，可以设置参数scale_pos_weight，通常设为sum(negative instances)/sum(positive instances); 若是多分类问题，可以通过xgboost.DMatrix(...,weight,...)或者在Scikit-Learn API中调用fit(...,sample_weight,...)
特征重要性可以通过该特征被选中作为节点分裂特征的次数来度量
xgboost应用示例可参见文章Xgboost应用及调参示例