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：

1. for k=1 to K: Initialize \(F_{0k}(x)=0\)
2. 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：

1. for k=1 to K: Initialize \(F_{0k}(x)=0\)
2. 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