# 前向分步算法

**前向分步算法 forward algorithm**

因为学习的是加法模型，如果能从前向后，每一步只学习一个基函数及其系数，逐渐逼近优化目标函数式(2)，那么就可以简化优化的复杂度。\
具体地，每步只需要优化如下损失函数：

$$
\min\_{\beta,\gamma}\sum\_{i=1}^NL(y\_i,\beta b(x\_i;\gamma))
$$

**输入：**\
训练数据集T\
损失函数L(y, f(x))\
基函数集$${b(x;\gamma)}$$\
**输出：**\
加法模型f(x)

**步骤：**\
1\. 令$$f\_0(x)=0$$\
2\. 假设当前是第m个基函数，损失函数为：

$$
L(y\_i,f\_{m-1}(x\_i)+\beta b(x\_i, \gamma))
$$

1. 极小化损失函数，得到参数$$\beta\_m, \gamma\_m$$  

   $$
   (\beta\_m, \gamma\_m) = \arg\min\_{\beta, \gamma}\sum\_{i=1}^NL(y\_i,f\_{m-1}(x\_i)+\beta b(x\_i, \gamma))
   $$
2. 更新$$f\_m$$  

   $$
   f\_m(x) = f\_{m-1}(x) + \beta\_mb(x;\gamma\_m)
   $$

   5.2-4步进行M次，共得到M个$$\beta$$和$$\gamma$$  
3. 得到加法模型  

   $$
   f(x) = f\_M(x) = \sum\_{m=1}^M\beta\_mb(x;\gamma\_m)
   $$


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