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$$H(x)=\sum_{t=1}^{T}\alpha _{t}h_{t}(x)$$

$$\iota _{exp}(H|D)=e^{-f(x)h(x)}$$

$$\frac{\alpha \iota _{exp}(H|D)}{\alpha H(x)} = e^{-H(x)}P(f(x)=1|x) + e^{H(x)}P(f(x)=-1|x)$$

$$H(x)=\frac{1}{2}ln\frac{P(f(x)=1|x)}{P(f(x)=-1|x)}$$

$$\frac{\partial \iota _{exp}(\alpha _{t}h_t|D_t)}{\partial \alpha _{t}} = \frac{\partial e^{(-f(x)\alpha _th_{t}(x))}}{\partial \alpha _{t}}\\=e^{-\alpha _{t}}P(f(x)=h_{t}(x)) + e^{\alpha _{t}}P(f(x)\neq h_{t}(x))\\=e^{-\alpha _{t}}(1-\epsilon _{t})+e^{\alpha _{t}}\epsilon _{t}$$

$$\alpha _{t}=\frac{1}{2}ln\left ( \frac{1-\epsilon _{t}}{\epsilon _{t}} \right )$$

Adboost算法在获得$H_{t-1}$之后样本分布进行调整，使下一轮的基学习器$h_t$能纠正$H_{t-1}$的一些错误，理想的$h_t$能纠正$H_{t-1}$的全部错误，即最小化：

$$\iota _{exp}(H_{t-1}+h_{t}|D)=e^{-f(x)(H_{t-1}(x)+h_{t}(x))}=e^{-f(x)H_{t-1}(x)h_{t}(x)}$$

$$Z_m=\frac{e^{-f(x)(H_{t-1}(x))}}{e^{-f(x)H_{t1}(x)}}=e^{-\alpha _tf(x)h_t(x)}$$

1) 初始化样本集权重为

$$D(1) = (w_{11}, w_{12}, ...w_{1m}) ;\;\; w_{1i}=\frac{1}{m};\;\; i =1,2...m$$

2) 对于k=1,2，...K:

• a) 使用具有权重$D_k$的样本集来训练数据，得到弱分类器$G_k(x)$
• b)计算$h_k(x)$的分类误差率

$$\epsilon _k = P(h_k(x_i) \neq y_i) = \sum\limits_{i=1}^{m}D_{ki}I(h_k(x_i) \neq y_i)$$

• c) 计算弱分类器的系数

$$\alpha_k = \frac{1}{2}log\frac{1-\epsilon _k}{\epsilon _k}$$

• d) 更新样本集的权重分布

$$D_{k+1,i} = \frac{D_{ki}}{Z_K}exp(-\alpha_kf_i(x)h_k(x_i)) \;\; i =1,2,...m$$

3) 构建最终分类器为：

$$f(x) = sign(\sum\limits_{k=1}^{K}\alpha_kh_k(x))$$

$$\alpha_k = \frac{1}{2}log\frac{1-\epsilon _k}{\epsilon _k} + log(R-1)$$

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