调整chapter8

docs/chapter8/chapter8.md

@@ -62,35 +62,12 @@ $$
 \end{aligned}
 $$
 
-## 8.16
-由式(8.13)可知
-$$
-\begin{aligned} \ell_{\exp }\left(H_{t-1}+h_{t} | \mathcal{D}\right) & \simeq \mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}}\left[e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})}\left(1-f(\boldsymbol{x}) h_{t}(\boldsymbol{x})+\frac{f^{2}(\boldsymbol{x}) h_{t}^{2}(\boldsymbol{x})}{2}\right)\right] \\ &=\mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}}\left[e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})}\left(1-f(\boldsymbol{x}) h_{t}(\boldsymbol{x})+\frac{1}{2}\right)\right] \quad \text { . } \end{aligned}
-$$
-
-求得的$h_t$使得损失函数最小，所以得出(8.14)
-$$
-\begin{aligned}
-h_{t}(\boldsymbol{x})&=\underset{h}{\arg \min } \ell_{\exp }\left(H_{t-1}+h | \mathcal{D}\right)
-\\&
-\begin{array}{l}{ =\arg \min _{h} \mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}}\left[e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})}\left(1-f(\boldsymbol{x}) h(\boldsymbol{x})+\frac{1}{2}\right)\right]} \\ {=\arg \max _{h} \mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}}\left[e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})} f(\boldsymbol{x}) h(\boldsymbol{x})\right]} \\ {=\underset{\boldsymbol{h}}{\arg \max } \mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}}\left[\frac{e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})}}{\mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}\left[e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})}\right.}} f(\boldsymbol{x}) h(\boldsymbol{x})\right]}\end{array} \end{aligned}
-$$
-
-最后一行式子里$\mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}}\left[e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})}\right]$因为是基于学习器$H_{t-1}$的指数损失函数的期望，所以是大于0的常数，所以这里不影响损失函数最小时，参数h的值。
-
-式(8.15)
-$$
-\mathcal{D}_{t}(\boldsymbol{x})=\frac{\mathcal{D}(\boldsymbol{x}) e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})}}{\mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}}\left[e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})}\right]}
-$$
-
-这里的$\mathcal{D}$是x的概率分布
-
 ## 8.16
 $$
 \begin{aligned} h_{t}(\boldsymbol{x}) &=\underset{h}{\arg \max } \mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}}\left[\frac{e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})}}{\mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}}\left[e^{-f(\boldsymbol{x}) H_{t-1}(\boldsymbol{x})}\right]} f(\boldsymbol{x}) h(\boldsymbol{x})\right] \\ &=\underset{\boldsymbol{h}}{\arg \max } \mathbb{E}_{\boldsymbol{x} \sim \mathcal{D}_{t}}[f(\boldsymbol{x}) h(\boldsymbol{x})] \end{aligned}
 $$
 
-假设x的概率分布是f(x)
+[推导]：假设x的概率分布是f(x)
 (注:本书中概率分布全都是$\mathcal{D(x)}$)
 
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