---
# System prepended metadata

title: 給教授看的理論

---

# 給教授看的理論

## 隱藏層神經元

### 圖示

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$F(x)=weight*x$

### Activation Function (激勵函數)

- 讓資料可以被微分
- 表示狀態(激活和不激活)

#### Relu
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### Cost Function (成本函數)

其實就是**均方誤差**，又稱Error函數

### Backpropagation (反向傳播)

利用更新**權重**來影響 ANN 的結果，本次專題藉由**隨機梯度下降**反向傳播

### Stochastic Gradient Descent 隨機梯度下降法



#### $/frac{dError}{dWeight_i}$
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w是權重，$\eta$是學習率
將$Error$對$Weight_{i}$微分，求出梯度g

將$Error$對$Weight_{i}$微分:

$\frac{dError}{dWeight_{i}}=\frac{\sum (x_i-f(x_i))^2}{dWeight_{i}}=\frac{\sum x_i^2+f(x_i)^2-2x_if(x_i)}{dWeight_{i}}$
$=(\sum\frac{x_i^2+f(x_i)^2-2x_if(x_i)}{df(x_i)})*\frac{df(x_i)}{dWeight_{i}}=\sum(2f(x_i)-2)*\frac{df(x_i)}{dWeight_{i}}$

其中對每一次的batch而言，$(\sum2f(x_i)-2)$是常數。
$\frac{df(x_i)}{dWeight_{i}}$可以透過自動微分快速取得。

$Weight_i=Weight_i+g*\eta;$

#### 多項輸入值

~~照做~~

#### 自動微分

自動微分大量使用鏈鎖法則和$\frac{d(a+b+c)}{de}=\frac{da+db+dc}{de}$
下面展示Error Singal的概念和為何和微分結果一樣:


