# 关于代价函数的两个假设

反向传播算法的目标：**求出代价函数C在任意w、b处的偏导**$$\partial C / \partial w$$**和**$$\partial C / \partial b$$。

反向传播函数对代价函数有两个假设，以第一章所使用的二次代价函数为例：

$$
\begin{eqnarray}
C = \frac{1}{2n} \sum\_x |y(x)-a^L(x)|^2,
\tag{26}\end{eqnarray}
$$

## 假设一

假设一：代价函数C可以写成每个样本的代价函数之和的形式

$$
C = \frac{1}{n} \sum\_x C\_x
$$

例如对于公式26来说，有

$$
C\_x = \frac{1}{2} |y-a^L |^2 \tag{1}
$$

问：为什么会有这样的假设？ 答：因为反向传播算法是针对每个样本单独计算偏导的。

## 假设二

假设二：代价函数$$C\_x$$可以写成神经元视角的形式。\
\&#xNAN;*神经元视角是我发明的词，我不知道怎么表达这种形式。参见*[*link*](https://windmising.gitbook.io/nielsen-nndl/introduction-1/assumptions)

例如公式（1）可以写成

$$
\begin{eqnarray}
C = \frac{1}{2} |y-a^L|^2 = \frac{1}{2} \sum\_j (y\_j-a^L\_j)^2,
\tag{27}\end{eqnarray}
$$


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