# 使用cross-entropy分类手写数字

本节使用network2.py。\
network2.py不仅包含cross-entropy，还有其他一些改进。\
使用`help(network2.Network.SGD)`可以获取文档信息。

通过代码验证发现使用cross-entropy的识别率要高一些。\
但就此认定cross-entropy要优于二次代价函数是不严谨的。\
还需要结合超参数的选择。

## 代码

### cross entropy代价函数

`class CrossEntropyCost`来实现cross entropy代价函数。

因为代价函数涉及到了两处不同的计算改变，因此将这两处的函数封装到一个类中。

1. cost的计算  

```python
@staticmethod
def fn(a, y):
    return np.sum(np.nan_to_num(-y*np.log(a)-(1-y)*np.log(1-a)))
```

1. error的计算

```python
@staticmethod
def delta(z, a, y):
    return (a-y)
```

### 二次代价函数

相应的，二次代价函数也封装成了一个类

```python
class QuadraticCost(object):

    @staticmethod
    def fn(a, y):
        return 0.5*np.linalg.norm(a-y)**2

    @staticmethod
    def delta(z, a, y):
        return (a-y) * sigmoid_prime(z)
```


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