《neural network and deep learning》题解——ch03 再看手写识别问题题解与源码分析

http://blog.csdn.net/u011239443/article/details/77649026

完整代码：https://github.com/xiaoyesoso/neural-networks-and-deep-learning/blob/master/src/network2.py

我们之前根据《neural network and deep learning》题解——ch02 反向传播讲解了ch02 Network源码分析。这篇是对ch02 Network源码分析的改进。这里我们结合《机器学习技法》学习笔记12——神经网络重新讲解下。

交叉熵代价函数

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)

class CrossEntropyCost(object):
    @staticmethod
    def fn(a, y):
        return np.sum(np.nan_to_num(-y * np.log(a) - (1 - y) * np.log(1 - a)))

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

这边我们把损失函数封装成两个类，静态函数 fn 返回的是损失，delta返回的是ch02 反向传播中的δ。该delta对应《机器学习技法》学习笔记12——神经网络中就是：

我们在Network中使用的就是二次代价函数，这里我们就只讲解另外的交叉熵代价函数：

对应代码：

np.sum(np.nan_to_num(-y * np.log(a) - (1 - y) * np.log(1 - a)))

接下来我们来看看关于delta的问题：

看看 network.py 中的 Network.cost_derivative ⽅法。这个⽅法是为⼆次代价函数写的。怎样修改可以⽤于交叉熵代价函数上？你能不能想到可能在交叉熵函数上遇到的问题？在 network2.py 中，我们已经去掉了Network.cost_derivative ⽅法，将其集成进了‘CrossEntropyCost.delta‘ ⽅法中。请问，这样是如何解决你已经发现的问题的？

对应《机器学习技法》学习笔记12——神经网络中，cost_derivative就是$\partial \frac{e_n}{\partial x^L}$，有链式法则得到：
${\delta }^L=\frac{\partial e_n}{\partial x^L}\frac{\partial x^L}{\partial s^L}$
network中也是的cost_derivative也是用在求δ。
而CrossEntropyCost.delta是：

return (a - y)

代码 中的 a 就是上式中的x，z 就是上式中的 s。
我们对CrossEntropyCost关于a求导，得到：
$-(\frac{y}{a}-\frac{1-y}{1-a})=\frac{-y(1-a)+a(1-y)}{a(1-a)}=\frac{-y+a}{a(1-a)}$
所以 CrossEntropyCost 的 cost_derivative 是 $\frac{-y+a}{a(1-a)}$
由 http://blog.csdn.net/u011239443/article/details/75091283#t0 可知：
$\frac{\partial a}{\partial z}=a(1-a)$
所以：
$\delta =\frac{\partial e_n}{\partial a}\frac{\partial a}{\partial z}=\frac{-y+a}{a(1-a)}a(1-a)=a-y$

初始化

和Network基本上一样，只不过封装成了一个default_weight_initializer函数

    def __init__(self, sizes, cost=CrossEntropyCost):
        self.num_layers = len(sizes)
        self.sizes = sizes
        self.default_weight_initializer()
        self.cost = cost

    def default_weight_initializer(self):
        self.biases = [np.random.rand(y, 1) for y in self.sizes[1:]]
        self.weights = [np.random.rand(y, x) / np.sqrt(x) for x, y in zip(self.sizes[:-1], self.sizes[1:])]

随机梯度下降

和Network基本上一样，各个monitor是代表是否需要检测该对应的指标。

    def SGD(self, training_data, epochs, mini_batch_size, eta,
            lmbda=0.0,
            evaluation_data=None,
            monitor_evaluation_cost=False,
            monitor_evaluation_accuracy=False,
            monitor_training_cost=False,
            monitor_training_accuray=False):
        if evaluation_data:
            n_data = len(evaluation_data)
        n = len(training_data)
        evaluation_cost, evaluation_accurary = [], []
        training_cost, training_accuray = [], []
        for j in xrange(epochs):
            random.shuffle(training_data)
            mini_batches = [training_data[k:k + mini_batch_size] for k in range(0, n, mini_batch_size)]
            for mini_batch in mini_batches:
                self.update_mini_batch(mini_batch, eta, lmbda, len(training_data))

            print "Epoch %s training complete" %(j+1)

            if monitor_training_cost:
                cost = self.total_cost(training_data, lmbda)
                training_cost.append(cost)
                print "Cost on train: {}".format(cost)

            if monitor_training_accuray:
                acc = self.accuracy(training_data,covert=True)
                training_accuray.append(acc)
                print "Acc on train: {} / {}".format(acc,n)

            if monitor_evaluation_cost:
                cost = self.total_cost(evaluation_data, lmbda,convert=True)
                evaluation_cost.append(cost)
                print "Cost on evaluation: {}".format(cost)

            if monitor_evaluation_accuracy:
                acc = self.accuracy(evaluation_data)
                evaluation_accurary.append(acc)
                print "Acc on evaluation: {} / {}".format(acc, n_data)

            print

        return evaluation_cost,evaluation_accurary,training_cost,training_accuray

反向传播

    def backprop(self, x, y):
        nabla_b = [np.zeros(b.shape) for b in self.biases]
        nabla_w = [np.zeros(w.shape) for w in self.weights]

        activation = x
        activations = [x]
        zs = []
        for b, w in zip(self.biases, self.weights):
            z = np.dot(w, activation) + b
            zs.append(z)
            activation = sigmoid(z)
            activations.append(activation)

        delta = (self.cost).delta(zs[-1], activations[-1], y)
        nabla_b[-1] = delta
        nabla_w[-1] = np.dot(delta, activations[-2].transpose())

        for l in xrange(2, self.num_layers):
            z = zs[-l]
            sp = sigmoid_prime(z)
            delta = np.dot(self.weights[-l + 1].transpose(), delta) * sp
            nabla_b[-l] = delta
            nabla_w[-l] = np.dot(delta, activations[-l - 1].transpose())
        return (nabla_b, nabla_w)

    def update_mini_batch(self, mini_batch, eta, lmbda, n):
        nabla_b = [np.zeros(b.shape) for b in self.biases]
        nabla_w = [np.zeros(w.shape) for w in self.weights]

        for x, y in mini_batch:
            delta_nabla_b, delta_nabla_w = self.backprop(x, y)
            nabla_b = [nb + dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]
            nabla_w = [nw + dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]

        self.weights = [(1 - eta * (lmbda / n)) * w - (eta / len(mini_batch)) * nw for w, nw in
                        zip(self.weights, nabla_w)]
        self.biases = [b - (eta / len(mini_batch)) * nb for b, nb in zip(self.biases, nabla_b)]

我们可以看到基本上和Network中一样，前面已经讲解过δ。这里的代码也可以和《机器学习技法》学习笔记12——神经网络中的公式对应：

L2规范化

主要区别是在最后两行更新的时候加入了L2规范化：

求偏导数得：

则：

L1规范化

这里引出了我们这节的另外一个问题：

更改上⾯的代码来实现 L1 规范化

求导得到：

则：

对应的代码应该写为：

  self.weights = [(1 - eta * (lmbda / n)*np.sign(w)) * w - (eta / len(mini_batch)) * nw for w, nw in
                        zip(self.weights, nabla_w)]
  self.biases = [b - (eta / len(mini_batch)) * nb for b, nb in zip(self.biases, nabla_b)]

测评

有些label，我们需要对其进行二元化处理，然后使用：

def vectorized_result(j):
    e = np.zeros((10, 1))
    e[j] = 1.0
    return e

计算损失率

这会加入L2 规范化

    def total_cost(self, data, lmbda, convert=False):
        cost = 0.0
        for x, y in data:
            a = self.feedforward(x)
            if convert:
                y = vectorized_result(y)
            cost += self.cost.fn(a, y) / len(data)
        cost += 0.5 * (lmbda / len(data)) * sum(np.linalg.norm(w) ** 2 for w in self.weights)
        return cost

回到我们之前的L1规范化实现的问题，这里代码可改成：

cost +=  (lmbda / len(data)) * sum(np.linalg.norm(w) for w in self.weights)

计算准确率

和Network中基本上一致

    def accuracy(self,data,covert=False):
        if covert:
            results = [(np.argmax(self.feedforward(x)),np.argmax(y)) for (x,y) in data]
        else:
            results = [(np.argmax(self.feedforward(x)),y) for (x,y) in data]
        return sum(int(x==y) for (x,y) in results)