《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∂en∂xL，有链式法则得到：
δL=∂en∂xL∂xL∂sL
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：
−(ya−1−y1−a)=−y(1−a)+a(1−y)a(1−a)=−y+aa(1−a)
所以 CrossEntropyCost 的 cost_derivative 是 −y+aa(1−a)
由 http://blog.csdn.net/u011239443/article/details/75091283#t0 可知：
∂a∂z=a(1−a)
所以：
δ=∂en∂a∂a∂z=−y+aa(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)