over 2 years ago



$$w = (2,2), b=-3$$


由於數學上階梯函數(step function)數學微分後會遇到無窮大,所以使用sigmoid函數代替。

直覺上,經過神經元的反應(activation)後得到的結果不一定與真實結果一致(y)。此時就能給出成本函數(cost function)如下,
$$ \frac{1}{2} | y - \sigma(z)|^2$$
給定一組訓練資料(training data), 最小化成本函數(cost function)
$$ \frac{1}{n} \sum_{x=1}^{n} C_x$$ 一般來說是一張「網」

圖中,$$C_x = \frac{1}{2} \sum_j (y_j-a^L_j)^2 = \frac{1}{2}\left[ \left( y_1 - a_1^3\right)^2 + \left(y_2-a_2^3\right)^2\right] $$

在第層的個神經元因為受到微小的改變而引起的誤差,$$\delta^l_j \equiv \frac{\partial C}{\partial z_j^l}= \left( a^l - y\right) \sigma'(z^l) \tag{1}$$前一層的誤差 可由後一層逆推,
\delta^l = \left( \left( w^{l+1}\right)^T \delta^{l+1}\right) \odot \sigma'(z^l) \tag{2}
\frac{\partial C}{\partial b_j^l} = \delta_j^l \tag{3}
\frac{\partial C}{\partial w_{jk}^l} = \delta^l_j \left( a_k^{l-1} \right) \tag{4}


ANN 演算法實作

  1. 輸入一組待訓練資料(training data)

  2. 對每一個樣本 x, 決定對應的神經元活化後結果(activation) 按照以下流程計算

    • 往前計算經過的每層神經元 z值 - 對每一層(layer) l=2,3,...L 計算 $$ a^{x,l} = \sigma(z^{x,l})$$
    • 輸出誤差(output error: 計算向量 $$ \sigma^{x,L} = (a^{x,L} - y^{L}) \odot \sigma'(z^{x,L}) $$
    • 背傳播(back-propagation)求解每一層的誤差值 -- For each L,L-1,L-2 ...2 $$ \begin{eqnarray} \delta^l = ((w^{l+1})^T \delta^{l+1}) \odot \sigma'(z^l) \end{eqnarray}$$
  3. 隨機梯度下降法(Stochastic Gradient Descent) - 對每一層 l = L,L-1,... 2 更新
    w^l \rightarrow
    w^l-\frac{\eta}{m} \sum_x \delta^{x,l} (a^{x,l-1})^T ,
    b^l \rightarrow b^l-\frac{\eta}{m}
    \sum_x \delta^{x,l}

A module to implement the stochastic gradient descent learning
algorithm for a feedforward neural network.  Gradients are calculated
using backpropagation.  Note that I have focused on making the code
simple, easily readable, and easily modifiable.  It is not optimized,
and omits many desirable features.

#### Libraries

# Standard library

import random

# Third-party libraries

import numpy as np

class Network(object):

    def __init__(self, sizes):
        """The list ``sizes`` contains the number of neurons in the
        respective layers of the network.  For example, if the list
        was [2, 3, 1] then it would be a three-layer network, with the
        first layer containing 2 neurons, the second layer 3 neurons,
        and the third layer 1 neuron.  The biases and weights for the
        network are initialized randomly, using a Gaussian
        distribution with mean 0, and variance 1.  Note that the first
        layer is assumed to be an input layer, and by convention we
        won't set any biases for those neurons, since biases are only
        ever used in computing the outputs from later layers."""
        self.num_layers = len(sizes)
        self.sizes = sizes
        self.biases = [np.random.randn(y, 1) for y in sizes[1:]]
        self.weights = [np.random.randn(y, x)
                        for x, y in zip(sizes[:-1], sizes[1:])]

    def feedforward(self, a):
        """Return the output of the network if ``a`` is input."""
        for b, w in zip(self.biases, self.weights):
            a = sigmoid(np.dot(w, a)+b)
        return a

    def SGD(self, training_data, epochs, mini_batch_size, eta,
        """Train the neural network using mini-batch stochastic
        gradient descent.  The ``training_data`` is a list of tuples
        ``(x, y)`` representing the training inputs and the desired
        outputs.  The other non-optional parameters are
        self-explanatory.  If ``test_data`` is provided then the
        network will be evaluated against the test data after each
        epoch, and partial progress printed out.  This is useful for
        tracking progress, but slows things down substantially."""
        if test_data: n_test = len(test_data)
        n = len(training_data)
        for j in xrange(epochs):
            mini_batches = [
                for k in xrange(0, n, mini_batch_size)]
            for mini_batch in mini_batches:
                self.update_mini_batch(mini_batch, eta)
            if test_data:
                print "Epoch {0}: {1} / {2}".format(
                    j, self.evaluate(test_data), n_test)
                print "Epoch {0} complete".format(j)

    def update_mini_batch(self, mini_batch, eta):
        """Update the network's weights and biases by applying
        gradient descent using backpropagation to a single mini batch.
        The ``mini_batch`` is a list of tuples ``(x, y)``, and ``eta``
        is the learning rate."""
        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 = [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)]

    def backprop(self, x, y):
        """Return a tuple ``(nabla_b, nabla_w)`` representing the
        gradient for the cost function C_x.  ``nabla_b`` and
        ``nabla_w`` are layer-by-layer lists of numpy arrays, similar
        to ``self.biases`` and ``self.weights``."""
        nabla_b = [np.zeros(b.shape) for b in self.biases]
        nabla_w = [np.zeros(w.shape) for w in self.weights]
        # feedforward

        activation = x
        activations = [x] # list to store all the activations, layer by layer

        zs = [] # list to store all the z vectors, layer by layer

        for b, w in zip(self.biases, self.weights):
            z = np.dot(w, activation)+b
            # print "z shape",z.shape

            activation = sigmoid(z)
        # backward pass

        delta = self.cost_derivative(activations[-1], y) *sigmoid_prime(zs[-1])
        nabla_b[-1] = delta
        nabla_w[-1] = np.dot(delta, activations[-2].transpose())
        # Note that the variable l in the loop below is used a little

        # differently to the notation in Chapter 2 of the book.  Here,

        # l = 1 means the last layer of neurons, l = 2 is the

        # second-last layer, and so on.  It's a renumbering of the

        # scheme in the book, used here to take advantage of the fact

        # that Python can use negative indices in lists.

        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 evaluate(self, test_data):
        """Return the number of test inputs for which the neural
        network outputs the correct result. Note that the neural
        network's output is assumed to be the index of whichever
        neuron in the final layer has the highest activation."""
        test_results = [(np.argmax(self.feedforward(x)), y)
                        for (x, y) in test_data]
        return sum(int(x == y) for (x, y) in test_results)

    def cost_derivative(self, output_activations, y):
        """Return the vector of partial derivatives \partial C_x /
        \partial a for the output activations."""
        return (output_activations-y)

#### Miscellaneous functions

def sigmoid(z):
    """The sigmoid function."""
    return 1.0/(1.0+np.exp(-z))

def sigmoid_prime(z):
    """Derivative of the sigmoid function."""
    return sigmoid(z)*(1-sigmoid(z))
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