Chapter 1 - Neural Network and Deep Learning

Nerual Network and Deep Learning: Chapter 1

Concepts

Goal: understand the core concepts

Perceptrons

A perceptron take several binary inputs and produces a single output

$$\text{output} = \begin{cases} 1, &\quad w\cdot x + b > 0 \\ 0, &\quad w\cdot x +b \leq 0 \end{cases}$$
NAND: $w = \{-2,-2\}^t, b = 3$

Sigmoid Neuron

inputs and output between 0 and 1.

$$\text{output} = \sigma(w\cdot x+b)$$
$$\text{sigmoid(logistic) function: }\sigma(z) \equiv\frac{1}{1+e^{-z}}$$
( $\sigma$ is one of the most commonly used activation function because of its lovely derivative property)

Neural Networks

(Sometimes called multilayer perceptrons or MLPs despite made up of sigmoid neurons)
feedforward networks: no loop allowed
(feedback networks: e.g. recurrent neural networks)

input layer: 784 (28x28)
1 hidden layer: x (eg. 15)
output layer: 10 (one-hot) (in this case, it doesn’t work well to use 4 output binary digits to represent 0~9)

Gradient Descent

Define a cost function:
mean squared error or MSE:

$$\text{cost}(w,b)\equiv \frac 1{2n}\sum_x\|y(x)-a\|^2$$
(if just use accuracy, it’s not smooth)

We have:

$$\Delta C\approx \nabla C \cdot \Delta v$$
$$\nabla C \equiv \left(\frac{\partial C}{\partial v_1}, \dots, \frac{\partial C}{\partial v_m} \right)^T$$
We can choose $\Delta v = -\eta \nabla C$ so that $\Delta C$ is negative (approx.).

Then in this neural network we have:

$$w_k'=w_k-\eta\frac {\partial C}{\partial w_k}$$
$$b_l'=b_l-\eta\frac {\partial C}{\partial b_l}$$

Use randomly chosen mini-batches of size $m$:

$$w_k'=w_k-\frac\eta m\sum_j{\frac {\partial C_{X_j}}{\partial w_k}}$$

$$b_l'=b_l-\frac\eta m \sum_j{\frac {\partial C_{X_j}}{\partial b_l}}$$

Implementation

View full data and code here.

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,
            test_data=None):
        """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):
            random.shuffle(training_data)
            mini_batches = [
                training_data[k:k+mini_batch_size]
                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)
            else:
                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``."""
        ''' 本文中省略 '''

    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))