In [1]:

    

%matplotlib inline

"""The sigmoid function (or the logistic curve) is a 
function that takes any real number, z, and outputs a number (0,1).
It is useful in neural networks for assigning weights on a relative scale.
The value z is the weighted sum of parameters involved in the learning algorithm."""

import numpy
import matplotlib.pyplot as plt
import math as mt

z = numpy.arange(-5, 5, .1)
sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))
sigma = sigma_fn(z)

fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(z, sigma)
ax.set_ylim([-0.1, 1.1])
ax.set_xlim([-5,5])
ax.grid(True)
ax.set_xlabel('z')
ax.set_title('sigmoid function')

plt.show()

"""Step Function"""
z = numpy.arange(-5, 5, .02)
step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)
step = step_fn(z)

fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(z, step)
ax.set_ylim([-0.5, 1.5])
ax.set_xlim([-5,5])
ax.grid(True)
ax.set_xlabel('z')
ax.set_title('step function')

plt.show()

"""Sine Function"""
z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)
t = numpy.sin(z)

fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(z, t)
ax.set_ylim([-1.0, 1.0])
ax.set_xlim([-2*mt.pi,2*mt.pi])
ax.grid(True)
ax.set_xlabel('z')
ax.set_title('sine function')

plt.show()

"""Plots a graph of the squashing function used by a rectified linear
unit"""
z = numpy.arange(-2, 2, .1)
zero = numpy.zeros(len(z))
y = numpy.max([zero, z], axis=0)

fig = plt.figure()
ax = fig.add_subplot(111)
ax.plot(z, y)
ax.set_ylim([-2.0, 2.0])
ax.set_xlim([-2.0, 2.0])
ax.grid(True)
ax.set_xlabel('z')
ax.set_title('Rectified linear unit')

plt.show()


    

In [2]:

    

from scipy import optimize

class Neural_Network(object):
    def __init__(self, Lambda=0):        
        #Define Hyperparameters
        self.inputLayerSize = 2
        self.outputLayerSize = 1
        self.hiddenLayerSize = 3
        
        #Weights (parameters)
        self.W1 = np.random.randn(self.inputLayerSize,self.hiddenLayerSize)
        self.W2 = np.random.randn(self.hiddenLayerSize,self.outputLayerSize)
        
        #Regularization Parameter:
        self.Lambda = Lambda
        
    def forward(self, X):
        #Propogate inputs though network
        self.z2 = np.dot(X, self.W1)
        self.a2 = self.sigmoid(self.z2)
        self.z3 = np.dot(self.a2, self.W2)
        yHat = self.sigmoid(self.z3) 
        return yHat
        
    def sigmoid(self, z):
        #Apply sigmoid activation function to scalar, vector, or matrix
        return 1/(1+np.exp(-z))
    
    def sigmoidPrime(self,z):
        #Gradient of sigmoid
        return np.exp(-z)/((1+np.exp(-z))**2)
    
    def costFunction(self, X, y):
        #Compute cost for given X,y, use weights already stored in class.
        self.yHat = self.forward(X)
        J = 0.5*sum((y-self.yHat)**2)/X.shape[0] + (self.Lambda/2)*(np.sum(self.W1**2)+np.sum(self.W2**2))
        return J
        
    def costFunctionPrime(self, X, y):
        #Compute derivative with respect to W and W2 for a given X and y:
        self.yHat = self.forward(X)
        
        delta3 = np.multiply(-(y-self.yHat), self.sigmoidPrime(self.z3))
        #Add gradient of regularization term:
        dJdW2 = np.dot(self.a2.T, delta3)/X.shape[0] + self.Lambda*self.W2
        
        delta2 = np.dot(delta3, self.W2.T)*self.sigmoidPrime(self.z2)
        #Add gradient of regularization term:
        dJdW1 = np.dot(X.T, delta2)/X.shape[0] + self.Lambda*self.W1
        
        return dJdW1, dJdW2
    
    #Helper functions for interacting with other methods/classes
    def getParams(self):
        #Get W1 and W2 Rolled into vector:
        params = np.concatenate((self.W1.ravel(), self.W2.ravel()))
        return params
    
    def setParams(self, params):
        #Set W1 and W2 using single parameter vector:
        W1_start = 0
        W1_end = self.hiddenLayerSize*self.inputLayerSize
        self.W1 = np.reshape(params[W1_start:W1_end], \
                             (self.inputLayerSize, self.hiddenLayerSize))
        W2_end = W1_end + self.hiddenLayerSize*self.outputLayerSize
        self.W2 = np.reshape(params[W1_end:W2_end], \
                             (self.hiddenLayerSize, self.outputLayerSize))
        
    def computeGradients(self, X, y):
        dJdW1, dJdW2 = self.costFunctionPrime(X, y)
        return np.concatenate((dJdW1.ravel(), dJdW2.ravel()))
    
        
class trainer(object):
    def __init__(self, N):
        #Make Local reference to network:
        self.N = N
        
    def callbackF(self, params):
        self.N.setParams(params)
        self.J.append(self.N.costFunction(self.X, self.y))
        self.testJ.append(self.N.costFunction(self.testX, self.testY))
        
    def costFunctionWrapper(self, params, X, y):
        self.N.setParams(params)
        cost = self.N.costFunction(X, y)
        grad = self.N.computeGradients(X,y)
        return cost, grad
        
    def train(self, trainX, trainY, testX, testY):
        #Make an internal variable for the callback function:
        self.X = trainX
        self.y = trainY
        
        self.testX = testX
        self.testY = testY

        #Make empty list to store training costs:
        self.J = []
        self.testJ = []
        
        params0 = self.N.getParams()

        options = {'maxiter': 200, 'disp' : True}
        _res = optimize.minimize(self.costFunctionWrapper, params0, jac=True, method='BFGS', \
                                 args=(trainX, trainY), options=options, callback=self.callbackF)

        self.N.setParams(_res.x)
        self.optimizationResults = _res


    

In [3]:

    

import numpy as np

#sigmoid
def nonlin(x, deriv=False):
    if (deriv==True):
        return x*(1-x)
    return 1/(1+np.exp(-x))

#input data
x=np.array([[0,0,1],[0,1,1],[1,0,1],[1,1,1]])

#output data
y=np.array([0,1,1,0]).T

#seed random numbers to make calculation
np.random.seed(1)

#initialize weights with mean=0
syn0=2*np.random.random((3,4))-1

for iter in range(10000):
    #forward propogation
    l0=x
    l1=nonlin(np.dot(l0,syn0))
    l1_error=y-l1
    #multiply error by slope of sigmoid at values of l1
    l1_delta=l1_error*nonlin(l1,True)
    #update weights
    syn0+=np.dot(l0.T, l1_delta)
    
print("Output after training: ",l1 )


    

    




Output after training:  [[6.55109972e-03 9.93684857e-01 9.93925710e-01 6.62304973e-03]
 [1.71082162e-03 9.97516440e-01 9.97766376e-01 1.82685927e-03]
 [2.05800960e-03 9.98268211e-01 9.97548919e-01 1.77362990e-03]
 [5.35659849e-04 9.99320839e-01 9.99100767e-01 4.87503198e-04]]

In [4]:

    

import numpy as np
import random
class Network(object):
    
    def _init_(self, sizes):
        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:])]

#sizes is the number of neurons in each layer
#for example, say n_1st_layer=3, n_2nd_layer=3, n_3rd_layer=1, then net=Network([3,3,1])

#The biases and weights are initialized randomly, using Gaussian distributions of mean=0, stdev=1
#z is a vector (or a np.array)

    def feedforward(self,a):
        #returns output w/ 'a' as an input
        for b, w in zip(self.biases, self.weights):
            a=sigmoid(np.dot(w,b)+b)
        return a
    
#Apply a Stochastic Gradient Descent (SGD) method:
    def SGD(self, training_data, epochs, mini_batch_size, eta, test_data=None):
        """Trains network using batches incorporating SGD. The network will be evaluated against the
        test data after each epoch, with partial progress being printed out (this is useful for tracking,
        but slows the process.)"""
        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(o,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):
        #updates w and b using backpropagation to a single mini batch. 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
            zs.append(z)
            activation = sigmoid(z)
            activations.append(activation)
        # 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)
    
    
    
#Functions
def sigmoid(z):
        return 1.0/(1.0+np.exp(-z))

def sigmoid_prime(z):
    return sigmoid(z)*(1-sigmoid(z))

network=Network()


    

In [5]:

    

# %load neural-networks-and-deep-learning/src/mnist_loader.py
"""
mnist_loader
~~~~~~~~~~~~

A library to load the MNIST image data.  For details of the data
structures that are returned, see the doc strings for ``load_data``
and ``load_data_wrapper``.  In practice, ``load_data_wrapper`` is the
function usually called by our neural network code.
"""

#### Libraries
# Standard library
import pickle
import gzip

# Third-party libraries
import numpy as np

def load_data():
    """Return the MNIST data as a tuple containing the training data,
    the validation data, and the test data.

    The ``training_data`` is returned as a tuple with two entries.
    The first entry contains the actual training images.  This is a
    numpy ndarray with 50,000 entries.  Each entry is, in turn, a
    numpy ndarray with 784 values, representing the 28 * 28 = 784
    pixels in a single MNIST image.

    The second entry in the ``training_data`` tuple is a numpy ndarray
    containing 50,000 entries.  Those entries are just the digit
    values (0...9) for the corresponding images contained in the first
    entry of the tuple.

    The ``validation_data`` and ``test_data`` are similar, except
    each contains only 10,000 images.

    This is a nice data format, but for use in neural networks it's
    helpful to modify the format of the ``training_data`` a little.
    That's done in the wrapper function ``load_data_wrapper()``, see
    below.
    """
    f = gzip.open('../data/mnist.pkl.gz', 'rb')
    training_data, validation_data, test_data = cPickle.load(f)
    f.close()
    return (training_data, validation_data, test_data)

def load_data_wrapper():
    """Return a tuple containing ``(training_data, validation_data,
    test_data)``. Based on ``load_data``, but the format is more
    convenient for use in our implementation of neural networks.

    In particular, ``training_data`` is a list containing 50,000
    2-tuples ``(x, y)``.  ``x`` is a 784-dimensional numpy.ndarray
    containing the input image.  ``y`` is a 10-dimensional
    numpy.ndarray representing the unit vector corresponding to the
    correct digit for ``x``.

    ``validation_data`` and ``test_data`` are lists containing 10,000
    2-tuples ``(x, y)``.  In each case, ``x`` is a 784-dimensional
    numpy.ndarry containing the input image, and ``y`` is the
    corresponding classification, i.e., the digit values (integers)
    corresponding to ``x``.

    Obviously, this means we're using slightly different formats for
    the training data and the validation / test data.  These formats
    turn out to be the most convenient for use in our neural network
    code."""
    tr_d, va_d, te_d = load_data()
    training_inputs = [np.reshape(x, (784, 1)) for x in tr_d[0]]
    training_results = [vectorized_result(y) for y in tr_d[1]]
    training_data = zip(training_inputs, training_results)
    validation_inputs = [np.reshape(x, (784, 1)) for x in va_d[0]]
    validation_data = zip(validation_inputs, va_d[1])
    test_inputs = [np.reshape(x, (784, 1)) for x in te_d[0]]
    test_data = zip(test_inputs, te_d[1])
    return (training_data, validation_data, test_data)

def vectorized_result(j):
    """Return a 10-dimensional unit vector with a 1.0 in the jth
    position and zeroes elsewhere.  This is used to convert a digit
    (0...9) into a corresponding desired output from the neural
    network."""
    e = np.zeros((10, 1))
    e[j] = 1.0
    return e

net=network.Network([784,30,30])
net.SGD(training_data,30,10,3,test_data=test_data)


    

    




---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-5-4ac16fae6ccb> in <module>()
     86     return e
     87 
---> 88 net=network.Network([784,30,30])
     89 net.SGD(training_data,30,10,3,test_data=test_data)

NameError: name 'network' is not defined

In [ ]: