In [33]:
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 [35]:
# %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)
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