{
"cells": [
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"source": [
"***\n",
"The code below is adapted from **Michael Nielsen's book:**\n",
"\n",
"http://neuralnetworksanddeeplearning.com/chap1.html\n",
"\n",
"I adapted it to run in *Python 3* and I added a couple of visualization and plotting functions. The code implements an MLP for classification of digit images. The MNIST data for handwritten digits is used for training and testing.\n",
"\n",
"The MNIST data comes in two parts. The first part contains 60,000 images to be used as training data. These images are scanned handwriting samples from 250 people, half of whom were US Census Bureau employees, and half of whom were high school students. The images are greyscale and 28 by 28 pixels in size. The second part of the MNIST data set is 10,000 images to be used as test data. Again, these are 28 by 28 greyscale images. We'll use the test data to evaluate how well our neural network has learned to recognize digits. To make this a good test of performance, the test data was taken from a different set of 250 people than the original training data (albeit still a group split between Census Bureau employees and high school students). This helps give us confidence that our system can recognize digits from people whose writing it didn't see during training.\n",
"\n",
"Download the MNIST data using:\n",
"\n",
"`git clone` https://github.com/mnielsen/neural-networks-and-deep-learning.git\n",
"\n",
"Or from: \n",
"\n",
"https://github.com/mnielsen/neural-networks-and-deep-learning/archive/master.zip"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"\"\"\"\n",
"mnist_loader\n",
"~~~~~~~~~~~~\n",
"\n",
"A library to load the MNIST image data. For details of the data\n",
"structures that are returned, see the doc strings for ``load_data``\n",
"and ``load_data_wrapper``. In practice, ``load_data_wrapper`` is the\n",
"function usually called by our neural network code.\n",
"\n",
"\n",
"\"\"\"\n",
"\n",
"#### Libraries\n",
"# Standard library\n",
"#import cPickle\n",
"import _pickle as cPickle\n",
"import gzip\n",
"\n",
"# Third-party libraries\n",
"import numpy as np\n",
"\n",
"def load_data():\n",
" \"\"\"Return the MNIST data as a tuple containing the training data,\n",
" the validation data, and the test data.\n",
"\n",
" The ``training_data`` is returned as a tuple with two entries.\n",
" The first entry contains the actual training images. This is a\n",
" numpy ndarray with 50,000 entries. Each entry is, in turn, a\n",
" numpy ndarray with 784 values, representing the 28 * 28 = 784\n",
" pixels in a single MNIST image.\n",
"\n",
" The second entry in the ``training_data`` tuple is a numpy ndarray\n",
" containing 50,000 entries. Those entries are just the digit\n",
" values (0...9) for the corresponding images contained in the first\n",
" entry of the tuple.\n",
"\n",
" The ``validation_data`` and ``test_data`` are similar, except\n",
" each contains only 10,000 images.\n",
"\n",
" This is a nice data format, but for use in neural networks it's\n",
" helpful to modify the format of the ``training_data`` a little.\n",
" That's done in the wrapper function ``load_data_wrapper()``, see\n",
" below.\n",
" \"\"\"\n",
" f = gzip.open('neural-networks-and-deep-learning/data/mnist.pkl.gz', 'rb')\n",
" \n",
" training_data, validation_data, test_data = cPickle.load(f, encoding='latin1')\n",
" f.close()\n",
" return (training_data, validation_data, test_data)\n",
"\n",
"def load_data_wrapper():\n",
" \"\"\"Return a tuple containing ``(training_data, validation_data,\n",
" test_data)``. Based on ``load_data``, but the format is more\n",
" convenient for use in our implementation of neural networks.\n",
"\n",
" In particular, ``training_data`` is a list containing 50,000\n",
" 2-tuples ``(x, y)``. ``x`` is a 784-dimensional numpy.ndarray\n",
" containing the input image. ``y`` is a 10-dimensional\n",
" numpy.ndarray representing the unit vector corresponding to the\n",
" correct digit for ``x``.\n",
"\n",
" ``validation_data`` and ``test_data`` are lists containing 10,000\n",
" 2-tuples ``(x, y)``. In each case, ``x`` is a 784-dimensional\n",
" numpy.ndarry containing the input image, and ``y`` is the\n",
" corresponding classification, i.e., the digit values (integers)\n",
" corresponding to ``x``.\n",
"\n",
" Obviously, this means we're using slightly different formats for\n",
" the training data and the validation / test data. These formats\n",
" turn out to be the most convenient for use in our neural network\n",
" code.\"\"\"\n",
" tr_d, va_d, te_d = load_data()\n",
" training_inputs = [np.reshape(x, (784, 1)) for x in tr_d[0]]\n",
" training_results = [vectorized_result(y) for y in tr_d[1]]\n",
" training_data = list(zip(training_inputs, training_results))\n",
" validation_inputs = [np.reshape(x, (784, 1)) for x in va_d[0]]\n",
" validation_data = list(zip(validation_inputs, va_d[1]))\n",
" test_inputs = [np.reshape(x, (784, 1)) for x in te_d[0]]\n",
" test_data = list(zip(test_inputs, te_d[1]))\n",
" return (training_data, validation_data, test_data)\n",
"\n",
"def vectorized_result(j):\n",
" \"\"\"Return a 10-dimensional unit vector with a 1.0 in the jth\n",
" position and zeroes elsewhere. This is used to convert a digit\n",
" (0...9) into a corresponding desired output from the neural\n",
" network.\"\"\"\n",
" e = np.zeros((10, 1))\n",
" e[j] = 1.0\n",
" return e\n"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"training_data, validation_data, test_data = load_data_wrapper()"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"\"\"\"\n",
"network.py\n",
"~~~~~~~~~~\n",
"\n",
"A module to implement the stochastic gradient descent learning\n",
"algorithm for a feedforward neural network. Gradients are calculated\n",
"using backpropagation. Note that I have focused on making the code\n",
"simple, easily readable, and easily modifiable. It is not optimized,\n",
"and omits many desirable features.\n",
"\"\"\"\n",
"\n",
"#### Libraries\n",
"# Standard library\n",
"import random\n",
"\n",
"# Third-party libraries\n",
"import numpy as np\n",
"\n",
"import matplotlib.pyplot as plt\n",
"from scipy.interpolate import make_interp_spline, BSpline\n",
"\n",
"\n",
"class Network(object):\n",
"\n",
" def __init__(self, sizes):\n",
" \"\"\"The list ``sizes`` contains the number of neurons in the\n",
" respective layers of the network. For example, if the list\n",
" was [2, 3, 1] then it would be a three-layer network, with the\n",
" first layer containing 2 neurons, the second layer 3 neurons,\n",
" and the third layer 1 neuron. The biases and weights for the\n",
" network are initialized randomly, using a Gaussian\n",
" distribution with mean 0, and variance 1. Note that the first\n",
" layer is assumed to be an input layer, and by convention we\n",
" won't set any biases for those neurons, since biases are only\n",
" ever used in computing the outputs from later layers.\"\"\"\n",
" self.num_layers = len(sizes)\n",
" self.sizes = sizes\n",
" self.biases = [np.random.randn(y, 1) for y in sizes[1:]]\n",
" print(\"Zip:\", sizes[:-1], sizes[1:], sizes[0:-1], \n",
" list(zip(sizes[:-1], sizes[1:])))\n",
" self.weights = [np.random.randn(y, x)\n",
" for x, y in zip(sizes[:-1], sizes[1:])]\n",
" print(\"Weight matrix: \", type(self.weights), len(self.weights))\n",
" for i in range(len(self.weights)):\n",
" print(\"\\tWeight row[{:d}]: {} {} {}x{}\".format(i, type(self.weights[i]), \n",
" self.weights[i].size, \n",
" self.sizes[i], self.sizes[i+1]))\n",
"\n",
" def display_weights(self, layer):\n",
" #print(\"Shape layer:\", self.weights[layer].shape)\n",
" #print(type(self.weights[layer][0:2]), self.weights[layer][0].shape)\n",
" # layers are internally numbered from 0, but the input starts from 1\n",
" fig, ax = plt.subplots()\n",
" #ax.imshow(activations, cmap=plt.cm.rainbow, interpolation='nearest')\n",
" #ax.imshow(activations, cmap=plt.cm.viridis, interpolation='nearest')\n",
" #ax.imshow(activations, cmap=plt.cm.plasma, interpolation='nearest')\n",
" #ax.imshow(activations, cmap=plt.cm.YlGnBu, interpolation='nearest')\n",
" ax.imshow(self.weights[layer-1], cmap=plt.cm.gray, interpolation='nearest')\n",
" plt.show()\n",
" \n",
" def display_learning_curve(self, smooth=False):\n",
" \n",
" plt.figure(figsize=(7,5))\n",
" plt.locator_params(axis='x', nbins=len(self.learning_curve))\n",
" #figure.suptitle('Example with only one row of subplots', fontsize=22)\n",
" \n",
" if smooth:\n",
" n_pts = list(range(1,len(self.learning_curve)+1))\n",
" n_pts_spline = np.linspace(min(n_pts), max(n_pts),300) #300 represents number of new pts\n",
"\n",
" spl = make_interp_spline(n_pts, self.learning_curve, k=3) #BSpline object\n",
" learning_curve_spline = spl(n_pts_spline)\n",
" \n",
" plt.plot(n_pts_spline, learning_curve_spline, color='red', linestyle='-', linewidth=1)\n",
" else:\n",
" plt.plot(range(1,len(self.learning_curve)+1), self.learning_curve, color='red', linestyle='-', linewidth=1)\n",
" \n",
" plt.xlabel('Epoch', fontsize=14)\n",
" plt.ylabel('Classification accuracy', fontsize=14)\n",
" plt.show()\n",
" \n",
" def feedforward(self, a):\n",
" \"\"\"Return the output of the network if ``a`` is input.\"\"\"\n",
" for b, w in zip(self.biases, self.weights):\n",
" a = sigmoid(np.dot(w, a)+b)\n",
" return a\n",
"\n",
" def SGD(self, training_data, epochs=10, \n",
" mini_batch_size=20, eta=0.5, test_data=None):\n",
" \"\"\"Train the neural network using mini-batch stochastic\n",
" gradient descent. The ``training_data`` is a list of tuples\n",
" ``(x, y)`` representing the training inputs and the desired\n",
" outputs. The other non-optional parameters are\n",
" self-explanatory. If ``test_data`` is provided then the\n",
" network will be evaluated against the test data after each\n",
" epoch, and partial progress printed out. This is useful for\n",
" tracking progress, but slows things down substantially.\"\"\"\n",
" if test_data: n_test = len(test_data)\n",
" n = len(training_data)\n",
" self.learning_curve = []\n",
" for j in range(epochs):\n",
" random.shuffle(training_data)\n",
" mini_batches = [\n",
" training_data[k:k+mini_batch_size]\n",
" for k in range(0, n, mini_batch_size)]\n",
" for mini_batch in mini_batches:\n",
" self.update_mini_batch(mini_batch, eta)\n",
" if test_data:\n",
" correct = self.evaluate(test_data)\n",
" print(\"Epoch {0}: {1} / {2}\".format(j, correct, n_test))\n",
" self.learning_curve.append(correct / n_test)\n",
" else:\n",
" print(\"Epoch {0} complete\".format(j))\n",
"\n",
" def update_mini_batch(self, mini_batch, eta):\n",
" \"\"\"Update the network's weights and biases by applying\n",
" gradient descent using backpropagation to a single mini batch.\n",
" The ``mini_batch`` is a list of tuples ``(x, y)``, and ``eta``\n",
" is the learning rate.\"\"\"\n",
" nabla_b = [np.zeros(b.shape) for b in self.biases]\n",
" nabla_w = [np.zeros(w.shape) for w in self.weights]\n",
" for x, y in mini_batch:\n",
" delta_nabla_b, delta_nabla_w = self.backprop(x, y)\n",
" nabla_b = [nb+dnb for nb, dnb in zip(nabla_b, delta_nabla_b)]\n",
" nabla_w = [nw+dnw for nw, dnw in zip(nabla_w, delta_nabla_w)]\n",
" self.weights = [w-(eta/len(mini_batch))*nw\n",
" for w, nw in zip(self.weights, nabla_w)]\n",
" self.biases = [b-(eta/len(mini_batch))*nb\n",
" for b, nb in zip(self.biases, nabla_b)]\n",
"\n",
" def backprop(self, x, y):\n",
" \"\"\"Return a tuple ``(nabla_b, nabla_w)`` representing the\n",
" gradient for the cost function C_x. ``nabla_b`` and\n",
" ``nabla_w`` are layer-by-layer lists of numpy arrays, similar\n",
" to ``self.biases`` and ``self.weights``.\"\"\"\n",
" nabla_b = [np.zeros(b.shape) for b in self.biases]\n",
" nabla_w = [np.zeros(w.shape) for w in self.weights]\n",
" # feedforward\n",
" activation = x\n",
" activations = [x] # list to store all the activations, layer by layer\n",
" zs = [] # list to store all the z vectors, layer by layer\n",
" for b, w in zip(self.biases, self.weights):\n",
" z = np.dot(w, activation)+b\n",
" zs.append(z)\n",
" activation = sigmoid(z)\n",
" activations.append(activation)\n",
" # backward pass\n",
" delta = self.cost_derivative(activations[-1], y) * \\\n",
" sigmoid_prime(zs[-1])\n",
" nabla_b[-1] = delta\n",
" nabla_w[-1] = np.dot(delta, activations[-2].transpose())\n",
" # Note that the variable l in the loop below is used a little\n",
" # differently to the notation in Chapter 2 of the book. Here,\n",
" # l = 1 means the last layer of neurons, l = 2 is the\n",
" # second-last layer, and so on. It's a renumbering of the\n",
" # scheme in the book, used here to take advantage of the fact\n",
" # that Python can use negative indices in lists.\n",
" for l in range(2, self.num_layers):\n",
" z = zs[-l]\n",
" sp = sigmoid_prime(z)\n",
" delta = np.dot(self.weights[-l+1].transpose(), delta) * sp\n",
" nabla_b[-l] = delta\n",
" nabla_w[-l] = np.dot(delta, activations[-l-1].transpose())\n",
" return (nabla_b, nabla_w)\n",
"\n",
" def evaluate(self, test_data):\n",
" \"\"\"Return the number of test inputs for which the neural\n",
" network outputs the correct result. Note that the neural\n",
" network's output is assumed to be the index of whichever\n",
" neuron in the final layer has the highest activation.\"\"\"\n",
" test_results = [(np.argmax(self.feedforward(x)), y)\n",
" for (x, y) in test_data]\n",
" return sum(int(x == y) for (x, y) in test_results)\n",
"\n",
" def cost_derivative(self, output_activations, y):\n",
" \"\"\"Return the vector of partial derivatives \\partial C_x /\n",
" \\partial a for the output activations.\"\"\"\n",
" return (output_activations-y)\n",
"\n",
"#### Miscellaneous functions\n",
"def sigmoid(z):\n",
" \"\"\"The sigmoid function.\"\"\"\n",
" return 1.0/(1.0+np.exp(-z))\n",
"\n",
"def sigmoid_prime(z):\n",
" \"\"\"Derivative of the sigmoid function.\"\"\"\n",
" return sigmoid(z)*(1-sigmoid(z))\n"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Zip: [784, 30] [30, 10] [784, 30] [(784, 30), (30, 10)]\n",
"Weight matrix: 2\n",
"\tWeight row[0]: 23520 784x30\n",
"\tWeight row[1]: 300 30x10\n"
]
},
{
"data": {
"image/png": 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szuHJdEfh3gKMbvP9KGBbN9xvt3H3baW/dwKv0To8VG52lMYXj48z7uzh9nQZd9/h7sfcvQX4FzI/f2ZWSWtRe97dXy3dXDbnr73+lds57Ex3FO6lwHfMbKyZ9QFmAW90w/12CzPrX3qTBDPrD1wKfHLyf5WlN4CflL7+CfDbHmxLlzpe0EquIePzZ2YG/ApocvfH2vyoLM5fR/0rp3MY0S0TcEofzZkDVABz3f3vC7/TbmJm42i9ygboDbyQe//M7EXg+7SuurYDeBB4HXgZqAM2AX/h7tm9yddB375P63+xHdgIzD4+HpwbM7sI+E9gFdBSuvnvaB0HLofz11H/bqBMzmGEZk6KiGRGMydFRDKjwi0ikhkVbhGRzKhwi4hkRoVbRCQzKtwiIplR4RYRyYwKt4hIZv4XUpGGDfyP3qUAAAAASUVORK5CYII=\n",
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