First commit

fig/backprop_magnitude_nabla.py

@@ -0,0 +1,52 @@
+"""
+backprop_magnitude_nabla
+~~~~~~~~~~~~~~~~~~~~~~~~
+
+Using backprop2 I constructed a 784-30-30-30-30-30-10 network to classify
+MNIST data.  I ran ten mini-batches of size 100, with eta = 0.01 and
+lambda = 0.05, using:
+
+net.SGD(otd[:1000], 1, 100, 0.01, 0.05,
+
+I obtained the following norms for the (unregularized) nabla_w for the
+respective mini-batches:
+
+[0.90845722175923671, 2.8852730656073566, 10.696793986223632, 37.75701921183488, 157.7365422527995, 304.43990075227839]
+[0.22493835119537842, 0.6555126517964851, 2.6036801277234076, 11.408825365731225, 46.882319190445472, 70.499637502698221]
+[0.11935180022357521, 0.19756069137133489, 0.8152794148335869, 3.4590802543293977, 15.470507965493903, 31.032396017142556]
+[0.15130005837653659, 0.39687135985664701, 1.4810006139254532, 4.392519005642268, 16.831939776937311, 34.082104455938733]
+[0.11594085276308999, 0.17177668061395848, 0.72204558746599512, 3.05062409378366, 14.133001132214286, 29.776204839994385]
+[0.10790389807606221, 0.20707152756018626, 0.96348134037828603, 3.9043824079499561, 15.986873430586924, 39.195258080490895]
+[0.088613291101645356, 0.129173436407863, 0.4242933114455002, 1.6154682713449411, 7.5451567587160069, 20.180545544006566]
+[0.086175380639289575, 0.12571016850457151, 0.44231149185805047, 1.8435833504677326, 7.61973813981073, 19.474539356281781]
+[0.095372080184163904, 0.15854489503205446, 0.70244235144444678, 2.6294803575724157, 10.427062019753425, 24.309420272033819]
+[0.096453131000155692, 0.13574642196947601, 0.53551377709415471, 2.0247466793066895, 9.4503978546018068, 21.73772148470092]
+
+Note that results are listed in order of layer.  They clearly show how
+the magnitude of nabla_w decreases as we go back through layers.
+
+In this program I take min-batches 7, 8, 9 as representative and plot
+them.  I omit the results from the first and final layers since they
+correspond to 784 input neurons and 10 output neurons, not 30 as in
+the other layers, making it difficult to compare results.
+
+Note that I haven't attempted to preserve the whole workflow here. It
+involved some minor hacking around with backprop2, which messed up
+that code.  That's why I've simply put the results in by hand below.
+"""
+
+# Third-party libraries
+import matplotlib.pyplot as plt
+
+nw1 = [0.129173436407863, 0.4242933114455002, 
+       1.6154682713449411, 7.5451567587160069]
+nw2 = [0.12571016850457151, 0.44231149185805047, 
+       1.8435833504677326, 7.61973813981073]
+nw3 = [0.15854489503205446, 0.70244235144444678, 
+       2.6294803575724157, 10.427062019753425]
+plt.plot(range(1, 5), nw1, "ro-", range(1, 5), nw2, "go-", 
+         range(1, 5), nw3, "bo-")
+plt.xlabel('Layer $l$')
+plt.ylabel(r"$\Vert\nabla C^l_w\Vert$")
+plt.xticks([1, 2, 3, 4])
+plt.show()

fig/false_minima.py

@@ -0,0 +1,40 @@
+"""
+false_minimum
+~~~~~~~~~~~~~
+
+Plots a function of two variables with many false minima."""
+
+#### Libraries
+# Third party libraries
+from matplotlib.ticker import LinearLocator
+# Note that axes3d is not explicitly used in the code, but is needed
+# to register the 3d plot type correctly
+from mpl_toolkits.mplot3d import axes3d 
+import matplotlib.pyplot as plt
+import numpy
+
+fig = plt.figure()
+ax = fig.gca(projection='3d')
+X = numpy.arange(-5, 5, 0.1)
+Y = numpy.arange(-5, 5, 0.1)
+X, Y = numpy.meshgrid(X, Y)
+Z = numpy.sin(X)*numpy.sin(Y)+0.2*X
+
+colortuple = ('w', 'b')
+colors = numpy.empty(X.shape, dtype=str)
+for x in xrange(len(X)):
+    for y in xrange(len(Y)):
+        colors[x, y] = colortuple[(x + y) % 2]
+
+surf = ax.plot_surface(X, Y, Z, rstride=1, cstride=1, facecolors=colors,
+        linewidth=0)
+
+ax.set_xlim3d(-5, 5)
+ax.set_ylim3d(-5, 5)
+ax.set_zlim3d(-2, 2)
+ax.w_xaxis.set_major_locator(LinearLocator(3))
+ax.w_yaxis.set_major_locator(LinearLocator(3))
+ax.w_zaxis.set_major_locator(LinearLocator(3))
+
+plt.show()
+

fig/generate_gradient.py

@@ -0,0 +1,119 @@
+"""generate_gradient.py
+~~~~~~~~~~~~~~~~~~~~~~~
+
+Use network2 to figure out the average starting values of the gradient
+error terms \delta^l_j = \partial C / \partial z^l_j = \partial C /
+\partial b^l_j.
+
+"""
+
+#### Libraries
+# Standard library
+import json
+import math
+import random
+import shutil
+import sys
+sys.path.append("../src/")
+
+# My library
+import mnist_loader
+import network2
+
+# Third-party libraries
+import matplotlib.pyplot as plt
+import numpy as np
+
+def main():
+    # Load the data
+    full_td, _, _ = mnist_loader.load_data_wrapper()
+    td = full_td[:1000] # Just use the first 1000 items of training data
+    epochs = 500 # Number of epochs to train for
+
+    print "\nTwo hidden layers:"
+    net = network2.Network([784, 30, 30, 10])
+    initial_norms(td, net)
+    abbreviated_gradient = [
+        ag[:6] for ag in get_average_gradient(net, td)[:-1]] 
+    print "Saving the averaged gradient for the top six neurons in each "+\
+        "layer.\nWARNING: This will affect the look of the book, so be "+\
+        "sure to check the\nrelevant material (early chapter 5)."
+    f = open("initial_gradient.json", "w")
+    json.dump(abbreviated_gradient, f)
+    f.close()
+    shutil.copy("initial_gradient.json", "../../js/initial_gradient.json")
+    training(td, net, epochs, "norms_during_training_2_layers.json")
+    plot_training(
+        epochs, "norms_during_training_2_layers.json", 2)
+
+    print "\nThree hidden layers:"
+    net = network2.Network([784, 30, 30, 30, 10])
+    initial_norms(td, net)
+    training(td, net, epochs, "norms_during_training_3_layers.json")
+    plot_training(
+        epochs, "norms_during_training_3_layers.json", 3)
+
+    print "\nFour hidden layers:"
+    net = network2.Network([784, 30, 30, 30, 30, 10])
+    initial_norms(td, net)
+    training(td, net, epochs, 
+             "norms_during_training_4_layers.json")
+    plot_training(
+        epochs, "norms_during_training_4_layers.json", 4)
+
+def initial_norms(training_data, net):
+    average_gradient = get_average_gradient(net, training_data)
+    norms = [list_norm(avg) for avg in average_gradient[:-1]]
+    print "Average gradient for the hidden layers: "+str(norms)
+    
+def training(training_data, net, epochs, filename):
+    norms = []
+    for j in range(epochs):
+        average_gradient = get_average_gradient(net, training_data)
+        norms.append([list_norm(avg) for avg in average_gradient[:-1]])
+        print "Epoch: %s" % j
+        net.SGD(training_data, 1, 1000, 0.1, lmbda=5.0)
+    f = open(filename, "w")
+    json.dump(norms, f)
+    f.close()
+
+def plot_training(epochs, filename, num_layers):
+    f = open(filename, "r")
+    norms = json.load(f)
+    f.close()
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    colors = ["#2A6EA6", "#FFA933", "#FF5555", "#55FF55", "#5555FF"]
+    for j in range(num_layers):
+        ax.plot(np.arange(epochs), 
+                [n[j] for n in norms], 
+                color=colors[j],
+                label="Hidden layer %s" % (j+1,))
+    ax.set_xlim([0, epochs])
+    ax.grid(True)
+    ax.set_xlabel('Number of epochs of training')
+    ax.set_title('Speed of learning: %s hidden layers' % num_layers)
+    ax.set_yscale('log')
+    plt.legend(loc="upper right")
+    fig_filename = "training_speed_%s_layers.png" % num_layers
+    plt.savefig(fig_filename)
+    shutil.copy(fig_filename, "../../images/"+fig_filename)
+    plt.show()
+
+def get_average_gradient(net, training_data):
+    nabla_b_results = [net.backprop(x, y)[0] for x, y in training_data]
+    gradient = list_sum(nabla_b_results)
+    return [(np.reshape(g, len(g))/len(training_data)).tolist() 
+            for g in gradient]
+
+def zip_sum(a, b): 
+    return [x+y for (x, y) in zip(a, b)]
+
+def list_sum(l):
+    return reduce(zip_sum, l)
+
+def list_norm(l):
+    return math.sqrt(sum([x*x for x in l]))
+
+if __name__ == "__main__":
+    main()

fig/initial_gradient.json

@@ -0,0 +1 @@
+[[-0.003970677333144113, -0.0031684316985881185, 0.008103235909196014, 0.012598010584130365, -0.026465907331998335, 0.0017583319323150341], [0.04152906589960523, 0.044025552524932406, -0.009669682279354514, 0.046736871369353235, 0.03877302528270452, 0.012336459551975156]]

fig/misleading_gradient.py

@@ -0,0 +1,43 @@
+"""
+misleading_gradient
+~~~~~~~~~~~~~~~~~~~
+
+Plots a function which misleads the gradient descent algorithm."""
+
+#### Libraries
+# Third party libraries
+from matplotlib.ticker import LinearLocator
+# Note that axes3d is not explicitly used in the code, but is needed
+# to register the 3d plot type correctly
+from mpl_toolkits.mplot3d import axes3d 
+import matplotlib.pyplot as plt
+import numpy
+
+fig = plt.figure()
+ax = fig.gca(projection='3d')
+X = numpy.arange(-1, 1, 0.025)
+Y = numpy.arange(-1, 1, 0.025)
+X, Y = numpy.meshgrid(X, Y)
+Z = X**2 + 10*Y**2
+
+colortuple = ('w', 'b')
+colors = numpy.empty(X.shape, dtype=str)
+for x in xrange(len(X)):
+    for y in xrange(len(Y)):
+        colors[x, y] = colortuple[(x + y) % 2]
+
+surf = ax.plot_surface(X, Y, Z, rstride=1, cstride=1, facecolors=colors,
+        linewidth=0)
+
+ax.set_xlim3d(-1, 1)
+ax.set_ylim3d(-1, 1)
+ax.set_zlim3d(0, 12)
+ax.w_xaxis.set_major_locator(LinearLocator(3))
+ax.w_yaxis.set_major_locator(LinearLocator(3))
+ax.w_zaxis.set_major_locator(LinearLocator(3))
+ax.text(0.05, -1.8, 0, "$w_1$", fontsize=20)
+ax.text(1.5, -0.25, 0, "$w_2$", fontsize=20)
+ax.text(1.79, 0, 9.62, "$C$", fontsize=20)
+
+plt.show()
+

fig/misleading_gradient_contours.py

@@ -0,0 +1,21 @@
+"""
+misleading_gradient_contours
+~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+
+Plots the contours of the function from misleading_gradient.py"""
+
+#### Libraries
+# Third party libraries
+import matplotlib.pyplot as plt
+import numpy
+
+X = numpy.arange(-1, 1, 0.02)
+Y = numpy.arange(-1, 1, 0.02)
+X, Y = numpy.meshgrid(X, Y)
+Z = X**2 + 10*Y**2
+
+plt.figure()
+CS = plt.contour(X, Y, Z, levels=[0.5, 1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0])
+plt.xlabel("$w_1$", fontsize=16)
+plt.ylabel("$w_2$", fontsize=16)
+plt.show()

fig/mnist.py

@@ -0,0 +1,234 @@
+"""
+mnist
+~~~~~
+
+Draws images based on the MNIST data."""
+
+#### Libraries
+# Standard library
+import cPickle
+import sys
+
+# My library
+sys.path.append('../src/')
+import mnist_loader
+
+# Third-party libraries
+import matplotlib
+import matplotlib.pyplot as plt
+import numpy as np
+
+def main():
+    training_set, validation_set, test_set = mnist_loader.load_data()
+    images = get_images(training_set)
+    plot_rotated_image(images[0])
+
+#### Plotting
+def plot_images_together(images):
+    """ Plot a single image containing all six MNIST images, one after
+    the other.  Note that we crop the sides of the images so that they
+    appear reasonably close together."""
+    fig = plt.figure()
+    images = [image[:, 3:25] for image in images]
+    image = np.concatenate(images, axis=1)
+    ax = fig.add_subplot(1, 1, 1)
+    ax.matshow(image, cmap = matplotlib.cm.binary)
+    plt.xticks(np.array([]))
+    plt.yticks(np.array([]))
+    plt.show()
+
+def plot_10_by_10_images(images):
+    """ Plot 100 MNIST images in a 10 by 10 table. Note that we crop
+    the images so that they appear reasonably close together.  The
+    image is post-processed to give the appearance of being continued."""
+    fig = plt.figure()
+    images = [image[3:25, 3:25] for image in images]
+    #image = np.concatenate(images, axis=1)
+    for x in range(10):
+        for y in range(10):
+            ax = fig.add_subplot(10, 10, 10*y+x)
+            ax.matshow(images[10*y+x], cmap = matplotlib.cm.binary)
+            plt.xticks(np.array([]))
+            plt.yticks(np.array([]))
+    plt.show()
+
+def plot_images_separately(images):
+    "Plot the six MNIST images separately."
+    fig = plt.figure()
+    for j in xrange(1, 7):
+        ax = fig.add_subplot(1, 6, j)
+        ax.matshow(images[j-1], cmap = matplotlib.cm.binary)
+        plt.xticks(np.array([]))
+        plt.yticks(np.array([]))
+    plt.show()
+
+def plot_mnist_digit(image):
+    """ Plot a single MNIST image."""
+    fig = plt.figure()
+    ax = fig.add_subplot(1, 1, 1)
+    ax.matshow(image, cmap = matplotlib.cm.binary)
+    plt.xticks(np.array([]))
+    plt.yticks(np.array([]))
+    plt.show()
+
+def plot_2_and_1(images):
+    "Plot a 2 and a 1 image from the MNIST set."
+    fig = plt.figure()
+    ax = fig.add_subplot(1, 2, 1)
+    ax.matshow(images[5], cmap = matplotlib.cm.binary)
+    plt.xticks(np.array([]))
+    plt.yticks(np.array([]))
+    ax = fig.add_subplot(1, 2, 2)
+    ax.matshow(images[3], cmap = matplotlib.cm.binary)
+    plt.xticks(np.array([]))
+    plt.yticks(np.array([]))
+    plt.show()
+
+def plot_top_left(image):
+    "Plot the top left of ``image``."
+    image[14:,:] = np.zeros((14,28))
+    image[:,14:] = np.zeros((28,14))
+    fig = plt.figure()
+    ax = fig.add_subplot(1, 1, 1)
+    ax.matshow(image, cmap = matplotlib.cm.binary)
+    plt.xticks(np.array([]))
+    plt.yticks(np.array([]))
+    plt.show()
+
+def plot_bad_images(images):
+    """This takes a list of images misclassified by a pretty good
+    neural network --- one achieving over 93 percent accuracy --- and
+    turns them into a figure."""
+    bad_image_indices = [8, 18, 33, 92, 119, 124, 149, 151, 193, 233, 241, 247, 259, 300, 313, 321, 324, 341, 349, 352, 359, 362, 381, 412, 435, 445, 449, 478, 479, 495, 502, 511, 528, 531, 547, 571, 578, 582, 597, 610, 619, 628, 629, 659, 667, 691, 707, 717, 726, 740, 791, 810, 844, 846, 898, 938, 939, 947, 956, 959, 965, 982, 1014, 1033, 1039, 1044, 1050, 1055, 1107, 1112, 1124, 1147, 1181, 1191, 1192, 1198, 1202, 1204, 1206, 1224, 1226, 1232, 1242, 1243, 1247, 1256, 1260, 1263, 1283, 1289, 1299, 1310, 1319, 1326, 1328, 1357, 1378, 1393, 1413, 1422, 1435, 1467, 1469, 1494, 1500, 1522, 1523, 1525, 1527, 1530, 1549, 1553, 1609, 1611, 1634, 1641, 1676, 1678, 1681, 1709, 1717, 1722, 1730, 1732, 1737, 1741, 1754, 1759, 1772, 1773, 1790, 1808, 1813, 1823, 1843, 1850, 1857, 1868, 1878, 1880, 1883, 1901, 1913, 1930, 1938, 1940, 1952, 1969, 1970, 1984, 2001, 2009, 2016, 2018, 2035, 2040, 2043, 2044, 2053, 2063, 2098, 2105, 2109, 2118, 2129, 2130, 2135, 2148, 2161, 2168, 2174, 2182, 2185, 2186, 2189, 2224, 2229, 2237, 2266, 2272, 2293, 2299, 2319, 2325, 2326, 2334, 2369, 2371, 2380, 2381, 2387, 2393, 2395, 2406, 2408, 2414, 2422, 2433, 2450, 2488, 2514, 2526, 2548, 2574, 2589, 2598, 2607, 2610, 2631, 2648, 2654, 2695, 2713, 2720, 2721, 2730, 2770, 2771, 2780, 2863, 2866, 2896, 2907, 2925, 2927, 2939, 2995, 3005, 3023, 3030, 3060, 3073, 3102, 3108, 3110, 3114, 3115, 3117, 3130, 3132, 3157, 3160, 3167, 3183, 3189, 3206, 3240, 3254, 3260, 3280, 3329, 3330, 3333, 3383, 3384, 3475, 3490, 3503, 3520, 3525, 3559, 3567, 3573, 3597, 3598, 3604, 3629, 3664, 3702, 3716, 3718, 3725, 3726, 3727, 3751, 3752, 3757, 3763, 3766, 3767, 3769, 3776, 3780, 3798, 3806, 3808, 3811, 3817, 3821, 3838, 3848, 3853, 3855, 3869, 3876, 3902, 3906, 3926, 3941, 3943, 3951, 3954, 3962, 3976, 3985, 3995, 4000, 4002, 4007, 4017, 4018, 4065, 4075, 4078, 4093, 4102, 4139, 4140, 4152, 4154, 4163, 4165, 4176, 4199, 4201, 4205, 4207, 4212, 4224, 4238, 4248, 4256, 4284, 4289, 4297, 4300, 4306, 4344, 4355, 4356, 4359, 4360, 4369, 4405, 4425, 4433, 4435, 4449, 4487, 4497, 4498, 4500, 4521, 4536, 4548, 4563, 4571, 4575, 4601, 4615, 4620, 4633, 4639, 4662, 4690, 4722, 4731, 4735, 4737, 4739, 4740, 4761, 4798, 4807, 4814, 4823, 4833, 4837, 4874, 4876, 4879, 4880, 4886, 4890, 4910, 4950, 4951, 4952, 4956, 4963, 4966, 4968, 4978, 4990, 5001, 5020, 5054, 5067, 5068, 5078, 5135, 5140, 5143, 5176, 5183, 5201, 5210, 5331, 5409, 5457, 5495, 5600, 5601, 5617, 5623, 5634, 5642, 5677, 5678, 5718, 5734, 5735, 5749, 5752, 5771, 5787, 5835, 5842, 5845, 5858, 5887, 5888, 5891, 5906, 5913, 5936, 5937, 5945, 5955, 5957, 5972, 5973, 5985, 5987, 5997, 6035, 6042, 6043, 6045, 6053, 6059, 6065, 6071, 6081, 6091, 6112, 6124, 6157, 6166, 6168, 6172, 6173, 6347, 6370, 6386, 6390, 6391, 6392, 6421, 6426, 6428, 6505, 6542, 6555, 6556, 6560, 6564, 6568, 6571, 6572, 6597, 6598, 6603, 6608, 6625, 6651, 6694, 6706, 6721, 6725, 6740, 6746, 6768, 6783, 6785, 6796, 6817, 6827, 6847, 6870, 6872, 6926, 6945, 7002, 7035, 7043, 7089, 7121, 7130, 7198, 7216, 7233, 7248, 7265, 7426, 7432, 7434, 7494, 7498, 7691, 7777, 7779, 7797, 7800, 7809, 7812, 7821, 7849, 7876, 7886, 7897, 7902, 7905, 7917, 7921, 7945, 7999, 8020, 8059, 8081, 8094, 8095, 8115, 8246, 8256, 8262, 8272, 8273, 8278, 8279, 8293, 8322, 8339, 8353, 8408, 8453, 8456, 8502, 8520, 8522, 8607, 9009, 9010, 9013, 9015, 9019, 9022, 9024, 9026, 9036, 9045, 9046, 9128, 9214, 9280, 9316, 9342, 9382, 9433, 9446, 9506, 9540, 9544, 9587, 9614, 9634, 9642, 9645, 9700, 9716, 9719, 9729, 9732, 9738, 9740, 9741, 9742, 9744, 9745, 9749, 9752, 9768, 9770, 9777, 9779, 9792, 9808, 9831, 9839, 9856, 9858, 9867, 9879, 9883, 9888, 9890, 9893, 9905, 9944, 9970, 9982]
+    n = len(bad_image_indices)
+    bad_images = [images[j] for j in bad_image_indices]
+    fig = plt.figure(figsize=(10, 15))
+    for j in xrange(1, n+1):
+        ax = fig.add_subplot(25, 125, j)
+        ax.matshow(bad_images[j-1], cmap = matplotlib.cm.binary)
+        ax.set_title(str(bad_image_indices[j-1]))
+        plt.xticks(np.array([]))
+        plt.yticks(np.array([]))
+    plt.subplots_adjust(hspace = 1.2)
+    plt.show()
+
+def plot_really_bad_images(images):
+    """This takes a list of the worst images from plot_bad_images and
+    turns them into a figure."""
+    really_bad_image_indices = [
+        324, 582, 659, 726, 846, 956, 1124, 1393,
+        1773, 1868, 2018, 2109, 2654, 4199, 4201, 4620, 5457, 5642]
+    n = len(really_bad_image_indices)
+    really_bad_images = [images[j] for j in really_bad_image_indices]
+    fig = plt.figure(figsize=(10, 2))
+    for j in xrange(1, n+1):
+        ax = fig.add_subplot(2, 9, j)
+        ax.matshow(really_bad_images[j-1], cmap = matplotlib.cm.binary)
+        #ax.set_title(str(really_bad_image_indices[j-1]))
+        plt.xticks(np.array([]))
+        plt.yticks(np.array([]))
+    plt.show()
+
+def plot_features(image):
+    "Plot the top right, bottom left, and bottom right of ``image``."
+    image_1, image_2, image_3 = np.copy(image), np.copy(image), np.copy(image)
+    image_1[:,:14] = np.zeros((28,14))
+    image_1[14:,:] = np.zeros((14,28))
+    image_2[:,14:] = np.zeros((28,14))
+    image_2[:14,:] = np.zeros((14,28))
+    image_3[:14,:] = np.zeros((14,28))
+    image_3[:,:14] = np.zeros((28,14))
+    fig = plt.figure()
+    ax = fig.add_subplot(1, 3, 1)
+    ax.matshow(image_1, cmap = matplotlib.cm.binary)
+    plt.xticks(np.array([]))
+    plt.yticks(np.array([]))
+    ax = fig.add_subplot(1, 3, 2)
+    ax.matshow(image_2, cmap = matplotlib.cm.binary)
+    plt.xticks(np.array([]))
+    plt.yticks(np.array([]))
+    ax = fig.add_subplot(1, 3, 3)
+    ax.matshow(image_3, cmap = matplotlib.cm.binary)
+    plt.xticks(np.array([]))
+    plt.yticks(np.array([]))
+    plt.show()
+
+def plot_rotated_image(image):
+    """ Plot an MNIST digit and a version rotated by 10 degrees."""
+    # Do the initial plot
+    fig = plt.figure()
+    ax = fig.add_subplot(1, 1, 1)
+    ax.matshow(image, cmap = matplotlib.cm.binary)
+    plt.xticks(np.array([]))
+    plt.yticks(np.array([]))
+    plt.show()
+    # Set up the rotated image.  There are fast matrix techniques
+    # for doing this, but we'll do a pedestrian approach
+    rot_image = np.zeros((28,28))
+    theta = 15*np.pi/180 # 15 degrees
+    def to_xy(j, k):
+        # Converts from matrix indices to x, y co-ords, using the
+        # 13, 14 matrix entry as the origin
+        return (k-13, -j+14) # x range: -13..14, y range: -13..14
+    def to_jk(x, y):
+        # Converts from x, y co-ords to matrix indices
+        return (-y+14, x+13)
+    def image_value(image, x, y):
+        # returns the value of the image at co-ordinate x, y
+        # (Note that this would be better done as a closure, if Pythong
+        # supported closures, so that image didn't need to be passed)
+        j, k = to_jk(x, y)
+        return image[j, k]
+    # Element by element, figure out what should be in the rotated
+    # image.  We simply take each matrix entry, figure out the
+    # corresponding x, y co-ordinates, rotate backward, and then
+    # average the nearby matrix elements.  It's not perfect, and it's
+    # not fast, but it works okay.
+    for j in range(28):
+        for k in range(28):
+            x, y = to_xy(j, k)
+            # rotate by -theta
+            x1 = np.cos(theta)*x + np.sin(theta)*y
+            y1 = -np.sin(theta)*x + np.cos(theta)*y
+            # Nearest integer x entries are x2 and x2+1. delta_x 
+            # measures how to interpolate
+            x2 = np.floor(x1)
+            delta_x = x1-x2
+            # Similarly for y
+            y2 = np.floor(y1)
+            delta_y = y1-y2
+            # Check if we're out of bounds, and if so continue to next entry
+            # This will miss a boundary row and layer, but that's okay,
+            # MNIST digits usually don't go that near the boundary.
+            if x2 < -13 or x2 > 13 or y2 < -13 or y2 > 13: continue
+            # If we're in bounds, average the nearby entries.
+            value \
+                = (1-delta_x)*(1-delta_y)*image_value(image, x2, y2)+\
+                (1-delta_x)*delta_y*image_value(image, x2, y2+1)+\
+                delta_x*(1-delta_y)*image_value(image, x2+1, y2)+\
+                delta_x*delta_y*image_value(image, x2+1, y2+1)
+            # Rescale the value by a hand-set fudge factor.  This
+            # seems to be necessary because the averaging doesn't
+            # quite work right.  The fudge-factor should probably be
+            # theta-dependent, but I've set it by hand.  
+            rot_image[j, k] = 1.3*value
+    plot_mnist_digit(rot_image)
+
+#### Miscellanea
+def load_data():
+    """ Return the MNIST data as a tuple containing the training data,
+    the validation data, and the test data."""
+    f = open('../data/mnist.pkl', 'rb')
+    training_set, validation_set, test_set = cPickle.load(f)
+    f.close()
+    return (training_set, validation_set, test_set)
+
+def get_images(training_set):
+    """ Return a list containing the images from the MNIST data
+    set. Each image is represented as a 2-d numpy array."""
+    flattened_images = training_set[0]
+    return [np.reshape(f, (-1, 28)) for f in flattened_images]
+
+#### Main
+if __name__ == "__main__":
+    main()

fig/more_data.json

@@ -0,0 +1 @@
+[69.09, 76.37, 85.29, 88.85, 91.27, 93.24, 94.89, 95.85, 95.97]

fig/more_data.py

@@ -0,0 +1,122 @@
+"""more_data
+~~~~~~~~~~~~
+
+Plot graphs to illustrate the performance of MNIST when different size
+training sets are used.
+
+"""
+
+# Standard library
+import json
+import random
+import sys
+
+# My library
+sys.path.append('../src/')
+import mnist_loader
+import network2
+
+# Third-party libraries
+import matplotlib.pyplot as plt
+import numpy as np
+from sklearn import svm
+
+# The sizes to use for the different training sets
+SIZES = [100, 200, 500, 1000, 2000, 5000, 10000, 20000, 50000] 
+
+def main():
+    run_networks()
+    run_svms()
+    make_plots()
+                       
+def run_networks():
+    # Make results more easily reproducible
+    random.seed(12345678)
+    np.random.seed(12345678)
+    training_data, validation_data, test_data = mnist_loader.load_data_wrapper()
+    net = network2.Network([784, 30, 10], cost=network2.CrossEntropyCost())
+    accuracies = []
+    for size in SIZES:
+        print "\n\nTraining network with data set size %s" % size
+        net.large_weight_initializer()
+        num_epochs = 1500000 / size 
+        net.SGD(training_data[:size], num_epochs, 10, 0.5, lmbda = size*0.0001)
+        accuracy = net.accuracy(validation_data) / 100.0
+        print "Accuracy was %s percent" % accuracy
+        accuracies.append(accuracy)
+    f = open("more_data.json", "w")
+    json.dump(accuracies, f)
+    f.close()
+
+def run_svms():
+    svm_training_data, svm_validation_data, svm_test_data \
+        = mnist_loader.load_data()
+    accuracies = []
+    for size in SIZES:
+        print "\n\nTraining SVM with data set size %s" % size
+        clf = svm.SVC()
+        clf.fit(svm_training_data[0][:size], svm_training_data[1][:size])
+        predictions = [int(a) for a in clf.predict(svm_validation_data[0])]
+        accuracy = sum(int(a == y) for a, y in 
+                       zip(predictions, svm_validation_data[1])) / 100.0
+        print "Accuracy was %s percent" % accuracy
+        accuracies.append(accuracy)
+    f = open("more_data_svm.json", "w")
+    json.dump(accuracies, f)
+    f.close()
+
+def make_plots():
+    f = open("more_data.json", "r")
+    accuracies = json.load(f)
+    f.close()
+    f = open("more_data_svm.json", "r")
+    svm_accuracies = json.load(f)
+    f.close()
+    make_linear_plot(accuracies)
+    make_log_plot(accuracies)
+    make_combined_plot(accuracies, svm_accuracies)
+
+def make_linear_plot(accuracies):
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    ax.plot(SIZES, accuracies, color='#2A6EA6')
+    ax.plot(SIZES, accuracies, "o", color='#FFA933')
+    ax.set_xlim(0, 50000)
+    ax.set_ylim(60, 100)
+    ax.grid(True)
+    ax.set_xlabel('Training set size')
+    ax.set_title('Accuracy (%) on the validation data')
+    plt.show()
+
+def make_log_plot(accuracies):
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    ax.plot(SIZES, accuracies, color='#2A6EA6')
+    ax.plot(SIZES, accuracies, "o", color='#FFA933')
+    ax.set_xlim(100, 50000)
+    ax.set_ylim(60, 100)
+    ax.set_xscale('log')
+    ax.grid(True)
+    ax.set_xlabel('Training set size')
+    ax.set_title('Accuracy (%) on the validation data')
+    plt.show()
+
+def make_combined_plot(accuracies, svm_accuracies):
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    ax.plot(SIZES, accuracies, color='#2A6EA6')
+    ax.plot(SIZES, accuracies, "o", color='#2A6EA6', 
+            label='Neural network accuracy (%)')
+    ax.plot(SIZES, svm_accuracies, color='#FFA933')
+    ax.plot(SIZES, svm_accuracies, "o", color='#FFA933',
+            label='SVM accuracy (%)')
+    ax.set_xlim(100, 50000)
+    ax.set_ylim(25, 100)
+    ax.set_xscale('log')
+    ax.grid(True)
+    ax.set_xlabel('Training set size')
+    plt.legend(loc="lower right")
+    plt.show()
+
+if __name__ == "__main__":
+    main()

fig/more_data_svm.json

@@ -0,0 +1 @@
+[25.07, 48.93, 75.13, 83.87, 88.49, 91.46, 92.45, 93.47, 94.48]

fig/multiple_eta.json

@@ -0,0 +1 @@
+[[[], [], [0.87809508908377998, 0.67406552530098141, 0.59798920430275404, 0.55533015743656189, 0.51751101003208144, 0.4942033354556824, 0.47255041042913526, 0.46069879353359433, 0.44304475294352064, 0.43099562372228112, 0.42310993427766375, 0.41408265298981006, 0.40573464183982105, 0.40110722961828227, 0.39162028064538967, 0.38705015774740958, 0.38116357043417587, 0.37603986695304614, 0.37297012040237154, 0.37057334627661631, 0.36551756338853658, 0.36335674264586654, 0.35745296185579917, 0.35535960956849127, 0.35365591135061097, 0.35011353300568238, 0.34946519495897871, 0.34604661988238178, 0.34386077098862522, 0.33919980880230349], []], [[], [], [0.49501954654296704, 0.4063145129425576, 0.40482383242804637, 0.37156577828840276, 0.37380111172151681, 0.37152751786000143, 0.35371985224004426, 0.3557161388797867, 0.34323780090168027, 0.3433514311156789, 0.3367645441708797, 0.34532085892085329, 0.33506383267050244, 0.34760988079085842, 0.34921493732996928, 0.33853424834583179, 0.32837282561262077, 0.33175599401109612, 0.33132920379429243, 0.33024353325326034, 0.32736756892399654, 0.3259638557593546, 0.32004264784244907, 0.33424319076405928, 0.33878125802305081, 0.32521839878261177, 0.32679267619514646, 0.32488571435373748, 0.33056367198473002, 0.33879633130932685], []], [[], [], [0.92489293305102116, 0.83919130289246469, 0.88748421594232696, 0.79625231780396133, 0.78117959228699174, 1.1365919079387048, 0.78787239608336346, 0.76778614131217449, 0.73689525303227721, 0.80127437393519696, 0.74433665287336681, 0.73725544607013882, 0.80249602203179993, 0.85190338199210014, 0.79872168623645712, 0.80243104440756152, 0.80649160680410659, 0.81467254023600921, 0.82526467696100858, 0.75042379852601759, 0.93658673378777402, 0.88236662906752283, 0.86121396033520892, 0.72492681699401829, 0.80405009868466648, 0.83959963179208197, 0.83387510808276821, 0.88282498566307899, 0.88583473645177979, 0.86068501713490919], []]]

fig/multiple_eta.py

@@ -0,0 +1,73 @@
+"""multiple_eta
+~~~~~~~~~~~~~~~
+
+This program shows how different values for the learning rate affect
+training.  In particular, we'll plot out how the cost changes using
+three different values for eta.
+
+"""
+
+# Standard library
+import json
+import random
+import sys
+
+# My library
+sys.path.append('../src/')
+import mnist_loader
+import network2
+
+# Third-party libraries
+import matplotlib.pyplot as plt
+import numpy as np
+
+# Constants
+LEARNING_RATES = [0.025, 0.25, 2.5]
+COLORS = ['#2A6EA6', '#FFCD33', '#FF7033']
+NUM_EPOCHS = 30
+
+def main():
+    run_networks()
+    make_plot()
+
+def run_networks():
+    """Train networks using three different values for the learning rate,
+    and store the cost curves in the file ``multiple_eta.json``, where
+    they can later be used by ``make_plot``.
+
+    """
+    # Make results more easily reproducible
+    random.seed(12345678)
+    np.random.seed(12345678)
+    training_data, validation_data, test_data = mnist_loader.load_data_wrapper()
+    results = []
+    for eta in LEARNING_RATES:
+        print "\nTrain a network using eta = "+str(eta)
+        net = network2.Network([784, 30, 10])
+        results.append(
+            net.SGD(training_data, NUM_EPOCHS, 10, eta, lmbda=5.0,
+                    evaluation_data=validation_data, 
+                    monitor_training_cost=True))
+    f = open("multiple_eta.json", "w")
+    json.dump(results, f)
+    f.close()
+
+def make_plot():
+    f = open("multiple_eta.json", "r")
+    results = json.load(f)
+    f.close()
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    for eta, result, color in zip(LEARNING_RATES, results, COLORS):
+        _, _, training_cost, _ = result
+        ax.plot(np.arange(NUM_EPOCHS), training_cost, "o-",
+                label="$\eta$ = "+str(eta),
+                color=color)
+    ax.set_xlim([0, NUM_EPOCHS])
+    ax.set_xlabel('Epoch')
+    ax.set_ylabel('Cost')
+    plt.legend(loc='upper right')
+    plt.show()
+
+if __name__ == "__main__":
+    main()

fig/overfitting.py

@@ -0,0 +1,179 @@
+"""
+overfitting
+~~~~~~~~~~~
+
+Plot graphs to illustrate the problem of overfitting.  
+"""
+
+# Standard library
+import json
+import random
+import sys
+
+# My library
+sys.path.append('../src/')
+import mnist_loader
+import network2
+
+# Third-party libraries
+import matplotlib.pyplot as plt
+import numpy as np
+
+
+def main(filename, num_epochs,
+         training_cost_xmin=200, 
+         test_accuracy_xmin=200, 
+         test_cost_xmin=0, 
+         training_accuracy_xmin=0,
+         training_set_size=1000, 
+         lmbda=0.0):
+    """``filename`` is the name of the file where the results will be
+    stored.  ``num_epochs`` is the number of epochs to train for.
+    ``training_set_size`` is the number of images to train on.
+    ``lmbda`` is the regularization parameter.  The other parameters
+    set the epochs at which to start plotting on the x axis.
+    """
+    run_network(filename, num_epochs, training_set_size, lmbda)
+    make_plots(filename, num_epochs, 
+               training_cost_xmin,
+               test_accuracy_xmin,
+               test_cost_xmin, 
+               training_accuracy_xmin,
+               training_set_size)
+                       
+def run_network(filename, num_epochs, training_set_size=1000, lmbda=0.0):
+    """Train the network for ``num_epochs`` on ``training_set_size``
+    images, and store the results in ``filename``.  Those results can
+    later be used by ``make_plots``.  Note that the results are stored
+    to disk in large part because it's convenient not to have to
+    ``run_network`` each time we want to make a plot (it's slow).
+
+    """
+    # Make results more easily reproducible
+    random.seed(12345678)
+    np.random.seed(12345678)
+    training_data, validation_data, test_data = mnist_loader.load_data_wrapper()
+    net = network2.Network([784, 30, 10], cost=network2.CrossEntropyCost())
+    net.large_weight_initializer()
+    test_cost, test_accuracy, training_cost, training_accuracy \
+        = net.SGD(training_data[:training_set_size], num_epochs, 10, 0.5,
+                  evaluation_data=test_data, lmbda = lmbda,
+                  monitor_evaluation_cost=True, 
+                  monitor_evaluation_accuracy=True, 
+                  monitor_training_cost=True, 
+                  monitor_training_accuracy=True)
+    f = open(filename, "w")
+    json.dump([test_cost, test_accuracy, training_cost, training_accuracy], f)
+    f.close()
+
+def make_plots(filename, num_epochs, 
+               training_cost_xmin=200, 
+               test_accuracy_xmin=200, 
+               test_cost_xmin=0, 
+               training_accuracy_xmin=0,
+               training_set_size=1000):
+    """Load the results from ``filename``, and generate the corresponding
+    plots. """
+    f = open(filename, "r")
+    test_cost, test_accuracy, training_cost, training_accuracy \
+        = json.load(f)
+    f.close()
+    plot_training_cost(training_cost, num_epochs, training_cost_xmin)
+    plot_test_accuracy(test_accuracy, num_epochs, test_accuracy_xmin)
+    plot_test_cost(test_cost, num_epochs, test_cost_xmin)
+    plot_training_accuracy(training_accuracy, num_epochs, 
+                           training_accuracy_xmin, training_set_size)
+    plot_overlay(test_accuracy, training_accuracy, num_epochs,
+                 min(test_accuracy_xmin, training_accuracy_xmin),
+                 training_set_size)
+
+def plot_training_cost(training_cost, num_epochs, training_cost_xmin):
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    ax.plot(np.arange(training_cost_xmin, num_epochs), 
+            training_cost[training_cost_xmin:num_epochs],
+            color='#2A6EA6')
+    ax.set_xlim([training_cost_xmin, num_epochs])
+    ax.grid(True)
+    ax.set_xlabel('Epoch')
+    ax.set_title('Cost on the training data')
+    plt.show()
+
+def plot_test_accuracy(test_accuracy, num_epochs, test_accuracy_xmin):
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    ax.plot(np.arange(test_accuracy_xmin, num_epochs), 
+            [accuracy/100.0 
+             for accuracy in test_accuracy[test_accuracy_xmin:num_epochs]],
+            color='#2A6EA6')
+    ax.set_xlim([test_accuracy_xmin, num_epochs])
+    ax.grid(True)
+    ax.set_xlabel('Epoch')
+    ax.set_title('Accuracy (%) on the test data')
+    plt.show()
+
+def plot_test_cost(test_cost, num_epochs, test_cost_xmin):
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    ax.plot(np.arange(test_cost_xmin, num_epochs), 
+            test_cost[test_cost_xmin:num_epochs],
+            color='#2A6EA6')
+    ax.set_xlim([test_cost_xmin, num_epochs])
+    ax.grid(True)
+    ax.set_xlabel('Epoch')
+    ax.set_title('Cost on the test data')
+    plt.show()
+
+def plot_training_accuracy(training_accuracy, num_epochs, 
+                           training_accuracy_xmin, training_set_size):
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    ax.plot(np.arange(training_accuracy_xmin, num_epochs), 
+            [accuracy*100.0/training_set_size 
+             for accuracy in training_accuracy[training_accuracy_xmin:num_epochs]],
+            color='#2A6EA6')
+    ax.set_xlim([training_accuracy_xmin, num_epochs])
+    ax.grid(True)
+    ax.set_xlabel('Epoch')
+    ax.set_title('Accuracy (%) on the training data')
+    plt.show()
+
+def plot_overlay(test_accuracy, training_accuracy, num_epochs, xmin,
+                 training_set_size):
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    ax.plot(np.arange(xmin, num_epochs), 
+            [accuracy/100.0 for accuracy in test_accuracy], 
+            color='#2A6EA6',
+            label="Accuracy on the test data")
+    ax.plot(np.arange(xmin, num_epochs), 
+            [accuracy*100.0/training_set_size 
+             for accuracy in training_accuracy], 
+            color='#FFA933',
+            label="Accuracy on the training data")
+    ax.grid(True)
+    ax.set_xlim([xmin, num_epochs])
+    ax.set_xlabel('Epoch')
+    ax.set_ylim([90, 100])
+    plt.legend(loc="lower right")
+    plt.show()
+
+if __name__ == "__main__":
+    filename = raw_input("Enter a file name: ")
+    num_epochs = int(raw_input(
+        "Enter the number of epochs to run for: "))
+    training_cost_xmin = int(raw_input(
+        "training_cost_xmin (suggest 200): "))
+    test_accuracy_xmin = int(raw_input(
+        "test_accuracy_xmin (suggest 200): "))
+    test_cost_xmin = int(raw_input(
+        "test_cost_xmin (suggest 0): "))
+    training_accuracy_xmin = int(raw_input(
+        "training_accuracy_xmin (suggest 0): "))
+    training_set_size = int(raw_input(
+        "Training set size (suggest 1000): "))
+    lmbda = float(raw_input(
+        "Enter the regularization parameter, lambda (suggest: 5.0): "))
+    main(filename, num_epochs, training_cost_xmin, 
+         test_accuracy_xmin, test_cost_xmin, training_accuracy_xmin,
+         training_set_size, lmbda)

fig/overfitting_full.json

@@ -0,0 +1 @@
+[[0.56135590058630858, 0.47806921271034553, 0.457510836259925, 0.42504920544144992, 0.39449553344420019, 0.39810448800345, 0.37017079712250733, 0.37403997639944547, 0.36290253019659285, 0.4006868170859208, 0.36817548958488616, 0.37299310675826219, 0.36871967242261605, 0.37146610246666006, 0.35704621996697938, 0.35821464151288968, 0.38622103466509744, 0.37010939716781127, 0.36539832104327125, 0.35511546847032671, 0.3828088676932585, 0.36160025922354638, 0.37028708356461698, 0.37605182846277163, 0.36634313696187393, 0.36129044456360238, 0.37531885586439506, 0.36415225595876555, 0.35707895858237054, 0.36631987373588193], [9136, 9275, 9307, 9377, 9450, 9429, 9468, 9488, 9494, 9424, 9483, 9483, 9505, 9499, 9508, 9508, 9445, 9524, 9524, 9524, 9494, 9527, 9518, 9505, 9533, 9529, 9512, 9530, 9532, 9531], [0.55994588582554705, 0.44664870303435988, 0.42455329174078477, 0.38578320429266705, 0.33992291017592285, 0.33162477096795895, 0.3137480626518645, 0.30028971890544093, 0.27353890048167528, 0.30236927117202678, 0.26487026303889277, 0.2661714884193439, 0.24734280015146709, 0.26355551438395558, 0.23088530423416964, 0.22618350577327287, 0.25137541006767478, 0.23085585354651994, 0.21417931191800957, 0.20049587923059808, 0.23713128948069295, 0.20327728799861464, 0.21953883029836488, 0.20264436321820509, 0.19643949703516961, 0.18467980669870671, 0.18788606162530633, 0.18535916502880764, 0.18466759834259142, 0.17218286758911475], [45708, 46605, 46797, 47190, 47543, 47570, 47638, 47838, 48061, 47825, 48160, 48195, 48265, 48156, 48439, 48449, 48267, 48433, 48598, 48697, 48380, 48648, 48500, 48669, 48734, 48796, 48802, 48837, 48810, 48932]]

fig/pca_limitations.py

@@ -0,0 +1,32 @@
+"""
+pca_limitations
+~~~~~~~~~~~~~~~
+
+Plot graphs to illustrate the limitations of PCA.
+"""
+
+# Third-party libraries
+from mpl_toolkits.mplot3d import Axes3D
+import matplotlib.pyplot as plt
+import numpy as np
+
+# Plot just the data
+fig = plt.figure()
+ax = fig.gca(projection='3d')
+z = np.linspace(-2, 2, 20)
+theta = np.linspace(-4 * np.pi, 4 * np.pi, 20)
+x = np.sin(theta)+0.03*np.random.randn(20)
+y = np.cos(theta)+0.03*np.random.randn(20)
+ax.plot(x, y, z, 'ro')
+plt.show()
+
+# Plot the data and the helix together
+fig = plt.figure()
+ax = fig.gca(projection='3d')
+z_helix = np.linspace(-2, 2, 100)
+theta_helix = np.linspace(-4 * np.pi, 4 * np.pi, 100)
+x_helix = np.sin(theta_helix)
+y_helix = np.cos(theta_helix)
+ax.plot(x, y, z, 'ro')
+ax.plot(x_helix, y_helix, z_helix, '')
+plt.show()

fig/regularized_full.json

@@ -0,0 +1 @@
+[[4.3072791918656037, 2.9331304641086344, 2.1348073553576041, 1.6588303607817259, 1.330889938797851, 1.1963223601928472, 1.1170765304219505, 1.0170754480838433, 0.99110935015398149, 1.0071179800661803, 0.96280080386971378, 0.99226609521675169, 0.96023984363523895, 0.97253784945751276, 0.93966545596520334, 0.95330563342376551, 0.96378529404233837, 0.97367336858037301, 0.94435985290781166, 0.94622931411839994, 0.98392022263201184, 0.94091005661041272, 0.9496551347987412, 0.94714964684453073, 0.95026655456196552, 0.92915894672179755, 0.95831053042987979, 1.0153994919718721, 0.92940339906358749, 0.97682851862658082], [9212, 9341, 9375, 9424, 9532, 9537, 9504, 9541, 9578, 9538, 9579, 9530, 9590, 9543, 9607, 9597, 9576, 9546, 9600, 9634, 9544, 9606, 9614, 9607, 9621, 9637, 9620, 9511, 9649, 9561], [1.2925405259017666, 0.92479539229795305, 0.72611252037165497, 0.61618944188425839, 0.49142410439713557, 0.46552608507795468, 0.46074829841290343, 0.40775149802551902, 0.39671750686791218, 0.42031570708192345, 0.38057096091326847, 0.40768033915334978, 0.3895210257834103, 0.40585871820346864, 0.36003072887701948, 0.37700037701783806, 0.39300003862768451, 0.40774598935627593, 0.37194215157507704, 0.3662415845761452, 0.40722309031673021, 0.36476961463606117, 0.36988528906574514, 0.36112644707329011, 0.380710641602238, 0.35700998663848571, 0.37724740623797381, 0.44991741876110503, 0.35820321110078079, 0.39226034353556583], [45919, 46835, 47204, 47434, 47989, 47930, 47839, 48157, 48218, 48105, 48313, 48089, 48282, 48111, 48463, 48362, 48243, 48123, 48416, 48533, 48123, 48483, 48435, 48548, 48434, 48524, 48417, 47797, 48561, 48235]]

fig/replaced_by_d3/README.md

@@ -0,0 +1,6 @@
+# Replaced by d3 directory
+
+This directory contains python code which generated png figures which
+were later replaced by d3 in the live version of the site.  They've
+been preserved here on the off chance that they may be of use at some
+point in the future.

fig/replaced_by_d3/relu.py

@@ -0,0 +1,24 @@
+"""
+relu
+~~~~
+
+Plots a graph of the squashing function used by a rectified linear
+unit."""
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+z = np.arange(-2, 2, .1)
+zero = np.zeros(len(z))
+y = np.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()

fig/replaced_by_d3/sigmoid.py

@@ -0,0 +1,23 @@
+"""
+sigmoid
+~~~~~~~
+
+Plots a graph of the sigmoid function."""
+
+import numpy
+import matplotlib.pyplot as plt
+
+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([-0.5, 1.5])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('sigmoid function')
+
+plt.show()

fig/replaced_by_d3/step.py

@@ -0,0 +1,23 @@
+"""
+step
+~~~~~~~
+
+Plots a graph of a step function."""
+
+import numpy
+import matplotlib.pyplot as plt
+
+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()

fig/replaced_by_d3/tanh.py

@@ -0,0 +1,22 @@
+"""
+tanh
+~~~~
+
+Plots a graph of the tanh function."""
+
+import numpy as np
+import matplotlib.pyplot as plt
+
+z = np.arange(-5, 5, .1)
+t = np.tanh(z)
+
+fig = plt.figure()
+ax = fig.add_subplot(111)
+ax.plot(z, t)
+ax.set_ylim([-1.0, 1.0])
+ax.set_xlim([-5,5])
+ax.grid(True)
+ax.set_xlabel('z')
+ax.set_title('tanh function')
+
+plt.show()

fig/serialize_images_to_json.py

@@ -0,0 +1,46 @@
+"""
+serialize_images_to_json
+~~~~~~~~~~~~~~~~~~~~~~~~
+
+Utility to serialize parts of the training and validation data to JSON, 
+for use with Javascript.  """
+
+#### Libraries
+# Standard library
+import json 
+import sys
+
+# My library
+sys.path.append('../src/')
+import mnist_loader
+
+# Third-party libraries
+import numpy as np
+
+
+# Number of training and validation data images to serialize
+NTD = 1000
+NVD = 100
+
+training_data, validation_data, test_data = mnist_loader.load_data_wrapper()
+
+def make_data_integer(td):
+    # This will be slow, due to the loop.  It'd be better if numpy did
+    # this directly.  But numpy.rint followed by tolist() doesn't
+    # convert to a standard Python int.
+    return [int(x) for x in (td*256).reshape(784).tolist()]
+
+data = {"training": [
+    {"x": [x[0] for x in training_data[j][0].tolist()],
+     "y": [y[0] for y in training_data[j][1].tolist()]}
+    for j in xrange(NTD)],
+        "validation": [
+    {"x": [x[0] for x in validation_data[j][0].tolist()],
+     "y": validation_data[j][1]}
+            for j in xrange(NVD)]}
+
+f = open("data_1000.json", "w")
+json.dump(data, f)
+f.close()
+
+

fig/valley.py

@@ -0,0 +1,43 @@
+"""
+valley
+~~~~~~
+
+Plots a function of two variables to minimize.  The function is a
+fairly generic valley function."""
+
+#### Libraries
+# Third party libraries
+from matplotlib.ticker import LinearLocator
+# Note that axes3d is not explicitly used in the code, but is needed
+# to register the 3d plot type correctly
+from mpl_toolkits.mplot3d import axes3d 
+import matplotlib.pyplot as plt
+import numpy
+
+fig = plt.figure()
+ax = fig.gca(projection='3d')
+X = numpy.arange(-1, 1, 0.1)
+Y = numpy.arange(-1, 1, 0.1)
+X, Y = numpy.meshgrid(X, Y)
+Z = X**2 + Y**2
+
+colortuple = ('w', 'b')
+colors = numpy.empty(X.shape, dtype=str)
+for x in xrange(len(X)):
+    for y in xrange(len(Y)):
+        colors[x, y] = colortuple[(x + y) % 2]
+
+surf = ax.plot_surface(X, Y, Z, rstride=1, cstride=1, facecolors=colors,
+        linewidth=0)
+
+ax.set_xlim3d(-1, 1)
+ax.set_ylim3d(-1, 1)
+ax.set_zlim3d(0, 2)
+ax.w_xaxis.set_major_locator(LinearLocator(3))
+ax.w_yaxis.set_major_locator(LinearLocator(3))
+ax.w_zaxis.set_major_locator(LinearLocator(3))
+ax.text(1.79, 0, 1.62, "$C$", fontsize=20)
+ax.text(0.05, -1.8, 0, "$v_1$", fontsize=20)
+ax.text(1.5, -0.25, 0, "$v_2$", fontsize=20)
+
+plt.show()

fig/valley2.py

@@ -0,0 +1,48 @@
+"""valley2.py
+~~~~~~~~~~~~~
+
+Plots a function of two variables to minimize.  The function is a
+fairly generic valley function.
+
+Note that this is a duplicate of valley.py, but omits labels on the
+axis.  It's bad practice to duplicate in this way, but I had
+considerable trouble getting matplotlib to update a graph in the way I
+needed (adding or removing labels), so finally fell back on this as a
+kludge solution.
+
+"""
+
+#### Libraries
+# Third party libraries
+from matplotlib.ticker import LinearLocator
+# Note that axes3d is not explicitly used in the code, but is needed
+# to register the 3d plot type correctly
+from mpl_toolkits.mplot3d import axes3d 
+import matplotlib.pyplot as plt
+import numpy
+
+fig = plt.figure()
+ax = fig.gca(projection='3d')
+X = numpy.arange(-1, 1, 0.1)
+Y = numpy.arange(-1, 1, 0.1)
+X, Y = numpy.meshgrid(X, Y)
+Z = X**2 + Y**2
+
+colortuple = ('w', 'b')
+colors = numpy.empty(X.shape, dtype=str)
+for x in xrange(len(X)):
+    for y in xrange(len(Y)):
+        colors[x, y] = colortuple[(x + y) % 2]
+
+surf = ax.plot_surface(X, Y, Z, rstride=1, cstride=1, facecolors=colors,
+        linewidth=0)
+
+ax.set_xlim3d(-1, 1)
+ax.set_ylim3d(-1, 1)
+ax.set_zlim3d(0, 2)
+ax.w_xaxis.set_major_locator(LinearLocator(3))
+ax.w_yaxis.set_major_locator(LinearLocator(3))
+ax.w_zaxis.set_major_locator(LinearLocator(3))
+ax.text(1.79, 0, 1.62, "$C$", fontsize=20)
+
+plt.show()

fig/weight_initialization.py

@@ -0,0 +1,89 @@
+"""weight_initialization 
+~~~~~~~~~~~~~~~~~~~~~~~~
+
+This program shows how weight initialization affects training.  In
+particular, we'll plot out how the classification accuracies improve
+using either large starting weights, whose standard deviation is 1, or
+the default starting weights, whose standard deviation is 1 over the
+square root of the number of input neurons.
+
+"""
+
+# Standard library
+import json
+import random
+import sys
+
+# My library
+sys.path.append('../src/')
+import mnist_loader
+import network2
+
+# Third-party libraries
+import matplotlib.pyplot as plt
+import numpy as np
+
+def main(filename, n, eta):
+    run_network(filename, n, eta)
+    make_plot(filename)
+                       
+def run_network(filename, n, eta):
+    """Train the network using both the default and the large starting
+    weights.  Store the results in the file with name ``filename``,
+    where they can later be used by ``make_plots``.
+
+    """
+    # Make results more easily reproducible
+    random.seed(12345678)
+    np.random.seed(12345678)
+    training_data, validation_data, test_data = mnist_loader.load_data_wrapper()
+    net = network2.Network([784, n, 10], cost=network2.CrossEntropyCost)
+    print "Train the network using the default starting weights."
+    default_vc, default_va, default_tc, default_ta \
+        = net.SGD(training_data, 30, 10, eta, lmbda=5.0,
+                  evaluation_data=validation_data, 
+                  monitor_evaluation_accuracy=True)
+    print "Train the network using the large starting weights."
+    net.large_weight_initializer()
+    large_vc, large_va, large_tc, large_ta \
+        = net.SGD(training_data, 30, 10, eta, lmbda=5.0,
+                  evaluation_data=validation_data, 
+                  monitor_evaluation_accuracy=True)
+    f = open(filename, "w")
+    json.dump({"default_weight_initialization":
+               [default_vc, default_va, default_tc, default_ta],
+               "large_weight_initialization":
+               [large_vc, large_va, large_tc, large_ta]}, 
+              f)
+    f.close()
+
+def make_plot(filename):
+    """Load the results from the file ``filename``, and generate the
+    corresponding plot.
+
+    """
+    f = open(filename, "r")
+    results = json.load(f)
+    f.close()
+    default_vc, default_va, default_tc, default_ta = results[
+        "default_weight_initialization"]
+    large_vc, large_va, large_tc, large_ta = results[
+        "large_weight_initialization"]
+    # Convert raw classification numbers to percentages, for plotting
+    default_va = [x/100.0 for x in default_va]
+    large_va = [x/100.0 for x in large_va]
+    fig = plt.figure()
+    ax = fig.add_subplot(111)
+    ax.plot(np.arange(0, 30, 1), large_va, color='#2A6EA6',
+            label="Old approach to weight initialization")
+    ax.plot(np.arange(0, 30, 1), default_va, color='#FFA933', 
+            label="New approach to weight initialization")
+    ax.set_xlim([0, 30])
+    ax.set_xlabel('Epoch')
+    ax.set_ylim([85, 100])
+    ax.set_title('Classification accuracy')
+    plt.legend(loc="lower right")
+    plt.show()
+
+if __name__ == "__main__":
+    main()

fig/weight_initialization_100.json

@@ -0,0 +1 @@
+{"default_weight_initialization": [[], [9295, 9481, 9547, 9592, 9664, 9673, 9702, 9719, 9726, 9726, 9732, 9732, 9730, 9734, 9745, 9751, 9757, 9761, 9764, 9766, 9758, 9767, 9756, 9752, 9777, 9775, 9770, 9770, 9771, 9781], [], []], "large_weight_initialization": [[], [8994, 9181, 9260, 9364, 9427, 9449, 9497, 9512, 9560, 9578, 9603, 9616, 9626, 9629, 9644, 9671, 9674, 9679, 9700, 9708, 9707, 9717, 9729, 9720, 9719, 9745, 9751, 9754, 9755, 9742], [], []]}

fig/weight_initialization_30.json

@@ -0,0 +1 @@
+{"default_weight_initialization": [[], [9270, 9414, 9470, 9504, 9537, 9550, 9587, 9594, 9596, 9594, 9616, 9595, 9622, 9630, 9636, 9641, 9625, 9652, 9637, 9634, 9642, 9639, 9649, 9646, 9646, 9653, 9646, 9653, 9640, 9650], [], []], "large_weight_initialization": [[], [8643, 9044, 9141, 9231, 9299, 9327, 9385, 9416, 9433, 9449, 9476, 9489, 9500, 9535, 9521, 9548, 9564, 9573, 9585, 9592, 9596, 9615, 9607, 9605, 9606, 9622, 9637, 9648, 9635, 9637], [], []]}

requirements.txt

@@ -0,0 +1,4 @@
+numpy==1.13.3
+scikit-learn==0.19.0
+scipy==0.19.1
+Theano==0.7.0

src/conv.py

@@ -0,0 +1,297 @@
+"""conv.py
+~~~~~~~~~~
+
+Code for many of the experiments involving convolutional networks in
+Chapter 6 of the book 'Neural Networks and Deep Learning', by Michael
+Nielsen.  The code essentially duplicates (and parallels) what is in
+the text, so this is simply a convenience, and has not been commented
+in detail.  Consult the original text for more details.
+
+"""
+
+from collections import Counter
+
+import matplotlib
+matplotlib.use('Agg')
+import matplotlib.pyplot as plt
+import numpy as np
+import theano
+import theano.tensor as T
+
+import network3
+from network3 import sigmoid, tanh, ReLU, Network
+from network3 import ConvPoolLayer, FullyConnectedLayer, SoftmaxLayer
+
+training_data, validation_data, test_data = network3.load_data_shared()
+mini_batch_size = 10
+
+def shallow(n=3, epochs=60):
+    nets = []
+    for j in range(n):
+        print "A shallow net with 100 hidden neurons"
+        net = Network([
+            FullyConnectedLayer(n_in=784, n_out=100),
+            SoftmaxLayer(n_in=100, n_out=10)], mini_batch_size)
+        net.SGD(
+            training_data, epochs, mini_batch_size, 0.1, 
+            validation_data, test_data)
+        nets.append(net)
+    return nets 
+
+def basic_conv(n=3, epochs=60):
+    for j in range(n):
+        print "Conv + FC architecture"
+        net = Network([
+            ConvPoolLayer(image_shape=(mini_batch_size, 1, 28, 28), 
+                          filter_shape=(20, 1, 5, 5), 
+                          poolsize=(2, 2)),
+            FullyConnectedLayer(n_in=20*12*12, n_out=100),
+            SoftmaxLayer(n_in=100, n_out=10)], mini_batch_size)
+        net.SGD(
+            training_data, epochs, mini_batch_size, 0.1, validation_data, test_data)
+    return net 
+
+def omit_FC():
+    for j in range(3):
+        print "Conv only, no FC"
+        net = Network([
+            ConvPoolLayer(image_shape=(mini_batch_size, 1, 28, 28), 
+                          filter_shape=(20, 1, 5, 5), 
+                          poolsize=(2, 2)),
+            SoftmaxLayer(n_in=20*12*12, n_out=10)], mini_batch_size)
+        net.SGD(training_data, 60, mini_batch_size, 0.1, validation_data, test_data)
+    return net 
+
+def dbl_conv(activation_fn=sigmoid):
+    for j in range(3):
+        print "Conv + Conv + FC architecture"
+        net = Network([
+            ConvPoolLayer(image_shape=(mini_batch_size, 1, 28, 28), 
+                          filter_shape=(20, 1, 5, 5), 
+                          poolsize=(2, 2),
+                          activation_fn=activation_fn),
+            ConvPoolLayer(image_shape=(mini_batch_size, 20, 12, 12), 
+                          filter_shape=(40, 20, 5, 5), 
+                          poolsize=(2, 2),
+                          activation_fn=activation_fn),
+            FullyConnectedLayer(
+                n_in=40*4*4, n_out=100, activation_fn=activation_fn),
+            SoftmaxLayer(n_in=100, n_out=10)], mini_batch_size)
+        net.SGD(training_data, 60, mini_batch_size, 0.1, validation_data, test_data)
+    return net 
+
+# The following experiment was eventually omitted from the chapter,
+# but I've left it in here, since it's an important negative result:
+# basic l2 regularization didn't help much.  The reason (I believe) is
+# that using convolutional-pooling layers is already a pretty strong
+# regularizer.
+def regularized_dbl_conv():
+    for lmbda in [0.00001, 0.0001, 0.001, 0.01, 0.1, 1.0, 10.0, 100.0]:
+        for j in range(3):
+            print "Conv + Conv + FC num %s, with regularization %s" % (j, lmbda)
+            net = Network([
+                ConvPoolLayer(image_shape=(mini_batch_size, 1, 28, 28), 
+                              filter_shape=(20, 1, 5, 5), 
+                              poolsize=(2, 2)),
+                ConvPoolLayer(image_shape=(mini_batch_size, 20, 12, 12), 
+                              filter_shape=(40, 20, 5, 5), 
+                              poolsize=(2, 2)),
+                FullyConnectedLayer(n_in=40*4*4, n_out=100),
+                SoftmaxLayer(n_in=100, n_out=10)], mini_batch_size)
+            net.SGD(training_data, 60, mini_batch_size, 0.1, validation_data, test_data, lmbda=lmbda)
+
+def dbl_conv_relu():
+    for lmbda in [0.0, 0.00001, 0.0001, 0.001, 0.01, 0.1, 1.0, 10.0, 100.0]:
+        for j in range(3):
+            print "Conv + Conv + FC num %s, relu, with regularization %s" % (j, lmbda)
+            net = Network([
+                ConvPoolLayer(image_shape=(mini_batch_size, 1, 28, 28), 
+                              filter_shape=(20, 1, 5, 5), 
+                              poolsize=(2, 2), 
+                              activation_fn=ReLU),
+                ConvPoolLayer(image_shape=(mini_batch_size, 20, 12, 12), 
+                              filter_shape=(40, 20, 5, 5), 
+                              poolsize=(2, 2), 
+                              activation_fn=ReLU),
+                FullyConnectedLayer(n_in=40*4*4, n_out=100, activation_fn=ReLU),
+                SoftmaxLayer(n_in=100, n_out=10)], mini_batch_size)
+            net.SGD(training_data, 60, mini_batch_size, 0.03, validation_data, test_data, lmbda=lmbda)
+
+#### Some subsequent functions may make use of the expanded MNIST
+#### data.  That can be generated by running expand_mnist.py.
+
+def expanded_data(n=100):
+    """n is the number of neurons in the fully-connected layer.  We'll try
+    n=100, 300, and 1000.
+
+    """
+    expanded_training_data, _, _ = network3.load_data_shared(
+        "../data/mnist_expanded.pkl.gz")
+    for j in range(3):
+        print "Training with expanded data, %s neurons in the FC layer, run num %s" % (n, j)
+        net = Network([
+            ConvPoolLayer(image_shape=(mini_batch_size, 1, 28, 28), 
+                          filter_shape=(20, 1, 5, 5), 
+                          poolsize=(2, 2), 
+                          activation_fn=ReLU),
+            ConvPoolLayer(image_shape=(mini_batch_size, 20, 12, 12), 
+                          filter_shape=(40, 20, 5, 5), 
+                          poolsize=(2, 2), 
+                          activation_fn=ReLU),
+            FullyConnectedLayer(n_in=40*4*4, n_out=n, activation_fn=ReLU),
+            SoftmaxLayer(n_in=n, n_out=10)], mini_batch_size)
+        net.SGD(expanded_training_data, 60, mini_batch_size, 0.03, 
+                validation_data, test_data, lmbda=0.1)
+    return net 
+
+def expanded_data_double_fc(n=100):
+    """n is the number of neurons in both fully-connected layers.  We'll
+    try n=100, 300, and 1000.
+
+    """
+    expanded_training_data, _, _ = network3.load_data_shared(
+        "../data/mnist_expanded.pkl.gz")
+    for j in range(3):
+        print "Training with expanded data, %s neurons in two FC layers, run num %s" % (n, j)
+        net = Network([
+            ConvPoolLayer(image_shape=(mini_batch_size, 1, 28, 28), 
+                          filter_shape=(20, 1, 5, 5), 
+                          poolsize=(2, 2), 
+                          activation_fn=ReLU),
+            ConvPoolLayer(image_shape=(mini_batch_size, 20, 12, 12), 
+                          filter_shape=(40, 20, 5, 5), 
+                          poolsize=(2, 2), 
+                          activation_fn=ReLU),
+            FullyConnectedLayer(n_in=40*4*4, n_out=n, activation_fn=ReLU),
+            FullyConnectedLayer(n_in=n, n_out=n, activation_fn=ReLU),
+            SoftmaxLayer(n_in=n, n_out=10)], mini_batch_size)
+        net.SGD(expanded_training_data, 60, mini_batch_size, 0.03, 
+                validation_data, test_data, lmbda=0.1)
+
+def double_fc_dropout(p0, p1, p2, repetitions):
+    expanded_training_data, _, _ = network3.load_data_shared(
+        "../data/mnist_expanded.pkl.gz")
+    nets = []
+    for j in range(repetitions):
+        print "\n\nTraining using a dropout network with parameters ",p0,p1,p2
+        print "Training with expanded data, run num %s" % j
+        net = Network([
+            ConvPoolLayer(image_shape=(mini_batch_size, 1, 28, 28), 
+                          filter_shape=(20, 1, 5, 5), 
+                          poolsize=(2, 2), 
+                          activation_fn=ReLU),
+            ConvPoolLayer(image_shape=(mini_batch_size, 20, 12, 12), 
+                          filter_shape=(40, 20, 5, 5), 
+                          poolsize=(2, 2), 
+                          activation_fn=ReLU),
+            FullyConnectedLayer(
+                n_in=40*4*4, n_out=1000, activation_fn=ReLU, p_dropout=p0),
+            FullyConnectedLayer(
+                n_in=1000, n_out=1000, activation_fn=ReLU, p_dropout=p1),
+            SoftmaxLayer(n_in=1000, n_out=10, p_dropout=p2)], mini_batch_size)
+        net.SGD(expanded_training_data, 40, mini_batch_size, 0.03, 
+                validation_data, test_data)
+        nets.append(net)
+    return nets
+
+def ensemble(nets): 
+    """Takes as input a list of nets, and then computes the accuracy on
+    the test data when classifications are computed by taking a vote
+    amongst the nets.  Returns a tuple containing a list of indices
+    for test data which is erroneously classified, and a list of the
+    corresponding erroneous predictions.
+
+    Note that this is a quick-and-dirty kluge: it'd be more reusable
+    (and faster) to define a Theano function taking the vote.  But
+    this works.
+
+    """
+    
+    test_x, test_y = test_data
+    for net in nets:
+        i = T.lscalar() # mini-batch index
+        net.test_mb_predictions = theano.function(
+            [i], net.layers[-1].y_out,
+            givens={
+                net.x: 
+                test_x[i*net.mini_batch_size: (i+1)*net.mini_batch_size]
+            })
+        net.test_predictions = list(np.concatenate(
+            [net.test_mb_predictions(i) for i in xrange(1000)]))
+    all_test_predictions = zip(*[net.test_predictions for net in nets])
+    def plurality(p): return Counter(p).most_common(1)[0][0]
+    plurality_test_predictions = [plurality(p) 
+                                  for p in all_test_predictions]
+    test_y_eval = test_y.eval()
+    error_locations = [j for j in xrange(10000) 
+                       if plurality_test_predictions[j] != test_y_eval[j]]
+    erroneous_predictions = [plurality(all_test_predictions[j])
+                             for j in error_locations]
+    print "Accuracy is {:.2%}".format((1-len(error_locations)/10000.0))
+    return error_locations, erroneous_predictions
+
+def plot_errors(error_locations, erroneous_predictions=None):
+    test_x, test_y = test_data[0].eval(), test_data[1].eval()
+    fig = plt.figure()
+    error_images = [np.array(test_x[i]).reshape(28, -1) for i in error_locations]
+    n = min(40, len(error_locations))
+    for j in range(n):
+        ax = plt.subplot2grid((5, 8), (j/8, j % 8))
+        ax.matshow(error_images[j], cmap = matplotlib.cm.binary)
+        ax.text(24, 5, test_y[error_locations[j]])
+        if erroneous_predictions:
+            ax.text(24, 24, erroneous_predictions[j])
+        plt.xticks(np.array([]))
+        plt.yticks(np.array([]))
+    plt.tight_layout()
+    return plt
+    
+def plot_filters(net, layer, x, y):
+
+    """Plot the filters for net after the (convolutional) layer number
+    layer.  They are plotted in x by y format.  So, for example, if we
+    have 20 filters after layer 0, then we can call show_filters(net, 0, 5, 4) to
+    get a 5 by 4 plot of all filters."""
+    filters = net.layers[layer].w.eval()
+    fig = plt.figure()
+    for j in range(len(filters)):
+        ax = fig.add_subplot(y, x, j)
+        ax.matshow(filters[j][0], cmap = matplotlib.cm.binary)
+        plt.xticks(np.array([]))
+        plt.yticks(np.array([]))
+    plt.tight_layout()
+    return plt
+
+
+#### Helper method to run all experiments in the book
+
+def run_experiments():
+
+    """Run the experiments described in the book.  Note that the later
+    experiments require access to the expanded training data, which
+    can be generated by running expand_mnist.py.
+
+    """
+    shallow()
+    basic_conv()
+    omit_FC()
+    dbl_conv(activation_fn=sigmoid)
+    # omitted, but still interesting: regularized_dbl_conv()
+    dbl_conv_relu()
+    expanded_data(n=100)
+    expanded_data(n=300)
+    expanded_data(n=1000)
+    expanded_data_double_fc(n=100)    
+    expanded_data_double_fc(n=300)
+    expanded_data_double_fc(n=1000)
+    nets = double_fc_dropout(0.5, 0.5, 0.5, 5)
+    # plot the erroneous digits in the ensemble of nets just trained
+    error_locations, erroneous_predictions = ensemble(nets)
+    plt = plot_errors(error_locations, erroneous_predictions)
+    plt.savefig("ensemble_errors.png")
+    # plot the filters learned by the first of the nets just trained
+    plt = plot_filters(nets[0], 0, 5, 4)
+    plt.savefig("net_full_layer_0.png")
+    plt = plot_filters(nets[0], 1, 8, 5)
+    plt.savefig("net_full_layer_1.png")
+

src/expand_mnist.py

@@ -0,0 +1,60 @@
+"""expand_mnist.py
+~~~~~~~~~~~~~~~~~~
+
+Take the 50,000 MNIST training images, and create an expanded set of
+250,000 images, by displacing each training image up, down, left and
+right, by one pixel.  Save the resulting file to
+../data/mnist_expanded.pkl.gz.
+
+Note that this program is memory intensive, and may not run on small
+systems.
+
+"""
+
+from __future__ import print_function
+
+#### Libraries
+
+# Standard library
+import cPickle
+import gzip
+import os.path
+import random
+
+# Third-party libraries
+import numpy as np
+
+print("Expanding the MNIST training set")
+
+if os.path.exists("../data/mnist_expanded.pkl.gz"):
+    print("The expanded training set already exists.  Exiting.")
+else:
+    f = gzip.open("../data/mnist.pkl.gz", 'rb')
+    training_data, validation_data, test_data = cPickle.load(f)
+    f.close()
+    expanded_training_pairs = []
+    j = 0 # counter
+    for x, y in zip(training_data[0], training_data[1]):
+        expanded_training_pairs.append((x, y))
+        image = np.reshape(x, (-1, 28))
+        j += 1
+        if j % 1000 == 0: print("Expanding image number", j)
+        # iterate over data telling us the details of how to
+        # do the displacement
+        for d, axis, index_position, index in [
+                (1,  0, "first", 0),
+                (-1, 0, "first", 27),
+                (1,  1, "last",  0),
+                (-1, 1, "last",  27)]:
+            new_img = np.roll(image, d, axis)
+            if index_position == "first": 
+                new_img[index, :] = np.zeros(28)
+            else: 
+                new_img[:, index] = np.zeros(28)
+            expanded_training_pairs.append((np.reshape(new_img, 784), y))
+    random.shuffle(expanded_training_pairs)
+    expanded_training_data = [list(d) for d in zip(*expanded_training_pairs)]
+    print("Saving expanded data. This may take a few minutes.")
+    f = gzip.open("../data/mnist_expanded.pkl.gz", "w")
+    cPickle.dump((expanded_training_data, validation_data, test_data), f)
+    f.close()

src/mnist_average_darkness.py

@@ -0,0 +1,64 @@
+"""
+mnist_average_darkness
+~~~~~~~~~~~~~~~~~~~~~~
+
+A naive classifier for recognizing handwritten digits from the MNIST
+data set.  The program classifies digits based on how dark they are
+--- the idea is that digits like "1" tend to be less dark than digits
+like "8", simply because the latter has a more complex shape.  When
+shown an image the classifier returns whichever digit in the training
+data had the closest average darkness.
+
+The program works in two steps: first it trains the classifier, and
+then it applies the classifier to the MNIST test data to see how many
+digits are correctly classified.
+
+Needless to say, this isn't a very good way of recognizing handwritten
+digits!  Still, it's useful to show what sort of performance we get
+from naive ideas."""
+
+#### Libraries
+# Standard library
+from collections import defaultdict
+
+# My libraries
+import mnist_loader
+
+def main():
+    training_data, validation_data, test_data = mnist_loader.load_data()
+    # training phase: compute the average darknesses for each digit,
+    # based on the training data
+    avgs = avg_darknesses(training_data)
+    # testing phase: see how many of the test images are classified
+    # correctly
+    num_correct = sum(int(guess_digit(image, avgs) == digit)
+                      for image, digit in zip(test_data[0], test_data[1]))
+    print "Baseline classifier using average darkness of image."
+    print "%s of %s values correct." % (num_correct, len(test_data[1]))
+
+def avg_darknesses(training_data):
+    """ Return a defaultdict whose keys are the digits 0 through 9.
+    For each digit we compute a value which is the average darkness of
+    training images containing that digit.  The darkness for any
+    particular image is just the sum of the darknesses for each pixel."""
+    digit_counts = defaultdict(int)
+    darknesses = defaultdict(float)
+    for image, digit in zip(training_data[0], training_data[1]):
+        digit_counts[digit] += 1
+        darknesses[digit] += sum(image)
+    avgs = defaultdict(float)
+    for digit, n in digit_counts.iteritems():
+        avgs[digit] = darknesses[digit] / n
+    return avgs
+
+def guess_digit(image, avgs):
+    """Return the digit whose average darkness in the training data is
+    closest to the darkness of ``image``.  Note that ``avgs`` is
+    assumed to be a defaultdict whose keys are 0...9, and whose values
+    are the corresponding average darknesses across the training data."""
+    darkness = sum(image)
+    distances = {k: abs(v-darkness) for k, v in avgs.iteritems()}
+    return min(distances, key=distances.get)
+
+if __name__ == "__main__":
+    main()

src/mnist_loader.py

@@ -0,0 +1,85 @@
+"""
+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 cPickle
+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

src/mnist_svm.py

@@ -0,0 +1,28 @@
+"""
+mnist_svm
+~~~~~~~~~
+
+A classifier program for recognizing handwritten digits from the MNIST
+data set, using an SVM classifier."""
+
+#### Libraries
+# My libraries
+import mnist_loader 
+
+# Third-party libraries
+from sklearn import svm
+
+def svm_baseline():
+    training_data, validation_data, test_data = mnist_loader.load_data()
+    # train
+    clf = svm.SVC()
+    clf.fit(training_data[0], training_data[1])
+    # test
+    predictions = [int(a) for a in clf.predict(test_data[0])]
+    num_correct = sum(int(a == y) for a, y in zip(predictions, test_data[1]))
+    print "Baseline classifier using an SVM."
+    print "%s of %s values correct." % (num_correct, len(test_data[1]))
+
+if __name__ == "__main__":
+    svm_baseline()
+    

src/network.py

@@ -0,0 +1,141 @@
+"""
+network.py
+~~~~~~~~~~
+
+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,
+            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``."""
+        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)
+
+#### 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))

src/network2.py

@@ -0,0 +1,332 @@
+"""network2.py
+~~~~~~~~~~~~~~
+
+An improved version of network.py, implementing the stochastic
+gradient descent learning algorithm for a feedforward neural network.
+Improvements include the addition of the cross-entropy cost function,
+regularization, and better initialization of network weights.  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 json
+import random
+import sys
+
+# Third-party libraries
+import numpy as np
+
+
+#### Define the quadratic and cross-entropy cost functions
+
+class QuadraticCost(object):
+
+    @staticmethod
+    def fn(a, y):
+        """Return the cost associated with an output ``a`` and desired output
+        ``y``.
+
+        """
+        return 0.5*np.linalg.norm(a-y)**2
+
+    @staticmethod
+    def delta(z, a, y):
+        """Return the error delta from the output layer."""
+        return (a-y) * sigmoid_prime(z)
+
+
+class CrossEntropyCost(object):
+
+    @staticmethod
+    def fn(a, y):
+        """Return the cost associated with an output ``a`` and desired output
+        ``y``.  Note that np.nan_to_num is used to ensure numerical
+        stability.  In particular, if both ``a`` and ``y`` have a 1.0
+        in the same slot, then the expression (1-y)*np.log(1-a)
+        returns nan.  The np.nan_to_num ensures that that is converted
+        to the correct value (0.0).
+
+        """
+        return np.sum(np.nan_to_num(-y*np.log(a)-(1-y)*np.log(1-a)))
+
+    @staticmethod
+    def delta(z, a, y):
+        """Return the error delta from the output layer.  Note that the
+        parameter ``z`` is not used by the method.  It is included in
+        the method's parameters in order to make the interface
+        consistent with the delta method for other cost classes.
+
+        """
+        return (a-y)
+
+
+#### Main Network class
+class Network(object):
+
+    def __init__(self, sizes, cost=CrossEntropyCost):
+        """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
+        ``self.default_weight_initializer`` (see docstring for that
+        method).
+
+        """
+        self.num_layers = len(sizes)
+        self.sizes = sizes
+        self.default_weight_initializer()
+        self.cost=cost
+
+    def default_weight_initializer(self):
+        """Initialize each weight using a Gaussian distribution with mean 0
+        and standard deviation 1 over the square root of the number of
+        weights connecting to the same neuron.  Initialize the biases
+        using a Gaussian distribution with mean 0 and standard
+        deviation 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.biases = [np.random.randn(y, 1) for y in self.sizes[1:]]
+        self.weights = [np.random.randn(y, x)/np.sqrt(x)
+                        for x, y in zip(self.sizes[:-1], self.sizes[1:])]
+
+    def large_weight_initializer(self):
+        """Initialize the weights using a Gaussian distribution with mean 0
+        and standard deviation 1.  Initialize the biases using a
+        Gaussian distribution with mean 0 and standard deviation 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.
+
+        This weight and bias initializer uses the same approach as in
+        Chapter 1, and is included for purposes of comparison.  It
+        will usually be better to use the default weight initializer
+        instead.
+
+        """
+        self.biases = [np.random.randn(y, 1) for y in self.sizes[1:]]
+        self.weights = [np.random.randn(y, x)
+                        for x, y in zip(self.sizes[:-1], self.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,
+            lmbda = 0.0,
+            evaluation_data=None,
+            monitor_evaluation_cost=False,
+            monitor_evaluation_accuracy=False,
+            monitor_training_cost=False,
+            monitor_training_accuracy=False):
+        """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, as is the
+        regularization parameter ``lmbda``.  The method also accepts
+        ``evaluation_data``, usually either the validation or test
+        data.  We can monitor the cost and accuracy on either the
+        evaluation data or the training data, by setting the
+        appropriate flags.  The method returns a tuple containing four
+        lists: the (per-epoch) costs on the evaluation data, the
+        accuracies on the evaluation data, the costs on the training
+        data, and the accuracies on the training data.  All values are
+        evaluated at the end of each training epoch.  So, for example,
+        if we train for 30 epochs, then the first element of the tuple
+        will be a 30-element list containing the cost on the
+        evaluation data at the end of each epoch. Note that the lists
+        are empty if the corresponding flag is not set.
+
+        """
+        if evaluation_data: n_data = len(evaluation_data)
+        n = len(training_data)
+        evaluation_cost, evaluation_accuracy = [], []
+        training_cost, training_accuracy = [], []
+        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, lmbda, len(training_data))
+            print "Epoch %s training complete" % j
+            if monitor_training_cost:
+                cost = self.total_cost(training_data, lmbda)
+                training_cost.append(cost)
+                print "Cost on training data: {}".format(cost)
+            if monitor_training_accuracy:
+                accuracy = self.accuracy(training_data, convert=True)
+                training_accuracy.append(accuracy)
+                print "Accuracy on training data: {} / {}".format(
+                    accuracy, n)
+            if monitor_evaluation_cost:
+                cost = self.total_cost(evaluation_data, lmbda, convert=True)
+                evaluation_cost.append(cost)
+                print "Cost on evaluation data: {}".format(cost)
+            if monitor_evaluation_accuracy:
+                accuracy = self.accuracy(evaluation_data)
+                evaluation_accuracy.append(accuracy)
+                print "Accuracy on evaluation data: {} / {}".format(
+                    self.accuracy(evaluation_data), n_data)
+            print
+        return evaluation_cost, evaluation_accuracy, \
+            training_cost, training_accuracy
+
+    def update_mini_batch(self, mini_batch, eta, lmbda, n):
+        """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)``, ``eta`` is the
+        learning rate, ``lmbda`` is the regularization parameter, and
+        ``n`` is the total size of the training data set.
+
+        """
+        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 = [(1-eta*(lmbda/n))*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).delta(zs[-1], activations[-1], y)
+        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 accuracy(self, data, convert=False):
+        """Return the number of inputs in ``data`` for which the neural
+        network outputs the correct result. The neural network's
+        output is assumed to be the index of whichever neuron in the
+        final layer has the highest activation.
+
+        The flag ``convert`` should be set to False if the data set is
+        validation or test data (the usual case), and to True if the
+        data set is the training data. The need for this flag arises
+        due to differences in the way the results ``y`` are
+        represented in the different data sets.  In particular, it
+        flags whether we need to convert between the different
+        representations.  It may seem strange to use different
+        representations for the different data sets.  Why not use the
+        same representation for all three data sets?  It's done for
+        efficiency reasons -- the program usually evaluates the cost
+        on the training data and the accuracy on other data sets.
+        These are different types of computations, and using different
+        representations speeds things up.  More details on the
+        representations can be found in
+        mnist_loader.load_data_wrapper.
+
+        """
+        if convert:
+            results = [(np.argmax(self.feedforward(x)), np.argmax(y))
+                       for (x, y) in data]
+        else:
+            results = [(np.argmax(self.feedforward(x)), y)
+                        for (x, y) in data]
+        return sum(int(x == y) for (x, y) in results)
+
+    def total_cost(self, data, lmbda, convert=False):
+        """Return the total cost for the data set ``data``.  The flag
+        ``convert`` should be set to False if the data set is the
+        training data (the usual case), and to True if the data set is
+        the validation or test data.  See comments on the similar (but
+        reversed) convention for the ``accuracy`` method, above.
+        """
+        cost = 0.0
+        for x, y in data:
+            a = self.feedforward(x)
+            if convert: y = vectorized_result(y)
+            cost += self.cost.fn(a, y)/len(data)
+        cost += 0.5*(lmbda/len(data))*sum(
+            np.linalg.norm(w)**2 for w in self.weights)
+        return cost
+
+    def save(self, filename):
+        """Save the neural network to the file ``filename``."""
+        data = {"sizes": self.sizes,
+                "weights": [w.tolist() for w in self.weights],
+                "biases": [b.tolist() for b in self.biases],
+                "cost": str(self.cost.__name__)}
+        f = open(filename, "w")
+        json.dump(data, f)
+        f.close()
+
+#### Loading a Network
+def load(filename):
+    """Load a neural network from the file ``filename``.  Returns an
+    instance of Network.
+
+    """
+    f = open(filename, "r")
+    data = json.load(f)
+    f.close()
+    cost = getattr(sys.modules[__name__], data["cost"])
+    net = Network(data["sizes"], cost=cost)
+    net.weights = [np.array(w) for w in data["weights"]]
+    net.biases = [np.array(b) for b in data["biases"]]
+    return net
+
+#### Miscellaneous functions
+def vectorized_result(j):
+    """Return a 10-dimensional unit vector with a 1.0 in the j'th 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
+
+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))

src/network3.py

@@ -0,0 +1,314 @@
+"""network3.py
+~~~~~~~~~~~~~~
+
+A Theano-based program for training and running simple neural
+networks.
+
+Supports several layer types (fully connected, convolutional, max
+pooling, softmax), and activation functions (sigmoid, tanh, and
+rectified linear units, with more easily added).
+
+When run on a CPU, this program is much faster than network.py and
+network2.py.  However, unlike network.py and network2.py it can also
+be run on a GPU, which makes it faster still.
+
+Because the code is based on Theano, the code is different in many
+ways from network.py and network2.py.  However, where possible I have
+tried to maintain consistency with the earlier programs.  In
+particular, the API is similar to network2.py.  Note that I have
+focused on making the code simple, easily readable, and easily
+modifiable.  It is not optimized, and omits many desirable features.
+
+This program incorporates ideas from the Theano documentation on
+convolutional neural nets (notably,
+http://deeplearning.net/tutorial/lenet.html ), from Misha Denil's
+implementation of dropout (https://github.com/mdenil/dropout ), and
+from Chris Olah (http://colah.github.io ).
+
+Written for Theano 0.6 and 0.7, needs some changes for more recent
+versions of Theano.
+
+"""
+
+#### Libraries
+# Standard library
+import cPickle
+import gzip
+
+# Third-party libraries
+import numpy as np
+import theano
+import theano.tensor as T
+from theano.tensor.nnet import conv
+from theano.tensor.nnet import softmax
+from theano.tensor import shared_randomstreams
+from theano.tensor.signal import downsample
+
+# Activation functions for neurons
+def linear(z): return z
+def ReLU(z): return T.maximum(0.0, z)
+from theano.tensor.nnet import sigmoid
+from theano.tensor import tanh
+
+
+#### Constants
+GPU = True
+if GPU:
+    print "Trying to run under a GPU.  If this is not desired, then modify "+\
+        "network3.py\nto set the GPU flag to False."
+    try: theano.config.device = 'gpu'
+    except: pass # it's already set
+    theano.config.floatX = 'float32'
+else:
+    print "Running with a CPU.  If this is not desired, then the modify "+\
+        "network3.py to set\nthe GPU flag to True."
+
+#### Load the MNIST data
+def load_data_shared(filename="../data/mnist.pkl.gz"):
+    f = gzip.open(filename, 'rb')
+    training_data, validation_data, test_data = cPickle.load(f)
+    f.close()
+    def shared(data):
+        """Place the data into shared variables.  This allows Theano to copy
+        the data to the GPU, if one is available.
+
+        """
+        shared_x = theano.shared(
+            np.asarray(data[0], dtype=theano.config.floatX), borrow=True)
+        shared_y = theano.shared(
+            np.asarray(data[1], dtype=theano.config.floatX), borrow=True)
+        return shared_x, T.cast(shared_y, "int32")
+    return [shared(training_data), shared(validation_data), shared(test_data)]
+
+#### Main class used to construct and train networks
+class Network(object):
+
+    def __init__(self, layers, mini_batch_size):
+        """Takes a list of `layers`, describing the network architecture, and
+        a value for the `mini_batch_size` to be used during training
+        by stochastic gradient descent.
+
+        """
+        self.layers = layers
+        self.mini_batch_size = mini_batch_size
+        self.params = [param for layer in self.layers for param in layer.params]
+        self.x = T.matrix("x")
+        self.y = T.ivector("y")
+        init_layer = self.layers[0]
+        init_layer.set_inpt(self.x, self.x, self.mini_batch_size)
+        for j in xrange(1, len(self.layers)):
+            prev_layer, layer  = self.layers[j-1], self.layers[j]
+            layer.set_inpt(
+                prev_layer.output, prev_layer.output_dropout, self.mini_batch_size)
+        self.output = self.layers[-1].output
+        self.output_dropout = self.layers[-1].output_dropout
+
+    def SGD(self, training_data, epochs, mini_batch_size, eta,
+            validation_data, test_data, lmbda=0.0):
+        """Train the network using mini-batch stochastic gradient descent."""
+        training_x, training_y = training_data
+        validation_x, validation_y = validation_data
+        test_x, test_y = test_data
+
+        # compute number of minibatches for training, validation and testing
+        num_training_batches = size(training_data)/mini_batch_size
+        num_validation_batches = size(validation_data)/mini_batch_size
+        num_test_batches = size(test_data)/mini_batch_size
+
+        # define the (regularized) cost function, symbolic gradients, and updates
+        l2_norm_squared = sum([(layer.w**2).sum() for layer in self.layers])
+        cost = self.layers[-1].cost(self)+\
+               0.5*lmbda*l2_norm_squared/num_training_batches
+        grads = T.grad(cost, self.params)
+        updates = [(param, param-eta*grad)
+                   for param, grad in zip(self.params, grads)]
+
+        # define functions to train a mini-batch, and to compute the
+        # accuracy in validation and test mini-batches.
+        i = T.lscalar() # mini-batch index
+        train_mb = theano.function(
+            [i], cost, updates=updates,
+            givens={
+                self.x:
+                training_x[i*self.mini_batch_size: (i+1)*self.mini_batch_size],
+                self.y:
+                training_y[i*self.mini_batch_size: (i+1)*self.mini_batch_size]
+            })
+        validate_mb_accuracy = theano.function(
+            [i], self.layers[-1].accuracy(self.y),
+            givens={
+                self.x:
+                validation_x[i*self.mini_batch_size: (i+1)*self.mini_batch_size],
+                self.y:
+                validation_y[i*self.mini_batch_size: (i+1)*self.mini_batch_size]
+            })
+        test_mb_accuracy = theano.function(
+            [i], self.layers[-1].accuracy(self.y),
+            givens={
+                self.x:
+                test_x[i*self.mini_batch_size: (i+1)*self.mini_batch_size],
+                self.y:
+                test_y[i*self.mini_batch_size: (i+1)*self.mini_batch_size]
+            })
+        self.test_mb_predictions = theano.function(
+            [i], self.layers[-1].y_out,
+            givens={
+                self.x:
+                test_x[i*self.mini_batch_size: (i+1)*self.mini_batch_size]
+            })
+        # Do the actual training
+        best_validation_accuracy = 0.0
+        for epoch in xrange(epochs):
+            for minibatch_index in xrange(num_training_batches):
+                iteration = num_training_batches*epoch+minibatch_index
+                if iteration % 1000 == 0:
+                    print("Training mini-batch number {0}".format(iteration))
+                cost_ij = train_mb(minibatch_index)
+                if (iteration+1) % num_training_batches == 0:
+                    validation_accuracy = np.mean(
+                        [validate_mb_accuracy(j) for j in xrange(num_validation_batches)])
+                    print("Epoch {0}: validation accuracy {1:.2%}".format(
+                        epoch, validation_accuracy))
+                    if validation_accuracy >= best_validation_accuracy:
+                        print("This is the best validation accuracy to date.")
+                        best_validation_accuracy = validation_accuracy
+                        best_iteration = iteration
+                        if test_data:
+                            test_accuracy = np.mean(
+                                [test_mb_accuracy(j) for j in xrange(num_test_batches)])
+                            print('The corresponding test accuracy is {0:.2%}'.format(
+                                test_accuracy))
+        print("Finished training network.")
+        print("Best validation accuracy of {0:.2%} obtained at iteration {1}".format(
+            best_validation_accuracy, best_iteration))
+        print("Corresponding test accuracy of {0:.2%}".format(test_accuracy))
+
+#### Define layer types
+
+class ConvPoolLayer(object):
+    """Used to create a combination of a convolutional and a max-pooling
+    layer.  A more sophisticated implementation would separate the
+    two, but for our purposes we'll always use them together, and it
+    simplifies the code, so it makes sense to combine them.
+
+    """
+
+    def __init__(self, filter_shape, image_shape, poolsize=(2, 2),
+                 activation_fn=sigmoid):
+        """`filter_shape` is a tuple of length 4, whose entries are the number
+        of filters, the number of input feature maps, the filter height, and the
+        filter width.
+
+        `image_shape` is a tuple of length 4, whose entries are the
+        mini-batch size, the number of input feature maps, the image
+        height, and the image width.
+
+        `poolsize` is a tuple of length 2, whose entries are the y and
+        x pooling sizes.
+
+        """
+        self.filter_shape = filter_shape
+        self.image_shape = image_shape
+        self.poolsize = poolsize
+        self.activation_fn=activation_fn
+        # initialize weights and biases
+        n_out = (filter_shape[0]*np.prod(filter_shape[2:])/np.prod(poolsize))
+        self.w = theano.shared(
+            np.asarray(
+                np.random.normal(loc=0, scale=np.sqrt(1.0/n_out), size=filter_shape),
+                dtype=theano.config.floatX),
+            borrow=True)
+        self.b = theano.shared(
+            np.asarray(
+                np.random.normal(loc=0, scale=1.0, size=(filter_shape[0],)),
+                dtype=theano.config.floatX),
+            borrow=True)
+        self.params = [self.w, self.b]
+
+    def set_inpt(self, inpt, inpt_dropout, mini_batch_size):
+        self.inpt = inpt.reshape(self.image_shape)
+        conv_out = conv.conv2d(
+            input=self.inpt, filters=self.w, filter_shape=self.filter_shape,
+            image_shape=self.image_shape)
+        pooled_out = downsample.max_pool_2d(
+            input=conv_out, ds=self.poolsize, ignore_border=True)
+        self.output = self.activation_fn(
+            pooled_out + self.b.dimshuffle('x', 0, 'x', 'x'))
+        self.output_dropout = self.output # no dropout in the convolutional layers
+
+class FullyConnectedLayer(object):
+
+    def __init__(self, n_in, n_out, activation_fn=sigmoid, p_dropout=0.0):
+        self.n_in = n_in
+        self.n_out = n_out
+        self.activation_fn = activation_fn
+        self.p_dropout = p_dropout
+        # Initialize weights and biases
+        self.w = theano.shared(
+            np.asarray(
+                np.random.normal(
+                    loc=0.0, scale=np.sqrt(1.0/n_out), size=(n_in, n_out)),
+                dtype=theano.config.floatX),
+            name='w', borrow=True)
+        self.b = theano.shared(
+            np.asarray(np.random.normal(loc=0.0, scale=1.0, size=(n_out,)),
+                       dtype=theano.config.floatX),
+            name='b', borrow=True)
+        self.params = [self.w, self.b]
+
+    def set_inpt(self, inpt, inpt_dropout, mini_batch_size):
+        self.inpt = inpt.reshape((mini_batch_size, self.n_in))
+        self.output = self.activation_fn(
+            (1-self.p_dropout)*T.dot(self.inpt, self.w) + self.b)
+        self.y_out = T.argmax(self.output, axis=1)
+        self.inpt_dropout = dropout_layer(
+            inpt_dropout.reshape((mini_batch_size, self.n_in)), self.p_dropout)
+        self.output_dropout = self.activation_fn(
+            T.dot(self.inpt_dropout, self.w) + self.b)
+
+    def accuracy(self, y):
+        "Return the accuracy for the mini-batch."
+        return T.mean(T.eq(y, self.y_out))
+
+class SoftmaxLayer(object):
+
+    def __init__(self, n_in, n_out, p_dropout=0.0):
+        self.n_in = n_in
+        self.n_out = n_out
+        self.p_dropout = p_dropout
+        # Initialize weights and biases
+        self.w = theano.shared(
+            np.zeros((n_in, n_out), dtype=theano.config.floatX),
+            name='w', borrow=True)
+        self.b = theano.shared(
+            np.zeros((n_out,), dtype=theano.config.floatX),
+            name='b', borrow=True)
+        self.params = [self.w, self.b]
+
+    def set_inpt(self, inpt, inpt_dropout, mini_batch_size):
+        self.inpt = inpt.reshape((mini_batch_size, self.n_in))
+        self.output = softmax((1-self.p_dropout)*T.dot(self.inpt, self.w) + self.b)
+        self.y_out = T.argmax(self.output, axis=1)
+        self.inpt_dropout = dropout_layer(
+            inpt_dropout.reshape((mini_batch_size, self.n_in)), self.p_dropout)
+        self.output_dropout = softmax(T.dot(self.inpt_dropout, self.w) + self.b)
+
+    def cost(self, net):
+        "Return the log-likelihood cost."
+        return -T.mean(T.log(self.output_dropout)[T.arange(net.y.shape[0]), net.y])
+
+    def accuracy(self, y):
+        "Return the accuracy for the mini-batch."
+        return T.mean(T.eq(y, self.y_out))
+
+
+#### Miscellanea
+def size(data):
+    "Return the size of the dataset `data`."
+    return data[0].get_value(borrow=True).shape[0]
+
+def dropout_layer(layer, p_dropout):
+    srng = shared_randomstreams.RandomStreams(
+        np.random.RandomState(0).randint(999999))
+    mask = srng.binomial(n=1, p=1-p_dropout, size=layer.shape)
+    return layer*T.cast(mask, theano.config.floatX)