SDS-2.2, Scalable Data Science

This is used in a non-profit educational setting with kind permission of Adam Breindel. This is not licensed by Adam for use in a for-profit setting. Please contact Adam directly at [email protected] to request or report such use cases or abuses. A few minor modifications and additional mathematical statistical pointers have been added by Raazesh Sainudiin when teaching PhD students in Uppsala University.

Archived YouTube video (no sound, sorry) of this live unedited lab-lecture:

Convolutional Neural Networks

aka CNN, ConvNet

As a baseline, let's start a lab running with what we already know.

We'll take our deep feed-forward multilayer perceptron network, with ReLU activations and reasonable initializations, and apply it to learning the MNIST digits.

The main part of the code looks like the following (full code you can run is in the next cell):

# imports, setup, load data sets

model = Sequential()
model.add(Dense(20, input_dim=784, kernel_initializer='normal', activation='relu'))
model.add(Dense(15, kernel_initializer='normal', activation='relu'))
model.add(Dense(10, kernel_initializer='normal', activation='softmax'))
model.compile(loss='categorical_crossentropy', optimizer='adam', metrics=['categorical_accuracy'])

categorical_labels = to_categorical(y_train, num_classes=10)

history = model.fit(X_train, categorical_labels, epochs=100, batch_size=100)

# print metrics, plot errors

Note the changes, which are largely about building a classifier instead of a regression model: * Output layer has one neuron per category, with softmax activation * Loss function is cross-entropy loss * Accuracy metric is categorical accuracy

Let's hold pointers into wikipedia for these new concepts.

The following is from: https://www.quora.com/How-does-Keras-calculate-accuracy.

Categorical accuracy:

def categorical_accuracy(y_true, y_pred):
 return K.cast(K.equal(K.argmax(y_true, axis=-1),
 K.argmax(y_pred, axis=-1)),
 K.floatx())

K.argmax(y_true) takes the highest value to be the prediction and matches against the comparative set.

Watch (1:39) *

Watch (1:54) *

from keras.models import Sequential
from keras.layers import Dense
from keras.utils import to_categorical
import sklearn.datasets
import datetime
import matplotlib.pyplot as plt
import numpy as np

train_libsvm = "/dbfs/databricks-datasets/mnist-digits/data-001/mnist-digits-train.txt"
test_libsvm = "/dbfs/databricks-datasets/mnist-digits/data-001/mnist-digits-test.txt"

X_train, y_train = sklearn.datasets.load_svmlight_file(train_libsvm, n_features=784)
X_train = X_train.toarray()

X_test, y_test = sklearn.datasets.load_svmlight_file(test_libsvm, n_features=784)
X_test = X_test.toarray()

model = Sequential()
model.add(Dense(20, input_dim=784, kernel_initializer='normal', activation='relu'))
model.add(Dense(15, kernel_initializer='normal', activation='relu'))
model.add(Dense(10, kernel_initializer='normal', activation='softmax'))
model.compile(loss='categorical_crossentropy', optimizer='adam', metrics=['categorical_accuracy'])

categorical_labels = to_categorical(y_train, num_classes=10)
start = datetime.datetime.today()

history = model.fit(X_train, categorical_labels, epochs=40, batch_size=100, validation_split=0.1, verbose=2)

scores = model.evaluate(X_test, to_categorical(y_test, num_classes=10))

print
for i in range(len(model.metrics_names)):
    print("%s: %f" % (model.metrics_names[i], scores[i]))

print ("Start: " + str(start))
end = datetime.datetime.today()
print ("End: " + str(end))
print ("Elapse: " + str(end-start))

Using TensorFlow backend.
Train on 54000 samples, validate on 6000 samples
Epoch 1/40
1s - loss: 0.4799 - categorical_accuracy: 0.8438 - val_loss: 0.1881 - val_categorical_accuracy: 0.9443
Epoch 2/40
1s - loss: 0.2151 - categorical_accuracy: 0.9355 - val_loss: 0.1650 - val_categorical_accuracy: 0.9503
Epoch 3/40
1s - loss: 0.1753 - categorical_accuracy: 0.9484 - val_loss: 0.1367 - val_categorical_accuracy: 0.9587
Epoch 4/40
1s - loss: 0.1533 - categorical_accuracy: 0.9543 - val_loss: 0.1321 - val_categorical_accuracy: 0.9627
Epoch 5/40
1s - loss: 0.1386 - categorical_accuracy: 0.9587 - val_loss: 0.1321 - val_categorical_accuracy: 0.9618
Epoch 6/40
1s - loss: 0.1287 - categorical_accuracy: 0.9612 - val_loss: 0.1332 - val_categorical_accuracy: 0.9613
Epoch 7/40
1s - loss: 0.1199 - categorical_accuracy: 0.9643 - val_loss: 0.1339 - val_categorical_accuracy: 0.9602
Epoch 8/40
1s - loss: 0.1158 - categorical_accuracy: 0.9642 - val_loss: 0.1220 - val_categorical_accuracy: 0.9658
Epoch 9/40
1s - loss: 0.1099 - categorical_accuracy: 0.9660 - val_loss: 0.1342 - val_categorical_accuracy: 0.9648
Epoch 10/40
1s - loss: 0.1056 - categorical_accuracy: 0.9675 - val_loss: 0.1344 - val_categorical_accuracy: 0.9622
Epoch 11/40
1s - loss: 0.0989 - categorical_accuracy: 0.9691 - val_loss: 0.1266 - val_categorical_accuracy: 0.9643
Epoch 12/40
1s - loss: 0.0982 - categorical_accuracy: 0.9699 - val_loss: 0.1226 - val_categorical_accuracy: 0.9650
Epoch 13/40
1s - loss: 0.0945 - categorical_accuracy: 0.9717 - val_loss: 0.1274 - val_categorical_accuracy: 0.9648
Epoch 14/40
1s - loss: 0.0915 - categorical_accuracy: 0.9711 - val_loss: 0.1374 - val_categorical_accuracy: 0.9637
Epoch 15/40
1s - loss: 0.0885 - categorical_accuracy: 0.9719 - val_loss: 0.1541 - val_categorical_accuracy: 0.9590
Epoch 16/40
1s - loss: 0.0845 - categorical_accuracy: 0.9736 - val_loss: 0.1307 - val_categorical_accuracy: 0.9658
Epoch 17/40
1s - loss: 0.0835 - categorical_accuracy: 0.9742 - val_loss: 0.1451 - val_categorical_accuracy: 0.9613
Epoch 18/40
1s - loss: 0.0801 - categorical_accuracy: 0.9749 - val_loss: 0.1352 - val_categorical_accuracy: 0.9647
Epoch 19/40
1s - loss: 0.0780 - categorical_accuracy: 0.9754 - val_loss: 0.1343 - val_categorical_accuracy: 0.9638
Epoch 20/40
1s - loss: 0.0800 - categorical_accuracy: 0.9746 - val_loss: 0.1306 - val_categorical_accuracy: 0.9682
Epoch 21/40
1s - loss: 0.0745 - categorical_accuracy: 0.9765 - val_loss: 0.1465 - val_categorical_accuracy: 0.9665
Epoch 22/40
1s - loss: 0.0722 - categorical_accuracy: 0.9769 - val_loss: 0.1390 - val_categorical_accuracy: 0.9657
Epoch 23/40
1s - loss: 0.0728 - categorical_accuracy: 0.9771 - val_loss: 0.1511 - val_categorical_accuracy: 0.9633
Epoch 24/40
1s - loss: 0.0717 - categorical_accuracy: 0.9772 - val_loss: 0.1515 - val_categorical_accuracy: 0.9638
Epoch 25/40
1s - loss: 0.0686 - categorical_accuracy: 0.9784 - val_loss: 0.1670 - val_categorical_accuracy: 0.9618
Epoch 26/40
1s - loss: 0.0666 - categorical_accuracy: 0.9789 - val_loss: 0.1446 - val_categorical_accuracy: 0.9655
Epoch 27/40
1s - loss: 0.0663 - categorical_accuracy: 0.9789 - val_loss: 0.1537 - val_categorical_accuracy: 0.9623
Epoch 28/40
1s - loss: 0.0671 - categorical_accuracy: 0.9792 - val_loss: 0.1541 - val_categorical_accuracy: 0.9677
Epoch 29/40
1s - loss: 0.0630 - categorical_accuracy: 0.9804 - val_loss: 0.1642 - val_categorical_accuracy: 0.9632
Epoch 30/40
1s - loss: 0.0660 - categorical_accuracy: 0.9797 - val_loss: 0.1585 - val_categorical_accuracy: 0.9648
Epoch 31/40
1s - loss: 0.0646 - categorical_accuracy: 0.9801 - val_loss: 0.1574 - val_categorical_accuracy: 0.9640
Epoch 32/40
1s - loss: 0.0640 - categorical_accuracy: 0.9789 - val_loss: 0.1506 - val_categorical_accuracy: 0.9657
Epoch 33/40
1s - loss: 0.0623 - categorical_accuracy: 0.9801 - val_loss: 0.1583 - val_categorical_accuracy: 0.9662
Epoch 34/40
1s - loss: 0.0630 - categorical_accuracy: 0.9803 - val_loss: 0.1626 - val_categorical_accuracy: 0.9630
Epoch 35/40
1s - loss: 0.0579 - categorical_accuracy: 0.9815 - val_loss: 0.1647 - val_categorical_accuracy: 0.9640
Epoch 36/40
1s - loss: 0.0605 - categorical_accuracy: 0.9815 - val_loss: 0.1693 - val_categorical_accuracy: 0.9635
Epoch 37/40
1s - loss: 0.0586 - categorical_accuracy: 0.9821 - val_loss: 0.1753 - val_categorical_accuracy: 0.9635
Epoch 38/40
1s - loss: 0.0583 - categorical_accuracy: 0.9812 - val_loss: 0.1703 - val_categorical_accuracy: 0.9653
Epoch 39/40
1s - loss: 0.0593 - categorical_accuracy: 0.9807 - val_loss: 0.1736 - val_categorical_accuracy: 0.9638
Epoch 40/40
1s - loss: 0.0538 - categorical_accuracy: 0.9826 - val_loss: 0.1817 - val_categorical_accuracy: 0.9635
   32/10000 [..............................] - ETA: 0s 1952/10000 [====>.........................] - ETA: 0s 3872/10000 [==========>...................] - ETA: 0s 5792/10000 [================>.............] - ETA: 0s 7712/10000 [======================>.......] - ETA: 0s 9632/10000 [===========================>..] - ETA: 0s
loss: 0.205790
categorical_accuracy: 0.959600
Start: 2017-12-07 09:30:17.311756
End: 2017-12-07 09:31:14.504506
Elapse: 0:00:57.192750

after about a minute we have:

...

Epoch 40/40
1s - loss: 0.0610 - categorical_accuracy: 0.9809 - val_loss: 0.1918 - val_categorical_accuracy: 0.9583

...

loss: 0.216120

categorical_accuracy: 0.955000

Start: 2017-12-06 07:35:33.948102

End: 2017-12-06 07:36:27.046130

Elapse: 0:00:53.098028

import matplotlib.pyplot as plt

fig, ax = plt.subplots()
fig.set_size_inches((5,5))
plt.plot(history.history['loss'])
plt.plot(history.history['val_loss'])
plt.title('model loss')
plt.ylabel('loss')
plt.xlabel('epoch')
plt.legend(['train', 'val'], loc='upper left')
display(fig)

What are the big takeaways from this experiment?

1. We get pretty impressive "apparent error" accuracy right from the start! A small network gets us to training accuracy 97% by epoch 20
2. The model appears to continue to learn if we let it run, although it does slow down and oscillate a bit.
3. Our test accuracy is about 95% after 5 epochs and never gets better ... it gets worse!
4. Therefore, we are overfitting very quickly... most of the "training" turns out to be a waste.
5. For what it's worth, we get 95% accuracy without much work.

This is not terrible compared to other, non-neural-network approaches to the problem. After all, we could probably tweak this a bit and do even better.

But we talked about using deep learning to solve "95%" problems or "98%" problems ... where one error in 20, or 50 simply won't work. If we can get to "multiple nines" of accuracy, then we can do things like automate mail sorting and translation, create cars that react properly (all the time) to street signs, and control systems for robots or drones that function autonomously.

Try two more experiments (try them separately):

1. Add a third, hidden layer.
2. Increase the size of the hidden layers.

Adding another layer slows things down a little (why?) but doesn't seem to make a difference in accuracy.

Adding a lot more neurons into the first topology slows things down significantly -- 10x as many neurons, and only a marginal increase in accuracy. Notice also (in the plot) that the learning clearly degrades after epoch 50 or so.

... We need a new approach!

... let's think about this:

What is layer 2 learning from layer 1? Combinations of pixels

Combinations of pixels contain information but...

There are a lot of them (combinations) and they are "fragile"

In fact, in our last experiment, we basically built a model that memorizes a bunch of "magic" pixel combinations.

What might be a better way to build features?

• When humans perform this task, we look not at arbitrary pixel combinations, but certain geometric patterns -- lines, curves, loops.
• These features are made up of combinations of pixels, but they are far from arbitrary
• We identify these features regardless of translation, rotation, etc.

Is there a way to get the network to do the same thing?

I.e., in layer one, identify pixels. Then in layer 2+, identify abstractions over pixels that are translation-invariant 2-D shapes?

We could look at where a "filter" that represents one of these features (e.g., and edge) matches the image.

How would this work?

Convolution

Convolution in the general mathematical sense is define as follows:

The convolution we deal with in deep learning is a simplified case. We want to compare two signals. Here are two visualizations, courtesy of Wikipedia, that help communicate how convolution emphasizes features:

Here's an animation (where we change ${\tau}$)

In one sense, the convolution captures and quantifies the pattern matching over space

If we perform this in two dimensions, we can achieve effects like highlighting edges:

The matrix here, also called a convolution kernel, is one of the functions we are convolving. Other convolution kernels can blur, "sharpen," etc.

So we'll drop in a number of convolution kernels, and the network will learn where to use them? Nope. Better than that.

We'll program in the idea of discrete convolution, and the network will learn what kernels extract meaningful features!

The values in a (fixed-size) convolution kernel matrix will be variables in our deep learning model. Although inuitively it seems like it would be hard to learn useful params, in fact, since those variables are used repeatedly across the image data, it "focuses" the error on a smallish number of parameters with a lot of influence -- so it should be vastly less expensive to train than just a huge fully connected layer like we discussed above.

This idea was developed in the late 1980s, and by 1989, Yann LeCun (at AT&T/Bell Labs) had built a practical high-accuracy system (used in the 1990s for processing handwritten checks and mail).

How do we hook this into our neural networks?

• First, we can preserve the geometric properties of our data by "shaping" the vectors as 2D instead of 1D.

• Then we'll create a layer whose value is not just activation applied to weighted sum of inputs, but instead it's the result of a dot-product (element-wise multiply and sum) between the kernel and a patch of the input vector (image).

  • This value will be our "pre-activation" and optionally feed into an activation function (or "detector")

If we perform this operation at lots of positions over the image, we'll get lots of outputs, as many as one for every input pixel.

• So we'll add another layer that "picks" the highest convolution pattern match from nearby pixels, which
  • makes our pattern match a little bit translation invariant (a fuzzy location match)
  • reduces the number of outputs significantly
• This layer is commonly called a pooling layer, and if we pick the "maximum match" then it's a "max pooling" layer.

The end result is that the kernel or filter together with max pooling creates a value in a subsequent layer which represents the appearance of a pattern in a local area in a prior layer.

Again, the network will be given a number of "slots" for these filters and will learn (by minimizing error) what filter values produce meaningful features. This is the key insight into how modern image-recognition networks are able to generalize -- i.e., learn to tell 6s from 7s or cats from dogs.

Ok, let's build our first ConvNet:

First, we want to explicity shape our data into a 2-D configuration. We'll end up with a 4-D tensor where the first dimension is the training examples, then each example is 28x28 pixels, and we'll explicitly say it's 1-layer deep. (Why? with color images, we typically process over 3 or 4 channels in this last dimension)

A step by step animation follows: * http://cs231n.github.io/assets/conv-demo/index.html

train_libsvm = "/dbfs/databricks-datasets/mnist-digits/data-001/mnist-digits-train.txt"
test_libsvm = "/dbfs/databricks-datasets/mnist-digits/data-001/mnist-digits-test.txt"

X_train, y_train = sklearn.datasets.load_svmlight_file(train_libsvm, n_features=784)
X_train = X_train.toarray()

X_test, y_test = sklearn.datasets.load_svmlight_file(test_libsvm, n_features=784)
X_test = X_test.toarray()

X_train = X_train.reshape( (X_train.shape[0], 28, 28, 1) )
X_train = X_train.astype('float32')
X_train /= 255
y_train = to_categorical(y_train, num_classes=10)

X_test = X_test.reshape( (X_test.shape[0], 28, 28, 1) )
X_test = X_test.astype('float32')
X_test /= 255
y_test = to_categorical(y_test, num_classes=10)

Now the model:

from keras.layers import Dense, Dropout, Activation, Flatten, Conv2D, MaxPooling2D

model = Sequential()

model.add(Conv2D(8, # number of kernels 
                (4, 4), # kernel size
                padding='valid', # no padding; output will be smaller than input
                input_shape=(28, 28, 1)))

model.add(Activation('relu'))

model.add(MaxPooling2D(pool_size=(2,2)))

model.add(Flatten())
model.add(Dense(128))
model.add(Activation('relu')) # alternative syntax for applying activation

model.add(Dense(10))
model.add(Activation('softmax'))

model.compile(loss='categorical_crossentropy', optimizer='adam', metrics=['accuracy'])

... and the training loop and output:

start = datetime.datetime.today()

history = model.fit(X_train, y_train, batch_size=128, epochs=8, verbose=2, validation_split=0.1)

scores = model.evaluate(X_test, y_test, verbose=1)

print
for i in range(len(model.metrics_names)):
    print("%s: %f" % (model.metrics_names[i], scores[i]))

Train on 54000 samples, validate on 6000 samples
Epoch 1/8
12s - loss: 0.3183 - acc: 0.9115 - val_loss: 0.1114 - val_acc: 0.9713
Epoch 2/8
13s - loss: 0.1021 - acc: 0.9700 - val_loss: 0.0714 - val_acc: 0.9803
Epoch 3/8
12s - loss: 0.0706 - acc: 0.9792 - val_loss: 0.0574 - val_acc: 0.9852
Epoch 4/8
15s - loss: 0.0518 - acc: 0.9851 - val_loss: 0.0538 - val_acc: 0.9857
Epoch 5/8
16s - loss: 0.0419 - acc: 0.9875 - val_loss: 0.0457 - val_acc: 0.9872
Epoch 6/8
16s - loss: 0.0348 - acc: 0.9896 - val_loss: 0.0473 - val_acc: 0.9867
Epoch 7/8
16s - loss: 0.0280 - acc: 0.9917 - val_loss: 0.0445 - val_acc: 0.9875
Epoch 8/8
14s - loss: 0.0225 - acc: 0.9934 - val_loss: 0.0485 - val_acc: 0.9868
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loss: 0.054931
acc: 0.983000

fig, ax = plt.subplots()
fig.set_size_inches((5,5))
plt.plot(history.history['loss'])
plt.plot(history.history['val_loss'])
plt.title('model loss')
plt.ylabel('loss')
plt.xlabel('epoch')
plt.legend(['train', 'val'], loc='upper left')
display(fig)

Our MNIST ConvNet

In our first convolutional MNIST experiment, we get to almost 99% validation accuracy in just a few epochs (a minutes or so on CPU)!

The training accuracy is effectively 100%, though, so we've almost completely overfit (i.e., memorized the training data) by this point and need to do a little work if we want to keep learning.

Let's add another convolutional layer:

model = Sequential()

model.add(Conv2D(8, # number of kernels 
                        (4, 4), # kernel size
                        padding='valid',
                        input_shape=(28, 28, 1)))

model.add(Activation('relu'))

model.add(Conv2D(8, (4, 4)))
model.add(Activation('relu'))

model.add(MaxPooling2D(pool_size=(2,2)))

model.add(Flatten())
model.add(Dense(128))
model.add(Activation('relu'))

model.add(Dense(10))
model.add(Activation('softmax'))

model.compile(loss='categorical_crossentropy', optimizer='adam', metrics=['accuracy'])

history = model.fit(X_train, y_train, batch_size=128, epochs=15, verbose=2, validation_split=0.1)

scores = model.evaluate(X_test, y_test, verbose=1)

print
for i in range(len(model.metrics_names)):
    print("%s: %f" % (model.metrics_names[i], scores[i]))

Train on 54000 samples, validate on 6000 samples
Epoch 1/15
21s - loss: 0.2698 - acc: 0.9242 - val_loss: 0.0763 - val_acc: 0.9790
Epoch 2/15
22s - loss: 0.0733 - acc: 0.9772 - val_loss: 0.0593 - val_acc: 0.9840
Epoch 3/15
21s - loss: 0.0522 - acc: 0.9838 - val_loss: 0.0479 - val_acc: 0.9867
Epoch 4/15
21s - loss: 0.0394 - acc: 0.9876 - val_loss: 0.0537 - val_acc: 0.9828
Epoch 5/15
21s - loss: 0.0301 - acc: 0.9907 - val_loss: 0.0434 - val_acc: 0.9885
Epoch 6/15
21s - loss: 0.0255 - acc: 0.9916 - val_loss: 0.0444 - val_acc: 0.9867
Epoch 7/15
21s - loss: 0.0192 - acc: 0.9941 - val_loss: 0.0416 - val_acc: 0.9892
Epoch 8/15
21s - loss: 0.0155 - acc: 0.9952 - val_loss: 0.0405 - val_acc: 0.9900
Epoch 9/15
21s - loss: 0.0143 - acc: 0.9953 - val_loss: 0.0490 - val_acc: 0.9860
Epoch 10/15
22s - loss: 0.0122 - acc: 0.9963 - val_loss: 0.0478 - val_acc: 0.9877
Epoch 11/15
21s - loss: 0.0087 - acc: 0.9972 - val_loss: 0.0508 - val_acc: 0.9877
Epoch 12/15
21s - loss: 0.0098 - acc: 0.9967 - val_loss: 0.0401 - val_acc: 0.9902
Epoch 13/15
21s - loss: 0.0060 - acc: 0.9984 - val_loss: 0.0450 - val_acc: 0.9897
Epoch 14/15
21s - loss: 0.0061 - acc: 0.9981 - val_loss: 0.0523 - val_acc: 0.9883
Epoch 15/15
21s - loss: 0.0069 - acc: 0.9977 - val_loss: 0.0534 - val_acc: 0.9885
   32/10000 [..............................] - ETA: 1s  448/10000 [>.............................] - ETA: 1s  864/10000 [=>............................] - ETA: 1s 1280/10000 [==>...........................] - ETA: 1s 1696/10000 [====>.........................] - ETA: 1s 2112/10000 [=====>........................] - ETA: 0s 2560/10000 [======>.......................] - ETA: 0s 2976/10000 [=======>......................] - ETA: 0s 3392/10000 [=========>....................] - ETA: 0s 3840/10000 [==========>...................] - ETA: 0s 4320/10000 [===========>..................] - ETA: 0s 4736/10000 [=============>................] - ETA: 0s 5152/10000 [==============>...............] - ETA: 0s 5568/10000 [===============>..............] - ETA: 0s 5952/10000 [================>.............] - ETA: 0s 6368/10000 [==================>...........] - ETA: 0s 6816/10000 [===================>..........] - ETA: 0s 7200/10000 [====================>.........] - ETA: 0s 7648/10000 [=====================>........] - ETA: 0s 8000/10000 [=======================>......] - ETA: 0s 8384/10000 [========================>.....] - ETA: 0s 8800/10000 [=========================>....] - ETA: 0s 9216/10000 [==========================>...] - ETA: 0s 9632/10000 [===========================>..] - ETA: 0s
loss: 0.043144
acc: 0.989100

While that's running, let's look at a number of "famous" convolutional networks!

LeNet (Yann LeCun, 1998)

AlexNet (2012)

Back to our labs: Still Overfitting

We're making progress on our test error -- about 99% -- but just a bit for all the additional time, due to the network overfitting the data.

There are a variety of techniques we can take to counter this -- forms of regularization.

Let's try a relatively simple solution solution that works surprisingly well: add a pair of Dropout filters, a layer that randomly omits a fraction of neurons from each training batch (thus exposing each neuron to only part of the training data).

We'll add more convolution kernels but shrink them to 3x3 as well.

model = Sequential()

model.add(Conv2D(32, # number of kernels 
                        (3, 3), # kernel size
                        padding='valid',
                        input_shape=(28, 28, 1)))

model.add(Activation('relu'))

model.add(Conv2D(32, (3, 3)))
model.add(Activation('relu'))

model.add(MaxPooling2D(pool_size=(2,2)))

model.add(Dropout(0.25)) # <- regularize
model.add(Flatten())
model.add(Dense(128))
model.add(Activation('relu'))

model.add(Dropout(0.5)) # <-regularize
model.add(Dense(10))
model.add(Activation('softmax'))

model.compile(loss='categorical_crossentropy', optimizer='adam', metrics=['accuracy'])
history = model.fit(X_train, y_train, batch_size=128, epochs=15, verbose=2)

scores = model.evaluate(X_test, y_test, verbose=2)

print
for i in range(len(model.metrics_names)):
    print("%s: %f" % (model.metrics_names[i], scores[i]))

Epoch 1/15
122s - loss: 0.2673 - acc: 0.9186
Epoch 2/15
123s - loss: 0.0903 - acc: 0.9718
Epoch 3/15
124s - loss: 0.0696 - acc: 0.9787
Epoch 4/15
130s - loss: 0.0598 - acc: 0.9821
Epoch 5/15
140s - loss: 0.0507 - acc: 0.9842
Epoch 6/15
148s - loss: 0.0441 - acc: 0.9864
Epoch 7/15
152s - loss: 0.0390 - acc: 0.9878
Epoch 8/15
153s - loss: 0.0353 - acc: 0.9888
Epoch 9/15
161s - loss: 0.0329 - acc: 0.9894
Epoch 10/15
171s - loss: 0.0323 - acc: 0.9894
Epoch 11/15
172s - loss: 0.0288 - acc: 0.9906
Epoch 12/15
71s - loss: 0.0255 - acc: 0.9914
Epoch 13/15
86s - loss: 0.0245 - acc: 0.9916
Epoch 14/15
91s - loss: 0.0232 - acc: 0.9922
Epoch 15/15
86s - loss: 0.0218 - acc: 0.9931

loss: 0.029627
acc: 0.991400

While that's running, let's look at some more recent ConvNet architectures:

VGG16 (2014)

GoogLeNet (2014)

"Inception" layer: parallel convolutions at different resolutions

Residual Networks (2015-)

Skip layers to improve training (error propagation). Residual layers learn from details at multiple previous layers.

ASIDE: Atrous / Dilated Convolutions

An atrous or dilated convolution is a convolution filter with "holes" in it. Effectively, it is a way to enlarge the filter spatially while not adding as many parameters or attending to every element in the input.

Why? Covering a larger input volume allows recognizing coarser-grained patterns; restricting the number of parameters is a way of regularizing or constraining the capacity of the model, making training easier.

Lab Wrapup

From the last lab, you should have a test accuracy of over 99.1%

For one more activity, try changing the optimizer to old-school "sgd" -- just to see how far we've come with these modern gradient descent techniques in the last few years.

Accuracy will end up noticeably worse ... about 96-97% test accuracy. Two key takeaways:

• Without a good optimizer, even a very powerful network design may not achieve results
• In fact, we could replace the word "optimizer" there with
  • initialization
  • activation
  • regularization
  • (etc.)
• All of these elements we've been working with operate together in a complex way to determine final performance

Of course this world evolves fast - see the new kid in the CNN block -- capsule networks

Hinton: “The pooling operation used in convolutional neural networks is a big mistake and the fact that it works so well is a disaster.”

Well worth the 8 minute read: * https://medium.com/ai%C2%B3-theory-practice-business/understanding-hintons-capsule-networks-part-i-intuition-b4b559d1159b

To understand deeper: * original paper: https://arxiv.org/abs/1710.09829