# Neural Network¶

In this tutorial, we'll create a simple neural network classifier in TensorFlow. The key advantage of this model over the Linear Classifier trained in the previous tutorial is that it can separate data which is NOT linearly separable.

We assume that you have the basic knowledge over the concept and you are just interested in the Tensorflow implementation of the Neural Nets. If you want to know more about the Neural Nets we suggest you to take this amazing course on machine learning or check out the following tutorials:

The structure of the neural network that we're going to implement is as follows. Like before, we're using images of handw-ritten digits of the MNIST data which has 10 classes (i.e. digits from 0 to 9). The implemented network has 2 hidden layers: the first one with 200 hidden units (neurons) and the second one (also known as classifier layer) with 10 (number of classes) neurons. Fig. 1- Sample Neural Network architecture with two layers implemented for classifying MNIST digits

## 0. Import the required libraries:¶

In :
# imports
import tensorflow as tf
import numpy as np
import matplotlib.pyplot as plt


## 1. Load the MNIST data¶

For this tutorial we use the MNIST dataset. MNIST is a dataset of handwritten digits. If you are into machine learning, you might have heard of this dataset by now. MNIST is kind of benchmark of datasets for deep learning and is easily accesible through Tensorflow

The dataset contains 55,000 examples for training, 5,000 examples for validation and 10,000 examples for testing. The digits have been size-normalized and centered in a fixed-size image (28x28 pixels) with values from 0 to 1. For simplicity, each image has been flattened and converted to a 1-D numpy array of 784 features (28*28). If you want to know more about the MNIST dataset you can check Yann Lecun's website.

### 1.1. Data dimension¶

Here, we specify the dimensions of the images which will be used in several places in the code below. Defining these variables makes it easier (compared with using hard-coded number all throughout the code) to modify them later. Ideally these would be inferred from the data that has been read, but here we will just write the numbers.

It's important to note that in a linear model, we have to flatten the input images into a vector. Here, each of the $28\times28$ images are flattened into a $1\times784$ vector.

In :
img_h = img_w = 28             # MNIST images are 28x28
img_size_flat = img_h * img_w  # 28x28=784, the total number of pixels
n_classes = 10                 # Number of classes, one class per digit


### 1.2. Helper functions to load the MNIST data¶

Here, we'll simply write a function (load_data) which has two modes: train (which loads the training and validation images and their corresponding labels) and test (which loads the test images and their corresponding labels).

Other than a function for loading the images and corresponding labels, we define two more functions:

1. randomize: which randomizes the order of images and their labels. This is important to make sure that the input images are sorted in a completely random order. Moreover, at the beginning of each epoch, we will re-randomize the order of data samples to make sure that the trained model is not sensitive to the order of data.

2. get_next_batch: which only selects a few number of images determined by the batch_size variable (if you don't know why, read about Stochastic Gradient Method)

In :
def load_data(mode='train'):
"""
:param mode: train or test
:return: images and the corresponding labels
"""
from tensorflow.examples.tutorials.mnist import input_data
if mode == 'train':
x_train, y_train, x_valid, y_valid = mnist.train.images, mnist.train.labels, \
mnist.validation.images, mnist.validation.labels
return x_train, y_train, x_valid, y_valid
elif mode == 'test':
x_test, y_test = mnist.test.images, mnist.test.labels
return x_test, y_test

def randomize(x, y):
""" Randomizes the order of data samples and their corresponding labels"""
permutation = np.random.permutation(y.shape)
shuffled_x = x[permutation, :]
shuffled_y = y[permutation]
return shuffled_x, shuffled_y

def get_next_batch(x, y, start, end):
x_batch = x[start:end]
y_batch = y[start:end]
return x_batch, y_batch


### 1.3. Load the data and display the sizes¶

Now we can use the defined helper function in train mode which loads the train and validation images and their corresponding labels. We'll also display their sizes:

In :
# Load MNIST data
x_train, y_train, x_valid, y_valid = load_data(mode='train')
print("Size of:")
print("- Training-set:\t\t{}".format(len(y_train)))
print("- Validation-set:\t{}".format(len(y_valid)))


Extracting MNIST_data/train-images-idx3-ubyte.gz
Extracting MNIST_data/train-labels-idx1-ubyte.gz
Extracting MNIST_data/t10k-images-idx3-ubyte.gz
Extracting MNIST_data/t10k-labels-idx1-ubyte.gz
Size of:
- Training-set:        55000
- Validation-set:    5000


To get a better sense of the data, let's checkout the shapes of the loaded arrays.

In :
print('x_train:\t{}'.format(x_train.shape))
print('y_train:\t{}'.format(y_train.shape))
print('x_train:\t{}'.format(x_valid.shape))
print('y_valid:\t{}'.format(y_valid.shape))


x_train:    (55000, 784)
y_train:    (55000, 10)
x_train:    (5000, 784)
y_valid:    (5000, 10)


As you can see, x_train and x_valid arrays contain 55000 and 5000 flattened images ( of size 28x28=784 values). y_train and y_valid contain the corresponding labels of the images in the training and validation set respectively.

Based on the dimesnion of the arrays, for each image, we have 10 values as its label. Why? This technique is called One-Hot Encoding. This means the labels have been converted from a single number to a vector whose length equals the number of possible classes. All elements of the vector are zero except for the $i^{th}$ element which is one and means the class is $i$. For example, the One-Hot encoded labels for the first 5 images in the validation set are:

In :
y_valid[:5, :]

Out:
array([[0., 0., 0., 0., 0., 1., 0., 0., 0., 0.],
[1., 0., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 1., 0., 0., 0., 0., 0.],
[0., 1., 0., 0., 0., 0., 0., 0., 0., 0.],
[0., 0., 0., 0., 0., 0., 0., 0., 0., 1.]]) 

where the 10 values in each row represents the label assigned to that partiular image.

## 2. Hyperparameters¶

Here, we have about 55,000 images in our training set. It takes a long time to calculate the gradient of the model using all these images. We therefore use Stochastic Gradient Descent which only uses a small batch of images in each iteration of the optimizer. Let's define some of the terms usually used in this context:

• epoch: one forward pass and one backward pass of all the training examples
• batch size: the number of training examples in one forward/backward pass. The higher the batch size, the more memory space you'll need.
• iteration: one forward pass and one backward pass of one batch of images the training examples
In :
# Hyper-parameters
epochs = 10             # Total number of training epochs
batch_size = 100        # Training batch size
display_freq = 100      # Frequency of displaying the training results
learning_rate = 0.001   # The optimization initial learning rate


Given the above definitions, each epoch consists of $55,000/100=550$ iterations.

## 3. Network configuration¶

In :
h1 = 200                # Number of units in the first hidden layer


## 4. Create network helper functions¶

### 4.1. Helper functions for creating new variables¶

As explained (and also illustrated in Fig. 1), we need to define two variables $\mathbf{W}$ and $\mathbf{b}$ to construt our linear model. These are generally called model parameters and as explained in our Tensor Types tutorial, we use Tensorflow Variables of proper size and initialization to define them.The following functions are written to be later used for generating the weight and bias variables of the desired shape:

In :
# weight and bais wrappers
def weight_variable(name, shape):
"""
Create a weight variable with appropriate initialization
:param name: weight name
:param shape: weight shape
:return: initialized weight variable
"""
initer = tf.truncated_normal_initializer(stddev=0.01)
return tf.get_variable('W_' + name,
dtype=tf.float32,
shape=shape,
initializer=initer)

def bias_variable(name, shape):
"""
Create a bias variable with appropriate initialization
:param name: bias variable name
:param shape: bias variable shape
:return: initialized bias variable
"""
initial = tf.constant(0., shape=shape, dtype=tf.float32)
return tf.get_variable('b_' + name,
dtype=tf.float32,
initializer=initial)


### 4.2. Helper-function for creating a fully-connected layer¶

In :
def fc_layer(x, num_units, name, use_relu=True):
"""
Create a fully-connected layer
:param x: input from previous layer
:param num_units: number of hidden units in the fully-connected layer
:param name: layer name
:param use_relu: boolean to add ReLU non-linearity (or not)
:return: The output array
"""
in_dim = x.get_shape()
W = weight_variable(name, shape=[in_dim, num_units])
b = bias_variable(name, [num_units])
layer = tf.matmul(x, W)
layer += b
if use_relu:
layer = tf.nn.relu(layer)
return layer


## 5. Create the network graph¶

### 5.1. Placeholders for the inputs (x) and corresponding labels (y)¶

First we need to define the proper tensors to feed in the input values to our model. As explained in the Tensor Types tutorial, placeholder variable is the suitable choice for the input images and corresponding labels. This allows us to change the inputs (images and labels) to the TensorFlow graph.

In :
# Create the graph for the linear model
# Placeholders for inputs (x) and outputs(y)
x = tf.placeholder(tf.float32, shape=[None, img_size_flat], name='X')
y = tf.placeholder(tf.float32, shape=[None, n_classes], name='Y')


Plceholder x is defined for the images; its data-type is set to float32 and the shape is set to [None, img_size_flat], where None means that the tensor may hold an arbitrary number of images with each image being a vector of length img_size_flat.

Next we have y which is the placeholder variable for the true labels associated with the images that were input in the placeholder variable x. The shape of this placeholder variable is [None, num_classes] which means it may hold an arbitrary number of labels and each label is a vector of length num_classes which is 10 in this case.

### 5.2. Create the network layers¶

In :
fc1 = fc_layer(x, h1, 'FC1', use_relu=True)
output_logits = fc_layer(fc1, n_classes, 'OUT', use_relu=False)


### 5.3. Define the loss function, optimizer, accuracy, and predicted class¶

In :
# Network predictions
cls_prediction = tf.argmax(output_logits, axis=1, name='predictions')

# Define the loss function, optimizer, and accuracy
loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(labels=y, logits=output_logits), name='loss')
correct_prediction = tf.equal(tf.argmax(output_logits, 1), tf.argmax(y, 1), name='correct_pred')
accuracy = tf.reduce_mean(tf.cast(correct_prediction, tf.float32), name='accuracy')


### 5.4. Initialize all variables¶

In :
# Create the op for initializing all variables
init = tf.global_variables_initializer()


## 6. Train¶

In :
sess = tf.InteractiveSession()
sess.run(init)
global_step = 0
# Number of training iterations in each epoch
num_tr_iter = int(len(y_train) / batch_size)
for epoch in range(epochs):
print('Training epoch: {}'.format(epoch + 1))
x_train, y_train = randomize(x_train, y_train)
for iteration in range(num_tr_iter):
global_step += 1
start = iteration * batch_size
end = (iteration + 1) * batch_size
x_batch, y_batch = get_next_batch(x_train, y_train, start, end)

# Run optimization op (backprop)
feed_dict_batch = {x: x_batch, y: y_batch}
sess.run(optimizer, feed_dict=feed_dict_batch)

if iteration % display_freq == 0:
# Calculate and display the batch loss and accuracy
loss_batch, acc_batch = sess.run([loss, accuracy],
feed_dict=feed_dict_batch)

print("iter {0:3d}:\t Loss={1:.2f},\tTraining Accuracy={2:.01%}".
format(iteration, loss_batch, acc_batch))

# Run validation after every epoch
feed_dict_valid = {x: x_valid[:1000], y: y_valid[:1000]}
loss_valid, acc_valid = sess.run([loss, accuracy], feed_dict=feed_dict_valid)
print('---------------------------------------------------------')
print("Epoch: {0}, validation loss: {1:.2f}, validation accuracy: {2:.01%}".
format(epoch + 1, loss_valid, acc_valid))
print('---------------------------------------------------------')


Training epoch: 1
iter   0:     Loss=2.28,    Training Accuracy=54.0%
iter 100:     Loss=0.58,    Training Accuracy=77.0%
iter 200:     Loss=0.44,    Training Accuracy=91.0%
iter 300:     Loss=0.24,    Training Accuracy=94.0%
iter 400:     Loss=0.28,    Training Accuracy=92.0%
iter 500:     Loss=0.16,    Training Accuracy=94.0%
---------------------------------------------------------
Epoch: 1, validation loss: 0.26, validation accuracy: 92.2%
---------------------------------------------------------
Training epoch: 2
iter   0:     Loss=0.18,    Training Accuracy=96.0%
iter 100:     Loss=0.21,    Training Accuracy=95.0%
iter 200:     Loss=0.07,    Training Accuracy=99.0%
iter 300:     Loss=0.13,    Training Accuracy=96.0%
iter 400:     Loss=0.25,    Training Accuracy=94.0%
iter 500:     Loss=0.19,    Training Accuracy=93.0%
---------------------------------------------------------
Epoch: 2, validation loss: 0.17, validation accuracy: 95.2%
---------------------------------------------------------
Training epoch: 3
iter   0:     Loss=0.12,    Training Accuracy=97.0%
iter 100:     Loss=0.04,    Training Accuracy=100.0%
iter 200:     Loss=0.11,    Training Accuracy=97.0%
iter 300:     Loss=0.19,    Training Accuracy=95.0%
iter 400:     Loss=0.11,    Training Accuracy=98.0%
iter 500:     Loss=0.20,    Training Accuracy=96.0%
---------------------------------------------------------
Epoch: 3, validation loss: 0.12, validation accuracy: 96.7%
---------------------------------------------------------
Training epoch: 4
iter   0:     Loss=0.09,    Training Accuracy=98.0%
iter 100:     Loss=0.09,    Training Accuracy=95.0%
iter 200:     Loss=0.08,    Training Accuracy=97.0%
iter 300:     Loss=0.05,    Training Accuracy=99.0%
iter 400:     Loss=0.15,    Training Accuracy=97.0%
iter 500:     Loss=0.04,    Training Accuracy=100.0%
---------------------------------------------------------
Epoch: 4, validation loss: 0.11, validation accuracy: 96.4%
---------------------------------------------------------
Training epoch: 5
iter   0:     Loss=0.15,    Training Accuracy=94.0%
iter 100:     Loss=0.09,    Training Accuracy=97.0%
iter 200:     Loss=0.04,    Training Accuracy=99.0%
iter 300:     Loss=0.05,    Training Accuracy=98.0%
iter 400:     Loss=0.05,    Training Accuracy=99.0%
iter 500:     Loss=0.02,    Training Accuracy=100.0%
---------------------------------------------------------
Epoch: 5, validation loss: 0.10, validation accuracy: 97.5%
---------------------------------------------------------
Training epoch: 6
iter   0:     Loss=0.07,    Training Accuracy=98.0%
iter 100:     Loss=0.15,    Training Accuracy=96.0%
iter 200:     Loss=0.04,    Training Accuracy=97.0%
iter 300:     Loss=0.05,    Training Accuracy=97.0%
iter 400:     Loss=0.03,    Training Accuracy=99.0%
iter 500:     Loss=0.05,    Training Accuracy=98.0%
---------------------------------------------------------
Epoch: 6, validation loss: 0.10, validation accuracy: 97.1%
---------------------------------------------------------
Training epoch: 7
iter   0:     Loss=0.05,    Training Accuracy=99.0%
iter 100:     Loss=0.02,    Training Accuracy=100.0%
iter 200:     Loss=0.02,    Training Accuracy=100.0%
iter 300:     Loss=0.02,    Training Accuracy=99.0%
iter 400:     Loss=0.06,    Training Accuracy=98.0%
iter 500:     Loss=0.07,    Training Accuracy=98.0%
---------------------------------------------------------
Epoch: 7, validation loss: 0.12, validation accuracy: 96.6%
---------------------------------------------------------
Training epoch: 8
iter   0:     Loss=0.04,    Training Accuracy=99.0%
iter 100:     Loss=0.02,    Training Accuracy=100.0%
iter 200:     Loss=0.04,    Training Accuracy=97.0%
iter 300:     Loss=0.05,    Training Accuracy=98.0%
iter 400:     Loss=0.05,    Training Accuracy=98.0%
iter 500:     Loss=0.04,    Training Accuracy=99.0%
---------------------------------------------------------
Epoch: 8, validation loss: 0.10, validation accuracy: 96.9%
---------------------------------------------------------
Training epoch: 9
iter   0:     Loss=0.06,    Training Accuracy=99.0%
iter 100:     Loss=0.03,    Training Accuracy=99.0%
iter 200:     Loss=0.03,    Training Accuracy=100.0%
iter 300:     Loss=0.02,    Training Accuracy=100.0%
iter 400:     Loss=0.02,    Training Accuracy=100.0%
iter 500:     Loss=0.05,    Training Accuracy=98.0%
---------------------------------------------------------
Epoch: 9, validation loss: 0.10, validation accuracy: 96.8%
---------------------------------------------------------
Training epoch: 10
iter   0:     Loss=0.02,    Training Accuracy=99.0%
iter 100:     Loss=0.01,    Training Accuracy=100.0%
iter 200:     Loss=0.01,    Training Accuracy=100.0%
iter 300:     Loss=0.03,    Training Accuracy=99.0%
iter 400:     Loss=0.02,    Training Accuracy=99.0%
iter 500:     Loss=0.01,    Training Accuracy=100.0%
---------------------------------------------------------
Epoch: 10, validation loss: 0.11, validation accuracy: 97.0%
---------------------------------------------------------


## 7. Test¶

### 7.1. Helper functions for plotting the results¶

In :
def plot_images(images, cls_true, cls_pred=None, title=None):
"""
Create figure with 3x3 sub-plots.
:param images: array of images to be plotted, (9, img_h*img_w)
:param cls_true: corresponding true labels (9,)
:param cls_pred: corresponding true labels (9,)
"""
fig, axes = plt.subplots(3, 3, figsize=(9, 9))
for i, ax in enumerate(axes.flat):
# Plot image.
ax.imshow(images[i].reshape(28, 28), cmap='binary')

# Show true and predicted classes.
if cls_pred is None:
ax_title = "True: {0}".format(cls_true[i])
else:
ax_title = "True: {0}, Pred: {1}".format(cls_true[i], cls_pred[i])

ax.set_title(ax_title)

# Remove ticks from the plot.
ax.set_xticks([])
ax.set_yticks([])

if title:
plt.suptitle(title, size=20)
plt.show(block=False)

def plot_example_errors(images, cls_true, cls_pred, title=None):
"""
Function for plotting examples of images that have been mis-classified
:param images: array of all images, (#imgs, img_h*img_w)
:param cls_true: corresponding true labels, (#imgs,)
:param cls_pred: corresponding predicted labels, (#imgs,)
"""
# Negate the boolean array.
incorrect = np.logical_not(np.equal(cls_pred, cls_true))

# Get the images from the test-set that have been
# incorrectly classified.
incorrect_images = images[incorrect]

# Get the true and predicted classes for those images.
cls_pred = cls_pred[incorrect]
cls_true = cls_true[incorrect]

# Plot the first 9 images.
plot_images(images=incorrect_images[0:9],
cls_true=cls_true[0:9],
cls_pred=cls_pred[0:9],
title=title)


### 6.2. Run the test and plot some results¶

In :
# Test the network after training
feed_dict_test = {x: x_test[:1000], y: y_test[:1000]}
loss_test, acc_test = sess.run([loss, accuracy], feed_dict=feed_dict_test)
print('---------------------------------------------------------')
print("Test loss: {0:.2f}, test accuracy: {1:.01%}".format(loss_test, acc_test))
print('---------------------------------------------------------')

# Plot some of the correct and misclassified examples
cls_pred = sess.run(cls_prediction, feed_dict=feed_dict_test)
cls_true = np.argmax(y_test[:1000], axis=1)
plot_images(x_test, cls_true, cls_pred, title='Correct Examples')
plot_example_errors(x_test[:1000], cls_true, cls_pred, title='Misclassified Examples')
plt.show()


Extracting MNIST_data/train-images-idx3-ubyte.gz
Extracting MNIST_data/train-labels-idx1-ubyte.gz
Extracting MNIST_data/t10k-images-idx3-ubyte.gz
Extracting MNIST_data/t10k-labels-idx1-ubyte.gz
---------------------------------------------------------
Test loss: 0.07, test accuracy: 97.8%
---------------------------------------------------------  