# Neural Network¶

## 1. Basic Usage of TensofFlow¶

• References
• TensorFlow is a programming system in which you represent computations as graphs.
• Nodes in the graph are called operations.
• An operation takes zero or more Tensors, performs some computation, and produces zero or more Tensors.
• A Tensor is a typed multi-dimensional array.
In [1]:
import tensorflow as tf

matrix1 = tf.constant([[3., 3.]])
matrix2 = tf.constant([[2.],[2.]])
print "matrix1 -", matrix1
print "matrix2 -", matrix2
print

matrix3 = tf.matmul(matrix1, matrix2)
print "matrix3 -", matrix3
matrix4 = tf.matmul(matrix2, matrix1)
print "matrix4 -", matrix4
print

matrix5 = tf.constant([[1., 1.], [2., 2.]])
print "matrix5 -", matrix5
matrix6 = tf.constant([10., 100.])
print "matrix6 -", matrix6
print

matrix7 = matrix5 + matrix6
print "matrix7 -", matrix7
matrix8 = matrix5 * matrix6
print "matrix8 -", matrix8

matrix8 = tf.constant([[1., 1.], [2., 2.]])
print "matrix8 -", matrix8
matrix9 = tf.ones([2])
print "matrix9 -", matrix9

matrix10 = matrix8 + matrix9  #broadcast
print "matrix10 -", matrix10

print

sess = tf.Session()
matrix3_result = sess.run(matrix3)
print "matrix3_result -\n", matrix3_result

matrix4_result = sess.run(matrix4)
print "matrix4_result -\n", matrix4_result

matrix7_result = sess.run(matrix7)
print "matrix7_result -\n", matrix7_result

matrix10_result = sess.run(matrix10)
print "matrix10_result -\n", matrix10_result
matrix1 - Tensor("Const:0", shape=(1, 2), dtype=float32)
matrix2 - Tensor("Const_1:0", shape=(2, 1), dtype=float32)

matrix3 - Tensor("MatMul:0", shape=(1, 1), dtype=float32)
matrix4 - Tensor("MatMul_1:0", shape=(2, 2), dtype=float32)

matrix5 - Tensor("Const_2:0", shape=(2, 2), dtype=float32)
matrix6 - Tensor("Const_3:0", shape=(2,), dtype=float32)

matrix7 - Tensor("add:0", shape=(2, 2), dtype=float32)
matrix8 - Tensor("mul:0", shape=(2, 2), dtype=float32)
matrix8 - Tensor("Const_4:0", shape=(2, 2), dtype=float32)
matrix9 - Tensor("ones:0", shape=(2,), dtype=float32)
matrix10 - Tensor("add_1:0", shape=(2, 2), dtype=float32)

matrix3_result -
[[ 12.]]
matrix4_result -
[[ 6.  6.]
[ 6.  6.]]
matrix7_result -
[[  11.  101.]
[  12.  102.]]
matrix10_result -
[[ 2.  2.]
[ 3.  3.]]

## 2. MNIST handwritten digits image set¶

In [2]:
from tensorflow.examples.tutorials.mnist import input_data
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
• Each image is 28 pixels by 28 pixels. We can interpret this as a big array of numbers:

• flatten 1-D tensor of size 28x28 = 784.

• Each entry in the tensor is a pixel intensity between 0 and 1, for a particular pixel in a particular image. $$[0, 0, 0, ..., 0.6, 0.7, 0.7, 0.5, ... 0.8, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 0.9, 0.3, ..., 0.4, 0.4, 0.4, ... 0, 0, 0]$$
In [3]:
print type(mnist.train.images), mnist.train.images.shape
print type(mnist.train.labels), mnist.train.labels.shape
<type 'numpy.ndarray'> (55000, 784)
<type 'numpy.ndarray'> (55000, 10)
• Number of train images is 55000.
• mnist.train.images is a tensor with a shape of [55000, 784].
• A one-hot vector is a vector which is 0 in most entries, and 1 in a single entry.
• In this case, the $n$th digit will be represented as a vector which is 1 in the nth entry.
• For example, 3 would be $[0,0,0,1,0,0,0,0,0,0]$.
• mnist.train.labels is a tensor with a shape of [55000, 10].
In [4]:
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

fig = plt.figure(figsize=(20, 5))
for i in range(5):
img = np.array(mnist.train.images[i])
img.shape = (28, 28)
plt.subplot(150 + (i+1))
plt.imshow(img)

## 3. Neural Network Model¶

• Input Layer to Output Layer
• $i=1...784$
• $j=1...10$ $$u_j = \sum_i W_{ji} x_i + b_j$$
• Presentation of Matrix and Vector
• Shape of ${\bf W} = 10 \times 784$
• Shape of ${\bf x} = 784 \times 1$
• Shape of ${\bf b} = 10 \times 1$
• Shape of ${\bf u} = 10 \times 1$ $${\bf u} = {\bf Wx + b}$$
In [5]:
batch_images, batch_labels = mnist.train.next_batch(1)
print batch_images.shape
print batch_images
print

print batch_labels.shape
print batch_labels
(1, 784)
[[ 0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
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0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.38039219  0.37647063
0.3019608   0.46274513  0.2392157   0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.35294119  0.5411765
0.92156869  0.92156869  0.92156869  0.92156869  0.92156869  0.92156869
0.98431379  0.98431379  0.97254908  0.99607849  0.96078438  0.92156869
0.74509805  0.08235294  0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.
0.54901963  0.98431379  0.99607849  0.99607849  0.99607849  0.99607849
0.99607849  0.99607849  0.99607849  0.99607849  0.99607849  0.99607849
0.99607849  0.99607849  0.99607849  0.99607849  0.74117649  0.09019608
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.88627458  0.99607849  0.81568635
0.78039223  0.78039223  0.78039223  0.78039223  0.54509807  0.2392157
0.2392157   0.2392157   0.2392157   0.2392157   0.50196081  0.8705883
0.99607849  0.99607849  0.74117649  0.08235294  0.          0.          0.
0.          0.          0.          0.          0.          0.
0.14901961  0.32156864  0.0509804   0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.13333334  0.83529419  0.99607849  0.99607849  0.45098042  0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.32941177  0.99607849  0.99607849  0.91764712  0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.32941177  0.99607849  0.99607849  0.91764712  0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.41568631  0.6156863   0.99607849  0.99607849  0.95294124  0.20000002
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.09803922  0.45882356  0.89411771
0.89411771  0.89411771  0.99215692  0.99607849  0.99607849  0.99607849
0.99607849  0.94117653  0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.26666668  0.4666667   0.86274517
0.99607849  0.99607849  0.99607849  0.99607849  0.99607849  0.99607849
0.99607849  0.99607849  0.99607849  0.55686277  0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.14509805  0.73333335  0.99215692
0.99607849  0.99607849  0.99607849  0.87450987  0.80784321  0.80784321
0.29411766  0.26666668  0.84313732  0.99607849  0.99607849  0.45882356
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.44313729
0.8588236   0.99607849  0.94901967  0.89019614  0.45098042  0.34901962
0.12156864  0.          0.          0.          0.          0.7843138
0.99607849  0.9450981   0.16078432  0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.66274512  0.99607849  0.6901961   0.24313727  0.          0.
0.          0.          0.          0.          0.          0.18823531
0.90588242  0.99607849  0.91764712  0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.07058824  0.48627454  0.          0.          0.
0.          0.          0.          0.          0.          0.
0.32941177  0.99607849  0.99607849  0.65098041  0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.54509807  0.99607849  0.9333334   0.22352943  0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.
0.82352948  0.98039222  0.99607849  0.65882355  0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.94901967  0.99607849  0.93725497  0.22352943  0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.
0.34901962  0.98431379  0.9450981   0.33725491  0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.
0.01960784  0.80784321  0.96470594  0.6156863   0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.01568628  0.45882356  0.27058825  0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.
0.          0.          0.          0.          0.          0.          0.        ]]

(1, 10)
[[ 0.  0.  0.  0.  0.  0.  0.  1.  0.  0.]]
• Transposed Matrix Operation in Tensorflow
• Shape of ${\bf x} = 1 \times 784$
• Shape of ${\bf W} = 784 \times 10$
• Shape of ${\bf b} = 1 \times 10$
• Shape of ${\bf u} = 1 \times 10$ $${\bf u} = {\bf xW + b}$$
In [6]:
batch_images, batch_labels = mnist.train.next_batch(100)
print batch_images.shape
print batch_images
print

print batch_labels.shape
print batch_labels
(100, 784)
[[ 0.  0.  0. ...,  0.  0.  0.]
[ 0.  0.  0. ...,  0.  0.  0.]
[ 0.  0.  0. ...,  0.  0.  0.]
...,
[ 0.  0.  0. ...,  0.  0.  0.]
[ 0.  0.  0. ...,  0.  0.  0.]
[ 0.  0.  0. ...,  0.  0.  0.]]

(100, 10)
[[ 0.  0.  0.  1.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  1.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  0.]
[ 0.  1.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  0.  1.  0.]
[ 0.  1.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  0.  0.  1.]
[ 0.  0.  0.  0.  0.  0.  0.  0.  1.  0.]
[ 1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  1.  0.  0.  0.  0.  0.  0.]
[ 0.  1.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  1.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  1.  0.  0.]
[ 1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  1.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  0.  0.  1.]
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  0.]
[ 1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  1.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  1.  0.  0.]
[ 0.  1.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  0.  0.  1.]
[ 0.  0.  0.  0.  0.  0.  0.  1.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  1.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  1.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  0.  1.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  0.  1.  0.]
[ 0.  0.  0.  1.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  1.  0.  0.  0.  0.  0.]
[ 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.  0.  0.  1.  0.  0.]
[ 0.  0.  0.  1.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  0.]
[ 0.  0.  0.  0.  1.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  1.  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.  1.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  0.  1.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  0.  0.  1.]
[ 0.  0.  0.  0.  0.  0.  0.  0.  0.  1.]
[ 0.  0.  0.  0.  1.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  1.  0.  0.  0.  0.  0.]
[ 1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  1.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  0.  1.  0.]
[ 0.  1.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  1.  0.  0.  0.  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.  1.  0.  0.]
[ 0.  1.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  1.  0.  0.]
[ 0.  0.  0.  0.  0.  1.  0.  0.  0.  0.]
[ 0.  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.  0.  0.  0.  1.]
[ 0.  0.  0.  0.  1.  0.  0.  0.  0.  0.]
[ 0.  0.  1.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  1.  0.  0.  0.  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.  1.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  0.]
[ 1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  1.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  1.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  1.  0.  0.  0.  0.  0.  0.  0.]
[ 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.  1.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  1.  0.  0.  0.  0.]
[ 1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  0.]
[ 0.  0.  0.  0.  1.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  1.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  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.  1.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  1.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  1.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  0.  0.  1.]
[ 0.  0.  1.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  1.  0.  0.  0.  0.  0.  0.  0.]
[ 1.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  1.  0.  0.  0.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  1.  0.  0.  0.]
[ 0.  0.  0.  0.  0.  0.  0.  1.  0.  0.]]
• Mini Batch (ex. batch size = 100)
• Shape of ${\bf x} = 100 \times 784$
• Shape of ${\bf W} = 784 \times 10$
• Shape of ${\bf b} = 100 \times 10$
• Shape of ${\bf u} = 100 \times 10$ $${\bf U} = {\bf XW + B}$$
In [7]:
import tensorflow as tf
x = tf.placeholder(tf.float32, [None, 784])
print "x -", x.get_shape()
x - (?, 784)
• we also need to add a new placeholder to input the correct answers (ground truth):
In [8]:
y = tf.placeholder(tf.float32, [None, 10])
• construct a single layer neural network
In [9]:
W = tf.Variable(tf.zeros([784, 10]))
b = tf.Variable(tf.zeros([10]))
print "W -", W.get_shape()
print "b -", b.get_shape()
W - (784, 10)
b - (10,)
In [10]:
u = tf.matmul(x, W) + b
print "u -", u.get_shape()
u - (?, 10)
• softmax
$${\bf z} = softmax({\bf u})$$
• Error functions
• Squarred error
• Using maximum likelihood estimation
• Cross entropy
In [11]:
error = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(u, y))

Suppose you have two tensors, where u contains computed scores for each class (for example, from u = W*x +b) and y contains one-hot encoded true labels.

u  = ... # Predicted label, e.g. u = tf.matmul(X, W) + b
y  = ... # True label, one-hot encoded
If you interpret the scores in u as unnormalized log probabilities, then they are logits. Additionally, the total cross-entropy loss computed in this manner:
z = tf.nn.softmax(u)
total_loss = tf.reduce_mean(-tf.reduce_sum(y * tf.log(z), [1]))
is essentially equivalent to the total cross-entropy loss computed with the function softmax_cross_entropy_with_logits():
total_loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(u, y))

## 4. Training¶

In [12]:
init = tf.initialize_all_variables()
In [13]:
sess = tf.Session()
sess.run(init)
In [14]:
batch_size = 100
total_batch = int(mnist.train.num_examples/batch_size)
for i in range(total_batch):
batch_images, batch_labels = mnist.train.next_batch(batch_size)
sess.run(total_loss, feed_dict={x: batch_images, y: batch_labels})

## 5. Evaluation¶

In [15]:
print type(mnist.test.images), mnist.test.images.shape
print type(mnist.test.labels), mnist.test.labels.shape
<type 'numpy.ndarray'> (10000, 784)
<type 'numpy.ndarray'> (10000, 10)
In [16]:
batch_x, batch_y = mnist.test.next_batch(10000)
prediction = sess.run(tf.argmax(u, 1), feed_dict={x:batch_x})
ground_truth = sess.run(tf.argmax(y, 1), feed_dict={y:batch_y})

print prediction
print ground_truth

sum = 0
diff_index_list = []
for i in range(10000):
if (prediction[i] == ground_truth[i]):
sum = sum + 1
else:
diff_index_list.append(i)
#print "%d - %d: %s" % (diff_a[i], diff_b[i], diff_a[i] == diff_b[i])

print sum / 10000.0
print len(diff_index_list)

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline

fig = plt.figure(figsize=(20, 5))
for i in range(5):
j = diff_index_list[i]
print "Error Index: %s, Prediction: %s, Ground Truth: %s" % (j, prediction[j], ground_truth[j])
img = np.array(mnist.test.images[j])
img.shape = (28, 28)
plt.subplot(150 + (i+1))
plt.imshow(img)
[7 2 1 ..., 4 5 6]
[7 2 1 ..., 4 5 6]
0.9116
884
Error Index: 8, Prediction: 6, Ground Truth: 5
Error Index: 33, Prediction: 6, Ground Truth: 4
Error Index: 63, Prediction: 2, Ground Truth: 3
Error Index: 66, Prediction: 7, Ground Truth: 6
Error Index: 77, Prediction: 7, Ground Truth: 2
In [17]:
prediction_and_ground_truth = tf.equal(tf.argmax(u, 1), tf.argmax(y, 1))
accuracy = tf.reduce_mean(tf.cast(prediction_and_ground_truth, tf.float32))
In [18]:
print(sess.run(accuracy, feed_dict={x: mnist.test.images, y: mnist.test.labels}))
0.9116

## 6. Single Layer Neural Network - All in one¶

In [24]:
import tensorflow as tf
from tensorflow.examples.tutorials.mnist import input_data

# Parameters
training_epochs = 40
learning_rate = 0.001
batch_size = 100

# Network Parameters
n_input = 784 # MNIST data input (img shape: 28*28)
n_classes = 10 # MNIST total classes (0-9 digits)

# tf Graph input
x = tf.placeholder(tf.float32, [None, n_input])
y = tf.placeholder(tf.float32, [None, n_classes])

# Construct model
W = tf.Variable(tf.zeros([n_input, n_classes]))
b = tf.Variable(tf.zeros([n_classes]))
u = tf.matmul(x, W) + b

# Define loss and target loss function
#z = tf.nn.softmax(u)
#error = tf.reduce_mean(-tf.reduce_sum(z_ * tf.log(z), reduction_indices=[1]))
error = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(u, y))

# Initializing the variables
init = tf.initialize_all_variables()

# Calculate accuracy with a Test model
prediction_ground_truth = tf.equal(tf.argmax(u, 1), tf.argmax(y, 1))
accuracy = tf.reduce_mean(tf.cast(prediction_ground_truth, tf.float32))

# Launch the tensorflow graph
with tf.Session() as sess:
sess.run(init)
total_batch = int(mnist.train.num_examples / batch_size)
print "Total batch: %d" % total_batch

# Training cycle
for epoch in range(training_epochs):
# Loop over all batches
for i in range(total_batch):
batch_images, batch_labels = mnist.train.next_batch(batch_size)
sess.run(total_loss, feed_dict={x: batch_images, y: batch_labels})
print "Epoch %d Finished - Accuracy: %f" % (epoch, accuracy.eval({x: mnist.test.images, y: mnist.test.labels}))

print("Optimization Finished!")
Total batch: 550
Epoch 0 Finished - Accuracy: 0.912200
Epoch 1 Finished - Accuracy: 0.917600
Epoch 2 Finished - Accuracy: 0.919600
Epoch 3 Finished - Accuracy: 0.922700
Epoch 4 Finished - Accuracy: 0.922400
Epoch 5 Finished - Accuracy: 0.923200
Epoch 6 Finished - Accuracy: 0.921100
Epoch 7 Finished - Accuracy: 0.921700
Epoch 8 Finished - Accuracy: 0.922900
Epoch 9 Finished - Accuracy: 0.921200
Epoch 10 Finished - Accuracy: 0.926000
Epoch 11 Finished - Accuracy: 0.923100
Epoch 12 Finished - Accuracy: 0.926600
Epoch 13 Finished - Accuracy: 0.925500
Epoch 14 Finished - Accuracy: 0.922900
Epoch 15 Finished - Accuracy: 0.922300
Epoch 16 Finished - Accuracy: 0.916700
Epoch 17 Finished - Accuracy: 0.923100
Epoch 18 Finished - Accuracy: 0.924400
Epoch 19 Finished - Accuracy: 0.925600
Epoch 20 Finished - Accuracy: 0.925500
Epoch 21 Finished - Accuracy: 0.924500
Epoch 22 Finished - Accuracy: 0.922600
Epoch 23 Finished - Accuracy: 0.925100
Epoch 24 Finished - Accuracy: 0.924500
Epoch 25 Finished - Accuracy: 0.924900
Epoch 26 Finished - Accuracy: 0.921500
Epoch 27 Finished - Accuracy: 0.924200
Epoch 28 Finished - Accuracy: 0.923700
Epoch 29 Finished - Accuracy: 0.919600
Epoch 30 Finished - Accuracy: 0.923600
Epoch 31 Finished - Accuracy: 0.923600
Epoch 32 Finished - Accuracy: 0.923000
Epoch 33 Finished - Accuracy: 0.923700
Epoch 34 Finished - Accuracy: 0.924800
Epoch 35 Finished - Accuracy: 0.923000
Epoch 36 Finished - Accuracy: 0.925400
Epoch 37 Finished - Accuracy: 0.922000
Epoch 38 Finished - Accuracy: 0.920100
Epoch 39 Finished - Accuracy: 0.924100
Optimization Finished!

## 7. Multi Layer Neural Network - All in one¶

In [25]:
# Parameters
training_epochs = 40
learning_rate = 0.001
batch_size = 100
display_step = 1

# Network Parameters
n_input = 784 # MNIST data input (img shape: 28*28)
n_hidden_1 = 256 # 1st layer number of features
n_hidden_2 = 256 # 2nd layer number of features
n_classes = 10 # MNIST total classes (0-9 digits)

# tf Graph input
x = tf.placeholder("float", [None, n_input])
y = tf.placeholder("float", [None, n_classes])

# Store layers weight & bias
weights = {
'W1': tf.Variable(tf.random_normal([n_input, n_hidden_1])),
'W2': tf.Variable(tf.random_normal([n_hidden_1, n_hidden_2])),
'out': tf.Variable(tf.random_normal([n_hidden_2, n_classes]))
}
biases = {
'b1': tf.Variable(tf.random_normal([n_hidden_1])),
'b2': tf.Variable(tf.random_normal([n_hidden_2])),
'out': tf.Variable(tf.random_normal([n_classes]))
}

# Create model
def multilayer_perceptron(x, weights, biases):
# Hidden layer with RELU activation
z_2 = tf.nn.relu(u_2)
# Hidden layer with RELU activation
z_3 = tf.nn.relu(u_3)
# Output layer with linear activation
return u_4

# Construct model
pred = multilayer_perceptron(x, weights, biases)

# Define loss and target loss function
# pred = tf.nn.softmax(pred)
# error = tf.reduce_mean(-tf.reduce_sum(y * tf.log(pred), reduction_indices=[1]))
error = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(pred, y))

# Initializing the variables
init = tf.initialize_all_variables()

# Calculate accuracy with a Test model
prediction_ground_truth = tf.equal(tf.argmax(pred, 1), tf.argmax(y, 1))
accuracy = tf.reduce_mean(tf.cast(prediction_ground_truth, tf.float32))

# Launch the graph
with tf.Session() as sess:
sess.run(init)
total_batch = int(mnist.train.num_examples/batch_size)
print "Total batch: %d" % total_batch

# Training cycle
for epoch in range(training_epochs):

# Loop over all batches
for i in range(total_batch):
batch_images, batch_labels = mnist.train.next_batch(batch_size)
sess.run(total_loss, feed_dict={x: batch_images, y: batch_labels})
print "Epoch %d Finished - Accuracy: %f" % (epoch, accuracy.eval({x: mnist.test.images, y: mnist.test.labels}))

print("Optimization Finished!")
Total batch: 550
Epoch 0 Finished - Accuracy: 0.822300
Epoch 1 Finished - Accuracy: 0.855200
Epoch 2 Finished - Accuracy: 0.870800
Epoch 3 Finished - Accuracy: 0.881200
Epoch 4 Finished - Accuracy: 0.887600
Epoch 5 Finished - Accuracy: 0.892100
Epoch 6 Finished - Accuracy: 0.898000
Epoch 7 Finished - Accuracy: 0.901100
Epoch 8 Finished - Accuracy: 0.905600
Epoch 9 Finished - Accuracy: 0.908900
Epoch 10 Finished - Accuracy: 0.910900
Epoch 11 Finished - Accuracy: 0.910400
Epoch 12 Finished - Accuracy: 0.912600
Epoch 13 Finished - Accuracy: 0.914200
Epoch 14 Finished - Accuracy: 0.914200
Epoch 15 Finished - Accuracy: 0.916800
Epoch 16 Finished - Accuracy: 0.917300
Epoch 17 Finished - Accuracy: 0.918700
Epoch 18 Finished - Accuracy: 0.917500
Epoch 19 Finished - Accuracy: 0.919200
Epoch 20 Finished - Accuracy: 0.920600
Epoch 21 Finished - Accuracy: 0.921000
Epoch 22 Finished - Accuracy: 0.919700
Epoch 23 Finished - Accuracy: 0.921100
Epoch 24 Finished - Accuracy: 0.922400
Epoch 25 Finished - Accuracy: 0.923000
Epoch 26 Finished - Accuracy: 0.923200
Epoch 27 Finished - Accuracy: 0.922200
Epoch 28 Finished - Accuracy: 0.922900
Epoch 29 Finished - Accuracy: 0.923100
Epoch 30 Finished - Accuracy: 0.924400
Epoch 31 Finished - Accuracy: 0.925000
Epoch 32 Finished - Accuracy: 0.924200
Epoch 33 Finished - Accuracy: 0.923600
Epoch 34 Finished - Accuracy: 0.924000
Epoch 35 Finished - Accuracy: 0.924600
Epoch 36 Finished - Accuracy: 0.926500
Epoch 37 Finished - Accuracy: 0.925400
Epoch 38 Finished - Accuracy: 0.927300
Epoch 39 Finished - Accuracy: 0.925700
Optimization Finished!
In [ ]: