Teaching ML

TF-3--graph

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Graph

A Graph in Tensorflow represents complicated computation dataflow consisting of Tensors.

A Tensor is a basic data structure in Tensorflow. There are several features of a Tensor.

  • Represents one of outputs of an Operation;
  • As a symbolic handle to one of the outputs of an Operation, Tensor provides a mean of computing the outputs in Tensorflow session instead of hold the real value;
  • A Tensor could also be fed as an input to another Operation, that enables Tensorflow to build a multi-step, complicated computation which is called a Graph;
  • After the Graph has been launched to a Session, the value of the Tensor can be computed by passing it to Session.run();

Exercise: Build a Softmax Regression in Tensorflow

Logistic Regression

Logistic Regression applies a sigmoid function on linear combination to break the constant gradient. As ranging between 0 and 1, sigmoid function is widely used in Neural Network for neural activation.

A sigmoid function is defined as $\normalsize \sigma(z) = {1 \over 1 + e^{-z}}$, where $\normalsize z = x^T * w + b$.

Softmax Regression

Logistic Regression could properly deal with 2-class classification problem. While in machine-learned neural networks, Softmax Regression is more common used because of the capability of multiple-class classfiction. Generally, Softmax Regression is a special case of Logistic Regression and is designed for filling the vacancy on its disadvantages.

A Softmax function is defined as: $\sigma(z)_j = \Large {{e^{z_j} \over \Sigma^k_{k=1} e^{z_k}}}$

The largest $\sigma(z)_j$ is then chosen as the predicted class.

Relationship between Logistic Regression and Softmax Regression

Let’s do some simple mathmatics.

When k = 2,
$ \begin{align*} \sigma(z) &= \normalsize{{1 \over e^{z_1} + e^{z_2}} \begin{bmatrix} e^{z_1} \\ e^{z_2} \end{bmatrix}} \\ &= \large\begin{bmatrix} {1 \over 1 + e^{(z_2 - z_1)}} \\ {1 \over 1 + e^{(z_1 - z_2)}}\end{bmatrix} \\ &= \large\begin{bmatrix} {1 \over 1 + e^{(z_2 - z_1)}} \\ \normalsize 1 - {1 \over 1 + e^{(z_2 - z_1)}}\end{bmatrix} \end{align*} $

Assume $Z = z_1 - z_2$, one of the $\sigma(z_1) = \large{1 \over 1 + e^{-Z}}$ while the other one $\sigma(z_1) = 1 - \large{1 \over 1 + e^{-Z}}$, which proves the function is consitent with Logistic Regression.

Now try to build a Softmax Regression in Tensorflow yourself. See Linear Regression sample for reference.

Necessary Headers

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from __future__ import absolute_import
from __future__ import print_function
from __future__ import division
import tensorflow as tf

MNIST data

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## MNIST data
from tensorflow.examples.tutorials.mnist import input_data
mnist = input_data.read_data_sets("/root/tensorflow/MNIST_data", one_hot=True)

Training Parameters

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## parameters
learning_rate = 0.01
training_epochs = 25
batch_size = 100
display_step = 1

Inputs

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## inputs
x = tf.placeholder(tf.float32, [None, 784]) # MNIST data image are of shape 28*28
y = tf.placeholder(tf.float32, [None, 10]) # MNIST data has 10 classes

Variables

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## variables
# initialize random uniform distributed weights with size of [784, 10] ranging from -1 to 1
W = tf.Variables(tf.random_uniform([784,10], -1.0, 1.0)) ###### write your code here ######
# initialize bias with size of [10] to zero
b = tf.Variables(tf.zeros([10])) ###### write your code here ######

Graph

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## graph
# comb = W * x + b (using a similar tensorflow function)
comb = ... ###### write your code here ######
# predicted value
pred = tf.nn.softmax(comb)
# entr equals to **negative** `tf.reduce_sum()` of y * log(pred), with reduction_indices = 1
entr = ... ###### write your code here ######
# cross entropy cost
cost = tf.reduce_mean(entr)
# optimizer
opti = tf.train.GradientDescentOptimizer(learning_rate)
# training_steps use optimizer to minimize the cost
training_steps = ... ###### write your code here ######
# initialization
init = tf.initialize_all_variables()

Run a Session

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## training
with tf.Session() as sess:
    sess.run(init)
    # training epochs
    for epoch in range(training_epochs):
        avg_cost = 0
        total_batch = int(mnist.train.num_examples / batch_size)
        # split the data into different batches and run
        for i in range(total_batch):
            batch_xs, batch_ys = mnist.train.next_batch(batch_size)
            # run training steps and cost both in session, which should be fed `x = batch_xs` and `y = batch_ys`
            _, cur_cost = ... ###### write your code here ######
            avg_cost += cur_cost / total_batch
        # show the average cost
        if (epoch+1) % display_step == 0:
            print("Epoch:", '%04d' % (epoch+1), "cost=", "{:.9f}".format(avg_cost))
    print("Optimization Finished!")
    # accuracy
    correct_prediction = tf.equal(tf.argmax(pred, 1), tf.argmax(y, 1))
    accuracy = tf.reduce_mean(tf.cast(correct_prediction, 'float'))
    print("Accuracy:", accuracy.eval({x: mnist.test.images, y: mnist.test.labels}))

Previous Chapter: Tensorflow Basics
Next Chapter: Summary and Tensorboard