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## Sections

### Logistic Regression using TensorFlow

This tutorial is about training a logistic regression by TensorFlow for binary classification.

### Introduction

In Linear Regression using TensorFlow post we described how to predict continuous-valued parameters by linearly modeling the system. What if the objective is to decide between two choices? The answer is simple: we are dealing with a classification problem. In this tutorial, the objective to decide whether the input image is digit "0" or digit "1" using Logistic Regression. In another word, whether it is digit "1" or not! The full source code is available in the associated GitHub repository.

### Dataset

The dataset that we work on that in this tutorial is the MNIST dataset. The main dataset consists of 55000 training and 10000 test images. The images are 28x28x1 which each of them represent a hand-written digit from 0 to 9. We create feature vectors of size 784 of each image. We only use 0 and 1 images for our setting.

### Logistic Regression

In linear regression the effort is to predict the outcome continuous value using the linear function of $y=W^{T}x$. On the other hand, in logistic regression we are determined to predict a binary label as $y\in\{0,1\}$ in which we use a different prediction process as opposed to linear regression. In logistic regression, the predicted output is the probability that the input sample belongs to a targeted class which is digit "1" in our case. In a binary-classification problem, obviously if the $P(x\in\{target\_class\})$ = M, then $P(x\in\{non\_target\_class\}) = 1 - M$. So the hypothesis can be created as follows:

$$P(y=1|x)=h_{W}(x)={{1}\over{1+exp(-W^{T}x)}}=Sigmoid(W^{T}x) \ \ \ (1)$$ $$P(y=0|x)=1 - P(y=1|x) = 1 - h_{W}(x) \ \ \ (2)$$

In the above equations, Sigmoid function maps the predicted output into probability space in which the values are in the range of $[0,1]$. The main objective is to find the model using which when the input sample is "1" the output becomes a high probability and becomes small otherwise. The important objective is to design the appropriate cost function to minimize the loss when the output is desired and vice versa. The cost function for a set of data such as $(x^{i},y^{i})$ can be defined as below:

$$Loss(W) = \sum_{i}{y^{(i)}log{1\over{h_{W}(x^{i})}}+(1-y^{(i)})log{1\over{1-h_{W}(x^{i})}}}$$

As it can be seen from the above equation, the loss function consists of two term and in each sample only one of them is non-zero considering the binary labels.

Up to now, we defined the formulation and optimization function of the logistic regression. In the next part, we show how to do it in code using mini-batch optimization.

### Description of the Overall Process

At first, we process the dataset and extract only "0" and "1" digits. The code implemented for logistic regression is heavily inspired by our Train a Convolutional Neural Network as a Classifier post. We refer to the aforementioned post for having a better understanding of the implementation details. In this tutorial, we only explain how we process dataset and how to implement logistic regression and the rest is clear from the CNN classifier post that we referred earlier.

### How to Do It in Code?

In this part, we explain how to extract desired samples from dataset and to implement logistic regression using Softmax.

## Process Dataset

At first, we need to extract "0" and "1" digits from MNIST dataset:

from tensorflow.examples.tutorials.mnist import input_data
mnist = input_data.read_data_sets("MNIST_data/", reshape=True, one_hot=False)

########################
### Data Processing ####
########################
# Organize the data and feed it to associated dictionaries.
data={}

data['train/image'] = mnist.train.images
data['train/label'] = mnist.train.labels
data['test/image'] = mnist.test.images
data['test/label'] = mnist.test.labels

# Get only the samples with zero and one label for training.
index_list_train = []
for sample_index in range(data['train/label'].shape):
label = data['train/label'][sample_index]
if label == 1 or label == 0:
index_list_train.append(sample_index)

# Reform the train data structure.
data['train/image'] = mnist.train.images[index_list_train]
data['train/label'] = mnist.train.labels[index_list_train]

# Get only the samples with zero and one label for test set.
index_list_test = []
for sample_index in range(data['test/label'].shape):
label = data['test/label'][sample_index]
if label == 1 or label == 0:
index_list_test.append(sample_index)

# Reform the test data structure.
data['test/image'] = mnist.test.images[index_list_test]
data['test/label'] = mnist.test.labels[index_list_test]

The code looks to be verbose but it's very simple actually. All we want is implemented in lines 28-32 in which the desired data samples are extracted. Next, we have to dig into logistic regression architecture.

## Logistic Regression Implementation

The logistic regression structure is simply feeding-forwarding the input features through a fully-connected layer in which the last layer only has two classes. The fully-connected architecture can be defined as below:

###############################################
########### Defining place holders ############
###############################################
image_place = tf.placeholder(tf.float32, shape=([None, num_features]), name='image')
label_place = tf.placeholder(tf.int32, shape=([None,]), name='gt')
label_one_hot = tf.one_hot(label_place, depth=FLAGS.num_classes, axis=-1)
dropout_param = tf.placeholder(tf.float32)

##################################################
########### Model + Loss + Accuracy ##############
##################################################
# A simple fully connected with two class and a Softmax is equivalent to Logistic Regression.
logits = tf.contrib.layers.fully_connected(inputs=image_place, num_outputs = FLAGS.num_classes, scope='fc')

The first few lines are defining place-holders in order to put the desired values on the graph. Please refer to this post for further details. The desired loss function can easily be implemented using TensorFlow using the following script:

# Define loss
with tf.name_scope('loss'):
loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(logits=logits, labels=label_one_hot))

# Accuracy
with tf.name_scope('accuracy'):
# Evaluate the model
correct_pred = tf.equal(tf.argmax(logits, 1), tf.argmax(label_one_hot, 1))

# Accuracy calculation
accuracy = tf.reduce_mean(tf.cast(correct_pred, tf.float32))

The tf.nn.softmax_cross_entropy_with_logits function does the work. It optimizes the previously defined cost function with a subtle difference. It generates two inputs in which even if the sample is digit "0", the correspondent probability will be high. So tf.nn.softmax_cross_entropy_with_logits function, for each class predict a probability and inherently on its own, makes the decision.

### Summary

In this tutorial, we described logistic regression and represented how to implement it in code. Instead of making a decision based on the output probability based on a targeted class, we extended the problem to a two class problem in which for each class we predict the probability. In the future posts, we will extend this problem to multi-class problem and we show it can be done with the similar approach.

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