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    Linear Classifier (Logistic Regression)

    Introduction

    In this tutorial, we'll create a simple linear classifier in TensorFlow. We will implement this model for classifying images of hand-written digits from the so-called MNIST data-set. The structure of the network is presented in the following figure.

    linear_classifier.png

    Fig. 1- Sample Logistic Regression structure implemented for classifying MNIST digits

    You should be familiar with basic linear algebra, Machine Learning and classification. To specifically learn about the linear classifiers, read this article.

    Technically, in a linear model we will use the simplest function to predict the label $\mathbf{y_i}$ of the image $\mathbf{x_i}$. We'll do so by using a linear mapping like $f(\mathbf{x_i}, \mathbf{W}, \mathbf{b})=\mathbf{W}\mathbf{x_i}+\mathbf{b}$ where $\mathbf{W}$ and $\mathbf{b}$ are called weight matrix and bias vector respectively.

    0. Import the required libraries:

    We will start with importing the required Python libraries.

    In [1]:
    # 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 ($28\times28$ 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 ($28\times28$).

    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 [2]:
    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

    In this section, we'll write the function which automatically loads the MNIST data and returns it in our desired shape and format. If you wanna learn more about loading your data, you may read our How to Load Your Data in TensorFlow tutorial which explains all the available methods to load your own data; no matter how big it is.

    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). You can replace this function to use your own dataset.

    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 [3]:
    def load_data(mode='train'):
        """
        Function to (download and) load the MNIST data
        :param mode: train or test
        :return: images and the corresponding labels
        """
        from tensorflow.examples.tutorials.mnist import input_data
        mnist = input_data.read_data_sets("MNIST_data/", one_hot=True)
        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[0])
        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 [4]:
    # 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)))
     
     
    Successfully downloaded train-images-idx3-ubyte.gz 9912422 bytes.
    Extracting MNIST_data/train-images-idx3-ubyte.gz
    Successfully downloaded train-labels-idx1-ubyte.gz 28881 bytes.
    Extracting MNIST_data/train-labels-idx1-ubyte.gz
    Successfully downloaded t10k-images-idx3-ubyte.gz 1648877 bytes.
    Extracting MNIST_data/t10k-images-idx3-ubyte.gz
    Successfully downloaded t10k-labels-idx1-ubyte.gz 4542 bytes.
    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 [5]:
    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 $28\times28=784$ 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 [6]:
    y_valid[:5, :]
     
    Out[6]:
    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 [7]:
    # 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. Helper functions for creating the network

    3.1. 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 [8]:
    def weight_variable(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',
                               dtype=tf.float32,
                               shape=shape,
                               initializer=initer)
    
    def bias_variable(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',
                               dtype=tf.float32,
                               initializer=initial)
     
     

    4. Create the network graph

    Now that we have defined all the helped functions to create our model, we can create our network.

    4.1. 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 [9]:
    # 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')
     
     

    Placeholder 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.

     

    4.2. Create the model structure

    After creating the proper input, we have to pass it to our model. Since we have a linear classifier, we will have output_logits$=\mathbf{W}\times \mathbf{x} + \mathbf{b}$ and we will use tf.nn.softmax to normalize the output_logits between $0$ and $1$ as probabilities (predictions) of samples.

    In [10]:
    # Create weight matrix initialized randomely from N~(0, 0.01)
    W = weight_variable(shape=[img_size_flat, n_classes])
    
    # Create bias vector initialized as zero
    b = bias_variable(shape=[n_classes])
    
    output_logits = tf.matmul(x, W) + b
     
     

    4.3. Define the loss function, optimizer, accuracy, and predicted class

    After creating the network, we have to calculate the loss and optimize it. Also, to evaluate our model, we have to calculate the correct_prediction and accuracy. We will also define cls_prediction to visualize our results.### 4.3. Define the loss function, optimizer, accuracy, and predicted class

    In [11]:
    # 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')
    optimizer = tf.train.AdamOptimizer(learning_rate=learning_rate, name='Adam-op').minimize(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')
    
    # Model predictions
    cls_prediction = tf.argmax(output_logits, axis=1, name='predictions')
     
     
    WARNING:tensorflow:From <ipython-input-11-c1f11071639f>:2: softmax_cross_entropy_with_logits (from tensorflow.python.ops.nn_ops) is deprecated and will be removed in a future version.
    Instructions for updating:
    
    Future major versions of TensorFlow will allow gradients to flow
    into the labels input on backprop by default.
    
    See tf.nn.softmax_cross_entropy_with_logits_v2.
    
     
     

    4.4. Initialize all variables

    As explained in the Tensor Types tutorial, we have to invoke a variable initializer operation to initialize all variables.

    In [12]:
    # Creating the op for initializing all variables
    init = tf.global_variables_initializer()
     
     

    5. Train

    After creating the graph, it is time to train our model. To train the model, As explained in the Graph_and_Session tutorial, we have to create a session and run the graph in our session.

    In [13]:
    # Create an interactive session (to keep the session in the other cells)
    sess = tf.InteractiveSession()
    # Initialize all variables
    sess.run(init)
    # 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))
        # Randomly shuffle the training data at the beginning of each epoch 
        x_train, y_train = randomize(x_train, y_train)
        for iteration in range(num_tr_iter):
            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=2.22,    Training Accuracy=24.0%
    iter 100:     Loss=0.78,    Training Accuracy=84.0%
    iter 200:     Loss=0.53,    Training Accuracy=89.0%
    iter 300:     Loss=0.42,    Training Accuracy=89.0%
    iter 400:     Loss=0.36,    Training Accuracy=90.0%
    iter 500:     Loss=0.33,    Training Accuracy=92.0%
    ---------------------------------------------------------
    Epoch: 1, validation loss: 0.40, validation accuracy: 88.7%
    ---------------------------------------------------------
    Training epoch: 2
    iter   0:     Loss=0.25,    Training Accuracy=95.0%
    iter 100:     Loss=0.35,    Training Accuracy=88.0%
    iter 200:     Loss=0.31,    Training Accuracy=91.0%
    iter 300:     Loss=0.38,    Training Accuracy=87.0%
    iter 400:     Loss=0.37,    Training Accuracy=91.0%
    iter 500:     Loss=0.35,    Training Accuracy=91.0%
    ---------------------------------------------------------
    Epoch: 2, validation loss: 0.35, validation accuracy: 90.1%
    ---------------------------------------------------------
    Training epoch: 3
    iter   0:     Loss=0.43,    Training Accuracy=89.0%
    iter 100:     Loss=0.29,    Training Accuracy=92.0%
    iter 200:     Loss=0.30,    Training Accuracy=89.0%
    iter 300:     Loss=0.30,    Training Accuracy=94.0%
    iter 400:     Loss=0.24,    Training Accuracy=92.0%
    iter 500:     Loss=0.21,    Training Accuracy=94.0%
    ---------------------------------------------------------
    Epoch: 3, validation loss: 0.32, validation accuracy: 90.5%
    ---------------------------------------------------------
    Training epoch: 4
    iter   0:     Loss=0.42,    Training Accuracy=88.0%
    iter 100:     Loss=0.33,    Training Accuracy=91.0%
    iter 200:     Loss=0.27,    Training Accuracy=92.0%
    iter 300:     Loss=0.20,    Training Accuracy=94.0%
    iter 400:     Loss=0.17,    Training Accuracy=95.0%
    iter 500:     Loss=0.30,    Training Accuracy=90.0%
    ---------------------------------------------------------
    Epoch: 4, validation loss: 0.31, validation accuracy: 90.7%
    ---------------------------------------------------------
    Training epoch: 5
    iter   0:     Loss=0.25,    Training Accuracy=94.0%
    iter 100:     Loss=0.25,    Training Accuracy=94.0%
    iter 200:     Loss=0.32,    Training Accuracy=94.0%
    iter 300:     Loss=0.27,    Training Accuracy=92.0%
    iter 400:     Loss=0.30,    Training Accuracy=91.0%
    iter 500:     Loss=0.32,    Training Accuracy=90.0%
    ---------------------------------------------------------
    Epoch: 5, validation loss: 0.30, validation accuracy: 91.3%
    ---------------------------------------------------------
    Training epoch: 6
    iter   0:     Loss=0.28,    Training Accuracy=92.0%
    iter 100:     Loss=0.40,    Training Accuracy=89.0%
    iter 200:     Loss=0.25,    Training Accuracy=92.0%
    iter 300:     Loss=0.29,    Training Accuracy=91.0%
    iter 400:     Loss=0.31,    Training Accuracy=92.0%
    iter 500:     Loss=0.23,    Training Accuracy=91.0%
    ---------------------------------------------------------
    Epoch: 6, validation loss: 0.29, validation accuracy: 90.9%
    ---------------------------------------------------------
    Training epoch: 7
    iter   0:     Loss=0.22,    Training Accuracy=97.0%
    iter 100:     Loss=0.16,    Training Accuracy=95.0%
    iter 200:     Loss=0.20,    Training Accuracy=95.0%
    iter 300:     Loss=0.33,    Training Accuracy=88.0%
    iter 400:     Loss=0.26,    Training Accuracy=91.0%
    iter 500:     Loss=0.23,    Training Accuracy=93.0%
    ---------------------------------------------------------
    Epoch: 7, validation loss: 0.29, validation accuracy: 91.5%
    ---------------------------------------------------------
    Training epoch: 8
    iter   0:     Loss=0.32,    Training Accuracy=92.0%
    iter 100:     Loss=0.28,    Training Accuracy=93.0%
    iter 200:     Loss=0.26,    Training Accuracy=94.0%
    iter 300:     Loss=0.21,    Training Accuracy=94.0%
    iter 400:     Loss=0.27,    Training Accuracy=93.0%
    iter 500:     Loss=0.24,    Training Accuracy=94.0%
    ---------------------------------------------------------
    Epoch: 8, validation loss: 0.29, validation accuracy: 91.8%
    ---------------------------------------------------------
    Training epoch: 9
    iter   0:     Loss=0.24,    Training Accuracy=93.0%
    iter 100:     Loss=0.40,    Training Accuracy=90.0%
    iter 200:     Loss=0.24,    Training Accuracy=94.0%
    iter 300:     Loss=0.27,    Training Accuracy=92.0%
    iter 400:     Loss=0.25,    Training Accuracy=93.0%
    iter 500:     Loss=0.11,    Training Accuracy=98.0%
    ---------------------------------------------------------
    Epoch: 9, validation loss: 0.29, validation accuracy: 91.3%
    ---------------------------------------------------------
    Training epoch: 10
    iter   0:     Loss=0.19,    Training Accuracy=95.0%
    iter 100:     Loss=0.29,    Training Accuracy=95.0%
    iter 200:     Loss=0.28,    Training Accuracy=86.0%
    iter 300:     Loss=0.47,    Training Accuracy=87.0%
    iter 400:     Loss=0.16,    Training Accuracy=98.0%
    iter 500:     Loss=0.19,    Training Accuracy=94.0%
    ---------------------------------------------------------
    Epoch: 10, validation loss: 0.28, validation accuracy: 91.4%
    ---------------------------------------------------------
     
     

    6. Test

    After the training is done, we have to test our model to see how good it performs on a new dataset. There are multiple approaches to for this purpose. We will use two different methods.

    6.1. Accuracy

    One way that we can evaluate our model is reporting the accuracy on the test set.

    In [14]:
    # Test the network after training
    # Accuracy
    x_test, y_test = load_data(mode='test')
    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('---------------------------------------------------------')
     
     
    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: 0.27, test accuracy: 91.7%
    ---------------------------------------------------------
     
     

    6.2. plot some results

    Another way to evaluate the model is to visualize the input and the model results and compare them with the true label of the input. This is advantages in numerous ways. For example, even if you get a decent accuracy, when you plot the results, you might see all the samples have been classified in one class. Another example is when you plot, you can have a rough idea on which examples your model failed. Let's define the helper functions to plot some correct and missclassified examples.

    6.2.1 Helper functions for plotting the results

    In [15]:
    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))
        fig.subplots_adjust(hspace=0.3, wspace=0.3)
        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)
     
     

    6.2.2 Visualize correct and missclassified examples

    In [16]:
    # 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()
     
     
     
     

    After we finished, we have to close the session to free the memory. We could have also used:

    with tf.Session as sess:
        ...
     

    Please check our Graph_and_Session tutorial if you do not know the differences between these two implementations.

    In [17]:
    sess.close()
     
     

    Thanks for reading! If you have any question or doubt, feel free to leave a comment in our website.

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