Lecture 1: Introduction to Neural Networks

by Long Nguyen

This notebook is supplemental to Lecture 1 of the video series "Introduction to Neural Networks". These lectures are adapted from Michael Nielsen's free online book "Neural Networks and Deep Learning".

The video lecture can be access at here.

The MNIST Dataset

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from mnist_loader import load_data_wrapper
import numpy as np
import matplotlib.pyplot as plt
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training_data, validation_data, test_data = load_data_wrapper()

The variables training_data, validation_data and test_data above are each a list of (image, label) tuples where image and label are numpy arrays. We won't use the validation_data for this course.

How many images are in training_data?

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Unpack the first image into two variables img1, lb1. What is the shape of each image? What is the shape of each label?

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img1 = training_data[0][0]      # array of pixels
lb1 =  training_data[0][1]      # one-hot vector label            

# print out the shape of img1, lb1

# or equivalently, unpacking the 2-tuple (image, label)
# img1, lb1 = training_data[0]
(784, 1)
(10, 1)

Both the image and its label are rank 2 numpy arrays of shape (781,1) and (10,1), respectively. A label is a one-hot encoding of the digit.

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The function plot_mnist_digit below draw an MNIST image using the matplotlib library.

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def plot_image(image):
    """ Plot a single MNIST image. 
        Argument image is a numpy.ndarray of shape (784,1)
    image = image.reshape(28,28)
    fig, axes = plt.subplots()
    axes.matshow(image, cmap=plt.cm.binary)

Call the function to plot the first image.

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def plot_images(images):
    """ Plot a list of MNIST images.
        Argument images is a list of (image, label) tuples. 
    fig, axes = plt.subplots(nrows=1, ncols=len(images))
    for j, ax in enumerate(axes):
        ax.matshow(images[j][0].reshape(28,28), cmap = plt.cm.binary)

Call the function above to plot first 10 images.

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The sigmoid function is defined as $$\sigma(x)=\frac{1}{1+e^{-x}}.$$ Implement the sigmoid function. Hint: Use np.exp() for the exponential function.

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def sigmoid(x):
    """Returns the output of the sigmoid or logistic function."""
    return 1/(1+np.exp(-x))

Given a vector $\vec{x}\in\mathbb{R}^n$, the sigmoid function $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ can be extended to a vector-valued function $\sigma:\mathbb{R}^n\rightarrow\mathbb{R}^n$ by applying $\sigma$ elementwise. That is, if $$\vec{x}=\left[ \begin{array}{cccc} x_{1} \\ x_{2} \\ \vdots \\ x_{m} \end{array} \right]$$ then $$\sigma(\vec{x})=\left[ \begin{array}{cccc} \sigma(x_{1}) \\ \sigma(x_{2}) \\ \vdots \\ \sigma(x_{m}) \end{array} \right].$$

Similarly, $\sigma$ can be applied to a $m\times n$ matrix elementwise.

For example, if $$\vec{x}=\left[ \begin{array}{cccc} 1 \\ 2 \\ 3 \end{array} \right]$$ then $$\sigma(\vec{x})=\left[ \begin{array}{cccc} \sigma(1) \\ \sigma(2) \\ \sigma(3) \end{array} \right]\approx\left[ \begin{array}{cccc} 0.73 \\ 0.88 \\ 0.95 \end{array} \right]$$

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x = np.array([[1],[2],[3]])

Define $f_1(\vec{x})=W_1\vec{x}+\vec{b}_1$ and $f_2(\vec{x})=W_2\vec{x}+\vec{b}_2$ for some $W_1, W_2, \vec{b}_1, \text{and } \vec{b}_2.$

Consider the classifier or score function $f=\sigma\circ f_2\circ\sigma\circ f_1:\mathbb{R}^{784}\rightarrow\mathbb{R}^{10}.$ This is a two-layer neural network. The score function takes a flattened MNIST image of shape (784,1) and output a one-hot vector of shape (10,1). The class with the highest score is the label predicted by the classifier.

The training a neural network amounts to producing a set parameters $W_1, W_2, \vec{b}_1, \text{and } \vec{b}_1$ whose score function $f(x; W_1, W_2, \vec{b}_1, \vec{b}_2)$ can accurately classify unseen images.

To demonstrate an example of such a score function, let's load up a set of parameters that has already been trained. As we will see in a later lecture, the training involve examining 50,000 images 30 times.
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with open("parameters.npy", mode="rb") as r:
    parameters = np.load(r)
    W1, B1, W2, B2 = parameters

Implement the score function with these set of parameters.

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def f(x, W1, W2, B1, B2):
    """Return the output of the network if ``x`` is input image and
    W1, W2, B1 and B2 are the learnable parameters or weights. """
    Z1 = np.dot(W1, x) + B1
    A1 = sigmoid(Z1)
    Z2 = np.dot(W2, A1) + B2
    A2 = sigmoid(Z2)
    return A2

Apply your score function above to the first two images. Does it classify them correctly?

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def predict(images, W1, W2, B1, B2):
    """Return the predictions for a list of images given the parameters.
    predictions = []  # empty list
    for im in images:
        a = f(im[0], W1, W2, B1, B2)
        predictions.append(np.argmax(a)) # add prediction to predictions list
    return predictions
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[5, 0, 4, 1, 9, 2, 1, 3, 1, 4]
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