FYS-STK4155/doc/pub/week43/ipynb/week43.ipynb

Week 43: Deep Learning: Constructing a Neural Network code and solving differential equations

Morten Hjorth-Jensen, Department of Physics, University of Oslo and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University

Date: October 21, 2024

Plans for week 43

Material for the lecture on Monday October 21, 2024.

• Building our own Feed-forward Neural Network with intro to Tensorflow

• Solving differential equations with Neural Networks

• Video of lecture at https://youtu.be/vkBNTn-MLqs

• Video os second part, solving differential equations with neural networks at https://youtu.be/2N8To65I2wQ

• Whiteboard notes on solving differential equations at https://github.com/CompPhysics/MachineLearning/blob/master/doc/HandWrittenNotes/2024/NotesOct21.pdf

Exercises and lab session week 43

Lab sessions on Tuesday and Wednesday.

• Exercise on writing your own neural network code

• The exercises this week will be continued next week as well

• Discussion of project 2

Mathematics of deep learning

Two recent books online.

1. The Modern Mathematics of Deep Learning, by Julius Berner, Philipp Grohs, Gitta Kutyniok, Philipp Petersen at https://arxiv.org/abs/2105.04026, published as Mathematical Aspects of Deep Learning, pp. 1-111. Cambridge University Press, 2022

2. Mathematical Introduction to Deep Learning: Methods, Implementations, and Theory, Arnulf Jentzen, Benno Kuckuck, Philippe von Wurstemberger at https://doi.org/10.48550/arXiv.2310.20360

Reminder on books with hands-on material and codes

• Sebastian Rashcka et al, Machine learning with Scikit-Learn and PyTorch at https://sebastianraschka.com/blog/2022/ml-pytorch-book.html

Reading recommendations

1. Rashkca et al., chapter 11, jupyter-notebook sent separately, from GitHub site at https://github.com/rasbt/machine-learning-book. See also chapters 12 and 13 on using Pytorch to make a Neural network code.

2. Goodfellow et al, chapter 6 and 7 contain most of the neural network background.

Using Automatic differentiation

In our discussions of ordinary differential equations and neural network codes we will also study the usage of Autograd, see for example https://www.youtube.com/watch?v=fRf4l5qaX1M&ab_channel=AlexSmola in computing gradients for deep learning. For the documentation of Autograd and examples see the lectures slides from week 39 and the Autograd documentation at https://github.com/HIPS/autograd.

Back propagation and automatic differentiation

For more details on the back propagation algorithm and automatic differentiation see

1. https://www.jmlr.org/papers/volume18/17-468/17-468.pdf

2. https://deepimaging.github.io/lectures/lecture_11_Backpropagation.pdf

3. Slides 12-44 at http://cs231n.stanford.edu/slides/2017/cs231n_2017_lecture4.pdf

Lecture Monday October 21

Setting up the back propagation algorithm and algorithm for a feed forward NN, initalizations

This is a reminder from where we ended last week.

The architecture (our model).

1. Set up your inputs and outputs (scalars, vectors, matrices or higher-order arrays)

2. Define the number of hidden layers and hidden nodes

3. Define activation functions for hidden layers and output layers

4. Define optimizer (plan learning rate, momentum, ADAgrad, RMSprop, ADAM etc) and array of initial learning rates

5. Define cost function and possible regularization terms with hyperparameters

6. Initialize weights and biases

7. Fix number of iterations for the feed forward part and back propagation part

Setting up the back propagation algorithm, part 1

Let us write this out in the form of an algorithm.

First, we set up the input data \boldsymbol{x} and the activations \boldsymbol{z}_1 of the input layer and compute the activation function and the pertinent outputs \boldsymbol{a}^1.

Secondly, we perform then the feed forward till we reach the output layer and compute all \boldsymbol{z}_l of the input layer and compute the activation function and the pertinent outputs \boldsymbol{a}^l for l=1,2,3,\dots,L.

Notation: The first hidden layer has l=1 as label and the final output layer has l=L.

Setting up the back propagation algorithm, part 2

Thereafter we compute the ouput error \boldsymbol{\delta}^L by computing all

\delta_j^L = \sigma'(z_j^L)\frac{\partial {\cal C}}{\partial (a_j^L)}.

Then we compute the back propagate error for each l=L-1,L-2,\dots,1 as

\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}\sigma'(z_j^l).

Setting up the Back propagation algorithm, part 3

Finally, we update the weights and the biases using gradient descent for each l=L-1,L-2,\dots,1 (the first hidden layer) and update the weights and biases according to the rules

w_{ij}^l\leftarrow  = w_{ij}^l- \eta \delta_j^la_i^{l-1},

b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,

with \eta being the learning rate.

Updating the gradients

With the back propagate error for each l=L-1,L-2,\dots,1 as

\delta_j^l = \sum_k \delta_k^{l+1}w_{kj}^{l+1}\sigma'(z_j^l),

we update the weights and the biases using gradient descent for each l=L-1,L-2,\dots,1 and update the weights and biases according to the rules

w_{ij}^l\leftarrow  = w_{ij}^l- \eta \delta_j^la_i^{l-1},

b_j^l \leftarrow b_j^l-\eta \frac{\partial {\cal C}}{\partial b_j^l}=b_j^l-\eta \delta_j^l,

Activation functions

A property that characterizes a neural network, other than its connectivity, is the choice of activation function(s). The following restrictions are imposed on an activation function for an FFNN to fulfill the universal approximation theorem

• Non-constant

• Bounded

• Monotonically-increasing

• Continuous

Activation functions, examples

Typical examples are the logistic Sigmoid

\sigma(x) = \frac{1}{1 + e^{-x}},

and the hyperbolic tangent function

\sigma(x) = \tanh(x)

The RELU function family

The ReLU activation function suffers from a problem known as the dying ReLUs: during training, some neurons effectively die, meaning they stop outputting anything other than 0.

In some cases, you may find that half of your network’s neurons are dead, especially if you used a large learning rate. During training, if a neuron’s weights get updated such that the weighted sum of the neuron’s inputs is negative, it will start outputting 0. When this happen, the neuron is unlikely to come back to life since the gradient of the ReLU function is 0 when its input is negative.

ELU function

To solve this problem, nowadays practitioners use a variant of the ReLU function, such as the leaky ReLU discussed above or the so-called exponential linear unit (ELU) function

ELU(z) = \left\{\begin{array}{cc} \alpha\left( \exp{(z)}-1\right) & z < 0,\\  z & z \ge 0.\end{array}\right.

Which activation function should we use?

In general it seems that the ELU activation function is better than the leaky ReLU function (and its variants), which is better than ReLU. ReLU performs better than \tanh which in turn performs better than the logistic function.

If runtime performance is an issue, then you may opt for the leaky ReLU function over the ELU function If you don’t want to tweak yet another hyperparameter, you may just use the default \alpha of 0.01 for the leaky ReLU, and 1 for ELU. If you have spare time and computing power, you can use cross-validation or bootstrap to evaluate other activation functions.

More on activation functions, output layers

In most cases you can use the ReLU activation function in the hidden layers (or one of its variants).

It is a bit faster to compute than other activation functions, and the gradient descent optimization does in general not get stuck.

For the output layer:

• For classification the softmax activation function is generally a good choice for classification tasks (when the classes are mutually exclusive).

• For regression tasks, you can simply use no activation function at all.

Setting up a Multi-layer perceptron model for classification

We are now gong to develop an example based on the MNIST data base. This is a classification problem and we need to use our cross-entropy function we discussed in connection with logistic regression. The cross-entropy defines our cost function for the classificaton problems with neural networks.

In binary classification with two classes (0, 1) we define the logistic/sigmoid function as the probability that a particular input is in class 0 or 1. This is possible because the logistic function takes any input from the real numbers and inputs a number between 0 and 1, and can therefore be interpreted as a probability. It also has other nice properties, such as a derivative that is simple to calculate.

For an input \boldsymbol{a} from the hidden layer, the probability that the input \boldsymbol{x} is in class 0 or 1 is just. We let \theta represent the unknown weights and biases to be adjusted by our equations). The variable x represents our activation values z. We have

P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) = \frac{1}{1 + \exp{(- \boldsymbol{x}})} ,

and

P(y = 1 \mid \boldsymbol{x}, \boldsymbol{\theta}) = 1 - P(y = 0 \mid \boldsymbol{x}, \boldsymbol{\theta}) ,

where y \in \{0, 1\} and \boldsymbol{\theta} represents the weights and biases of our network.

Defining the cost function

Our cost function is given as (see the Logistic regression lectures)

\mathcal{C}(\boldsymbol{\theta}) = - \ln P(\mathcal{D} \mid \boldsymbol{\theta}) = - \sum_{i=1}^n
y_i \ln[P(y_i = 0)] + (1 - y_i) \ln [1 - P(y_i = 0)] = \sum_{i=1}^n \mathcal{L}_i(\boldsymbol{\theta}) .

This last equality means that we can interpret our cost function as a sum over the loss function for each point in the dataset \mathcal{L}_i(\boldsymbol{\theta}).
The negative sign is just so that we can think about our algorithm as minimizing a positive number, rather than maximizing a negative number.

In multiclass classification it is common to treat each integer label as a so called one-hot vector:

y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) , and

y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,

i.e. a binary bit string of length C, where C = 10 is the number of classes in the MNIST dataset (numbers from 0 to 9)..

If \boldsymbol{x}_i is the $i$-th input (image), y_{ic} refers to the $c$-th component of the $i$-th output vector \boldsymbol{y}_i.
The probability of \boldsymbol{x}_i being in class c will be given by the softmax function:

P(y_{ic} = 1 \mid \boldsymbol{x}_i, \boldsymbol{\theta}) = \frac{\exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_c)}}
{\sum_{c'=0}^{C-1} \exp{((\boldsymbol{a}_i^{hidden})^T \boldsymbol{w}_{c'})}} ,

which reduces to the logistic function in the binary case.
The likelihood of this $C$-class classifier is now given as:

P(\mathcal{D} \mid \boldsymbol{\theta}) = \prod_{i=1}^n \prod_{c=0}^{C-1} [P(y_{ic} = 1)]^{y_{ic}} .

Again we take the negative log-likelihood to define our cost function:

\mathcal{C}(\boldsymbol{\theta}) = - \log{P(\mathcal{D} \mid \boldsymbol{\theta})}.

See the logistic regression lectures for a full definition of the cost function.

The back propagation equations need now only a small change, namely the definition of a new cost function. We are thus ready to use the same equations as before!

Example: binary classification problem

As an example of the above, relevant for project 2 as well, let us consider a binary class. As discussed in our logistic regression lectures, we defined a cost function in terms of the parameters \beta as

\mathcal{C}(\boldsymbol{\beta}) = - \sum_{i=1}^n \left(y_i\log{p(y_i \vert x_i,\boldsymbol{\beta})}+(1-y_i)\log{1-p(y_i \vert x_i,\boldsymbol{\beta})}\right),

where we had defined the logistic (sigmoid) function

p(y_i =1\vert x_i,\boldsymbol{\beta})=\frac{\exp{(\beta_0+\beta_1 x_i)}}{1+\exp{(\beta_0+\beta_1 x_i)}},

and

p(y_i =0\vert x_i,\boldsymbol{\beta})=1-p(y_i =1\vert x_i,\boldsymbol{\beta}).

The parameters \boldsymbol{\beta} were defined using a minimization method like gradient descent or Newton-Raphson's method.

Now we replace x_i with the activation z_i^l for a given layer l and the outputs as y_i=a_i^l=f(z_i^l), with z_i^l now being a function of the weights w_{ij}^l and biases b_i^l. We have then

a_i^l = y_i = \frac{\exp{(z_i^l)}}{1+\exp{(z_i^l)}},

with

z_i^l = \sum_{j}w_{ij}^l a_j^{l-1}+b_i^l,

where the superscript l-1 indicates that these are the outputs from layer l-1. Our cost function at the final layer l=L is now

\mathcal{C}(\boldsymbol{W}) = - \sum_{i=1}^n \left(t_i\log{a_i^L}+(1-t_i)\log{(1-a_i^L)}\right),

where we have defined the targets t_i. The derivatives of the cost function with respect to the output a_i^L are then easily calculated and we get

\frac{\partial \mathcal{C}(\boldsymbol{W})}{\partial a_i^L} = \frac{a_i^L-t_i}{a_i^L(1-a_i^L)}.

In case we use another activation function than the logistic one, we need to evaluate other derivatives.

The Softmax function

In case we employ the more general case given by the Softmax equation, we need to evaluate the derivative of the activation function with respect to the activation z_i^l, that is we need

\frac{\partial f(z_i^l)}{\partial w_{jk}^l} =
\frac{\partial f(z_i^l)}{\partial z_j^l} \frac{\partial z_j^l}{\partial w_{jk}^l}= \frac{\partial f(z_i^l)}{\partial z_j^l}a_k^{l-1}.

For the Softmax function we have

f(z_i^l) = \frac{\exp{(z_i^l)}}{\sum_{m=1}^K\exp{(z_m^l)}}.

Its derivative with respect to z_j^l gives

\frac{\partial f(z_i^l)}{\partial z_j^l}= f(z_i^l)\left(\delta_{ij}-f(z_j^l)\right),

which in case of the simply binary model reduces to having i=j.

Developing a code for doing neural networks with back propagation

One can identify a set of key steps when using neural networks to solve supervised learning problems:

1. Collect and pre-process data

2. Define model and architecture

3. Choose cost function and optimizer

4. Train the model

5. Evaluate model performance on test data

6. Adjust hyperparameters (if necessary, network architecture)

Collect and pre-process data

Here we will be using the MNIST dataset, which is readily available through the scikit-learn package. You may also find it for example here.
The MNIST (Modified National Institute of Standards and Technology) database is a large database of handwritten digits that is commonly used for training various image processing systems.
The MNIST dataset consists of 70 000 images of size 28\times 28 pixels, each labeled from 0 to 9.
The scikit-learn dataset we will use consists of a selection of 1797 images of size 8\times 8 collected and processed from this database.

To feed data into a feed-forward neural network we need to represent the inputs as a design/feature matrix X = (n_{inputs}, n_{features}). Each row represents an input, in this case a handwritten digit, and each column represents a feature, in this case a pixel. The correct answers, also known as labels or targets are represented as a 1D array of integers Y = (n_{inputs}) = (5, 3, 1, 8,...).

As an example, say we want to build a neural network using supervised learning to predict Body-Mass Index (BMI) from measurements of height (in m)
and weight (in kg). If we have measurements of 5 people the design/feature matrix could be for example:

$$ X = \begin{bmatrix} 1.85 & 81\ 1.71 & 65\ 1.95 & 103\ 1.55 & 42\ 1.63 & 56 \end{bmatrix} ,$$

and the targets would be:

Y = (23.7, 22.2, 27.1, 17.5, 21.1)

Since each input image is a 2D matrix, we need to flatten the image (i.e. "unravel" the 2D matrix into a 1D array) to turn the data into a design/feature matrix. This means we lose all spatial information in the image, such as locality and translational invariance. More complicated architectures such as Convolutional Neural Networks can take advantage of such information, and are most commonly applied when analyzing images.

In [1]:

%matplotlib inline

# import necessary packages
import numpy as np
import matplotlib.pyplot as plt
from sklearn import datasets


# ensure the same random numbers appear every time
np.random.seed(0)

# display images in notebook
%matplotlib inline
plt.rcParams['figure.figsize'] = (12,12)


# download MNIST dataset
digits = datasets.load_digits()

# define inputs and labels
inputs = digits.images
labels = digits.target

print("inputs = (n_inputs, pixel_width, pixel_height) = " + str(inputs.shape))
print("labels = (n_inputs) = " + str(labels.shape))


# flatten the image
# the value -1 means dimension is inferred from the remaining dimensions: 8x8 = 64
n_inputs = len(inputs)
inputs = inputs.reshape(n_inputs, -1)
print("X = (n_inputs, n_features) = " + str(inputs.shape))


# choose some random images to display
indices = np.arange(n_inputs)
random_indices = np.random.choice(indices, size=5)

for i, image in enumerate(digits.images[random_indices]):
    plt.subplot(1, 5, i+1)
    plt.axis('off')
    plt.imshow(image, cmap=plt.cm.gray_r, interpolation='nearest')
    plt.title("Label: %d" % digits.target[random_indices[i]])
plt.show()

Train and test datasets

Performing analysis before partitioning the dataset is a major error, that can lead to incorrect conclusions.

We will reserve 80 \% of our dataset for training and 20 \% for testing.

It is important that the train and test datasets are drawn randomly from our dataset, to ensure no bias in the sampling.
Say you are taking measurements of weather data to predict the weather in the coming 5 days. You don't want to train your model on measurements taken from the hours 00.00 to 12.00, and then test it on data collected from 12.00 to 24.00.

In [2]:

from sklearn.model_selection import train_test_split

# one-liner from scikit-learn library
train_size = 0.8
test_size = 1 - train_size
X_train, X_test, Y_train, Y_test = train_test_split(inputs, labels, train_size=train_size,
                                                    test_size=test_size)

# equivalently in numpy
def train_test_split_numpy(inputs, labels, train_size, test_size):
    n_inputs = len(inputs)
    inputs_shuffled = inputs.copy()
    labels_shuffled = labels.copy()
    
    np.random.shuffle(inputs_shuffled)
    np.random.shuffle(labels_shuffled)
    
    train_end = int(n_inputs*train_size)
    X_train, X_test = inputs_shuffled[:train_end], inputs_shuffled[train_end:]
    Y_train, Y_test = labels_shuffled[:train_end], labels_shuffled[train_end:]
    
    return X_train, X_test, Y_train, Y_test

#X_train, X_test, Y_train, Y_test = train_test_split_numpy(inputs, labels, train_size, test_size)

print("Number of training images: " + str(len(X_train)))
print("Number of test images: " + str(len(X_test)))

Define model and architecture

Our simple feed-forward neural network will consist of an input layer, a single hidden layer and an output layer. The activation y of each neuron is a weighted sum of inputs, passed through an activation function. In case of the simple perceptron model we have

z = \sum_{i=1}^n w_i a_i , y = f(z) ,

where f is the activation function, a_i represents input from neuron i in the preceding layer and w_i is the weight to input i.
The activation of the neurons in the input layer is just the features (e.g. a pixel value).

The simplest activation function for a neuron is the Heaviside function:

$$ f(z) = \begin{cases} 1, & z > 0\ 0, & \text{otherwise} \end{cases}

A feed-forward neural network with this activation is known as a *perceptron*.  
For a binary classifier (i.e. two classes, 0 or 1, dog or not-dog) we can also use this in our output layer.  
This activation can be generalized to $k$ classes (using e.g. the *one-against-all* strategy), 
and we call these architectures *multiclass perceptrons*.  

However, it is now common to use the terms Single Layer Perceptron (SLP) (1 hidden layer) and  
Multilayer Perceptron (MLP) (2 or more hidden layers) to refer to feed-forward neural networks with any activation function.  

Typical choices for activation functions include the sigmoid function, hyperbolic tangent, and Rectified Linear Unit (ReLU).  
We will be using the sigmoid function $\sigma(x)$:  

$$ f(x) = \sigma(x) = \frac{1}{1 + e^{-x}} ,$$

which is inspired by probability theory (see logistic regression) and was most commonly used until about 2011. See the discussion below concerning other activation functions.

Layers

• Input

Since each input image has 8x8 = 64 pixels or features, we have an input layer of 64 neurons.

• Hidden layer

We will use 50 neurons in the hidden layer receiving input from the neurons in the input layer.
Since each neuron in the hidden layer is connected to the 64 inputs we have 64x50 = 3200 weights to the hidden layer.

• Output

If we were building a binary classifier, it would be sufficient with a single neuron in the output layer, which could output 0 or 1 according to the Heaviside function. This would be an example of a hard classifier, meaning it outputs the class of the input directly. However, if we are dealing with noisy data it is often beneficial to use a soft classifier, which outputs the probability of being in class 0 or 1.

For a soft binary classifier, we could use a single neuron and interpret the output as either being the probability of being in class 0 or the probability of being in class 1. Alternatively we could use 2 neurons, and interpret each neuron as the probability of being in each class.

Since we are doing multiclass classification, with 10 categories, it is natural to use 10 neurons in the output layer. We number the neurons j = 0,1,...,9. The activation of each output neuron j will be according to the softmax function:

$$ P(\text{class $j$} \mid \text{input $\boldsymbol{a}$}) = \frac{\exp{(\boldsymbol{a}^T \boldsymbol{w}j)}} {\sum{c=0}^{9} \exp{(\boldsymbol{a}^T \boldsymbol{w}_c)}} ,$$

i.e. each neuron j outputs the probability of being in class j given an input from the hidden layer \boldsymbol{a}, with \boldsymbol{w}_j the weights of neuron j to the inputs.
The denominator is a normalization factor to ensure the outputs (probabilities) sum up to 1.
The exponent is just the weighted sum of inputs as before:

z_j = \sum_{i=1}^n w_ {ij} a_i+b_j.

Since each neuron in the output layer is connected to the 50 inputs from the hidden layer we have 50x10 = 500 weights to the output layer.

Weights and biases

Typically weights are initialized with small values distributed around zero, drawn from a uniform or normal distribution. Setting all weights to zero means all neurons give the same output, making the network useless.

Adding a bias value to the weighted sum of inputs allows the neural network to represent a greater range of values. Without it, any input with the value 0 will be mapped to zero (before being passed through the activation). The bias unit has an output of 1, and a weight to each neuron j, b_j:

z_j = \sum_{i=1}^n w_ {ij} a_i + b_j.

The bias weights \boldsymbol{b} are often initialized to zero, but a small value like 0.01 ensures all neurons have some output which can be backpropagated in the first training cycle.

In [3]:

# building our neural network

n_inputs, n_features = X_train.shape
n_hidden_neurons = 50
n_categories = 10

# we make the weights normally distributed using numpy.random.randn

# weights and bias in the hidden layer
hidden_weights = np.random.randn(n_features, n_hidden_neurons)
hidden_bias = np.zeros(n_hidden_neurons) + 0.01

# weights and bias in the output layer
output_weights = np.random.randn(n_hidden_neurons, n_categories)
output_bias = np.zeros(n_categories) + 0.01

Feed-forward pass

Denote F the number of features, H the number of hidden neurons and C the number of categories.
For each input image we calculate a weighted sum of input features (pixel values) to each neuron j in the hidden layer l:

z_{j}^{l} = \sum_{i=1}^{F} w_{ij}^{l} x_i + b_{j}^{l},

this is then passed through our activation function

a_{j}^{l} = f(z_{j}^{l}) .

We calculate a weighted sum of inputs (activations in the hidden layer) to each neuron j in the output layer:

z_{j}^{L} = \sum_{i=1}^{H} w_{ij}^{L} a_{i}^{l} + b_{j}^{L}.

Finally we calculate the output of neuron j in the output layer using the softmax function:

$$ a_{j}^{L} = \frac{\exp{(z_j^{L})}} {\sum_{c=0}^{C-1} \exp{(z_c^{L})}} .$$

Matrix multiplications

Since our data has the dimensions X = (n_{inputs}, n_{features}) and our weights to the hidden layer have the dimensions
W_{hidden} = (n_{features}, n_{hidden}), we can easily feed the network all our training data in one go by taking the matrix product

X W^{h} = (n_{inputs}, n_{hidden}),

and obtain a matrix that holds the weighted sum of inputs to the hidden layer for each input image and each hidden neuron.
We also add the bias to obtain a matrix of weighted sums to the hidden layer Z^{h}:

\boldsymbol{z}^{l} = \boldsymbol{X} \boldsymbol{W}^{l} + \boldsymbol{b}^{l} ,

meaning the same bias (1D array with size equal number of hidden neurons) is added to each input image.
This is then passed through the activation:

\boldsymbol{a}^{l} = f(\boldsymbol{z}^l) .

This is fed to the output layer:

\boldsymbol{z}^{L} = \boldsymbol{a}^{L} \boldsymbol{W}^{L} + \boldsymbol{b}^{L} .

Finally we receive our output values for each image and each category by passing it through the softmax function:

output = softmax (\boldsymbol{z}^{L}) = (n_{inputs}, n_{categories}) .

In [4]:

# setup the feed-forward pass, subscript h = hidden layer

def sigmoid(x):
    return 1/(1 + np.exp(-x))

def feed_forward(X):
    # weighted sum of inputs to the hidden layer
    z_h = np.matmul(X, hidden_weights) + hidden_bias
    # activation in the hidden layer
    a_h = sigmoid(z_h)
    
    # weighted sum of inputs to the output layer
    z_o = np.matmul(a_h, output_weights) + output_bias
    # softmax output
    # axis 0 holds each input and axis 1 the probabilities of each category
    exp_term = np.exp(z_o)
    probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
    
    return probabilities

probabilities = feed_forward(X_train)
print("probabilities = (n_inputs, n_categories) = " + str(probabilities.shape))
print("probability that image 0 is in category 0,1,2,...,9 = \n" + str(probabilities[0]))
print("probabilities sum up to: " + str(probabilities[0].sum()))
print()

# we obtain a prediction by taking the class with the highest likelihood
def predict(X):
    probabilities = feed_forward(X)
    return np.argmax(probabilities, axis=1)

predictions = predict(X_train)
print("predictions = (n_inputs) = " + str(predictions.shape))
print("prediction for image 0: " + str(predictions[0]))
print("correct label for image 0: " + str(Y_train[0]))

Choose cost function and optimizer

To measure how well our neural network is doing we need to introduce a cost function.
We will call the function that gives the error of a single sample output the loss function, and the function that gives the total error of our network across all samples the cost function. A typical choice for multiclass classification is the cross-entropy loss, also known as the negative log likelihood.

In multiclass classification it is common to treat each integer label as a so called one-hot vector:

y = 5 \quad \rightarrow \quad \boldsymbol{y} = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0) , y = 1 \quad \rightarrow \quad \boldsymbol{y} = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0) ,

i.e. a binary bit string of length C, where C = 10 is the number of classes in the MNIST dataset.

Let y_{ic} denote the $c$-th component of the $i$-th one-hot vector.
We define the cost function \mathcal{C} as a sum over the cross-entropy loss for each point \boldsymbol{x}_i in the dataset.

In the one-hot representation only one of the terms in the loss function is non-zero, namely the probability of the correct category c'
(i.e. the category c' such that y_{ic'} = 1). This means that the cross entropy loss only punishes you for how wrong you got the correct label. The probability of category c is given by the softmax function. The vector \boldsymbol{\theta} represents the parameters of our network, i.e. all the weights and biases.

Optimizing the cost function

The network is trained by finding the weights and biases that minimize the cost function. One of the most widely used classes of methods is gradient descent and its generalizations. The idea behind gradient descent is simply to adjust the weights in the direction where the gradient of the cost function is large and negative. This ensures we flow toward a local minimum of the cost function.
Each parameter \theta is iteratively adjusted according to the rule

\theta_{i+1} = \theta_i - \eta \nabla \mathcal{C}(\theta_i) ,

where \eta is known as the learning rate, which controls how big a step we take towards the minimum.
This update can be repeated for any number of iterations, or until we are satisfied with the result.

A simple and effective improvement is a variant called Batch Gradient Descent.
Instead of calculating the gradient on the whole dataset, we calculate an approximation of the gradient on a subset of the data called a minibatch.
If there are N data points and we have a minibatch size of M, the total number of batches is N/M.
We denote each minibatch B_k, with k = 1, 2,...,N/M. The gradient then becomes:

$$ \nabla \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \nabla \mathcal{L}i(\theta) \quad \rightarrow \quad \frac{1}{M} \sum{i \in B_k} \nabla \mathcal{L}_i(\theta) ,$$

i.e. instead of averaging the loss over the entire dataset, we average over a minibatch.

This has two important benefits:

1. Introducing stochasticity decreases the chance that the algorithm becomes stuck in a local minima.

2. It significantly speeds up the calculation, since we do not have to use the entire dataset to calculate the gradient.

The various optmization methods, with codes and algorithms, are discussed in our lectures on Gradient descent approaches.

Regularization

It is common to add an extra term to the cost function, proportional to the size of the weights. This is equivalent to constraining the size of the weights, so that they do not grow out of control. Constraining the size of the weights means that the weights cannot grow arbitrarily large to fit the training data, and in this way reduces overfitting.

We will measure the size of the weights using the so called L2-norm, meaning our cost function becomes:

$$ \mathcal{C}(\theta) = \frac{1}{N} \sum_{i=1}^N \mathcal{L}i(\theta) \quad \rightarrow \quad \frac{1}{N} \sum{i=1}^N \mathcal{L}i(\theta) + \lambda \lvert \lvert \boldsymbol{w} \rvert \rvert_2^2 = \frac{1}{N} \sum{i=1}^N \mathcal{L}(\theta) + \lambda \sum_{ij} w_{ij}^2,$$

i.e. we sum up all the weights squared. The factor \lambda is known as a regularization parameter.

In order to train the model, we need to calculate the derivative of the cost function with respect to every bias and weight in the network. In total our network has (64 + 1)\times 50=3250 weights in the hidden layer and (50 + 1)\times 10=510 weights to the output layer (+1 for the bias), and the gradient must be calculated for every parameter. We use the backpropagation algorithm discussed above. This is a clever use of the chain rule that allows us to calculate the gradient efficently.

Matrix multiplication

To more efficently train our network these equations are implemented using matrix operations.
The error in the output layer is calculated simply as, with \boldsymbol{t} being our targets,

\delta_L = \boldsymbol{t} - \boldsymbol{y} = (n_{inputs}, n_{categories}) .

The gradient for the output weights is calculated as

\nabla W_{L} = \boldsymbol{a}^T \delta_L = (n_{hidden}, n_{categories}) ,

where \boldsymbol{a} = (n_{inputs}, n_{hidden}). This simply means that we are summing up the gradients for each input.
Since we are going backwards we have to transpose the activation matrix.

The gradient with respect to the output bias is then

\nabla \boldsymbol{b}_{L} = \sum_{i=1}^{n_{inputs}} \delta_L = (n_{categories}) .

The error in the hidden layer is

\Delta_h = \delta_L W_{L}^T \circ f'(z_{h}) = \delta_L W_{L}^T \circ a_{h} \circ (1 - a_{h}) = (n_{inputs}, n_{hidden}) ,

where f'(a_{h}) is the derivative of the activation in the hidden layer. The matrix products mean that we are summing up the products for each neuron in the output layer. The symbol \circ denotes the Hadamard product, meaning element-wise multiplication.

This again gives us the gradients in the hidden layer:

\nabla W_{h} = X^T \delta_h = (n_{features}, n_{hidden}) , \nabla b_{h} = \sum_{i=1}^{n_{inputs}} \delta_h = (n_{hidden}) .

In [5]:

# to categorical turns our integer vector into a onehot representation
from sklearn.metrics import accuracy_score

# one-hot in numpy
def to_categorical_numpy(integer_vector):
    n_inputs = len(integer_vector)
    n_categories = np.max(integer_vector) + 1
    onehot_vector = np.zeros((n_inputs, n_categories))
    onehot_vector[range(n_inputs), integer_vector] = 1
    
    return onehot_vector

#Y_train_onehot, Y_test_onehot = to_categorical(Y_train), to_categorical(Y_test)
Y_train_onehot, Y_test_onehot = to_categorical_numpy(Y_train), to_categorical_numpy(Y_test)

def feed_forward_train(X):
    # weighted sum of inputs to the hidden layer
    z_h = np.matmul(X, hidden_weights) + hidden_bias
    # activation in the hidden layer
    a_h = sigmoid(z_h)
    
    # weighted sum of inputs to the output layer
    z_o = np.matmul(a_h, output_weights) + output_bias
    # softmax output
    # axis 0 holds each input and axis 1 the probabilities of each category
    exp_term = np.exp(z_o)
    probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
    
    # for backpropagation need activations in hidden and output layers
    return a_h, probabilities

def backpropagation(X, Y):
    a_h, probabilities = feed_forward_train(X)
    
    # error in the output layer
    error_output = probabilities - Y
    # error in the hidden layer
    error_hidden = np.matmul(error_output, output_weights.T) * a_h * (1 - a_h)
    
    # gradients for the output layer
    output_weights_gradient = np.matmul(a_h.T, error_output)
    output_bias_gradient = np.sum(error_output, axis=0)
    
    # gradient for the hidden layer
    hidden_weights_gradient = np.matmul(X.T, error_hidden)
    hidden_bias_gradient = np.sum(error_hidden, axis=0)

    return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient

print("Old accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))

eta = 0.01
lmbd = 0.01
for i in range(1000):
    # calculate gradients
    dWo, dBo, dWh, dBh = backpropagation(X_train, Y_train_onehot)
    
    # regularization term gradients
    dWo += lmbd * output_weights
    dWh += lmbd * hidden_weights
    
    # update weights and biases
    output_weights -= eta * dWo
    output_bias -= eta * dBo
    hidden_weights -= eta * dWh
    hidden_bias -= eta * dBh

print("New accuracy on training data: " + str(accuracy_score(predict(X_train), Y_train)))

Improving performance

As we can see the network does not seem to be learning at all. It seems to be just guessing the label for each image.
In order to obtain a network that does something useful, we will have to do a bit more work.

The choice of hyperparameters such as learning rate and regularization parameter is hugely influential for the performance of the network. Typically a grid-search is performed, wherein we test different hyperparameters separated by orders of magnitude. For example we could test the learning rates \eta = 10^{-6}, 10^{-5},...,10^{-1} with different regularization parameters \lambda = 10^{-6},...,10^{-0}.

Next, we haven't implemented minibatching yet, which introduces stochasticity and is though to act as an important regularizer on the weights. We call a feed-forward + backward pass with a minibatch an iteration, and a full training period going through the entire dataset (n/M batches) an epoch.

If this does not improve network performance, you may want to consider altering the network architecture, adding more neurons or hidden layers.
Andrew Ng goes through some of these considerations in this video. You can find a summary of the video here.

Full object-oriented implementation

It is very natural to think of the network as an object, with specific instances of the network being realizations of this object with different hyperparameters. An implementation using Python classes provides a clean structure and interface, and the full implementation of our neural network is given below.

In [6]:

class NeuralNetwork:
    def __init__(
            self,
            X_data,
            Y_data,
            n_hidden_neurons=50,
            n_categories=10,
            epochs=10,
            batch_size=100,
            eta=0.1,
            lmbd=0.0):

        self.X_data_full = X_data
        self.Y_data_full = Y_data

        self.n_inputs = X_data.shape[0]
        self.n_features = X_data.shape[1]
        self.n_hidden_neurons = n_hidden_neurons
        self.n_categories = n_categories

        self.epochs = epochs
        self.batch_size = batch_size
        self.iterations = self.n_inputs // self.batch_size
        self.eta = eta
        self.lmbd = lmbd

        self.create_biases_and_weights()

    def create_biases_and_weights(self):
        self.hidden_weights = np.random.randn(self.n_features, self.n_hidden_neurons)
        self.hidden_bias = np.zeros(self.n_hidden_neurons) + 0.01

        self.output_weights = np.random.randn(self.n_hidden_neurons, self.n_categories)
        self.output_bias = np.zeros(self.n_categories) + 0.01

    def feed_forward(self):
        # feed-forward for training
        self.z_h = np.matmul(self.X_data, self.hidden_weights) + self.hidden_bias
        self.a_h = sigmoid(self.z_h)

        self.z_o = np.matmul(self.a_h, self.output_weights) + self.output_bias

        exp_term = np.exp(self.z_o)
        self.probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)

    def feed_forward_out(self, X):
        # feed-forward for output
        z_h = np.matmul(X, self.hidden_weights) + self.hidden_bias
        a_h = sigmoid(z_h)

        z_o = np.matmul(a_h, self.output_weights) + self.output_bias
        
        exp_term = np.exp(z_o)
        probabilities = exp_term / np.sum(exp_term, axis=1, keepdims=True)
        return probabilities

    def backpropagation(self):
        error_output = self.probabilities - self.Y_data
        error_hidden = np.matmul(error_output, self.output_weights.T) * self.a_h * (1 - self.a_h)

        self.output_weights_gradient = np.matmul(self.a_h.T, error_output)
        self.output_bias_gradient = np.sum(error_output, axis=0)

        self.hidden_weights_gradient = np.matmul(self.X_data.T, error_hidden)
        self.hidden_bias_gradient = np.sum(error_hidden, axis=0)

        if self.lmbd > 0.0:
            self.output_weights_gradient += self.lmbd * self.output_weights
            self.hidden_weights_gradient += self.lmbd * self.hidden_weights

        self.output_weights -= self.eta * self.output_weights_gradient
        self.output_bias -= self.eta * self.output_bias_gradient
        self.hidden_weights -= self.eta * self.hidden_weights_gradient
        self.hidden_bias -= self.eta * self.hidden_bias_gradient

    def predict(self, X):
        probabilities = self.feed_forward_out(X)
        return np.argmax(probabilities, axis=1)

    def predict_probabilities(self, X):
        probabilities = self.feed_forward_out(X)
        return probabilities

    def train(self):
        data_indices = np.arange(self.n_inputs)

        for i in range(self.epochs):
            for j in range(self.iterations):
                # pick datapoints with replacement
                chosen_datapoints = np.random.choice(
                    data_indices, size=self.batch_size, replace=False
                )

                # minibatch training data
                self.X_data = self.X_data_full[chosen_datapoints]
                self.Y_data = self.Y_data_full[chosen_datapoints]

                self.feed_forward()
                self.backpropagation()

Evaluate model performance on test data

To measure the performance of our network we evaluate how well it does it data it has never seen before, i.e. the test data.
We measure the performance of the network using the accuracy score.
The accuracy is as you would expect just the number of images correctly labeled divided by the total number of images. A perfect classifier will have an accuracy score of 1.

\text{Accuracy} = \frac{\sum_{i=1}^n I(\tilde{y}_i = y_i)}{n} ,

where I is the indicator function, 1 if \tilde{y}_i = y_i and 0 otherwise.

In [7]:

epochs = 100
batch_size = 100

dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
                    n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
dnn.train()
test_predict = dnn.predict(X_test)

# accuracy score from scikit library
print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))

# equivalent in numpy
def accuracy_score_numpy(Y_test, Y_pred):
    return np.sum(Y_test == Y_pred) / len(Y_test)

#print("Accuracy score on test set: ", accuracy_score_numpy(Y_test, test_predict))

Adjust hyperparameters

We now perform a grid search to find the optimal hyperparameters for the network.
Note that we are only using 1 layer with 50 neurons, and human performance is estimated to be around 98\% (2\% error rate).

In [8]:

eta_vals = np.logspace(-5, 1, 7)
lmbd_vals = np.logspace(-5, 1, 7)
# store the models for later use
DNN_numpy = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)

# grid search
for i, eta in enumerate(eta_vals):
    for j, lmbd in enumerate(lmbd_vals):
        dnn = NeuralNetwork(X_train, Y_train_onehot, eta=eta, lmbd=lmbd, epochs=epochs, batch_size=batch_size,
                            n_hidden_neurons=n_hidden_neurons, n_categories=n_categories)
        dnn.train()
        
        DNN_numpy[i][j] = dnn
        
        test_predict = dnn.predict(X_test)
        
        print("Learning rate  = ", eta)
        print("Lambda = ", lmbd)
        print("Accuracy score on test set: ", accuracy_score(Y_test, test_predict))
        print()

Visualization

In [9]:

# visual representation of grid search
# uses seaborn heatmap, you can also do this with matplotlib imshow
import seaborn as sns

sns.set()

train_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))
test_accuracy = np.zeros((len(eta_vals), len(lmbd_vals)))

for i in range(len(eta_vals)):
    for j in range(len(lmbd_vals)):
        dnn = DNN_numpy[i][j]
        
        train_pred = dnn.predict(X_train) 
        test_pred = dnn.predict(X_test)

        train_accuracy[i][j] = accuracy_score(Y_train, train_pred)
        test_accuracy[i][j] = accuracy_score(Y_test, test_pred)

        
fig, ax = plt.subplots(figsize = (10, 10))
sns.heatmap(train_accuracy, annot=True, ax=ax, cmap="viridis")
ax.set_title("Training Accuracy")
ax.set_ylabel("$\eta$")
ax.set_xlabel("$\lambda$")
plt.show()

fig, ax = plt.subplots(figsize = (10, 10))
sns.heatmap(test_accuracy, annot=True, ax=ax, cmap="viridis")
ax.set_title("Test Accuracy")
ax.set_ylabel("$\eta$")
ax.set_xlabel("$\lambda$")
plt.show()

scikit-learn implementation

scikit-learn focuses more on traditional machine learning methods, such as regression, clustering, decision trees, etc. As such, it has only two types of neural networks: Multi Layer Perceptron outputting continuous values, MPLRegressor, and Multi Layer Perceptron outputting labels, MLPClassifier. We will see how simple it is to use these classes.

scikit-learn implements a few improvements from our neural network, such as early stopping, a varying learning rate, different optimization methods, etc. We would therefore expect a better performance overall.

In [10]:

from sklearn.neural_network import MLPClassifier
# store models for later use
DNN_scikit = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)

for i, eta in enumerate(eta_vals):
    for j, lmbd in enumerate(lmbd_vals):
        dnn = MLPClassifier(hidden_layer_sizes=(n_hidden_neurons), activation='logistic',
                            alpha=lmbd, learning_rate_init=eta, max_iter=epochs)
        dnn.fit(X_train, Y_train)
        
        DNN_scikit[i][j] = dnn
        
        print("Learning rate  = ", eta)
        print("Lambda = ", lmbd)
        print("Accuracy score on test set: ", dnn.score(X_test, Y_test))
        print()

Visualization

Warning:

Output truncated. This notebook contains too many cells to display efficiently.