Deep Learning from first principles in Python, R and Octave – Part 1

“You don’t perceive objects as they are. You perceive them as you are.”
“Your interpretation of physical objects has everything to do with the historical trajectory of your brain – and little to do with the objects themselves.”
“The brain generates its own reality, even before it receives information coming in from the eyes and the other senses. This is known as the internal model”

                          David Eagleman - The Brain: The Story of You

This is the first in the series of posts, I intend to write on Deep Learning. This post is inspired by the Deep Learning Specialization by Prof Andrew Ng on Coursera and Neural Networks for Machine Learning by Prof Geoffrey Hinton also on Coursera. In this post I implement Logistic regression with a 2 layer Neural Network i.e. a Neural Network that just has an input layer and an output layer and with no hidden layer.I am certain that any self-respecting Deep Learning/Neural Network would consider a Neural Network without hidden layers as no Neural Network at all!

This 2 layer network is implemented in Python, R and Octave languages. I have included Octave, into the mix, as Octave is a close cousin of Matlab. These implementations in Python, R and Octave are equivalent vectorized implementations. So, if you are familiar in any one of the languages, you should be able to look at the corresponding code in the other two. You can download this R Markdown file and Octave code from DeepLearning -Part 1

Check out my video presentation which discusses the derivations in detail
1. Elements of Neural Networks and Deep Le- Part 1
2. Elements of Neural Networks and Deep Learning – Part 2

To start with, Logistic Regression is performed using sklearn’s logistic regression package for the cancer data set also from sklearn. This is shown below

1. Logistic Regression

import numpy as np
import pandas as pd
import os
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LogisticRegression
from sklearn.datasets import make_classification, make_blobs

from sklearn.metrics import confusion_matrix
from matplotlib.colors import ListedColormap
from sklearn.datasets import load_breast_cancer
# Load the cancer data
(X_cancer, y_cancer) = load_breast_cancer(return_X_y = True)
X_train, X_test, y_train, y_test = train_test_split(X_cancer, y_cancer,
                                                   random_state = 0)
# Call the Logisitic Regression function
clf = LogisticRegression().fit(X_train, y_train)
print('Accuracy of Logistic regression classifier on training set: {:.2f}'
     .format(clf.score(X_train, y_train)))
print('Accuracy of Logistic regression classifier on test set: {:.2f}'
     .format(clf.score(X_test, y_test)))
## Accuracy of Logistic regression classifier on training set: 0.96
## Accuracy of Logistic regression classifier on test set: 0.96

To check on other classification algorithms, check my post Practical Machine Learning with R and Python – Part 2.

Checkout my book ‘Deep Learning from first principles: Second Edition – In vectorized Python, R and Octave’. My book starts with the implementation of a simple 2-layer Neural Network and works its way to a generic L-Layer Deep Learning Network, with all the bells and whistles. The derivations have been discussed in detail. The code has been extensively commented and included in its entirety in the Appendix sections. My book is available on Amazon as paperback ($14.99) and in kindle version($9.99/Rs449).

You may also like my companion book “Practical Machine Learning with R and Python:Second Edition- Machine Learning in stereo” available in Amazon in paperback($10.99) and Kindle($7.99/Rs449) versions. This book is ideal for a quick reference of the various ML functions and associated measurements in both R and Python which are essential to delve deep into Deep Learning.

2. Logistic Regression as a 2 layer Neural Network

In the following section Logistic Regression is implemented as a 2 layer Neural Network in Python, R and Octave. The same cancer data set from sklearn will be used to train and test the Neural Network in Python, R and Octave. This can be represented diagrammatically as below


The cancer data set has 30 input features, and the target variable ‘output’ is either 0 or 1. Hence the sigmoid activation function will be used in the output layer for classification.

This simple 2 layer Neural Network is shown below
At the input layer there are 30 features and the corresponding weights of these inputs which are initialized to small random values.
Z= w_{1}x_{1} +w_{2}x_{2} +..+ w_{30}x_{30} + b
where ‘b’ is the bias term

The Activation function is the sigmoid function which is a= 1/(1+e^{-z})
The Loss, when the sigmoid function is used in the output layer, is given by
L=-(ylog(a) + (1-y)log(1-a)) (1)

Gradient Descent

Forward propagation

In forward propagation cycle of the Neural Network the output Z and the output of activation function, the sigmoid function, is first computed. Then using the output ‘y’ for the given features, the ‘Loss’ is computed using equation (1) above.

Backward propagation

The backward propagation cycle determines how the ‘Loss’ is impacted for small variations from the previous layers upto the input layer. In other words, backward propagation computes the changes in the weights at the input layer, which will minimize the loss. Several cycles of gradient descent are performed in the path of steepest descent to find the local minima. In other words the set of weights and biases, at the input layer, which will result in the lowest loss is computed by gradient descent. The weights at the input layer are decreased by a parameter known as the ‘learning rate’. Too big a ‘learning rate’ can overshoot the local minima, and too small a ‘learning rate’ can take a long time to reach the local minima. This is done for ‘m’ training examples.

Chain rule of differentiation
Let y=f(u)
and u=g(x) then
\partial y/\partial x = \partial y/\partial u * \partial u/\partial x

Derivative of sigmoid
Let x= 1 + e^{-z}  then
\sigma = 1/x
\partial \sigma/\partial x = -1/x^{2}
\partial x/\partial z = -e^{-z}
Using the chain rule of differentiation we get
\partial \sigma/\partial z = \partial \sigma/\partial x * \partial x/\partial z
=-1/(1+e^{-z})^{2}* -e^{-z} = e^{-z}/(1+e^{-z})^{2}
Therefore \partial \sigma/\partial z = \sigma(1-\sigma)        -(2)

The 3 equations for the 2 layer Neural Network representation of Logistic Regression are
L=-(y*log(a) + (1-y)*log(1-a))      -(a)
a=1/(1+e^{-Z})      -(b)
Z= w_{1}x_{1} +w_{2}x_{2} +...+ w_{30}x_{30} +b = Z = \sum_{i} w_{i}*x_{i} + b -(c)

The back propagation step requires the computation of dL/dw_{i} and dL/db_{i}. In the case of regression it would be dE/dw_{i} and dE/db_{i} where dE is the Mean Squared Error function.
Computing the derivatives for back propagation we have
dL/da = -(y/a + (1-y)/(1-a))          -(d)
because d/dx(logx) = 1/x
Also from equation (2) we get
da/dZ = a (1-a)                                  – (e)
By chain rule
\partial L/\partial Z = \partial L/\partial a * \partial a/\partial Z
therefore substituting the results of (d) & (e) we get
\partial L/\partial Z = -(y/a + (1-y)/(1-a)) * a(1-a) = a-y         (f)
\partial L/\partial w_{i}= \partial L/\partial a * \partial a/\partial Z * \partial Z/\partial w_{i}                                                           -(g)
\partial Z/\partial w_{i} = x_{i}            – (h)
and from (f) we have  \partial L/\partial Z =a-y
Therefore  (g) reduces to
\partial L/\partial w_{i} = x_{i}* (a-y) -(i)
\partial L/\partial b = \partial L/\partial a * \partial a/\partial Z * \partial Z/\partial b -(j)
\partial Z/\partial b = 1 and using (f) in (j)
\partial L/\partial b = a-y

The gradient computes the weights at the input layer and the corresponding bias by using the values
of dw_{i} and db
w_{i} := w_{i} -\alpha * dw_{i}
b := b -\alpha * db
I found the computation graph representation in the book Deep Learning: Ian Goodfellow, Yoshua Bengio, Aaron Courville, very useful to visualize and also compute the backward propagation. For the 2 layer Neural Network of Logistic Regression the computation graph is shown below

3. Neural Network for Logistic Regression -Python code (vectorized)

import numpy as np
import pandas as pd
import os
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split

# Define the sigmoid function
def sigmoid(z):  
    return a

# Initialize
def initialize(dim):
    w = np.zeros(dim).reshape(dim,1)
    b = 0   
    return w

# Compute the loss
def computeLoss(numTraining,Y,A):
    loss=-1/numTraining *np.sum(Y*np.log(A) + (1-Y)*(np.log(1-A)))

# Execute the forward propagation
def forwardPropagation(w,b,X,Y):
    # Compute Z,X)+b
    # Determine the number of training samples
    # Compute the output of the sigmoid activation function 
    #Compute the loss
    loss = computeLoss(numTraining,Y,A)
    # Compute the gradients dZ, dw and db
    # Return the results as a dictionary
    gradients = {"dw": dw,
             "db": db}
    loss = np.squeeze(loss)
    return gradients,loss

# Compute Gradient Descent    
def gradientDescent(w, b, X, Y, numIerations, learningRate):
    idx =[]
    # Iterate 
    for i in range(numIerations):
        #Get the derivates
        dw = gradients["dw"]
        db = gradients["db"]
        w = w-learningRate*dw
        b = b-learningRate*db

        # Store the loss
        if i % 100 == 0:
        # Set params and grads
        params = {"w": w,
                  "b": b}  
        grads = {"dw": dw,
                 "db": db}
    return params, grads, losses,idx

# Predict the output for a training set 
def predict(w,b,X):
    # Compute the sigmoid
    for i in range(A.shape[1]):
        #If the value is > 0.5 then set as 1
        if(A[0][i] > 0.5):
        # Else set as 0

    return yPredicted

#Normalize the data   
def normalize(x):
    x_norm = None
    x_norm = np.linalg.norm(x,axis=1,keepdims=True)
    x= x/x_norm
    return x

# Run the 2 layer Neural Network on the cancer data set

from sklearn.datasets import load_breast_cancer
# Load the cancer data
(X_cancer, y_cancer) = load_breast_cancer(return_X_y = True)
# Create train and test sets
X_train, X_test, y_train, y_test = train_test_split(X_cancer, y_cancer,
                                                   random_state = 0)
# Normalize the data for better performance

# Create weight vectors of zeros. The size is the number of features in the data set=30

#Normalize the training data so that gradient descent performs better
#Transpose X_train so that we have a matrix as (features, numSamples)

# Reshape to remove the rank 1 array and then transpose

# Run gradient descent for 4000 times and compute the weights
parameters, grads, costs,idx = gradientDescent(w, b, X_train2, y_train2, numIerations=4000, learningRate=0.75)
w = parameters["w"]
b = parameters["b"]

# Normalize X_test
#Transpose X_train so that we have a matrix as (features, numSamples)

#Reshape y_test

# Predict the values for 
yPredictionTest = predict(w, b, X_test2)
yPredictionTrain = predict(w, b, X_train2)

# Print the accuracy
print("train accuracy: {} %".format(100 - np.mean(np.abs(yPredictionTrain - y_train2)) * 100))
print("test accuracy: {} %".format(100 - np.mean(np.abs(yPredictionTest - y_test)) * 100))

# Plot the Costs vs the number of iterations
fig1=plt.title("Gradient descent-Cost vs No of iterations")
fig1=plt.xlabel("No of iterations")
fig1.figure.savefig("fig1", bbox_inches='tight')
## train accuracy: 90.3755868545 %
## test accuracy: 89.5104895105 %

Note: It can be seen that the Accuracy on the training and test set is 90.37% and 89.51%. This is comparatively poorer than the 96% which the logistic regression of sklearn achieves! But this is mainly because of the absence of hidden layers which is the real power of neural networks.

4. Neural Network for Logistic Regression -R code (vectorized)

# Define the sigmoid function
sigmoid <- function(z){
    a <- 1/(1+ exp(-z))

# Compute the loss
computeLoss <- function(numTraining,Y,A){
    loss <- -1/numTraining* sum(Y*log(A) + (1-Y)*log(1-A))

# Compute forward propagation
forwardPropagation <- function(w,b,X,Y){
    # Compute Z
    Z <- t(w) %*% X +b
    #Set the number of samples
    numTraining <- ncol(X)
    # Compute the activation function
    #Compute the loss
    loss <- computeLoss(numTraining,Y,A)
    # Compute the gradients dZ, dw and db
    dw<-1/numTraining * X %*% t(dZ)
    fwdProp <- list("loss" = loss, "dw" = dw, "db" = db)

# Perform one cycle of Gradient descent
gradientDescent <- function(w, b, X, Y, numIerations, learningRate){
    losses <- NULL
    idx <- NULL
    # Loop through the number of iterations
    for(i in 1:numIerations){
        fwdProp <-forwardPropagation(w,b,X,Y)
        #Get the derivatives
        dw <- fwdProp$dw
        db <- fwdProp$db
        #Perform gradient descent
        w = w-learningRate*dw
        b = b-learningRate*db
        l <- fwdProp$loss
        # Stoe the loss
        if(i %% 100 == 0){
            idx <- c(idx,i)
            losses <- c(losses,l)  
    # Return the weights and losses
    gradDescnt <- list("w"=w,"b"=b,"dw"=dw,"db"=db,"losses"=losses,"idx"=idx)

# Compute the predicted value for input
predict <- function(w,b,X){
    # Create a ector of 0's
    Z <- t(w) %*% X +b
    # Compute sigmoid
    for(i in 1:dim(A)[2]){
        # If A > 0.5 set value as 1
        if(A[1,i] > 0.5)
        # Else set as 0


# Normalize the matrix
normalize <- function(x){
    #Create the norm of the matrix.Perform the Frobenius norm of the matrix 
    #Sweep by rows by norm. Note '1' in the function which performing on every row
    normalized<-sweep(x, 1, n, FUN="/")

# Run the 2 layer Neural Network on the cancer data set
# Read the data (from sklearn)
cancer <- read.csv("cancer.csv")
# Rename the target variable
names(cancer) <- c(seq(1,30),"output")
# Split as training and test sets
train_idx <- trainTestSplit(cancer,trainPercent=75,seed=5)
train <- cancer[train_idx, ]
test <- cancer[-train_idx, ]

# Set the features
X_train <-train[,1:30]
y_train <- train[,31]
X_test <- test[,1:30]
y_test <- test[,31]
# Create a matrix of 0's with the number of features
w <-matrix(rep(0,dim(X_train)[2]))
b <-0
X_train1 <- normalize(X_train)

# Reshape  then transpose

# Perform gradient descent
gradDescent= gradientDescent(w, b, X_train2, y_train2, numIerations=3000, learningRate=0.77)

# Normalize X_test
#Transpose X_train so that we have a matrix as (features, numSamples)

#Reshape y_test and take transpose

# Use the values of the weights generated from Gradient Descent
yPredictionTest = predict(gradDescent$w, gradDescent$b, X_test2)
yPredictionTrain = predict(gradDescent$w, gradDescent$b, X_train2)

sprintf("Train accuracy: %f",(100 - mean(abs(yPredictionTrain - y_train2)) * 100))
## [1] "Train accuracy: 90.845070"
sprintf("test accuracy: %f",(100 - mean(abs(yPredictionTest - y_test)) * 100))
## [1] "test accuracy: 87.323944"
df <-data.frame(gradDescent$idx, gradDescent$losses)
names(df) <- c("iterations","losses")
ggplot(df,aes(x=iterations,y=losses)) + geom_point() + geom_line(col="blue") +
    ggtitle("Gradient Descent - Losses vs No of Iterations") +
    xlab("No of iterations") + ylab("Losses")

4. Neural Network for Logistic Regression -Octave code (vectorized)

# Define sigmoid function
function a = sigmoid(z)
a = 1 ./ (1+ exp(-z));
# Compute the loss
function loss=computeLoss(numtraining,Y,A)
loss = -1/numtraining * sum((Y .* log(A)) + (1-Y) .* log(1-A));

# Perform forward propagation
function [loss,dw,db,dZ] = forwardPropagation(w,b,X,Y)
% Compute Z
Z = w' * X + b;
numtraining = size(X)(1,2);
# Compute sigmoid
A = sigmoid(Z);

#Compute loss. Note this is element wise product
loss =computeLoss(numtraining,Y,A);
# Compute the gradients dZ, dw and db
dZ = A-Y;
dw = 1/numtraining* X * dZ';
db =1/numtraining*sum(dZ);


# Compute Gradient Descent
function [w,b,dw,db,losses,index]=gradientDescent(w, b, X, Y, numIerations, learningRate)
#Initialize losses and idx
# Loop through the number of iterations
for i=1:numIerations,
[loss,dw,db,dZ] = forwardPropagation(w,b,X,Y);
# Perform Gradient descent
w = w - learningRate*dw;
b = b - learningRate*db;
if(mod(i,100) ==0)
# Append index and loss
index = [index i];
losses = [losses loss];


# Determine the predicted value for dataset
function yPredicted = predict(w,b,X)
m = size(X)(1,2);
# Compute Z
Z = w' * X + b;
# Compute sigmoid
A = sigmoid(Z);
for i=1:size(X)(1,2),
# Set predicted as 1 if A > 0,5
if(A(1,i) >= 0.5)

# Normalize by dividing each value by the sum of squares
function normalized = normalize(x)
# Compute Frobenius norm. Square the elements, sum rows and then find square root
a = sqrt(sum(x .^ 2,2));
# Perform element wise division
normalized = x ./ a;

# Split into train and test sets
function [X_train,y_train,X_test,y_test] = trainTestSplit(dataset,trainPercent)
# Create a random index
ix = randperm(length(dataset));
# Split into training
trainSize = floor(trainPercent/100 * length(dataset));
# And test
X_train = train(:,1:30);
y_train = train(:,31);
X_test = test(:,1:30);
y_test = test(:,31);

[X_train,y_train,X_test,y_test] = trainTestSplit(cancer,75);
[w1,b1,dw,db,losses,idx]=gradientDescent(w, b, X_train2, y_train1, numIerations=3000, learningRate=0.75);
# Normalize X_test
#Transpose X_train so that we have a matrix as (features, numSamples)
# Use the values of the weights generated from Gradient Descent
yPredictionTest = predict(w1, b1, X_test2);
yPredictionTrain = predict(w1, b1, X_train2);

trainAccuracy=100-mean(abs(yPredictionTrain - y_train1))*100
testAccuracy=100- mean(abs(yPredictionTest - y_test1))*100
trainAccuracy = 90.845
testAccuracy = 89.510
title ('Gradient descent- Cost vs No of iterations');
xlabel ("No of iterations");
ylabel ("Cost");

This post starts with a simple 2 layer Neural Network implementation of Logistic Regression. Clearly the performance of this simple Neural Network is comparatively poor to the highly optimized sklearn’s Logistic Regression. This is because the above neural network did not have any hidden layers. Deep Learning & Neural Networks achieve extraordinary performance because of the presence of deep hidden layers

The Deep Learning journey has begun… Don’t miss the bus!
Stay tuned for more interesting posts in Deep Learning!!

1. Deep Learning Specialization
2. Neural Networks for Machine Learning
3. Deep Learning, Ian Goodfellow, Yoshua Bengio and Aaron Courville
4. Neural Networks: The mechanics of backpropagation
5. Machine Learning

Also see
1. My book ‘Practical Machine Learning with R and Python’ on Amazon
2. Simplifying Machine Learning: Bias, Variance, regularization and odd facts – Part 4
3. The 3rd paperback & kindle editions of my books on Cricket, now on Amazon
4. Practical Machine Learning with R and Python – Part 4
5. Introducing QCSimulator: A 5-qubit quantum computing simulator in R
6. A Bluemix recipe with MongoDB and Node.js
7. My travels through the realms of Data Science, Machine Learning, Deep Learning and (AI)

To see all posts check Index of posts

Cloud Computing – Show me the money!

Published in Telecom Lead – Cloud Computing – Show me the money!

A lot has been said about the merits of cloud computing and how it is going to be the technological choice of most enterprises in the not so distant future. But the key question that is bound to keep cropping up in the higher echelons of the enterprise is whether the cloud makes good business sense. While most know that cloud computing adopts a pay-per-use model similar to regular utilities like electricity and water and does away with upfront infrastructure costs to the organization the nagging question to most senior management people is whether cloud computing is prudent choice in the long term.

This is not an easy question to answer and depends on a multitude of factors. The alternative to cloud computing is to have an in-house infrastructure of servers, hardware and software, software licenses, broadband links, firewalls etc. All these will form the Capital Expenditure (CAPEX) for the organization. In addition to these expenses will be the Operational Expenditures (OPEX) of real estate to house the equipment, power supply systems, cooling systems, maintenance personnel, annual maintenance contracts (AMC) etc which will be recurring expenses for the organization.

Cloud Computing does away completely with procurement of hardware, software, databases, licenses etc and an enterprise should be able to host their application in a couple of hours provided they know ahead of time the resources their application will need.

Hence as can be seen while the upfront costs and the running costs of maintaining a data center will be high in comparison to the zero upfront costs of the deploying on the cloud the steeper operational costs of the cloud will eventually catch up with the in-house infrastructure.

Depending on how well the application is designed the point at which the cumulative running costs of the cloud breaks even with in-house data center can be made to occur a couple of years down the line after the application is deployed.  Assuming that the break even happens in 3 years the advantage of cloud deployment is that the enterprise does not have to worry about equipment obsolescence, upgrading of software etc not to mention the depreciation of the equipment costs.

Moreover cloud technology is extremely useful to enterprises which are planning to deploy application in which there is difficulty in forecasting the type of traffic that will be hit their application. Where the traffic may be intermittent, bursty or seasonal then a cloud makes perfect business sense since can it scale up or scale down depending on the traffic.

Some typical applications which are prime candidates for the cloud are CRM software, office tools, testing tools, online retail stores, webmail etc.

One possible worry of the enterprise will be the security concerns while deploying to the public cloud. In such situations the organization can take a hybrid strategy where their sensitive data are hosted in in-house data centers and their main application is hosted on a public cloud.

Hence in most situation cloud deployments do have a definite edge for certain key application of the enterprise.

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The Business of Cloud Computing

Cloud Computing is the spanking new paradigm in the world of computing. The key differentiator in this technology is that the enterprise only pays for the amount of resources used – be it CPUs, memory or databases. While it does away with Capital Expenditure for organizations by providing a utility model of pricing it results in recurring Operating Expenses for the organization. However the important thing is that the cloud grows and shrinks according to demand and hence the cost to the organization is dependent on the traffic it generates. While web based applications are prime candidates for the cloud other equally eligible candidates are batch processing jobs, nightly builds or CPU intensive analytics. Except for the case of web application, for other types of applications, a reasonable estimate can be made on the resources needed and appropriate choice be made on the cloud.

This article looks at web applications where the traffic on the site can be seasonal and can vary during periods of the day. Besides web sites should be capable of handling bursty traffic with enormous loads at particular intervals.

The important consideration for web sites is to ensure that the application is truly optimized and exhibits the property of scaling horizontally. While it appears that scaling out will occur for any reasonably designed application the issue is that as the number of hits increase on the web site the response time increases steeply but the number of transactions per second plateaus at some particular load level and does not increase after that. It can be said that for a certain CPU instance configuration the peak transaction per second will reach a particular limit and cannot be increased any further. However the cloud also provides a key component namely the load balancer along with auto scaling which create a new instances when this threshold is reached.

What are the business considerations that need to be taken while designing for the cloud?

One needs to be conservative in choosing the instance type. While larger instances will provide a better performance they also cost more. Hence the instance type should be large enough and no larger. It would be wasteful of using extremely large instances where the last instance only uses a part of the total traffic while costing a lot more.

The analogy is that if 16 units if task have to be performed it is better to have a small CPU instance capable of handling 3 units of task requiring a total of 6 CPUs (6 * 3 = 18 > 16) rather than having a large CPU instance capable of handling 5 units of task requiring a total of 4 large CPUs (5 * 4= 20> 16). The second option would result in a waste processing power.

Assuming that the upfront cost to the organization for hosting the website in-house is ‘P’ and the cost amortized over a period of 1 years is ‘p’ per hour. Further if the instance cost is ‘c’ and ‘n’ is number of instances needed to support the projected demand and the revenue to the organization hosting the website is ‘r’ per 1000 hits then a cloud deployment will make business sense when

(rh– n * ch) – ph > 0 where h is the hour

As long as the right hand side is positive the organization will profit. However as the traffic increases and the throughput of website plateaus the enterprise will hit a ‘window of diminishing returns’.

However if the performance of the application is poor and the number of instances needed to support the traffic is disproportionately large then the above equation will be negative and will result in loss to the organization.

(rh – n * ch) – ph < 0

Hence deployment to the cloud besides requiring a strong technical background also needs a sound business sense in order to reap the benefits of the cloud.

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