# My book ‘Deep Learning from first principles:Second Edition’ now on Amazon

The second edition of my book ‘Deep Learning from first principles:Second Edition- In vectorized Python, R and Octave’, is now available on Amazon, in both paperback ($18.99) and kindle ($9.99/Rs449/-)  versions. Since this book is almost 70% code, all functions, and code snippets have been formatted to use the fixed-width font ‘Lucida Console’. In addition line numbers have been added to all code snippets. This makes the code more organized and much more readable. I have also fixed typos in the book

The book includes the following chapters

Table of Contents
Preface 4
Introduction 6
1. Logistic Regression as a Neural Network 8
2. Implementing a simple Neural Network 23
3. Building a L- Layer Deep Learning Network 48
4. Deep Learning network with the Softmax 85
5. MNIST classification with Softmax 103
6. Initialization, regularization in Deep Learning 121
7. Gradient Descent Optimization techniques 167
8. Gradient Check in Deep Learning 197
1. Appendix A 214
2. Appendix 1 – Logistic Regression as a Neural Network 220
3. Appendix 2 - Implementing a simple Neural Network 227
4. Appendix 3 - Building a L- Layer Deep Learning Network 240
5. Appendix 4 - Deep Learning network with the Softmax 259
6. Appendix 5 - MNIST classification with Softmax 269
7. Appendix 6 - Initialization, regularization in Deep Learning 302
8. Appendix 7 - Gradient Descent Optimization techniques 344
9. Appendix 8 – Gradient Check 405
References 475

To see posts click Index of Posts

# Deep Learning from first principles in Python, R and Octave – Part 3

“Once upon a time, I, Chuang Tzu, dreamt I was a butterfly, fluttering hither and thither, to all intents and purposes a butterfly. I was conscious only of following my fancies as a butterfly, and was unconscious of my individuality as a man. Suddenly, I awoke, and there I lay, myself again. Now I do not know whether I was then a man dreaming I was a butterfly, or whether I am now a butterfly dreaming that I am a man.”
from The Brain: The Story of you – David Eagleman

“Thought is a great big vector of neural activity”
Prof Geoffrey Hinton

# Introduction

This is the third part in my series on Deep Learning from first principles in Python, R and Octave. In the first part Deep Learning from first principles in Python, R and Octave-Part 1, I implemented logistic regression as a 2 layer neural network. The 2nd part Deep Learning from first principles in Python, R and Octave-Part 2, dealt with the implementation of 3 layer Neural Networks with 1 hidden layer to perform classification tasks, where the 2 classes cannot be separated by a linear boundary. In this third part, I implement a multi-layer, Deep Learning (DL) network of arbitrary depth (any number of hidden layers) and arbitrary height (any number of activation units in each hidden layer). The implementations of these Deep Learning networks, in all the 3 parts, are based on vectorized versions in Python, R and Octave. The implementation in the 3rd part is for a L-layer Deep Netwwork, but without any regularization, early stopping, momentum or learning rate adaptation techniques. However even the barebones multi-layer DL, is a handful and has enough hyperparameters to fine-tune and adjust.

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 ($18.99) and in kindle version($9.99/Rs449).

The implementation of the vectorized L-layer Deep Learning network in Python, R and Octave were both exhausting, and exacting!! Keeping track of the indices, layer number and matrix dimensions required quite bit of focus. While the implementation was demanding, it was also very exciting to get the code to work. The trick was to be able to shift gears between the slight quirkiness between the languages. Here are some of challenges I faced.

1. Python and Octave allow multiple return values to be unpacked in a single statement. With R, unpacking multiple return values from a list, requires the list returned, to be unpacked separately. I did see that there is a package gsubfn, which does this.  I hope this feature becomes a base R feature.
2. Python and R allow dissimilar elements to be saved and returned from functions using dictionaries or lists respectively. However there is no real equivalent in Octave. The closest I got to this functionality in Octave, was the ‘cell array’. But the cell array can be accessed only by the index, and not with the key as in a Python dictionary or R list. This makes things just a bit more difficult in Octave.
3. Python and Octave include implicit broadcasting. In R, broadcasting is not implicit, but R has a nifty function, the sweep(), with which we can broadcast either by columns or by rows
4. The closest equivalent of Python’s dictionary, or R’s list, in Octave is the cell array. However I had to manage separate cell arrays for weights and biases and during gradient descent and separate gradients dW and dB
5. In Python the rank-1 numpy arrays can be annoying at times. This issue is not present in R and Octave.

Though the number of lines of code for Deep Learning functions in Python, R and Octave are about ~350 apiece, they have been some of the most difficult code I have implemented. The current vectorized implementation supports the relu, sigmoid and tanh activation functions as of now. I will be adding other activation functions like the ‘leaky relu’, ‘softmax’ and others, to the implementation in the weeks to come.

While testing with different hyper-parameters namely i) the number of hidden layers, ii) the number of activation units in each layer, iii) the activation function and iv) the number iterations, I found the L-layer Deep Learning Network to be very sensitive to these hyper-parameters. It is not easy to tune the parameters. Adding more hidden layers, or more units per layer, does not help and mostly results in gradient descent getting stuck in some local minima. It does take a fair amount of trial and error and very close observation on how the DL network performs for logical changes. We then can zero in on the most the optimal solution. Feel free to download/fork my code from Github DeepLearning-Part 3 and play around with the hyper-parameters for your own problems.

#### Derivation of a Multi Layer Deep Learning Network

Note: A detailed discussion of the derivation below is available in my video presentation Neural Network 4
Lets take a simple 3 layer Neural network with 3 hidden layers and an output layer

In the forward propagation cycle the equations are

$Z_{1} = W_{1}A_{0} +b_{1}$  and  $A_{1} = g(Z_{1})$
$Z_{2} = W_{2}A_{1} +b_{2}$  and  $A_{2} = g(Z_{2})$
$Z_{3} = W_{3}A_{2} +b_{3}$  and $A_{3} = g(Z_{3})$

The loss function is given by
$L = -(ylogA3 + (1-y)log(1-A3))$
and $dL/dA3 = -(Y/A_{3} + (1-Y)/(1-A_{3}))$

For a binary classification the output activation function is the sigmoid function given by
$A_{3} = 1/(1+ e^{-Z3})$. It can be shown that
$dA_{3}/dZ_{3} = A_{3}(1-A_3)$ see equation 2 in Part 1

$\partial L/\partial Z_{3} = \partial L/\partial A_{3}* \partial A_{3}/\partial Z_{3} = A3-Y$ see equation (f) in  Part 1
and since
$\partial L/\partial A_{2} = \partial L/\partial Z_{3} * \partial Z_{3}/\partial A_{2} = (A_{3} -Y) * W_{3}$ because $\partial Z_{3}/\partial A_{2} = W_{3}$ -(1a)
and $\partial L/\partial Z_{2} =\partial L/\partial A_{2} * \partial A_{2}/\partial Z_{2} = (A_{3} -Y) * W_{3} *g'(Z_{2})$ -(1b)
$\partial L/\partial W_{2} = \partial L/\partial Z_{2} * A_{1}$ -(1c)
since $\partial Z_{2}/\partial W_{2} = A_{1}$
and
$\partial L/\partial b_{2} = \partial L/\partial Z_{2}$ -(1d)
because
$\partial Z_{2}/\partial b_{2} =1$

Also

$\partial L/\partial A_{1} =\partial L/\partial Z_{2} * \partial Z_{2}/\partial A_{1} = \partial L/\partial Z_{2} * W_{2}$     – (2a)
$\partial L/\partial Z_{1} =\partial L/\partial A_{1} * \partial A_{1}/\partial Z_{1} = \partial L/\partial A_{1} * W_{2} *g'(Z_{1})$          – (2b)
$\partial L/\partial W_{1} = \partial L/\partial Z_{1} * A_{0}$ – (2c)
$\partial L/\partial b_{1} = \partial L/\partial Z_{1}$ – (2d)

Inspecting the above equations (1a – 1d & 2a-2d), our ‘Uber deep, bottomless’ brain  can easily discern the pattern in these equations. The equation for any layer ‘l’ is of the form
$Z_{l} = W_{l}A_{l-1} +b_{l}$     and  $A_{l} = g(Z_{l})$
The equation for the backward propagation have the general form
$\partial L/\partial A_{l} = \partial L/\partial Z_{l+1} * W^{l+1}$
$\partial L/\partial Z_{l}=\partial L/\partial A_{l} *g'(Z_{l})$
$\partial L/\partial W_{l} =\partial L/\partial Z_{l} *A^{l-1}$
$\partial L/\partial b_{l} =\partial L/\partial Z_{l}$

Some other important results The derivatives of the activation functions in the implemented Deep Learning network
g(z) = sigmoid(z) = $1/(1+e^{-z})$ = a g’(z) = a(1-a) – See Part 1
g(z) = tanh(z) = a g’(z) = $1 - a^{2}$
g(z) = relu(z) = z  when z>0 and 0 when z 0 and 0 when z <= 0
While it appears that there is a discontinuity for the derivative at 0 the small value at the discontinuity does not present a problem

The implementation of the multi layer vectorized Deep Learning Network for Python, R and Octave is included below. For all these implementations, initially I create the size and configuration of the the Deep Learning network with the layer dimennsions So for example layersDimension Vector ‘V’ of length L indicating ‘L’ layers where

V (in Python)= $[v_{0}, v_{1}, v_{2}$, … $v_{L-1}]$
V (in R)= $c(v_{1}, v_{2}, v_{3}$ , … $v_{L})$
V (in Octave)= [ $v_{1} v_{2} v_{3}$$v_{L}]$

In all of these implementations the first element is the number of input features to the Deep Learning network and the last element is always a ‘sigmoid’ activation function since all the problems deal with binary classification.

The number of elements between the first and the last element are the number of hidden layers and the magnitude of each $v_{i}$ is the number of activation units in each hidden layer, which is specified while actually executing the Deep Learning network using the function L_Layer_DeepModel(), in all the implementations Python, R and Octave

## 1a. Classification with Multi layer Deep Learning Network – Relu activation(Python)

In the code below a 4 layer Neural Network is trained to generate a non-linear boundary between the classes. In the code below the ‘Relu’ Activation function is used. The number of activation units in each layer is 9. The cost vs iterations is plotted in addition to the decision boundary. Further the accuracy, precision, recall and F1 score are also computed

import os
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.colors
import sklearn.linear_model

from sklearn.model_selection import train_test_split
from sklearn.datasets import make_classification, make_blobs
from matplotlib.colors import ListedColormap
import sklearn
import sklearn.datasets

#from DLfunctions import plot_decision_boundary
execfile("./DLfunctions34.py") #
os.chdir("C:\\software\\DeepLearning-Posts\\part3")

# Create clusters of 2 classes
X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9,
cluster_std = 1.3, random_state = 4)
#Create 2 classes
Y1=Y1.reshape(400,1)
Y1 = Y1 % 2
X2=X1.T
Y2=Y1.T
# Set the dimensions of DL Network
#  Below we have
#  2 - 2 input features
#  9,9 - 2 hidden layers with 9 activation units per layer and
#  1 - 1 sigmoid activation unit in the output layer as this is a binary classification
# The activation in the hidden layer is the 'relu' specified in L_Layer_DeepModel

layersDimensions = [2, 9, 9,1] #  4-layer model
parameters = L_Layer_DeepModel(X2, Y2, layersDimensions,hiddenActivationFunc='relu', learning_rate = 0.3,num_iterations = 2500, fig="fig1.png")
#Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), X2,Y2,str(0.3),"fig2.png")

# Compute the confusion matrix
yhat = predict(parameters,X2)
from sklearn.metrics import confusion_matrix
a=confusion_matrix(Y2.T,yhat.T)
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score
print('Accuracy: {:.2f}'.format(accuracy_score(Y2.T, yhat.T)))
print('Precision: {:.2f}'.format(precision_score(Y2.T, yhat.T)))
print('Recall: {:.2f}'.format(recall_score(Y2.T, yhat.T)))
print('F1: {:.2f}'.format(f1_score(Y2.T, yhat.T)))
## Accuracy: 0.90
## Precision: 0.91
## Recall: 0.87
## F1: 0.89

For more details on metrics like Accuracy, Recall, Precision etc. used in classification take a look at my post Practical Machine Learning with R and Python – Part 2. More details about these and other metrics besides implementation of the most common machine learning algorithms are available in my book My book ‘Practical Machine Learning with R and Python’ on Amazon

## 1b. Classification with Multi layer Deep Learning Network – Relu activation(R)

In the code below, binary classification is performed on the same data set as above using the Relu activation function. The DL network is same as above

library(ggplot2)
x <- z[,1:2]
y <- z[,3]
X1 <- t(x)
Y1 <- t(y)

# Set the dimensions of the Deep Learning network
# No of input features =2, 2 hidden layers with 9 activation units and 1 output layer
layersDimensions = c(2, 9, 9,1)
# Execute the Deep Learning Neural Network
retvals = L_Layer_DeepModel(X1, Y1, layersDimensions,
hiddenActivationFunc='relu',
learningRate = 0.3,
numIterations = 5000,
print_cost = True)
library(ggplot2)
source("DLfunctions33.R")
# Get the computed costs
costs <- retvals[['costs']]
# Create a sequence of iterations
numIterations=5000
iterations <- seq(0,numIterations,by=1000)
df <-data.frame(iterations,costs)
# Plot the Costs vs number of iterations
ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") +
xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations")

# Plot the decision boundary
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",0.3)

library(caret)
# Predict the output for the data values
yhat <-predict(retvals$parameters,X1,hiddenActivationFunc="relu") yhat[yhat==FALSE]=0 yhat[yhat==TRUE]=1 # Compute the confusion matrix confusionMatrix(yhat,Y1) ## Confusion Matrix and Statistics ## ## Reference ## Prediction 0 1 ## 0 201 10 ## 1 21 168 ## ## Accuracy : 0.9225 ## 95% CI : (0.8918, 0.9467) ## No Information Rate : 0.555 ## P-Value [Acc > NIR] : < 2e-16 ## ## Kappa : 0.8441 ## Mcnemar's Test P-Value : 0.07249 ## ## Sensitivity : 0.9054 ## Specificity : 0.9438 ## Pos Pred Value : 0.9526 ## Neg Pred Value : 0.8889 ## Prevalence : 0.5550 ## Detection Rate : 0.5025 ## Detection Prevalence : 0.5275 ## Balanced Accuracy : 0.9246 ## ## 'Positive' Class : 0 ##  ## 1c. Classification with Multi layer Deep Learning Network – Relu activation(Octave) Included below is the code for performing classification. Incidentally Octave does not seem to have implemented the confusion matrix, but confusionmat is available in Matlab. # Read the data data=csvread("data.csv"); X=data(:,1:2); Y=data(:,3); # Set layer dimensions layersDimensions = [2 9 7 1] #tanh=-0.5(ok), #relu=0.1 best! # Execute Deep Network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', learningRate = 0.1, numIterations = 10000); plotCostVsIterations(10000,costs); plotDecisionBoundary(data,weights, biases,hiddenActivationFunc="tanh")  ## 2a. Classification with Multi layer Deep Learning Network – Tanh activation(Python) Below the Tanh activation function is used to perform the same classification. I found the Tanh activation required a simpler Neural Network of 3 layers. # Tanh activation import os import numpy as np import matplotlib.pyplot as plt import matplotlib.colors import sklearn.linear_model from sklearn.model_selection import train_test_split from sklearn.datasets import make_classification, make_blobs from matplotlib.colors import ListedColormap import sklearn import sklearn.datasets #from DLfunctions import plot_decision_boundary os.chdir("C:\\software\\DeepLearning-Posts\\part3") execfile("./DLfunctions34.py") # Create the dataset X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9, cluster_std = 1.3, random_state = 4) #Create 2 classes Y1=Y1.reshape(400,1) Y1 = Y1 % 2 X2=X1.T Y2=Y1.T # Set the dimensions of the Neural Network layersDimensions = [2, 4, 1] # 3-layer model # Compute the DL network parameters = L_Layer_DeepModel(X2, Y2, layersDimensions, hiddenActivationFunc='tanh', learning_rate = .5,num_iterations = 2500,fig="fig3.png") #Plot the decision boundary plot_decision_boundary(lambda x: predict(parameters, x.T), X2,Y2,str(0.5),"fig4.png")  ## 2b. Classification with Multi layer Deep Learning Network – Tanh activation(R) R performs better with a Tanh activation than the Relu as can be seen below  #Set the dimensions of the Neural Network layersDimensions = c(2, 9, 9,1) library(ggplot2) # Read the data z <- as.matrix(read.csv("data.csv",header=FALSE)) x <- z[,1:2] y <- z[,3] X1 <- t(x) Y1 <- t(y) # Execute the Deep Model retvals = L_Layer_DeepModel(X1, Y1, layersDimensions, hiddenActivationFunc='tanh', learningRate = 0.3, numIterations = 5000, print_cost = True) # Get the costs costs <- retvals[['costs']] iterations <- seq(0,numIterations,by=1000) df <-data.frame(iterations,costs) # Plot Cost vs number of iterations ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") + xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations") #Plot the decision boundary plotDecisionBoundary(z,retvals,hiddenActivationFunc="tanh",0.3) ## 2c. Classification with Multi layer Deep Learning Network – Tanh activation(Octave) The code below uses the Tanh activation in the hidden layers for Octave # Read the data data=csvread("data.csv"); X=data(:,1:2); Y=data(:,3); # Set layer dimensions layersDimensions = [2 9 7 1] #tanh=-0.5(ok), #relu=0.1 best! # Execute Deep Network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='tanh', learningRate = 0.1, numIterations = 10000); plotCostVsIterations(10000,costs); plotDecisionBoundary(data,weights, biases,hiddenActivationFunc="tanh")  ## 3. Bernoulli’s Lemniscate To make things more interesting, I create a 2D figure of the Bernoulli’s lemniscate to perform non-linear classification. The Lemniscate is given by the equation $(x^{2} + y^{2})^{2}$ = $2a^{2}*(x^{2}-y^{2})$ ## 3a. Classifying a lemniscate with Deep Learning Network – Relu activation(Python) import os import numpy as np import matplotlib.pyplot as plt os.chdir("C:\\software\\DeepLearning-Posts\\part3") execfile("./DLfunctions33.py") x1=np.random.uniform(0,10,2000).reshape(2000,1) x2=np.random.uniform(0,10,2000).reshape(2000,1) X=np.append(x1,x2,axis=1) X.shape # Create a subset of values where squared is <0,4. Perform ravel() to flatten this vector # Create the equation # (x^{2} + y^{2})^2 - 2a^2*(x^{2}-y^{2}) <= 0 a=np.power(np.power(X[:,0]-5,2) + np.power(X[:,1]-5,2),2) b=np.power(X[:,0]-5,2) - np.power(X[:,1]-5,2) c= a - (b*np.power(4,2)) <=0 Y=c.reshape(2000,1) # Create a scatter plot of the lemniscate plt.scatter(X[:,0], X[:,1], c=Y, marker= 'o', s=15,cmap="viridis") Z=np.append(X,Y,axis=1) plt.savefig("fig50.png",bbox_inches='tight') plt.clf() # Set the data for classification X2=X.T Y2=Y.T # These settings work the best # Set the Deep Learning layer dimensions for a Relu activation layersDimensions = [2,7,4,1] #Execute the DL network parameters = L_Layer_DeepModel(X2, Y2, layersDimensions, hiddenActivationFunc='relu', learning_rate = 0.5,num_iterations = 10000, fig="fig5.png") #Plot the decision boundary plot_decision_boundary(lambda x: predict(parameters, x.T), X2, Y2,str(2.2),"fig6.png") # Compute the Confusion matrix yhat = predict(parameters,X2) from sklearn.metrics import confusion_matrix a=confusion_matrix(Y2.T,yhat.T) from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score print('Accuracy: {:.2f}'.format(accuracy_score(Y2.T, yhat.T))) print('Precision: {:.2f}'.format(precision_score(Y2.T, yhat.T))) print('Recall: {:.2f}'.format(recall_score(Y2.T, yhat.T))) print('F1: {:.2f}'.format(f1_score(Y2.T, yhat.T))) ## Accuracy: 0.93 ## Precision: 0.77 ## Recall: 0.76 ## F1: 0.76 We could get better performance by tuning further. Do play around if you fork the code. Note:: The lemniscate data is saved as a CSV and then read in R and also in Octave. I do this instead of recreating the lemniscate shape ## 3b. Classifying a lemniscate with Deep Learning Network – Relu activation(R code) The R decision boundary for the Bernoulli’s lemniscate is shown below Z <- as.matrix(read.csv("lemniscate.csv",header=FALSE)) Z1=data.frame(Z) # Create a scatter plot of the lemniscate ggplot(Z1,aes(x=V1,y=V2,col=V3)) +geom_point() #Set the data for the DL network X=Z[,1:2] Y=Z[,3] X1=t(X) Y1=t(Y) # Set the layer dimensions for the tanh activation function layersDimensions = c(2,5,4,1) # Execute the Deep Learning network with Tanh activation retvals = L_Layer_DeepModel(X1, Y1, layersDimensions, hiddenActivationFunc='tanh', learningRate = 0.3, numIterations = 20000, print_cost = True) # Plot cost vs iteration costs <- retvals[['costs']] numIterations = 20000 iterations <- seq(0,numIterations,by=1000) df <-data.frame(iterations,costs) ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") + xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations") #Plot the decision boundary plotDecisionBoundary(Z,retvals,hiddenActivationFunc="tanh",0.3) ## 3c. Classifying a lemniscate with Deep Learning Network – Relu activation(Octave code) Octave is used to generate the non-linear lemniscate boundary.  # Read the data data=csvread("lemniscate.csv"); X=data(:,1:2); Y=data(:,3); # Set the dimensions of the layers layersDimensions = [2 9 7 1] # Compute the DL network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', learningRate = 0.20, numIterations = 10000); plotCostVsIterations(10000,costs); plotDecisionBoundary(data,weights, biases,hiddenActivationFunc="relu")  ## 4a. Binary Classification using MNIST – Python code Finally I perform a simple classification using the MNIST handwritten digits, which according to Prof Geoffrey Hinton is “the Drosophila of Deep Learning”. The Python code for reading the MNIST data is taken from Alex Kesling’s github link MNIST. In the Python code below, I perform a simple binary classification between the handwritten digit ‘5’ and ‘not 5’ which is all other digits. I will perform the proper classification of all digits using the Softmax classifier some time later. import os import numpy as np import matplotlib.pyplot as plt os.chdir("C:\\software\\DeepLearning-Posts\\part3") execfile("./DLfunctions34.py") execfile("./load_mnist.py") training=list(read(dataset='training',path="./mnist")) test=list(read(dataset='testing',path="./mnist")) lbls=[] pxls=[] print(len(training)) # Select the first 10000 training data and the labels for i in range(10000): l,p=training[i] lbls.append(l) pxls.append(p) labels= np.array(lbls) pixels=np.array(pxls) # Sey y=1 when labels == 5 and 0 otherwise y=(labels==5).reshape(-1,1) X=pixels.reshape(pixels.shape[0],-1) # Create the necessary feature and target variable X1=X.T Y1=y.T # Create the layer dimensions. The number of features are 28 x 28 = 784 since the 28 x 28 # pixels is flattened to single vector of length 784. layersDimensions=[784, 15,9,7,1] # Works very well parameters = L_Layer_DeepModel(X1, Y1, layersDimensions, hiddenActivationFunc='relu', learning_rate = 0.1,num_iterations = 1000, fig="fig7.png") # Test data lbls1=[] pxls1=[] for i in range(800): l,p=test[i] lbls1.append(l) pxls1.append(p) testLabels=np.array(lbls1) testData=np.array(pxls1) ytest=(testLabels==5).reshape(-1,1) Xtest=testData.reshape(testData.shape[0],-1) Xtest1=Xtest.T Ytest1=ytest.T yhat = predict(parameters,Xtest1) from sklearn.metrics import confusion_matrix a=confusion_matrix(Ytest1.T,yhat.T) from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score print('Accuracy: {:.2f}'.format(accuracy_score(Ytest1.T, yhat.T))) print('Precision: {:.2f}'.format(precision_score(Ytest1.T, yhat.T))) print('Recall: {:.2f}'.format(recall_score(Ytest1.T, yhat.T))) print('F1: {:.2f}'.format(f1_score(Ytest1.T, yhat.T))) probs=predict_proba(parameters,Xtest1) from sklearn.metrics import precision_recall_curve precision, recall, thresholds = precision_recall_curve(Ytest1.T, probs.T) closest_zero = np.argmin(np.abs(thresholds)) closest_zero_p = precision[closest_zero] closest_zero_r = recall[closest_zero] plt.xlim([0.0, 1.01]) plt.ylim([0.0, 1.01]) plt.plot(precision, recall, label='Precision-Recall Curve') plt.plot(closest_zero_p, closest_zero_r, 'o', markersize = 12, fillstyle = 'none', c='r', mew=3) plt.xlabel('Precision', fontsize=16) plt.ylabel('Recall', fontsize=16) plt.savefig("fig8.png",bbox_inches='tight')   ## Accuracy: 0.99 ## Precision: 0.96 ## Recall: 0.89 ## F1: 0.92 In addition to plotting the Cost vs Iterations, I also plot the Precision-Recall curve to show how the Precision and Recall, which are complementary to each other vary with respect to the other. To know more about Precision-Recall, please check my post Practical Machine Learning with R and Python – Part 4. Check out my compact and minimal book “Practical Machine Learning with R and Python:Second edition- Machine Learning in stereo” available in Amazon in paperback($10.99) and kindle($7.99) versions. My book includes implementations of key ML algorithms and associated measures and metrics. The book is ideal for anybody who is familiar with the concepts and would like a quick reference to the different ML algorithms that can be applied to problems and how to select the best model. Pick your copy today!! A physical copy of the book is much better than scrolling down a webpage. Personally, I tend to use my own book quite frequently to refer to R, Python constructs, subsetting, machine Learning function calls and the necessary parameters etc. It is useless to commit any of this to memory, and a physical copy of a book is much easier to thumb through for the relevant code snippet. Pick up your copy today! ## 4b. Binary Classification using MNIST – R code In the R code below the same binary classification of the digit ‘5’ and the ‘not 5’ is performed. The code to read and display the MNIST data is taken from Brendan O’ Connor’s github link at MNIST source("mnist.R") load_mnist() #show_digit(train$x[2,]
layersDimensions=c(784, 7,7,3,1) # Works at 1500
x <- t(train$x) # Choose only 5000 training data x2 <- x[,1:5000] y <-train$y
# Set labels for all digits that are 'not 5' to 0
y[y!=5] <- 0
# Set labels of digit 5 as 1
y[y==5] <- 1
# Set the data
y1 <- as.matrix(y)
y2 <- t(y1)
# Choose the 1st 5000 data
y3 <- y2[,1:5000]

#Execute the Deep Learning Model
retvals = L_Layer_DeepModel(x2, y3, layersDimensions,
hiddenActivationFunc='tanh',
learningRate = 0.3,
numIterations = 3000, print_cost = True)
# Plot cost vs iteration
costs <- retvals[['costs']]
numIterations = 3000
iterations <- seq(0,numIterations,by=1000)
df <-data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") +
xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations")

# Compute probability scores
scores <- computeScores(retvals$parameters, x2,hiddenActivationFunc='relu') a=y3==1 b=y3==0 # Compute probabilities of class 0 and class 1 class1=scores[a] class0=scores[b] # Plot ROC curve pr <-pr.curve(scores.class0=class1, scores.class1=class0, curve=T) plot(pr) The AUC curve hugs the top left corner and hence the performance of the classifier is quite good. ## 4c. Binary Classification using MNIST – Octave code This code to load MNIST data was taken from Daniel E blog. Precision recall curves are available in Matlab but are yet to be implemented in Octave’s statistics package.  load('./mnist/mnist.txt.gz'); % load the dataset # Subset the 'not 5' digits a=(trainY != 5); # Subset '5' b=(trainY == 5); #make a copy of trainY #Set 'not 5' as 0 and '5' as 1 y=trainY; y(a)=0; y(b)=1; X=trainX(1:5000,:); Y=y(1:5000); # Set the dimensions of layer layersDimensions=[784, 7,7,3,1]; # Compute the DL network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', learningRate = 0.1, numIterations = 5000);  # Conclusion It was quite a challenge coding a Deep Learning Network in Python, R and Octave. The Deep Learning network implementation, in this post,is the base Deep Learning network, without any of the regularization methods included. Here are some key learning that I got while playing with different multi-layer networks on different problems a. Deep Learning Networks come with many levers, the hyper-parameters, – learning rate – activation unit – number of hidden layers – number of units per hidden layer – number of iterations while performing gradient descent b. Deep Networks are very sensitive. A change in any of the hyper-parameter makes it perform very differently c. Initially I thought adding more hidden layers, or more units per hidden layer will make the DL network better at learning. On the contrary, there is a performance degradation after the optimal DL configuration d. At a sub-optimal number of hidden layers or number of hidden units, gradient descent seems to get stuck at a local minima e. There were occasions when the cost came down, only to increase slowly as the number of iterations were increased. Probably early stopping would have helped. f. I also did come across situations of ‘exploding/vanishing gradient’, cost went to Inf/-Inf. Here I would think inclusion of ‘momentum method’ would have helped I intend to add the additional hyper-parameters of L1, L2 regularization, momentum method, early stopping etc. into the code in my future posts. Feel free to fork/clone the code from Github Deep Learning – Part 3, and take the DL network apart and play around with it. I will be continuing this series with more hyper-parameters to handle vanishing and exploding gradients, early stopping and regularization in the weeks to come. I also intend to add some more activation functions to this basic Multi-Layer Network. Hang around, there are more exciting things to come. Watch this space! To see all posts see Index of posts # Deep Learning from first principles in Python, R and Octave – Part 2 “What does the world outside your head really ‘look’ like? Not only is there no color, there’s also no sound: the compression and expansion of air is picked up by the ears, and turned into electrical signals. The brain then presents these signals to us as mellifluous tones and swishes and clatters and jangles. Reality is also odorless: there’s no such thing as smell outside our brains. Molecules floating through the air bind to receptors in our nose and are interpreted as different smells by our brain. The real world is not full of rich sensory events; instead, our brains light up the world with their own sensuality.” The Brain: The Story of You” by David Eagleman The world is Maya, illusory. The ultimate reality, the Brahman, is all-pervading and all-permeating, which is colourless, odourless, tasteless, nameless and formless Bhagavad Gita ## 1. Introduction This post is a follow-up post to my earlier post Deep Learning from first principles in Python, R and Octave-Part 1. In the first part, I implemented Logistic Regression, in vectorized Python,R and Octave, with a wannabe Neural Network (a Neural Network with no hidden layers). In this second part, I implement a regular, but somewhat primitive Neural Network (a Neural Network with just 1 hidden layer). The 2nd part implements classification of manually created datasets, where the different clusters of the 2 classes are not linearly separable. Neural Network perform really well in learning all sorts of non-linear boundaries between classes. Initially logistic regression is used perform the classification and the decision boundary is plotted. Vanilla logistic regression performs quite poorly. Using SVMs with a radial basis kernel would have performed much better in creating non-linear boundaries. To see R and Python implementations of SVMs take a look at my post Practical Machine Learning with R and Python – Part 4. 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 ($18.99) 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. Take a look at my video presentation which discusses the below derivation step-by- step Elements of Neural Networks and Deep Learning – Part 3 You can clone and fork this R Markdown file along with the vectorized implementations of the 3 layer Neural Network for Python, R and Octave from Github DeepLearning-Part2 ### 2. The 3 layer Neural Network A simple representation of a 3 layer Neural Network (NN) with 1 hidden layer is shown below. In the above Neural Network, there are 2 input features at the input layer, 3 hidden units at the hidden layer and 1 output layer as it deals with binary classification. The activation unit at the hidden layer can be a tanh, sigmoid, relu etc. At the output layer the activation is a sigmoid to handle binary classification # Superscript indicates layer 1 $z_{11} = w_{11}^{1}x_{1} + w_{21}^{1}x_{2} + b_{1}$ $z_{12} = w_{12}^{1}x_{1} + w_{22}^{1}x_{2} + b_{1}$ $z_{13} = w_{13}^{1}x_{1} + w_{23}^{1}x_{2} + b_{1}$ Also $a_{11} = tanh(z_{11})$ $a_{12} = tanh(z_{12})$ $a_{13} = tanh(z_{13})$ # Superscript indicates layer 2 $z_{21} = w_{11}^{2}a_{11} + w_{21}^{2}a_{12} + w_{31}^{2}a_{13} + b_{2}$ $a_{21} = sigmoid(z21)$ Hence $Z1= \begin{pmatrix} z11\\ z12\\ z13 \end{pmatrix} =\begin{pmatrix} w_{11}^{1} & w_{21}^{1} \\ w_{12}^{1} & w_{22}^{1} \\ w_{13}^{1} & w_{23}^{1} \end{pmatrix} * \begin{pmatrix} x1\\ x2 \end{pmatrix} + b_{1}$ And $A1= \begin{pmatrix} a11\\ a12\\ a13 \end{pmatrix} = \begin{pmatrix} tanh(z11)\\ tanh(z12)\\ tanh(z13) \end{pmatrix}$ Similarly $Z2= z_{21} = \begin{pmatrix} w_{11}^{2} & w_{21}^{2} & w_{31}^{2} \end{pmatrix} *\begin{pmatrix} z_{11}\\ z_{12}\\ z_{13} \end{pmatrix} +b_{2}$ and $A2 = a_{21} = sigmoid(z_{21})$ These equations can be written as $Z1 = W1 * X + b1$ $A1 = tanh(Z1)$ $Z2 = W2 * A1 + b2$ $A2 = sigmoid(Z2)$ I) Some important results (a memory refresher!) $d/dx(e^{x}) = e^{x}$ and $d/dx(e^{-x}) = -e^{-x}$ -(a) and $sinhx = (e^{x} - e^{-x})/2$ and $coshx = (e^{x} + e^{-x})/2$ Using (a) we can shown that $d/dx(sinhx) = coshx$ and $d/dx(coshx) = sinhx$ (b) Now $d/dx(f(x)/g(x)) = (g(x)*d/dx(f(x)) - f(x)*d/dx(g(x)))/g(x)^{2}$ -(c) Since $tanhx =z= sinhx/coshx$ and using (b) we get $tanhx = (coshx*d/dx(sinhx) - sinhx*d/dx(coshx))/(cosh^{2})$ Using the values of the derivatives of sinhx and coshx from (b) above we get $d/dx(tanhx) = (coshx^{2} - sinhx{2})/coshx{2} = 1 - tanhx^{2}$ Since $tanhx =z$ $d/dx(tanhx) = 1 - tanhx^{2}= 1 - z^{2}$ -(d) II) Derivatives $L=-(Ylog(A2) + (1-Y)log(1-A2))$ $dL/dA2 = -(Y/A2 + (1-Y)/(1-A2))$ Since $A2 = sigmoid(Z2)$ therefore $dA2/dZ2 = A2(1-A2)$ see Part1 $Z2 = W2A1 +b2$ $dZ2/dW2 = A1$ $dZ2/db2 = 1$ $A1 = tanh(Z1)$ and $dA1/dZ1 = 1 - A1^{2}$ $Z1 = W1X + b1$ $dZ1/dW1 = X$ $dZ1/db1 = 1$ III) Back propagation Using the derivatives from II) we can derive the following results using Chain Rule $\partial L/\partial Z2 = \partial L/\partial A2 * \partial A2/\partial Z2$ $= -(Y/A2 + (1-Y)/(1-A2)) * A2(1-A2) = A2 - Y$ $\partial L/\partial W2 = \partial L/\partial A2 * \partial A2/\partial Z2 * \partial Z2/\partial W2$ $= (A2-Y) *A1$ -(A) $\partial L/\partial b2 = \partial L/\partial A2 * \partial A2/\partial Z2 * \partial Z2/\partial b2 = (A2-Y)$ -(B) $\partial L/\partial Z1 = \partial L/\partial A2 * \partial A2/\partial Z2 * \partial Z2/\partial A1 *\partial A1/\partial Z1 = (A2-Y) * W2 * (1-A1^{2})$ $\partial L/\partial W1 = \partial L/\partial A2 * \partial A2/\partial Z2 * \partial Z2/\partial A1 *\partial A1/\partial Z1 *\partial Z1/\partial W1$ $=(A2-Y) * W2 * (1-A1^{2}) * X$ -(C) $\partial L/\partial b1 = \partial L/\partial A2 * \partial A2/\partial Z2 * \partial Z2/\partial A1 *dA1/dZ1 *dZ1/db1$ $= (A2-Y) * W2 * (1-A1^{2})$ -(D) IV) Gradient Descent The key computations in the backward cycle are $W1 = W1-learningRate * \partial L/\partial W1$ – From (C) $b1 = b1-learningRate * \partial L/\partial b1$ – From (D) $W2 = W2-learningRate * \partial L/\partial W2$ – From (A) $b2 = b2-learningRate * \partial L/\partial b2$ – From (B) The weights and biases (W1,b1,W2,b2) are updated for each iteration thus minimizing the loss/cost. These derivations can be represented pictorially using the computation graph (from the book Deep Learning by Ian Goodfellow, Joshua Bengio and Aaron Courville) ### 3. Manually create a data set that is not lineary separable Initially I create a dataset with 2 classes which has around 9 clusters that cannot be separated by linear boundaries. Note: This data set is saved as data.csv and is used for the R and Octave Neural networks to see how they perform on the same dataset. import numpy as np import matplotlib.pyplot as plt import matplotlib.colors import sklearn.linear_model from sklearn.model_selection import train_test_split from sklearn.datasets import make_classification, make_blobs from matplotlib.colors import ListedColormap import sklearn import sklearn.datasets colors=['black','gold'] cmap = matplotlib.colors.ListedColormap(colors) X, y = make_blobs(n_samples = 400, n_features = 2, centers = 7, cluster_std = 1.3, random_state = 4) #Create 2 classes y=y.reshape(400,1) y = y % 2 #Plot the figure plt.figure() plt.title('Non-linearly separable classes') plt.scatter(X[:,0], X[:,1], c=y, marker= 'o', s=50,cmap=cmap) plt.savefig('fig1.png', bbox_inches='tight') ### 4. Logistic Regression On the above created dataset, classification with logistic regression is performed, and the decision boundary is plotted. It can be seen that logistic regression performs quite poorly import numpy as np import matplotlib.pyplot as plt import matplotlib.colors import sklearn.linear_model from sklearn.model_selection import train_test_split from sklearn.datasets import make_classification, make_blobs from matplotlib.colors import ListedColormap import sklearn import sklearn.datasets #from DLfunctions import plot_decision_boundary execfile("./DLfunctions.py") # Since import does not work in Rmd!!! colors=['black','gold'] cmap = matplotlib.colors.ListedColormap(colors) X, y = make_blobs(n_samples = 400, n_features = 2, centers = 7, cluster_std = 1.3, random_state = 4) #Create 2 classes y=y.reshape(400,1) y = y % 2 # Train the logistic regression classifier clf = sklearn.linear_model.LogisticRegressionCV(); clf.fit(X, y); # Plot the decision boundary for logistic regression plot_decision_boundary_n(lambda x: clf.predict(x), X.T, y.T,"fig2.png")  ### 5. The 3 layer Neural Network in Python (vectorized) The vectorized implementation is included below. Note that in the case of Python a learning rate of 0.5 and 3 hidden units performs very well. ## Random data set with 9 clusters import numpy as np import matplotlib import matplotlib.pyplot as plt import sklearn.linear_model import pandas as pd from sklearn.datasets import make_classification, make_blobs execfile("./DLfunctions.py") # Since import does not work in Rmd!!! X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9, cluster_std = 1.3, random_state = 4) #Create 2 classes Y1=Y1.reshape(400,1) Y1 = Y1 % 2 X2=X1.T Y2=Y1.T #Perform gradient descent parameters,costs = computeNN(X2, Y2, numHidden = 4, learningRate=0.5, numIterations = 10000) plot_decision_boundary(lambda x: predict(parameters, x.T), X2, Y2,str(4),str(0.5),"fig3.png") ## Cost after iteration 0: 0.692669 ## Cost after iteration 1000: 0.246650 ## Cost after iteration 2000: 0.227801 ## Cost after iteration 3000: 0.226809 ## Cost after iteration 4000: 0.226518 ## Cost after iteration 5000: 0.226331 ## Cost after iteration 6000: 0.226194 ## Cost after iteration 7000: 0.226085 ## Cost after iteration 8000: 0.225994 ## Cost after iteration 9000: 0.225915 ### 6. The 3 layer Neural Network in R (vectorized) For this the dataset created by Python is saved to see how R performs on the same dataset. The vectorized implementation of a Neural Network was just a little more interesting as R does not have a similar package like ‘numpy’. While numpy handles broadcasting implicitly, in R I had to use the ‘sweep’ command to broadcast. The implementaion is included below. Note that since the initialization with random weights is slightly different, R performs best with a learning rate of 0.1 and with 6 hidden units source("DLfunctions2_1.R") z <- as.matrix(read.csv("data.csv",header=FALSE)) # x <- z[,1:2] y <- z[,3] x1 <- t(x) y1 <- t(y) #Perform gradient descent nn <-computeNN(x1, y1, 6, learningRate=0.1,numIterations=10000) # Good ## [1] 0.7075341 ## [1] 0.2606695 ## [1] 0.2198039 ## [1] 0.2091238 ## [1] 0.211146 ## [1] 0.2108461 ## [1] 0.2105351 ## [1] 0.210211 ## [1] 0.2099104 ## [1] 0.2096437 ## [1] 0.209409 plotDecisionBoundary(z,nn,6,0.1) ### 7. The 3 layer Neural Network in Octave (vectorized) This uses the same dataset that was generated using Python code. source("DL-function2.m") data=csvread("data.csv"); X=data(:,1:2); Y=data(:,3); # Make sure that the model parameters are correct. Take the transpose of X & Y #Perform gradient descent [W1,b1,W2,b2,costs]= computeNN(X', Y',4, learningRate=0.5, numIterations = 10000); ### 8a. Performance for different learning rates (Python) import numpy as np import matplotlib import matplotlib.pyplot as plt import sklearn.linear_model import pandas as pd from sklearn.datasets import make_classification, make_blobs execfile("./DLfunctions.py") # Since import does not work in Rmd!!! # Create data X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9, cluster_std = 1.3, random_state = 4) #Create 2 classes Y1=Y1.reshape(400,1) Y1 = Y1 % 2 X2=X1.T Y2=Y1.T # Create a list of learning rates learningRate=[0.5,1.2,3.0] df=pd.DataFrame() #Compute costs for each learning rate for lr in learningRate: parameters,costs = computeNN(X2, Y2, numHidden = 4, learningRate=lr, numIterations = 10000) print(costs) df1=pd.DataFrame(costs) df=pd.concat([df,df1],axis=1) #Set the iterations iterations=[0,1000,2000,3000,4000,5000,6000,7000,8000,9000] #Create data frame #Set index df1=df.set_index([iterations]) df1.columns=[0.5,1.2,3.0] fig=df1.plot() fig=plt.title("Cost vs No of Iterations for different learning rates") plt.savefig('fig4.png', bbox_inches='tight') ### 8b. Performance for different hidden units (Python) import numpy as np import matplotlib import matplotlib.pyplot as plt import sklearn.linear_model import pandas as pd from sklearn.datasets import make_classification, make_blobs execfile("./DLfunctions.py") # Since import does not work in Rmd!!! #Create data set X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9, cluster_std = 1.3, random_state = 4) #Create 2 classes Y1=Y1.reshape(400,1) Y1 = Y1 % 2 X2=X1.T Y2=Y1.T # Make a list of hidden unis numHidden=[3,5,7] df=pd.DataFrame() #Compute costs for different hidden units for numHid in numHidden: parameters,costs = computeNN(X2, Y2, numHidden = numHid, learningRate=1.2, numIterations = 10000) print(costs) df1=pd.DataFrame(costs) df=pd.concat([df,df1],axis=1) #Set the iterations iterations=[0,1000,2000,3000,4000,5000,6000,7000,8000,9000] #Set index df1=df.set_index([iterations]) df1.columns=[3,5,7] #Plot fig=df1.plot() fig=plt.title("Cost vs No of Iterations for different no of hidden units") plt.savefig('fig5.png', bbox_inches='tight') ### 9a. Performance for different learning rates (R) source("DLfunctions2_1.R") # Read data z <- as.matrix(read.csv("data.csv",header=FALSE)) # x <- z[,1:2] y <- z[,3] x1 <- t(x) y1 <- t(y) #Loop through learning rates and compute costs learningRate <-c(0.1,1.2,3.0) df <- NULL for(i in seq_along(learningRate)){ nn <- computeNN(x1, y1, 6, learningRate=learningRate[i],numIterations=10000) cost <- nn$costs
df <- cbind(df,cost)

}      

#Create dataframe
df <- data.frame(df)
iterations=seq(0,10000,by=1000)
df <- cbind(iterations,df)
names(df) <- c("iterations","0.5","1.2","3.0")
library(reshape2)
df1 <- melt(df,id="iterations")  # Melt the data
#Plot
ggplot(df1) + geom_line(aes(x=iterations,y=value,colour=variable),size=1)  +
xlab("Iterations") +
ylab('Cost') + ggtitle("Cost vs No iterations for  different learning rates")

### 9b. Performance  for different hidden units (R)

source("DLfunctions2_1.R")
# Loop through Num hidden units
numHidden <-c(4,6,9)
df <- NULL
for(i in seq_along(numHidden)){
nn <-  computeNN(x1, y1, numHidden[i], learningRate=0.1,numIterations=10000)
cost <- nn$costs df <- cbind(df,cost) }  df <- data.frame(df) iterations=seq(0,10000,by=1000) df <- cbind(iterations,df) names(df) <- c("iterations","4","6","9") library(reshape2) # Melt df1 <- melt(df,id="iterations") # Plot ggplot(df1) + geom_line(aes(x=iterations,y=value,colour=variable),size=1) + xlab("Iterations") + ylab('Cost') + ggtitle("Cost vs No iterations for different number of hidden units") ## 10a. Performance of the Neural Network for different learning rates (Octave) source("DL-function2.m") plotLRCostVsIterations() print -djph figa.jpg ## 10b. Performance of the Neural Network for different number of hidden units (Octave) source("DL-function2.m") plotHiddenCostVsIterations() print -djph figa.jpg ## 11. Turning the heat on the Neural Network In this 2nd part I create a a central region of positives and and the outside region as negatives. The points are generated using the equation of a circle (x – a)^{2} + (y -b) ^{2} = R^{2} . How does the 3 layer Neural Network perform on this? Here’s a look! Note: The same dataset is also used for R and Octave Neural Network constructions ## 12. Manually creating a circular central region import numpy as np import matplotlib.pyplot as plt import matplotlib.colors import sklearn.linear_model from sklearn.model_selection import train_test_split from sklearn.datasets import make_classification, make_blobs from matplotlib.colors import ListedColormap import sklearn import sklearn.datasets colors=['black','gold'] cmap = matplotlib.colors.ListedColormap(colors) x1=np.random.uniform(0,10,800).reshape(800,1) x2=np.random.uniform(0,10,800).reshape(800,1) X=np.append(x1,x2,axis=1) X.shape # Create (x-a)^2 + (y-b)^2 = R^2 # Create a subset of values where squared is <0,4. Perform ravel() to flatten this vector a=(np.power(X[:,0]-5,2) + np.power(X[:,1]-5,2) <= 6).ravel() Y=a.reshape(800,1) cmap = matplotlib.colors.ListedColormap(colors) plt.figure() plt.title('Non-linearly separable classes') plt.scatter(X[:,0], X[:,1], c=Y, marker= 'o', s=15,cmap=cmap) plt.savefig('fig6.png', bbox_inches='tight') ### 13a. Decision boundary with hidden units=4 and learning rate = 2.2 (Python) With the above hyper parameters the decision boundary is triangular import numpy as np import matplotlib.pyplot as plt import matplotlib.colors import sklearn.linear_model execfile("./DLfunctions.py") x1=np.random.uniform(0,10,800).reshape(800,1) x2=np.random.uniform(0,10,800).reshape(800,1) X=np.append(x1,x2,axis=1) X.shape # Create a subset of values where squared is <0,4. Perform ravel() to flatten this vector a=(np.power(X[:,0]-5,2) + np.power(X[:,1]-5,2) <= 6).ravel() Y=a.reshape(800,1) X2=X.T Y2=Y.T parameters,costs = computeNN(X2, Y2, numHidden = 4, learningRate=2.2, numIterations = 10000) plot_decision_boundary(lambda x: predict(parameters, x.T), X2, Y2,str(4),str(2.2),"fig7.png")  ## Cost after iteration 0: 0.692836 ## Cost after iteration 1000: 0.331052 ## Cost after iteration 2000: 0.326428 ## Cost after iteration 3000: 0.474887 ## Cost after iteration 4000: 0.247989 ## Cost after iteration 5000: 0.218009 ## Cost after iteration 6000: 0.201034 ## Cost after iteration 7000: 0.197030 ## Cost after iteration 8000: 0.193507 ## Cost after iteration 9000: 0.191949 ### 13b. Decision boundary with hidden units=12 and learning rate = 2.2 (Python) With the above hyper parameters the decision boundary is triangular import numpy as np import matplotlib.pyplot as plt import matplotlib.colors import sklearn.linear_model execfile("./DLfunctions.py") x1=np.random.uniform(0,10,800).reshape(800,1) x2=np.random.uniform(0,10,800).reshape(800,1) X=np.append(x1,x2,axis=1) X.shape # Create a subset of values where squared is <0,4. Perform ravel() to flatten this vector a=(np.power(X[:,0]-5,2) + np.power(X[:,1]-5,2) <= 6).ravel() Y=a.reshape(800,1) X2=X.T Y2=Y.T parameters,costs = computeNN(X2, Y2, numHidden = 12, learningRate=2.2, numIterations = 10000) plot_decision_boundary(lambda x: predict(parameters, x.T), X2, Y2,str(12),str(2.2),"fig8.png")  ## Cost after iteration 0: 0.693291 ## Cost after iteration 1000: 0.383318 ## Cost after iteration 2000: 0.298807 ## Cost after iteration 3000: 0.251735 ## Cost after iteration 4000: 0.177843 ## Cost after iteration 5000: 0.130414 ## Cost after iteration 6000: 0.152400 ## Cost after iteration 7000: 0.065359 ## Cost after iteration 8000: 0.050921 ## Cost after iteration 9000: 0.039719 ### 14a. Decision boundary with hidden units=9 and learning rate = 0.5 (R) When the number of hidden units is 6 and the learning rate is 0,1, is also a triangular shape in R source("DLfunctions2_1.R") z <- as.matrix(read.csv("data1.csv",header=FALSE)) # N x <- z[,1:2] y <- z[,3] x1 <- t(x) y1 <- t(y) nn <-computeNN(x1, y1, 9, learningRate=0.5,numIterations=10000) # Triangular ## [1] 0.8398838 ## [1] 0.3303621 ## [1] 0.3127731 ## [1] 0.3012791 ## [1] 0.3305543 ## [1] 0.3303964 ## [1] 0.2334615 ## [1] 0.1920771 ## [1] 0.2341225 ## [1] 0.2188118 ## [1] 0.2082687 plotDecisionBoundary(z,nn,6,0.1) ### 14b. Decision boundary with hidden units=8 and learning rate = 0.1 (R) source("DLfunctions2_1.R") z <- as.matrix(read.csv("data1.csv",header=FALSE)) # N x <- z[,1:2] y <- z[,3] x1 <- t(x) y1 <- t(y) nn <-computeNN(x1, y1, 8, learningRate=0.1,numIterations=10000) # Hemisphere ## [1] 0.7273279 ## [1] 0.3169335 ## [1] 0.2378464 ## [1] 0.1688635 ## [1] 0.1368466 ## [1] 0.120664 ## [1] 0.111211 ## [1] 0.1043362 ## [1] 0.09800573 ## [1] 0.09126161 ## [1] 0.0840379 plotDecisionBoundary(z,nn,8,0.1) ### 15a. Decision boundary with hidden units=12 and learning rate = 1.5 (Octave) source("DL-function2.m") data=csvread("data1.csv"); X=data(:,1:2); Y=data(:,3); # Make sure that the model parameters are correct. Take the transpose of X & Y [W1,b1,W2,b2,costs]= computeNN(X', Y',12, learningRate=1.5, numIterations = 10000); plotDecisionBoundary(data, W1,b1,W2,b2) print -djpg fige.jpg Conclusion: This post implemented a 3 layer Neural Network to create non-linear boundaries while performing classification. Clearly the Neural Network performs very well when the number of hidden units and learning rate are varied. To be continued… Watch this space!! To see all posts check Index of posts # 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 $\sigma=1/(1+e^{-z})$ 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) Finally $\partial L/\partial w_{i}= \partial L/\partial a * \partial a/\partial Z * \partial Z/\partial w_{i}$ -(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) Also $\partial L/\partial b = \partial L/\partial a * \partial a/\partial Z * \partial Z/\partial b$ -(j) Since $\partial Z/\partial b = 1$ 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): a=1/(1+np.exp(-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))) return(loss) # Execute the forward propagation def forwardPropagation(w,b,X,Y): # Compute Z Z=np.dot(w.T,X)+b # Determine the number of training samples numTraining=float(len(X)) # Compute the output of the sigmoid activation function A=sigmoid(Z) #Compute the loss loss = computeLoss(numTraining,Y,A) # Compute the gradients dZ, dw and db dZ=A-Y dw=1/numTraining*np.dot(X,dZ.T) db=1/numTraining*np.sum(dZ) # 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): losses=[] idx =[] # Iterate for i in range(numIerations): gradients,loss=forwardPropagation(w,b,X,Y) #Get the derivates dw = gradients["dw"] db = gradients["db"] w = w-learningRate*dw b = b-learningRate*db # Store the loss if i % 100 == 0: idx.append(i) losses.append(loss) # 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): size=X.shape[1] yPredicted=np.zeros((1,size)) Z=np.dot(w.T,X) # Compute the sigmoid A=sigmoid(Z) for i in range(A.shape[1]): #If the value is > 0.5 then set as 1 if(A[0][i] > 0.5): yPredicted[0][i]=1 else: # Else set as 0 yPredicted[0][i]=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 X_train1=normalize(X_train) # Create weight vectors of zeros. The size is the number of features in the data set=30 w=np.zeros((X_train.shape[1],1)) #w=np.zeros((30,1)) b=0 #Normalize the training data so that gradient descent performs better X_train1=normalize(X_train) #Transpose X_train so that we have a matrix as (features, numSamples) X_train2=X_train1.T # Reshape to remove the rank 1 array and then transpose y_train1=y_train.reshape(len(y_train),1) y_train2=y_train1.T # 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 X_test1=normalize(X_test) #Transpose X_train so that we have a matrix as (features, numSamples) X_test2=X_test1.T #Reshape y_test y_test1=y_test.reshape(len(y_test),1) y_test2=y_test1.T # 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.plot(idx,costs) fig1=plt.title("Gradient descent-Cost vs No of iterations") fig1=plt.xlabel("No of iterations") fig1=plt.ylabel("Cost") 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) source("RFunctions-1.R") # Define the sigmoid function sigmoid <- function(z){ a <- 1/(1+ exp(-z)) a } # Compute the loss computeLoss <- function(numTraining,Y,A){ loss <- -1/numTraining* sum(Y*log(A) + (1-Y)*log(1-A)) return(loss) } # 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 A=sigmoid(Z) #Compute the loss loss <- computeLoss(numTraining,Y,A) # Compute the gradients dZ, dw and db dZ<-A-Y dw<-1/numTraining * X %*% t(dZ) db<-1/numTraining*sum(dZ) fwdProp <- list("loss" = loss, "dw" = dw, "db" = db) return(fwdProp) } # 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

}

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

return(yPredicted)
}

# Normalize the matrix
normalize <- function(x){
#Create the norm of the matrix.Perform the Frobenius norm of the matrix
n<-as.matrix(sqrt(rowSums(x^2)))
#Sweep by rows by norm. Note '1' in the function which performing on every row
normalized<-sweep(x, 1, n, FUN="/")
return(normalized)
}

# Run the 2 layer Neural Network on the cancer data set
# Read the data (from sklearn)
# 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)
X_train2=t(X_train1)

# Reshape  then transpose
y_train1=as.matrix(y_train)
y_train2=t(y_train1)

# Normalize X_test
X_test1=normalize(X_test)
#Transpose X_train so that we have a matrix as (features, numSamples)
X_test2=t(X_test1)

#Reshape y_test and take transpose
y_test1=as.matrix(y_test)
y_test2=t(y_test1)

# 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)

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

end
 # Compute Gradient Descent function [w,b,dw,db,losses,index]=gradientDescent(w, b, X, Y, numIerations, learningRate) #Initialize losses and idx losses=[]; index=[]; # 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]; endif

end
end
 # Determine the predicted value for dataset function yPredicted = predict(w,b,X) m = size(X)(1,2); yPredicted=zeros(1,m); # 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) yPredicted(1,i)=1; else yPredicted(1,i)=0; endif end end
 # 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; end
 # 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)); train=dataset(ix(1:trainSize),:); # And test test=dataset(ix(trainSize+1:length(dataset)),:); X_train = train(:,1:30); y_train = train(:,31); X_test = test(:,1:30); y_test = test(:,31); end

 cancer=csvread("cancer.csv"); [X_train,y_train,X_test,y_test] = trainTestSplit(cancer,75); w=zeros(size(X_train)(1,2),1); b=0; X_train1=normalize(X_train); X_train2=X_train1'; y_train1=y_train'; [w1,b1,dw,db,losses,idx]=gradientDescent(w, b, X_train2, y_train1, numIerations=3000, learningRate=0.75); # Normalize X_test X_test1=normalize(X_test); #Transpose X_train so that we have a matrix as (features, numSamples) X_test2=X_test1'; y_test1=y_test'; # 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 graphics_toolkit('gnuplot') plot(idx,losses); title ('Gradient descent- Cost vs No of iterations'); xlabel ("No of iterations"); ylabel ("Cost");

Conclusion
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!!

To see all posts check Index of posts

# My travels through the realms of Data Science, Machine Learning, Deep Learning and (AI)

Then felt I like some watcher of the skies
When a new planet swims into his ken;
Or like stout Cortez when with eagle eyes
He star’d at the Pacific—and all his men
Look’d at each other with a wild surmise—
Silent, upon a peak in Darien.
On First Looking into Chapman’s Homer by John Keats

The above excerpt from John Keat’s poem captures the the exhilaration that one experiences, when discovering something for the first time. This also  summarizes to some extent my own as enjoyment while pursuing Data Science, Machine Learning and the like.

I decided to write this post, as occasionally youngsters approach me and ask me where they should start their adventure in Data Science & Machine Learning. There are other times, when the ‘not-so-youngsters’ want to know what their next step should be after having done some courses. This post includes my travels through the domains of Data Science, Machine Learning, Deep Learning and (soon to be done AI).

By no means, am I an authority in this field, which is ever-widening and almost bottomless, yet I would like to share some of my experiences in this fascinating field. I include a short review of the courses I have done below. I also include alternative routes through  courses which I did not do, but are probably equally good as well.  Feel free to pick and choose any course or set of courses. Alternatively, you may prefer to read books or attend bricks-n-mortar classes, In any case,  I hope the list below will provide you with some overall direction.

All my learning in the above domains have come from MOOCs and I restrict myself to the top 3 MOOCs, or in my opinion, ‘the original MOOCs’, namely Coursera, edX or Udacity, but may throw in some courses from other online sites if they are only available there. I would recommend these 3 MOOCs over the other numerous online courses and also over face-to-face classroom courses for the following reasons. These MOOCs

• Are taken by world class colleges and the lectures are delivered by top class Professors who have a great depth of knowledge and a wealth of experience
• The Professors, besides delivering quality content, also point out to important tips, tricks and traps
• You can revisit lectures in online courses
• Lectures are usually short between 8 -15 mins (Personally, my attention span is around 15-20 mins at a time!)

Here is a fair warning and something quite obvious. No amount of courses, lectures or books will help if you don’t put it to use through some language like Octave, R or Python.

The journey
My trip through Data Science, Machine Learning  started with an off-chance remark,about 3 years ago,  from an old friend of mine who spoke to me about having done a few  courses at Coursera, and really liked it.  He further suggested that I should try. This was the final push which set me sailing into this vast domain.

I have included the list of the courses I have done over the past 3 years (33 certifications completed and another 9 audited-listened only without doing the assignments). For each of the courses I have included a short review of the course, whether I think the course is mandatory, the language in which the course is based on, and finally whether I have done the course myself etc. I have also included alternative courses, which I may have not done, but which I think are equally good. Finally, I suggest some courses which I have heard of and which are very good and worth taking.

1. Machine Learning, Stanford, Prof Andrew Ng, Coursera
(Requirement: Mandatory, Language:Octave,Status:Completed)
This course provides an excellent foundation to build your Machine Learning citadel on. The course covers the mathematical details of linear, logistic and multivariate regression. There is also a good coverage of topics like Neural Networks, SVMs, Anamoly Detection, underfitting, overfitting, regularization etc. Prof Andrew Ng presents the material in a very lucid manner. It is a great course to start with. It would be a good idea to brush up  some basics of linear algebra, matrices and a little bit of calculus, specifically computing the local maxima/minima. You should be able to take this course even if you don’t know Octave as the Prof goes over the key aspects of the language.

2. Statistical Learning, Prof Trevor Hastie & Prof Robert Tibesherani, Online Stanford– (Requirement:Mandatory, Language:R, Status;Completed) –
The course includes linear and polynomial regression, logistic regression. Details also include cross-validation and the bootstrap methods, how to do model selection and regularization (ridge and lasso). It also touches on non-linear models, generalized additive models, boosting and SVMs. Some unsupervised learning methods are  also discussed. The 2 Professors take turns in delivering lectures with a slight touch of humor.

3a. Data Science Specialization: Prof Roger Peng, Prof Brian Caffo & Prof Jeff Leek, John Hopkins University (Requirement: Option A, Language: R Status: Completed)
This is a comprehensive 10 module specialization based on R. This Specialization gives a very broad overview of Data Science and Machine Learning. The modules cover R programming, Statistical Inference, Practical Machine Learning, how to build R products and R packages and finally has a very good Capstone project on NLP

3b. Applied Data Science with Python Specialization: University of Michigan (Requirement: Option B, Language: Python, Status: Not done)
In this specialization I only did  the Applied Machine Learning in Python (Prof Kevyn-Collin Thomson). This is a very good course that covers a lot of Machine Learning algorithms(linear, logistic, ridge, lasso regression, knn, SVMs etc. Also included are confusion matrices, ROC curves etc. This is based on Python’s Scikit Learn

3c. Machine Learning Specialization, University Of Washington (Requirement:Option C, Language:Python, Status : Not completed). This appears to be a very good Specialization in Python

4. Statistics with R Specialization, Duke University (Requirement: Useful and a must know, Language R, Status:Not Completed)
I audited (listened only) to the following 2 modules from this Specialization.
a.Inferential Statistics
b.Linear Regression and Modeling
Both these courses are taught by Prof Mine Cetikya-Rundel who delivers her lessons with extraordinary clarity.  Her lectures are filled with many examples which she walks you through in great detail

5.Bayesian Statistics: From Concept to Data Analysis: Univ of California, Santa Cruz (Requirement: Optional, Language : R, Status:Completed)
This is an interesting course and provides an alternative point of view to frequentist approach

6. Data Science and Engineering with Spark, University of California, Berkeley, Prof Antony Joseph, Prof Ameet Talwalkar, Prof Jon Bates
(Required: Mandatory for Big Data, Status:Completed, Language; pySpark)
This specialization contains 3 modules
a.Introduction to Apache Spark
b.Distributed Machine Learning with Apache Spark
c.Big Data Analysis with Apache Spark

This is an excellent course for those who want to make an entry into Distributed Machine Learning. The exercises are fairly challenging and your code will predominantly be made of map/reduce and lambda operations as you process data that is distributed across Spark RDDs. I really liked  the part where the Prof shows how a matrix multiplication on a single machine is of the order of O(nd^2+d^3) (which is the basis of Machine Learning) is reduced to O(nd^2) by taking outer products on data which is distributed.

7. Deep Learning Prof Andrew Ng, Younes Bensouda Mourri, Kian Katanforoosh : Requirement:Mandatory,Language:Python, Tensorflow Status:Completed)

This course had 5 Modules which start from the fundamentals of Neural Networks, their derivation and vectorized Python implementation. The specialization also covers regularization, optimization techniques, mini batch normalization, Convolutional Neural Networks, Recurrent Neural Networks, LSTMs applied to a wide variety of real world problems

The modules are
a. Neural Networks and Deep Learning
In this course Prof Andrew Ng explains differential calculus, linear algebra and vectorized Python implementations of Deep Learning algorithms. The derivation for back-propagation is done and then the Prof shows how to compute a multi-layered DL network
b.Improving Deep Neural Networks: Hyperparameter tuning, Regularization and Optimization
Deep Neural Networks can be very flexible, and come with a lots of knobs (hyper-parameters) to tune with. In this module, Prof Andrew Ng shows a systematic way to tune hyperparameters and by how much should one tune. The course also covers regularization(L1,L2,dropout), gradient descent optimization and batch normalization methods. The visualizations used to explain the momentum method, RMSprop, Adam,LR decay and batch normalization are really powerful and serve to clarify the concepts. As an added bonus,the module also includes a great introduction to Tensorflow.
c.Structuring Machine Learning Projects
A very good module with useful tips, tricks and traps that need to be considered while working on Machine Learning and Deep Learning projects
d. Convolutional Neural Networks
This domain has a lot of really cool ideas, where images represented as 3D volumes, are compressed and stretched longitudinally before applying a multi-layered deep learning neural network to this thin slice for performing classification,detection etc. The Prof provides a glimpse into this fascinating world of image classification, detection andl neural art transfer with frameworks like Keras and Tensorflow.
e. Sequence Models
In this module covers in good detail concepts like RNNs, GRUs, LSTMs, word embeddings, beam search and attention model.

8. Neural Networks for Machine Learning, Prof Geoffrey Hinton,University of Toronto
(Requirement: Mandatory, Language;Octave, Status:Completed)
This is a broad course which starts from the basic of Perceptrons, all the way to Boltzman Machines, RNNs, CNNS, LSTMs etc The course also covers regularization, learning rate decay, momentum method etc

9.Probabilistic Graphical Models, Stanford  Prof Daphne Koller(Language:Octave, Status: Partially completed)
This has 3 courses
a.Probabilistic Graphical Models 1: Representation – Done
b.Probabilistic Graphical Models 2: Inference – To do
c.Probabilistic Graphical Models 3: Learning – To do
This course discusses how a system, which can be represented as a complex interaction
of probability distributions, will behave. This is probably the toughest course I did.  I did manage to get through the 1st module, While I felt that grasped a few things, I did not wholly understand the import of this. However I feel this is an important domain and I will definitely revisit this in future

10. Mining Massive Data Sets Prof Jure Leskovec, Prof Anand Rajaraman and ProfJeff Ullman. Online Stanford, Status Partially done.
I did quickly audit this course, a year back, when it used to be in Coursera. It now seems to have moved to Stanford online. But this is a very good course that discusses key concepts of Mining Big Data of the order a few Petabytes

11. Introduction to Artificial Intelligence, Prof Sebastian Thrun & Prof Peter Norvig, Udacity
This is a really good course. I have started on this course a couple of times and somehow gave up. Will revisit to complete in future. Quite extensive in its coverage.Touches BFS,DFS, A-Star, PGM, Machine Learning etc.

12. Deep Learning (with TensorFlow), Vincent Vanhoucke, Principal Scientist at Google Brain.
Got started on this one and abandoned some time back. In my to do list though

My learning journey is based on Lao Tzu’s dictum of ‘A good traveler has no fixed plans and is not intent on arriving’. You could have a goal and try to plan your courses accordingly.
And so my journey continues…

I hope you find this list useful.

# Neural Networks: On Perceptrons and Sigmoid Neurons

Neural Networks had their beginnings in 1943 when Warren McCulloch, a neurophysiologist, and a young mathematician, Walter Pitts, wrote a paper on how neurons might work.  Much later in 1958, Frank Rosenblatt, a neuro-biologist proposed the Perceptron. The Perceptron is a computer model or computerized machine which is devised to represent or simulate the ability of the brain to recognize and discriminate. In machine learning, the perceptron is an algorithm for supervised learning of binary classifiers

Initially it was believed that  Perceptrons were capable of many things including “the ability to walk, talk, see, write, reproduce itself and be conscious of its existence.”

However, a subsequent paper by Marvin Minky and Seymour Papert of MIT, titled “Perceptrons” proved that the Perceptron was truly limited in its functionality. Specifically they showed that the Perceptron was incapable of producing XOR functionality. The Perceptron is only capable of classification where the data points are linearly separable.

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 ($18.99) and in kindle version($9.99/Rs449).

This post implements the simple learning algorithm of the ‘Linear Perceptron’ and the ‘Sigmoid Perceptron’.  The implementation has been done in Octave. This implementation is based on “Neural networks for Machine Learning” course by Prof Geoffrey Hinton at Coursera

Perceptron learning procedure
z = ∑wixi  + b
where wi is the ith weight and xi is the ith  feature

For every training case compute the activation output zi

• If the output classifies correctly, leave the weights alone
• If the output classifies a ‘0’ as a ‘1’, then subtract the the feature from the weight
• If the output classifies a ‘0’ as a ‘1’, then add the feature to the weight

This simple neural network is represented below

Sigmoid neuron learning procedure
zi = sigmoid(∑wixi  + b)
where sigmoid is
$sigmoid(z) = 1/1+e^{-z}$

Hence
$z_{i} = 1/1+e^{-(\sum w_{i}x_{i}+b)}$
For every training case compute the activation output zi

• If the output classifies correctly, leave the weights alone
• If the output incorrectly classifies a ‘0’ as a ‘1’ i.e. $z_{i} >sigmoid(0)$, then subtract the feature from the weight
• If the output incorrectly classifies a ‘1’ as ‘0’ i.e., i.e $z_{i} < sigmoid(0)$, then add the feature to the weight
• Iterate till errors <= 1

This is shown below

I have implemented the learning algorithm of the Perceptron and Sigmoid Neuron in Octave. The code is available at Github at Perceptron.

1. Perceptron execution

I performed the tests on 2 different datasets

Data 1

Data 2

2. Sigmoid Perceptron execution
Data 1 & Data 2

It can be seen that the Perceptron does work for simple linearly separable data. I will be implementing other more advanced Neural Networks in the months to come.

Watch this space!

# Video presentation on Machine Learning, Data Science, NLP and Big Data – Part 1

Here is the 1st part of my video presentation on “Machine Learning, Data Science, NLP and Big Data – Part 1”