Category: backpropagation

Understanding Neural Style Transfer with Tensorflow and Keras

Neural Style Transfer (NST)  is a fascinating area of Deep Learning and Convolutional Neural Networks. NST is an interesting technique, in which the style from an image, known as the ‘style image’ is transferred to another image ‘content image’ and we get a third a image which is a generated image which has the content of the original image and the style of another image.

NST can be used to reimagine how famous painters like Van Gogh, Claude Monet or a Picasso would have visualised a scenery or architecture. NST uses Convolutional Neural Networks (CNNs) to achieve this artistic style transfer from one image to another. NST was originally implemented by Gati et al., in their paper Neural Algorithm of Artistic Style. Convolutional Neural Networks have been very successful in image classification image recognition et cetera. CNN networks have also been have also generated very interesting pictures using Neural Style Transfer which will be shown in this post. An interesting aspect of CNN’s is that the first couple of layers in the CNN capture basic features of the image like edges and  pixel values. But as we go deeper into the CNN, the network captures higher level features of the input image.

To get started with Neural Style transfer  we will be using the VGG19 pre-trained network. The VGG19 CNN is a compact pre-trained your network which can be used for performing the NST. However, we could have also used Resnet or InceptionV3 networks for this purpose but these are very large networks. The idea of using a network trained on a different task and applying it to a new task is called transfer learning.

What needs to be done to transfer the style from one of the image to another image. This brings us to the question – What is ‘style’? What is it that distinguishes Van Gogh’s painting or Picasso’s cubist art. Convolutional Neural Networks capture basic features in the lower layers and much more complex features in the deeper layers.  Style can be computed by taking the correlation of the feature maps in a layer L. This is my interpretation of how style is captured.  Since style  is intrinsic to  the image, it  implies that the style feature would exist across all the filters in a layer. Hence, to pick up this style we would need to get the correlation of the filters across channels of a lawyer. This is computed mathematically, using the Gram matrix which calculates the correlation of the activation of a the filter by the style image and generated image

To transfer the style from one image to the content image we need to do two parallel operations while doing forward propagation
– Compute the content loss between the source image and the generated image
– Compute the style loss between the style image and the generated image
– Finally we need to compute the total loss

In order to get transfer the style from the ‘style’ image to the ‘content ‘image resulting in a  ‘generated’  image  the total loss has to be minimised. Therefore backward propagation with gradient descent  is done to minimise the total loss comprising of the content and style loss.

Initially we make the Generated Image ‘G’ the same as the source image ‘S’

The content loss at layer ‘l’

L_{content} = 1/2 \sum_{i}^{j} ( F^{l}_{i,j} - P^{l}_{i,j})^{2}

where F^{l}_{i,j} and P^{l}_{i,j} represent the activations at layer ‘l’ in a filter i, at position ‘j’. The intuition is that the activations will be same for similar source and generated image. We need to minimise the content loss so that the generated stylized image is as close to the original image as possible. An intermediate layer of VGG19 block5_conv2 is used

The Style layers that are are used are

style_layers = [‘block1_conv1’,
‘block2_conv1’,
‘block3_conv1’,
‘block4_conv1’,
‘block5_conv1’]
To compute the Style Loss the Gram matrix needs to be computed. The Gram Matrix is computed by unrolling the filters as shown below (source: Convolutional Neural Networks by Prof Andrew Ng, Coursera). The result is a matrix of size n_{c} x n_{c} where n_{c} is the number of channels
The above diagram shows the filters of height n_{H} and width n_{W} with n_{C} channels
The contribution of layer ‘l’ to style loss is given by
L^{'}_{style} = \frac{\sum_{i}^{j} (G^{2}_{i,j} - A^l{i,j})^2}{4N^{2}_{l}M^{2}_{l}}
where G_{i,j}  and A_{i,j} are the Gram matrices of the style and generated images respectively. By minimising the distance in the gram matrices of the style and generated image we can ensure that generated image is a stylized version of the original image similar to the style image
The total loss is given by
L_{total} = \alpha L_{content} + \beta L_{style}
Back propagation with gradient descent works to minimise the content loss between the source and generated image, while the style loss tries to minimise the discrepancies in the style of the style image and generated image. Running through forward and backpropagation through several epochs successfully transfers the style from the style image to the source image.
You can check the Notebook at Neural Style Transfer

Note: The code in this notebook is largely based on the Neural Style Transfer tutorial from Tensorflow, though I may have taken some changes from other blogs. I also made a few changes to the code in this tutorial, like removing the scaling factor, or the class definition (Personally, I belong to the old school (C language) and am not much in love with the ‘self.”..All references are included below

Note: Here is a interesting thought. Could we do a Neural Style Transfer in music? Imagine Carlos Santana playing ‘Hotel California’ or Brian May style in ‘Another brick in the wall’. While our first reaction would be that it may not sound good as we are used to style of these songs, we may be surprised by a possible style transfer. This is definitely music to the ears!

 

Here are few runs from this

A) Run 1

1. Neural Style Transfer – a) Content Image – My portrait.  b) Style Image – Wassily Kadinsky Oil on canvas, 1913, Vassily Kadinsky’s composition

 

2. Result of Neural Style Transfer

 

 

2) Run 2

a) Content Image – Portrait of my parents b) Style Image –  Vincent Van Gogh’s ,Starry Night Oil on canvas 1889

 

2. Result of Neural Style Transfer

 

 

Run 3

1.  Content Image – Caesar 2 (Masai Mara- 20 Jun 2018).  Style Image – The Great Wave at Kanagawa – Katsushika Hokosai, 1826-1833

 

Screenshot 2020-04-12 at 12.40.44 PM

2. Result of Neural Style Transfer

lkg

 

 

Run 4

1.   Content Image – Junagarh Fort , Rajasthan   Sep 2016              b) Style Image – Le Pont Japonais by Claude Monet, Oil on canvas, 1920

 

 

2. Result of Neural Style Transfer

 

Neural Style Transfer is a very ingenious idea which shows that we can segregate the style of a painting and transfer to another image.

References

1. A Neural Algorithm of Artistic Style, Leon A. Gatys, Alexander S. Ecker, Matthias Bethge
2. Neural style transfer
3. Neural Style Transfer: Creating Art with Deep Learning using tf.keras and eager execution
4. Convolutional Neural Network, DeepLearning.AI Specialization, Prof Andrew Ng
5. Intuitive Guide to Neural Style Transfer

See also

1. Big Data-5: kNiFi-ing through cricket data with yorkpy
2. Cricketr adds team analytics to its repertoire
3. Cricpy performs granular analysis of players
4. My book ‘Deep Learning from first principles:Second Edition’ now on Amazon
5. Programming Zen and now – Some essential tips-2
6. The Anomaly
7. Practical Machine Learning with R and Python – Part 5
8. Literacy in India – A deepR dive
9. “Is it an animal? Is it an insect?” in Android

To see all posts click Index of posts

Take 4+: Presentations on ‘Elements of Neural Networks and Deep Learning’ – Parts 1-8

“Lights, camera and … action – Take 4+!”

This post includes  a rework of all presentation of ‘Elements of Neural Networks and Deep  Learning Parts 1-8 ‘ since my earlier presentations had some missing parts, omissions and some occasional errors. So I have re-recorded all the presentations.
This series of presentation will do a deep-dive  into Deep Learning networks starting from the fundamentals. The equations required for performing learning in a L-layer Deep Learning network  are derived in detail, starting from the basics. Further, the presentations also discuss multi-class classification, regularization techniques, and gradient descent optimization methods in deep networks methods. Finally the presentations also touch on how  Deep Learning Networks can be tuned.

The corresponding implementations are available in vectorized R, Python and Octave are available in my book ‘Deep Learning from first principles:Second edition- In vectorized Python, R and Octave‘

1. Elements of Neural Networks and Deep Learning – Part 1
This presentation introduces Neural Networks and Deep Learning. A look at history of Neural Networks, Perceptrons and why Deep Learning networks are required and concluding with a simple toy examples of a Neural Network and how they compute. This part also includes a small digression on the basics of Machine Learning and how the algorithm learns from a data set

2. Elements of Neural Networks and Deep Learning – Part 2
This presentation takes logistic regression as an example and creates an equivalent 2 layer Neural network. The presentation also takes a look at forward & backward propagation and how the cost is minimized using gradient descent


The implementation of the discussed 2 layer Neural Network in vectorized R, Python and Octave are available in my post ‘Deep Learning from first principles in Python, R and Octave – Part 1‘

3. Elements of Neural Networks and Deep Learning – Part 3
This 3rd part, discusses a primitive neural network with an input layer, output layer and a hidden layer. The neural network uses tanh activation in the hidden layer and a sigmoid activation in the output layer. The equations for forward and backward propagation are derived.


To see the implementations for the above discussed video see my post ‘Deep Learning from first principles in Python, R and Octave – Part 2‘

4. Elements of Neural Network and Deep Learning – Part 4
This presentation is a continuation of my 3rd presentation in which I derived the equations for a simple 3 layer Neural Network with 1 hidden layer. In this video presentation, I discuss step-by-step the derivations for a L-Layer, multi-unit Deep Learning Network, with any activation function g(z)


The implementations of L-Layer, multi-unit Deep Learning Network in vectorized R, Python and Octave are available in my post Deep Learning from first principles in Python, R and Octave – Part 3

5. Elements of Neural Network and Deep Learning – Part 5
This presentation discusses multi-class classification using the Softmax function. The detailed derivation for the Jacobian of the Softmax is discussed, and subsequently the derivative of cross-entropy loss is also discussed in detail. Finally the final set of equations for a Neural Network with multi-class classification is derived.


The corresponding implementations in vectorized R, Python and Octave are available in the following posts
a. Deep Learning from first principles in Python, R and Octave – Part 4
b. Deep Learning from first principles in Python, R and Octave – Part 5

6. Elements of Neural Networks and Deep Learning – Part 6
This part discusses initialization methods specifically like He and Xavier. The presentation also focuses on how to prevent over-fitting using regularization. Lastly the dropout method of regularization is also discussed


The corresponding implementations in vectorized R, Python and Octave of the above discussed methods are available in my post Deep Learning from first principles in Python, R and Octave – Part 6

7. Elements of Neural Networks and Deep Learning – Part 7
This presentation introduces exponentially weighted moving average and shows how this is used in different approaches to gradient descent optimization. The key techniques discussed are learning rate decay, momentum method, rmsprop and adam.

The equivalent implementations of the gradient descent optimization techniques in R, Python and Octave can be seen in my post Deep Learning from first principles in Python, R and Octave – Part 7

8. Elements of Neural Networks and Deep Learning – Part 8
This last part touches on the method to adopt while tuning hyper-parameters in Deep Learning networks

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 concludes this series of presentations on “Elements of Neural Networks and Deep Learning’

Also
1. My book ‘Practical Machine Learning in R and Python: Third edition’ on Amazon
2. Introducing cricpy:A python package to analyze performances of cricketers
3. Natural language processing: What would Shakespeare say?
4. Big Data-2: Move into the big league:Graduate from R to SparkR
5. Presentation on Wireless Technologies – Part 1
6. Introducing cricketr! : An R package to analyze performances of cricketers

To see all posts click Index of posts

My presentations on ‘Elements of Neural Networks & Deep Learning’ -Parts 6,7,8

This is the final set of presentations in my series ‘Elements of Neural Networks and Deep Learning’. This set follows the earlier 2 sets of presentations namely
1. My presentations on ‘Elements of Neural Networks & Deep Learning’ -Part1,2,3
2. My presentations on ‘Elements of Neural Networks & Deep Learning’ -Parts 4,5

In this final set of presentations I discuss initialization methods, regularization techniques including dropout. Next I also discuss gradient descent optimization methods like momentum, rmsprop, adam etc. Lastly, I briefly also touch on hyper-parameter tuning approaches. The corresponding implementations are available in vectorized R, Python and Octave are available in my book ‘Deep Learning from first principles:Second edition- In vectorized Python, R and Octave‘

1. Elements of Neural Networks and Deep Learning – Part 6
This part discusses initialization methods specifically like He and Xavier. The presentation also focuses on how to prevent over-fitting using regularization. Lastly the dropout method of regularization is also discusses


The corresponding implementations in vectorized R, Python and Octave of the above discussed methods are available in my post Deep Learning from first principles in Python, R and Octave – Part 6

2. Elements of Neural Networks and Deep Learning – Part 7
This presentation introduces exponentially weighted moving average and shows how this is used in different approaches to gradient descent optimization. The key techniques discussed are learning rate decay, momentum method, rmsprop and adam.


The equivalent implementations of the gradient descent optimization techniques in R, Python and Octave can be seen in my post Deep Learning from first principles in Python, R and Octave – Part 7

3. Elements of Neural Networks and Deep Learning – Part 8
This last part touches upon hyper-parameter tuning in Deep Learning networks


This concludes this series of presentations on “Elements of Neural Networks and Deep Learning’

Important note: Do check out my later version of these videos at Take 4+: Presentations on ‘Elements of Neural Networks and Deep Learning’ – Parts 1-8 . These have more content and also include some corrections. Check it out!

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 and in kindle version($9.99/Rs449).

See also
1. My book ‘Practical Machine Learning in R and Python: Third edition’ on Amazon
2. Big Data-1: Move into the big league:Graduate from Python to Pyspark
3. My travels through the realms of Data Science, Machine Learning, Deep Learning and (AI)
4. Revisiting crimes against women in India
5. Introducing cricket package yorkr: Part 1- Beaten by sheer pace!
6. Deblurring with OpenCV: Weiner filter reloaded
7. Taking a closer look at Quantum gates and their operations

To see all posts click Index of posts

My presentations on ‘Elements of Neural Networks & Deep Learning’ -Parts 4,5

This is the next set of presentations on “Elements of Neural Networks and Deep Learning”.  In the 4th presentation I discuss and derive the generalized equations for a multi-unit, multi-layer Deep Learning network.  The 5th presentation derives the equations for a Deep Learning network when performing multi-class classification along with the derivations for cross-entropy loss. The corresponding implementations are available in vectorized R, Python and Octave are available in my book ‘Deep Learning from first principles:Second edition- In vectorized Python, R and Octave‘

Important note: Do check out my later version of these videos at Take 4+: Presentations on ‘Elements of Neural Networks and Deep Learning’ – Parts 1-8 . These have more content and also include some corrections. Check it out!

1. Elements of Neural Network and Deep Learning – Part 4
This presentation is a continuation of my 3rd presentation in which I derived the equations for a simple 3 layer Neural Network with 1 hidden layer. In this video presentation, I discuss step-by-step the derivations for a L-Layer, multi-unit Deep Learning Network, with any activation function g(z)


The implementations of L-Layer, multi-unit Deep Learning Network in vectorized R, Python and Octave are available in my post Deep Learning from first principles in Python, R and Octave – Part 3

2. Elements of Neural Network and Deep Learning – Part 5
This presentation discusses multi-class classification using the Softmax function. The detailed derivation for the Jacobian of the Softmax is discussed, and subsequently the derivative of cross-entropy loss is also discussed in detail. Finally the final set of equations for a Neural Network with multi-class classification is derived.


The corresponding implementations in vectorized R, Python and Octave are available in the following posts
a. Deep Learning from first principles in Python, R and Octave – Part 4
b. Deep Learning from first principles in Python, R and Octave – Part 5

To be continued. Watch this space!

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).

Also see
1. My book ‘Practical Machine Learning in R and Python: Third edition’ on Amazon
2. Big Data-2: Move into the big league:Graduate from R to SparkR
3. Introducing QCSimulator: A 5-qubit quantum computing simulator in R
4. My TEDx talk on the “Internet of Things
5. Rock N’ Roll with Bluemix, Cloudant & NodeExpress
6. GooglyPlus: yorkr analyzes IPL players, teams, matches with plots and tables
7. Literacy in India – A deepR dive
8. Fun simulation of a Chain in Android

To see all posts click Index of Posts

My presentations on ‘Elements of Neural Networks & Deep Learning’ -Part1,2,3

I will be uploading a series of presentations on ‘Elements of Neural Networks and Deep Learning’. In these video presentations I discuss the derivations of L -Layer Deep Learning Networks, starting from the basics. The corresponding implementations are available in vectorized R, Python and Octave are available in my book ‘Deep Learning from first principles:Second edition- In vectorized Python, R and Octave‘

1. Elements of Neural Networks and Deep Learning – Part 1
This presentation introduces Neural Networks and Deep Learning. A look at history of Neural Networks, Perceptrons and why Deep Learning networks are required and concluding with a simple toy examples of a Neural Network and how they compute

2. Elements of Neural Networks and Deep Learning – Part 2
This presentation takes logistic regression as an example and creates an equivalent 2 layer Neural network. The presentation also takes a look at forward & backward propagation and how the cost is minimized using gradient descent


The implementation of the discussed 2 layer Neural Network in vectorized R, Python and Octave are available in my post ‘Deep Learning from first principles in Python, R and Octave – Part 1‘

3. Elements of Neural Networks and Deep Learning – Part 3
This 3rd part, discusses a primitive neural network with an input layer, output layer and a hidden layer. The neural network uses tanh activation in the hidden layer and a sigmoid activation in the output layer. The equations for forward and backward propagation are derived.


To see the implementations for the above discussed video see my post ‘Deep Learning from first principles in Python, R and Octave – Part 2‘

Important note: Do check out my later version of these videos at Take 4+: Presentations on ‘Elements of Neural Networks and Deep Learning’ – Parts 1-8 . These have more content and also include some corrections. Check it out!

To be continued. Watch this space!

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
1. My book ‘Practical Machine Learning in R and Python: Third edition’ on Amazon
2. Introducing cricpy:A python package to analyze performances of cricketers
3. Natural language processing: What would Shakespeare say?
4. TWS-4: Gossip protocol: Epidemics and rumors to the rescue
5. Getting started with memcached-libmemcached
6. Simplifying ML: Impact of degree of polynomial degree on bias & variance and other insights

To see all posts click Index of posts

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

Untitled

 

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

Also see
1. My book ‘Practical Machine Learning in R and Python: Second edition’ on Amazon
2. The 3rd paperback & kindle editions of my books on Cricket, now on Amazon
3. De-blurring revisited with Wiener filter using OpenCV
4. TWS-4: Gossip protocol: Epidemics and rumors to the rescue
5. A Cloud medley with IBM Bluemix, Cloudant DB and Node.js
6. Practical Machine Learning with R and Python – Part 6
7. GooglyPlus: yorkr analyzes IPL players, teams, matches with plots and tables
8. Fun simulation of a Chain in Android

To see posts click Index of Posts

My book “Deep Learning from first principles” now on Amazon

Note: The 2nd edition of this book is now available on Amazon

My 4th book(self-published), “Deep Learning from first principles – In vectorized Python, R and Octave” (557 pages), is now available on Amazon in both paperback ($18.99) and kindle ($9.99/Rs449). The book starts with the most primitive 2-layer Neural Network and works  its way to a generic L-layer Deep Learning Network, with all the bells and whistles.  The book includes detailed derivations and vectorized implementations in Python, R and Octave.  The code has been extensively  commented and has been included in the Appendix section.

Pick up your copy today!!!

My other books
1. Practical Machine Learning with R and Python
2. Beaten by sheer pace – Cricket analytics with yorkr
3. Cricket analytics with cricketr

Deep Learning from first principles in Python, R and Octave – Part 8

1. Introduction

You don’t understand anything until you learn it more than one way. Marvin Minsky
No computer has ever been designed that is ever aware of what it’s doing; but most of the time, we aren’t either. Marvin Minsky
A wealth of information creates a poverty of attention. Herbert Simon

This post, Deep Learning from first Principles in Python, R and Octave-Part8, is my final post in my Deep Learning from first principles series. In this post, I discuss and implement a key functionality needed while building Deep Learning networks viz. ‘Gradient Checking’. Gradient Checking is an important method to check the correctness of your implementation, specifically the forward propagation and the backward propagation cycles of an implementation. In addition I also discuss some tips for tuning hyper-parameters of a Deep Learning network based on my experience.

My post in this  ‘Deep Learning Series’ so far were
1. Deep Learning from first principles in Python, R and Octave – Part 1 In part 1, I implement logistic regression as a neural network in vectorized Python, R and Octave
2. Deep Learning from first principles in Python, R and Octave – Part 2 In the second part I implement a simple Neural network with just 1 hidden layer and a sigmoid activation output function
3. Deep Learning from first principles in Python, R and Octave – Part 3 The 3rd part implemented a multi-layer Deep Learning Network with sigmoid activation output in vectorized Python, R and Octave
4. Deep Learning from first principles in Python, R and Octave – Part 4 The 4th part deals with multi-class classification. Specifically, I derive the Jacobian of the Softmax function and enhance my L-Layer DL network to include Softmax output function in addition to Sigmoid activation
5. Deep Learning from first principles in Python, R and Octave – Part 5 This post uses the Softmax classifier implemented to classify MNIST digits using a L-layer Deep Learning network
6. Deep Learning from first principles in Python, R and Octave – Part 6 The 6th part adds more bells and whistles to my L-Layer DL network, by including different initialization types namely He and Xavier. Besides L2 Regularization and random dropout is added.
7. Deep Learning from first principles in Python, R and Octave – Part 7 The 7th part deals with Stochastic Gradient Descent Optimization methods including momentum, RMSProp and Adam
8. Deep Learning from first principles in Python, R and Octave – Part 8 – This post implements a critical function for ensuring the correctness of a L-Layer Deep Learning network implementation using Gradient Checking

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- Machine Learning in stereo” available in Amazon in paperback($9.99) and Kindle($6.99) 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.

Gradient Checking is based on the following approach. One iteration of Gradient Descent computes and updates the parameters \theta by doing
\theta := \theta - \frac{d}{d\theta}J(\theta).
To minimize the cost we will need to minimize J(\theta)
Let g(\theta) be a function that computes the derivative \frac {d}{d\theta}J(\theta). Gradient Checking allows us to numerically evaluate the implementation of the function g(\theta) and verify its correctness.
We know the derivative of a function is given by
\frac {d}{d\theta}J(\theta) = lim->0 \frac {J(\theta +\epsilon) - J(\theta -\epsilon)} {2*\epsilon}
Note: The above derivative is based on the 2 sided derivative. The 1-sided derivative  is given by \frac {d}{d\theta}J(\theta) = lim->0 \frac {J(\theta +\epsilon) - J(\theta)} {\epsilon}
Gradient Checking is based on the 2-sided derivative because the error is of the order O(\epsilon^{2}) as opposed O(\epsilon) for the 1-sided derivative.
Hence Gradient Check uses the 2 sided derivative as follows.
g(\theta) = lim->0 \frac {J(\theta +\epsilon) - J(\theta -\epsilon)} {2*\epsilon}

In Gradient Check the following is done
A) Run one normal cycle of your implementation by doing the following
a) Compute the output activation by running 1 cycle of forward propagation
b) Compute the cost using the output activation
c) Compute the gradients using backpropation (grad)

B) Perform gradient check steps as below
a) Set \theta . Flatten all ‘weights’ and ‘bias’ matrices and vectors to a column vector.
b) Initialize \theta+ by bumping up \theta by adding \epsilon (\theta + \epsilon)
c) Perform forward propagation with \theta+
d) Compute cost with \theta+ i.e. J(\theta+)
e) Initialize  \theta- by bumping down \theta by subtracting \epsilon (\theta - \epsilon)
f) Perform forward propagation with \theta-
g) Compute cost with \theta- i.e.  J(\theta-)
h) Compute \frac {d} {d\theta} J(\theta) or ‘gradapprox’ as\frac {J(\theta+) - J(\theta-) } {2\epsilon} using the 2 sided derivative.
i) Compute L2norm or the Euclidean distance between ‘grad’ and ‘gradapprox’. If the
diference is of the order of 10^{-5} or 10^{-7} the implementation is correct. In the Deep Learning Specialization Prof Andrew Ng mentions that if the difference is of the order of 10^{-7} then the implementation is correct. A difference of 10^{-5} is also ok. Anything more than that is a cause of worry and you should look at your code more closely. To see more details click Gradient checking and advanced optimization

You can clone/download the code from Github at DeepLearning-Part8

After spending a better part of 3 days, I now realize how critical Gradient Check is for ensuring the correctness of you implementation. Initially I was getting very high difference and did not know how to understand the results or debug my implementation. After many hours of staring at the results, I  was able to finally arrive at a way, to localize issues in the implementation. In fact, I did catch a small bug in my Python code, which did not exist in the R and Octave implementations. I will demonstrate this below

1.1a Gradient Check – Sigmoid Activation – Python

import numpy as np
import matplotlib

exec(open("DLfunctions8.py").read())
exec(open("testcases.py").read())
#Load the data
train_X, train_Y, test_X, test_Y = load_dataset()
#Set layer dimensions
layersDimensions = [2,4,1]  
parameters = initializeDeepModel(layersDimensions)
#Perform forward prop
AL, caches, dropoutMat = forwardPropagationDeep(train_X, parameters, keep_prob=1, hiddenActivationFunc="relu",outputActivationFunc="sigmoid")
#Compute cost
cost = computeCost(AL, train_Y, outputActivationFunc="sigmoid") 
print("cost=",cost)
#Perform backprop and get gradients
gradients = backwardPropagationDeep(AL, train_Y, caches, dropoutMat, lambd=0, keep_prob=1,                                   hiddenActivationFunc="relu",outputActivationFunc="sigmoid")

epsilon = 1e-7
outputActivationFunc="sigmoid"

# Set-up variables
# Flatten parameters to a vector
parameters_values, _ = dictionary_to_vector(parameters)
#Flatten gradients to a vector
grad = gradients_to_vector(parameters,gradients)
num_parameters = parameters_values.shape[0]
#Initialize
J_plus = np.zeros((num_parameters, 1))
J_minus = np.zeros((num_parameters, 1))
gradapprox = np.zeros((num_parameters, 1))

# Compute gradapprox using 2 sided derivative
for i in range(num_parameters):
    # Compute J_plus[i]. 
    thetaplus = np.copy(parameters_values)                                   
    thetaplus[i][0] = thetaplus[i][0] + epsilon                                 
    AL, caches, dropoutMat = forwardPropagationDeep(train_X, vector_to_dictionary(parameters,thetaplus), keep_prob=1, 
                                              hiddenActivationFunc="relu",outputActivationFunc=outputActivationFunc)
    J_plus[i] = computeCost(AL, train_Y, outputActivationFunc=outputActivationFunc) 
    
    
    # Compute J_minus[i]. 
    thetaminus = np.copy(parameters_values)                                     
    thetaminus[i][0] = thetaminus[i][0] - epsilon                                     
    AL, caches, dropoutMat  = forwardPropagationDeep(train_X, vector_to_dictionary(parameters,thetaminus), keep_prob=1, 
                                              hiddenActivationFunc="relu",outputActivationFunc=outputActivationFunc)     
    J_minus[i] = computeCost(AL, train_Y, outputActivationFunc=outputActivationFunc)                            
       
    # Compute gradapprox[i]   
    gradapprox[i] = (J_plus[i] - J_minus[i])/(2*epsilon)

# Compare gradapprox to backward propagation gradients by computing difference. 
numerator = np.linalg.norm(grad-gradapprox)                                           
denominator = np.linalg.norm(grad) +  np.linalg.norm(gradapprox)                                         
difference =  numerator/denominator                                        

#Check the difference
if difference > 1e-5:
    print ("\033[93m" + "There is a mistake in the backward propagation! difference = " + str(difference) + "\033[0m")
else:
    print ("\033[92m" + "Your backward propagation works perfectly fine! difference = " + str(difference) + "\033[0m")
print(difference)
print("\n")    
# The technique below can be used to identify 
# which of the parameters are in error
# Covert grad to dictionary
m=vector_to_dictionary2(parameters,grad)
print("Gradients from backprop")
print(m)
print("\n")
# Convert gradapprox to dictionary
n=vector_to_dictionary2(parameters,gradapprox)
print("Gradapprox from gradient check")
print(n)

## (300, 2)
## (300,)
## cost= 0.6931455556341791
## [92mYour backward propagation works perfectly fine! difference = 1.1604150683743381e-06[0m
## 1.1604150683743381e-06
## 
## 
## Gradients from backprop
## {'dW1': array([[-6.19439955e-06, -2.06438046e-06],
##        [-1.50165447e-05,  7.50401672e-05],
##        [ 1.33435433e-04,  1.74112143e-04],
##        [-3.40909024e-05, -1.38363681e-04]]), 'db1': array([[ 7.31333221e-07],
##        [ 7.98425950e-06],
##        [ 8.15002817e-08],
##        [-5.69821155e-08]]), 'dW2': array([[2.73416304e-04, 2.96061451e-04, 7.51837363e-05, 1.01257729e-04]]), 'db2': array([[-7.22232235e-06]])}
## 
## 
## Gradapprox from gradient check
## {'dW1': array([[-6.19448937e-06, -2.06501483e-06],
##        [-1.50168766e-05,  7.50399742e-05],
##        [ 1.33435485e-04,  1.74112391e-04],
##        [-3.40910633e-05, -1.38363765e-04]]), 'db1': array([[ 7.31081862e-07],
##        [ 7.98472399e-06],
##        [ 8.16013923e-08],
##        [-5.71764858e-08]]), 'dW2': array([[2.73416290e-04, 2.96061509e-04, 7.51831930e-05, 1.01257891e-04]]), 'db2': array([[-7.22255589e-06]])}

1.1b Gradient Check – Softmax Activation – Python (Error!!)

In the code below I show, how I managed to spot a bug in your implementation

import numpy as np
exec(open("DLfunctions8.py").read())
N = 100 # number of points per class
D = 2 # dimensionality
K = 3 # number of classes
X = np.zeros((N*K,D)) # data matrix (each row = single example)
y = np.zeros(N*K, dtype='uint8') # class labels
for j in range(K):
  ix = range(N*j,N*(j+1))
  r = np.linspace(0.0,1,N) # radius
  t = np.linspace(j*4,(j+1)*4,N) + np.random.randn(N)*0.2 # theta
  X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]
  y[ix] = j


# Plot the data
#plt.scatter(X[:, 0], X[:, 1], c=y, s=40, cmap=plt.cm.Spectral)
layersDimensions = [2,3,3]
y1=y.reshape(-1,1).T
train_X=X.T
train_Y=y1

parameters = initializeDeepModel(layersDimensions)
#Compute forward prop
AL, caches, dropoutMat = forwardPropagationDeep(train_X, parameters, keep_prob=1, 
                                                hiddenActivationFunc="relu",outputActivationFunc="softmax")
#Compute cost
cost = computeCost(AL, train_Y, outputActivationFunc="softmax") 
print("cost=",cost)
#Compute gradients from backprop
gradients = backwardPropagationDeep(AL, train_Y, caches, dropoutMat, lambd=0, keep_prob=1, 
                                    hiddenActivationFunc="relu",outputActivationFunc="softmax")
# Note the transpose of the gradients for Softmax has to be taken
L= len(parameters)//2
print(L)
gradients['dW'+str(L)]=gradients['dW'+str(L)].T
gradients['db'+str(L)]=gradients['db'+str(L)].T
# Perform gradient check
gradient_check_n(parameters, gradients, train_X, train_Y, epsilon = 1e-7,outputActivationFunc="softmax")

cost= 1.0986187818144022
2
There is a mistake in the backward propagation! difference = 0.7100295155692544
0.7100295155692544


Gradients from backprop
{'dW1': array([[ 0.00050125,  0.00045194],
       [ 0.00096392,  0.00039641],
       [-0.00014276, -0.00045639]]), 'db1': array([[ 0.00070082],
       [-0.00224399],
       [ 0.00052305]]), 'dW2': array([[-8.40953794e-05, -9.52657769e-04, -1.10269379e-04],
       [-7.45469382e-04,  9.49795606e-04,  2.29045434e-04],
       [ 8.29564761e-04,  2.86216305e-06, -1.18776055e-04]]), 
     'db2': array([[-0.00253808],
       [-0.00505508],
       [ 0.00759315]])}


Gradapprox from gradient check
{'dW1': array([[ 0.00050125,  0.00045194],
       [ 0.00096392,  0.00039641],
       [-0.00014276, -0.00045639]]), 'db1': array([[ 0.00070082],
       [-0.00224399],
       [ 0.00052305]]), 'dW2': array([[-8.40960634e-05, -9.52657953e-04, -1.10268461e-04],
       [-7.45469242e-04,  9.49796908e-04,  2.29045671e-04],
       [ 8.29565305e-04,  2.86104473e-06, -1.18776100e-04]]), 
     'db2': array([[-8.46211989e-06],
       [-1.68487446e-05],
       [ 2.53108645e-05]])}

Gradient Check gives a high value of the difference of 0.7100295. Inspecting the Gradients and Gradapprox we can see there is a very big discrepancy in db2. After I went over my code I discovered that I my computation in the function layerActivationBackward for Softmax was

 
   # Erroneous code
   if activationFunc == 'softmax':
        dW = 1/numtraining * np.dot(A_prev,dZ)
        db = np.sum(dZ, axis=0, keepdims=True)
        dA_prev = np.dot(dZ,W)
instead of
   # Fixed code
   if activationFunc == 'softmax':
        dW = 1/numtraining * np.dot(A_prev,dZ)
        db = 1/numtraining *  np.sum(dZ, axis=0, keepdims=True)
        dA_prev = np.dot(dZ,W)

After fixing this error when I ran Gradient Check I get

1.1c Gradient Check – Softmax Activation – Python (Corrected!!)

import numpy as np
exec(open("DLfunctions8.py").read())
N = 100 # number of points per class
D = 2 # dimensionality
K = 3 # number of classes
X = np.zeros((N*K,D)) # data matrix (each row = single example)
y = np.zeros(N*K, dtype='uint8') # class labels
for j in range(K):
  ix = range(N*j,N*(j+1))
  r = np.linspace(0.0,1,N) # radius
  t = np.linspace(j*4,(j+1)*4,N) + np.random.randn(N)*0.2 # theta
  X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]
  y[ix] = j


# Plot the data
#plt.scatter(X[:, 0], X[:, 1], c=y, s=40, cmap=plt.cm.Spectral)
layersDimensions = [2,3,3]
y1=y.reshape(-1,1).T
train_X=X.T
train_Y=y1
#Set layer dimensions
parameters = initializeDeepModel(layersDimensions)
#Perform forward prop
AL, caches, dropoutMat = forwardPropagationDeep(train_X, parameters, keep_prob=1, 
                                                hiddenActivationFunc="relu",outputActivationFunc="softmax")
#Compute cost
cost = computeCost(AL, train_Y, outputActivationFunc="softmax") 
print("cost=",cost)
#Compute gradients from backprop
gradients = backwardPropagationDeep(AL, train_Y, caches, dropoutMat, lambd=0, keep_prob=1, 
                                    hiddenActivationFunc="relu",outputActivationFunc="softmax")
# Note the transpose of the gradients for Softmax has to be taken
L= len(parameters)//2
print(L)
gradients['dW'+str(L)]=gradients['dW'+str(L)].T
gradients['db'+str(L)]=gradients['db'+str(L)].T
#Perform gradient check
gradient_check_n(parameters, gradients, train_X, train_Y, epsilon = 1e-7,outputActivationFunc="softmax")
## cost= 1.0986193170234435
## 2
## [92mYour backward propagation works perfectly fine! difference = 5.268804859613151e-07[0m
## 5.268804859613151e-07
## 
## 
## Gradients from backprop
## {'dW1': array([[ 0.00053206,  0.00038987],
##        [ 0.00093941,  0.00038077],
##        [-0.00012177, -0.0004692 ]]), 'db1': array([[ 0.00072662],
##        [-0.00210198],
##        [ 0.00046741]]), 'dW2': array([[-7.83441270e-05, -9.70179498e-04, -1.08715815e-04],
##        [-7.70175008e-04,  9.54478237e-04,  2.27690198e-04],
##        [ 8.48519135e-04,  1.57012608e-05, -1.18974383e-04]]), 'db2': array([[-8.52190476e-06],
##        [-1.69954294e-05],
##        [ 2.55173342e-05]])}
## 
## 
## Gradapprox from gradient check
## {'dW1': array([[ 0.00053206,  0.00038987],
##        [ 0.00093941,  0.00038077],
##        [-0.00012177, -0.0004692 ]]), 'db1': array([[ 0.00072662],
##        [-0.00210198],
##        [ 0.00046741]]), 'dW2': array([[-7.83439980e-05, -9.70180603e-04, -1.08716369e-04],
##        [-7.70173925e-04,  9.54478718e-04,  2.27690089e-04],
##        [ 8.48520143e-04,  1.57018842e-05, -1.18973720e-04]]), 'db2': array([[-8.52096171e-06],
##        [-1.69964043e-05],
##        [ 2.55162558e-05]])}

1.2a Gradient Check – Sigmoid Activation – R

source("DLfunctions8.R")
z <- as.matrix(read.csv("circles.csv",header=FALSE)) 

x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
#Set layer dimensions
layersDimensions = c(2,5,1)
parameters = initializeDeepModel(layersDimensions)
#Perform forward prop
retvals = forwardPropagationDeep(X, parameters,keep_prob=1, hiddenActivationFunc="relu",
                                 outputActivationFunc="sigmoid")
AL <- retvals[['AL']]
caches <- retvals[['caches']]
dropoutMat <- retvals[['dropoutMat']]
#Compute cost
cost <- computeCost(AL, Y,outputActivationFunc="sigmoid",
                    numClasses=layersDimensions[length(layersDimensions)])
print(cost)
## [1] 0.6931447
# Backward propagation.
gradients = backwardPropagationDeep(AL, Y, caches, dropoutMat, lambd=0, keep_prob=1, hiddenActivationFunc="relu",
                                    outputActivationFunc="sigmoid",numClasses=layersDimensions[length(layersDimensions)])
epsilon = 1e-07
outputActivationFunc="sigmoid"
#Convert parameter list to vector
parameters_values = list_to_vector(parameters)
#Convert gradient list to vector
grad = gradients_to_vector(parameters,gradients)
num_parameters = dim(parameters_values)[1]
#Initialize
J_plus = matrix(rep(0,num_parameters),
                nrow=num_parameters,ncol=1)
J_minus = matrix(rep(0,num_parameters),
                 nrow=num_parameters,ncol=1)
gradapprox = matrix(rep(0,num_parameters),
                    nrow=num_parameters,ncol=1)

# Compute gradapprox
for(i in 1:num_parameters){
    # Compute J_plus[i]. 
    thetaplus = parameters_values                                   
    thetaplus[i][1] = thetaplus[i][1] + epsilon                                 
    retvals = forwardPropagationDeep(X, vector_to_list(parameters,thetaplus), keep_prob=1, 
                                                hiddenActivationFunc="relu",outputActivationFunc=outputActivationFunc)
    
    AL <- retvals[['AL']]
    J_plus[i] = computeCost(AL, Y, outputActivationFunc=outputActivationFunc) 


   # Compute J_minus[i]. 
    thetaminus = parameters_values                                     
    thetaminus[i][1] = thetaminus[i][1] - epsilon                                     
    retvals  = forwardPropagationDeep(X, vector_to_list(parameters,thetaminus), keep_prob=1, 
                                                 hiddenActivationFunc="relu",outputActivationFunc=outputActivationFunc)     
    AL <- retvals[['AL']]
    J_minus[i] = computeCost(AL, Y, outputActivationFunc=outputActivationFunc)                            

    # Compute gradapprox[i]   
    gradapprox[i] = (J_plus[i] - J_minus[i])/(2*epsilon)
}
# Compare gradapprox to backward propagation gradients by computing difference.
#Compute L2Norm
numerator = L2NormVec(grad-gradapprox)                                           
denominator = L2NormVec(grad) +  L2NormVec(gradapprox)                                         
difference =  numerator/denominator 
if(difference > 1e-5){
    cat("There is a mistake, the difference is too high",difference)
} else{
    cat("The implementations works perfectly", difference)
}
## The implementations works perfectly 1.279911e-06
# This can be used to check
print("Gradients from backprop")
## [1] "Gradients from backprop"
vector_to_list2(parameters,grad)
## $dW1
##               [,1]          [,2]
## [1,] -7.641588e-05 -3.427989e-07
## [2,] -9.049683e-06  6.906304e-05
## [3,]  3.401039e-06 -1.503914e-04
## [4,]  1.535226e-04 -1.686402e-04
## [5,] -6.029292e-05 -2.715648e-04
## 
## $db1
##               [,1]
## [1,]  6.930318e-06
## [2,] -3.283117e-05
## [3,]  1.310647e-05
## [4,] -3.454308e-05
## [5,] -2.331729e-08
## 
## $dW2
##              [,1]         [,2]         [,3]        [,4]         [,5]
## [1,] 0.0001612356 0.0001113475 0.0002435824 0.000362149 2.874116e-05
## 
## $db2
##              [,1]
## [1,] -1.16364e-05
print("Grad approx from gradient check")
## [1] "Grad approx from gradient check"
vector_to_list2(parameters,gradapprox)
## $dW1
##               [,1]          [,2]
## [1,] -7.641554e-05 -3.430589e-07
## [2,] -9.049428e-06  6.906253e-05
## [3,]  3.401168e-06 -1.503919e-04
## [4,]  1.535228e-04 -1.686401e-04
## [5,] -6.029288e-05 -2.715650e-04
## 
## $db1
##               [,1]
## [1,]  6.930012e-06
## [2,] -3.283096e-05
## [3,]  1.310618e-05
## [4,] -3.454237e-05
## [5,] -2.275957e-08
## 
## $dW2
##              [,1]         [,2]         [,3]         [,4]        [,5]
## [1,] 0.0001612355 0.0001113476 0.0002435829 0.0003621486 2.87409e-05
## 
## $db2
##              [,1]
## [1,] -1.16368e-05

1.2b Gradient Check – Softmax Activation – R

source("DLfunctions8.R")
Z <- as.matrix(read.csv("spiral.csv",header=FALSE)) 

# Setup the data
X <- Z[,1:2]
y <- Z[,3]
X <- t(X)
Y <- t(y)
layersDimensions = c(2, 3, 3)
parameters = initializeDeepModel(layersDimensions)
#Perform forward prop
retvals = forwardPropagationDeep(X, parameters,keep_prob=1, hiddenActivationFunc="relu",
                                 outputActivationFunc="softmax")
AL <- retvals[['AL']]
caches <- retvals[['caches']]
dropoutMat <- retvals[['dropoutMat']]
#Compute cost
cost <- computeCost(AL, Y,outputActivationFunc="softmax",
                    numClasses=layersDimensions[length(layersDimensions)])
print(cost)
## [1] 1.098618
# Backward propagation.
gradients = backwardPropagationDeep(AL, Y, caches, dropoutMat, lambd=0, keep_prob=1, hiddenActivationFunc="relu",
                                    outputActivationFunc="softmax",numClasses=layersDimensions[length(layersDimensions)])
# Need to take transpose of the last layer for Softmax
L=length(parameters)/2
gradients[[paste('dW',L,sep="")]]=t(gradients[[paste('dW',L,sep="")]])
gradients[[paste('db',L,sep="")]]=t(gradients[[paste('db',L,sep="")]])
#Perform gradient check
gradient_check_n(parameters, gradients, X, Y, 
                 epsilon = 1e-7,outputActivationFunc="softmax")
## The implementations works perfectly 3.903011e-07[1] "Gradients from backprop"
## $dW1
##              [,1]          [,2]
## [1,] 0.0007962367 -0.0001907606
## [2,] 0.0004444254  0.0010354412
## [3,] 0.0003078611  0.0007591255
## 
## $db1
##               [,1]
## [1,] -0.0017305136
## [2,]  0.0005393734
## [3,]  0.0012484550
## 
## $dW2
##               [,1]          [,2]          [,3]
## [1,] -3.515627e-04  7.487283e-04 -3.971656e-04
## [2,] -6.381521e-05 -1.257328e-06  6.507254e-05
## [3,] -1.719479e-04 -4.857264e-04  6.576743e-04
## 
## $db2
##               [,1]
## [1,] -5.536383e-06
## [2,] -1.824656e-05
## [3,]  2.378295e-05
## 
## [1] "Grad approx from gradient check"
## $dW1
##              [,1]          [,2]
## [1,] 0.0007962364 -0.0001907607
## [2,] 0.0004444256  0.0010354406
## [3,] 0.0003078615  0.0007591250
## 
## $db1
##               [,1]
## [1,] -0.0017305135
## [2,]  0.0005393741
## [3,]  0.0012484547
## 
## $dW2
##               [,1]          [,2]          [,3]
## [1,] -3.515632e-04  7.487277e-04 -3.971656e-04
## [2,] -6.381451e-05 -1.257883e-06  6.507239e-05
## [3,] -1.719469e-04 -4.857270e-04  6.576739e-04
## 
## $db2
##               [,1]
## [1,] -5.536682e-06
## [2,] -1.824652e-05
## [3,]  2.378209e-05

1.3a Gradient Check – Sigmoid Activation – Octave

source("DL8functions.m")
################## Circles
data=csvread("circles.csv");

X=data(:,1:2);
Y=data(:,3);
#Set layer dimensions
layersDimensions = [2 5  1]; #tanh=-0.5(ok), #relu=0.1 best!
[weights biases] = initializeDeepModel(layersDimensions);
#Perform forward prop
[AL forward_caches activation_caches droputMat] = forwardPropagationDeep(X', weights, biases,keep_prob=1, 
                 hiddenActivationFunc="relu", outputActivationFunc="sigmoid");
#Compute cost
cost = computeCost(AL, Y',outputActivationFunc=outputActivationFunc,numClasses=layersDimensions(size(layersDimensions)(2)));
disp(cost);
#Compute gradients from cost
[gradsDA gradsDW gradsDB] = backwardPropagationDeep(AL, Y', activation_caches,forward_caches, droputMat, lambd=0, keep_prob=1, 
                                 hiddenActivationFunc="relu", outputActivationFunc="sigmoid",
                                 numClasses=layersDimensions(size(layersDimensions)(2)));
epsilon = 1e-07;
outputActivationFunc="sigmoid";
# Convert paramter cell array to vector
parameters_values = cellArray_to_vector(weights, biases);
#Convert gradient cell array to vector
grad = gradients_to_vector(gradsDW,gradsDB);
num_parameters = size(parameters_values)(1);
#Initialize
J_plus = zeros(num_parameters, 1);
J_minus = zeros(num_parameters, 1);
gradapprox = zeros(num_parameters, 1);
# Compute gradapprox
for i = 1:num_parameters
    # Compute J_plus[i]. 
    thetaplus = parameters_values;                                   
    thetaplus(i,1) = thetaplus(i,1) + epsilon;      
    [weights1 biases1] =vector_to_cellArray(weights, biases,thetaplus);    
    [AL forward_caches activation_caches droputMat] = forwardPropagationDeep(X', weights1, biases1, keep_prob=1, 
                                              hiddenActivationFunc="relu",outputActivationFunc=outputActivationFunc);
    J_plus(i) = computeCost(AL, Y', outputActivationFunc=outputActivationFunc); 
        
    # Compute J_minus[i]. 
    thetaminus = parameters_values;                                  
    thetaminus(i,1) = thetaminus(i,1) - epsilon ;    
    [weights1 biases1] = vector_to_cellArray(weights, biases,thetaminus);    
    [AL forward_caches activation_caches droputMat]  = forwardPropagationDeep(X',weights1, biases1, keep_prob=1, 
                                              hiddenActivationFunc="relu",outputActivationFunc=outputActivationFunc);     
    J_minus(i) = computeCost(AL, Y', outputActivationFunc=outputActivationFunc);                            
      
    # Compute gradapprox[i]   
    gradapprox(i) = (J_plus(i) - J_minus(i))/(2*epsilon);

endfor

#Compute L2Norm
numerator = L2NormVec(grad-gradapprox);                                           
denominator = L2NormVec(grad) +  L2NormVec(gradapprox);                                         
difference =  numerator/denominator;
disp(difference);
#Check difference
if difference > 1e-04
   printf("There is a mistake in the implementation ");
   disp(difference);
else
   printf("The implementation works perfectly");
      disp(difference);
endif
[weights1 biases1] = vector_to_cellArray(weights, biases,grad);  
printf("Gradients from back propagation"); 
disp(weights1);
disp(biases1); 
[weights2 biases2] = vector_to_cellArray(weights, biases,gradapprox); 
printf("Gradients from gradient check");
disp(weights2);
disp(biases2); 

0.69315
1.4893e-005
The implementation works perfectly 1.4893e-005
Gradients from back propagation
{
[1,1] =
5.0349e-005 2.1323e-005
8.8632e-007 1.8231e-006
9.3784e-005 1.0057e-004
1.0875e-004 -1.9529e-007
5.4502e-005 3.2721e-005
[1,2] =
1.0567e-005 6.0615e-005 4.6004e-005 1.3977e-004 1.0405e-004
}
{
[1,1] =
-1.8716e-005
1.1309e-009
4.7686e-005
1.2051e-005
-1.4612e-005
[1,2] = 9.5808e-006
}
Gradients from gradient check
{
[1,1] =
5.0348e-005 2.1320e-005
8.8485e-007 1.8219e-006
9.3784e-005 1.0057e-004
1.0875e-004 -1.9762e-007
5.4502e-005 3.2723e-005
[1,2] =
[1,2] =
1.0565e-005 6.0614e-005 4.6007e-005 1.3977e-004 1.0405e-004
}
{
[1,1] =
-1.8713e-005
1.1102e-009
4.7687e-005
1.2048e-005
-1.4609e-005
[1,2] = 9.5790e-006
}

1.3b Gradient Check – Softmax Activation – Octave

source("DL8functions.m")
data=csvread("spiral.csv");

# Setup the data
X=data(:,1:2);
Y=data(:,3);
# Set the layer dimensions
layersDimensions = [2 3  3]; 
[weights biases] = initializeDeepModel(layersDimensions);
# Run forward prop
[AL forward_caches activation_caches droputMat] = forwardPropagationDeep(X', weights, biases,keep_prob=1, 
                 hiddenActivationFunc="relu", outputActivationFunc="softmax");
# Compute cost
cost = computeCost(AL, Y',outputActivationFunc=outputActivationFunc,numClasses=layersDimensions(size(layersDimensions)(2)));
disp(cost);
# Perform backward prop
[gradsDA gradsDW gradsDB] = backwardPropagationDeep(AL, Y', activation_caches,forward_caches, droputMat, lambd=0, keep_prob=1, 
                                 hiddenActivationFunc="relu", outputActivationFunc="softmax",
                                 numClasses=layersDimensions(size(layersDimensions)(2)));

#Take transpose of last layer for Softmax                                
L=size(weights)(2);
gradsDW{L}= gradsDW{L}';
gradsDB{L}= gradsDB{L}';   
#Perform gradient check
difference= gradient_check_n(weights, biases, gradsDW,gradsDB, X, Y, epsilon = 1e-7,
                  outputActivationFunc="softmax",numClasses=layersDimensions(size(layersDimensions)(2)));

 1.0986
The implementation works perfectly  2.0021e-005
Gradients from back propagation
{
  [1,1] =
    -7.1590e-005  4.1375e-005
    -1.9494e-004  -5.2014e-005
    -1.4554e-004  5.1699e-005
  [1,2] =
    3.3129e-004  1.9806e-004  -1.5662e-005
    -4.9692e-004  -3.7756e-004  -8.2318e-005
    1.6562e-004  1.7950e-004  9.7980e-005
}
{
  [1,1] =
    -3.0856e-005
    -3.3321e-004
    -3.8197e-004
  [1,2] =
    1.2046e-006
    2.9259e-007
    -1.4972e-006
}
Gradients from gradient check
{
  [1,1] =
    -7.1586e-005  4.1377e-005
    -1.9494e-004  -5.2013e-005
    -1.4554e-004  5.1695e-005
    3.3129e-004  1.9806e-004  -1.5664e-005
    -4.9692e-004  -3.7756e-004  -8.2316e-005
    1.6562e-004  1.7950e-004  9.7979e-005
}
{
  [1,1] =
    -3.0852e-005
    -3.3321e-004
    -3.8197e-004
  [1,2] =
    1.1902e-006
    2.8200e-007
    -1.4644e-006
}

2.1 Tip for tuning hyperparameters

Deep Learning Networks come with a large number of hyper parameters which require tuning. The hyper parameters are

1. \alpha -learning rate
2. Number of layers
3. Number of hidden units
4. Number of iterations
5. Momentum – \beta – 0.9
6. RMSProp – \beta_{1} – 0.9
7. Adam – \beta_{1},\beta_{2} and \epsilon
8. learning rate decay
9. mini batch size
10. Initialization method – He, Xavier
11. Regularization

– Among the above the most critical is learning rate \alpha . Rather than just trying out random values, it may help to try out values on a logarithmic scale. So we could try out values -0.01,0.1,1.0,10 etc. If we find that the cost is between 0.01 and 0.1 we could use a technique similar to binary search or bisection, so we can try 0.01, 0.05. If we need to be bigger than 0.01 and 0.05 we could try 0.25  and then keep halving the distance etc.
– The performance of Momentum and RMSProp are very good and work well with values 0.9. Even with this, it is better to try out values of 1-\beta in the logarithmic range. So 1-\beta could 0.001,0.01,0.1 and hence \beta would be 0.999,0.99 or 0.9
– Increasing the number of hidden units or number of hidden layers need to be done gradually. I have noticed that increasing number of hidden layers heavily does not improve performance and sometimes degrades it.
– Sometimes, I tend to increase the number of iterations if I think I see a steady decrease in the cost for a certain learning rate
– It may also help to add learning rate decay if you see there is an oscillation while it decreases.
– Xavier and He initializations also help in a fast convergence and are worth trying out.

3.1 Final thoughts

As I come to a close in this Deep Learning Series from first principles in Python, R and Octave, I must admit that I learnt a lot in the process.

* Building a L-layer, vectorized Deep Learning Network in Python, R and Octave was extremely challenging but very rewarding
* One benefit of building vectorized versions in Python, R and Octave was that I was looking at each function that I was implementing thrice, and hence I was able to fix any bugs in any of the languages
* In addition since I built the generic L-Layer DL network with all the bells and whistles, layer by layer I further had an opportunity to look at all the functions in each successive post.
* Each language has its advantages and disadvantages. From the performance perspective I think Python is the best, followed by Octave and then R
* Interesting, I noticed that even if small bugs creep into your implementation, the DL network does learn and does generate a valid set of weights and biases, however this may not be an optimum solution. In one case of an inadvertent bug, I was not updating the weights in the final layer of the DL network. Yet, using all the other layers, the DL network was able to come with a reasonable solution (maybe like random dropout, remaining units can still learn the data!)
* Having said that, the Gradient Check method discussed and implemented in this post can be very useful in ironing out bugs.
Feel free to clone/download the code from Github at DeepLearning-Part8

Conclusion

These last couple of months when I was writing the posts and the also churning up the code in Python, R and Octave were  very hectic. There have been times when I found that implementations of some function to be extremely demanding and I almost felt like giving up. Other times, I have spent quite some time on an intractable DL network which would not respond to changes in hyper-parameters. All in all, it was a great learning experience. I would suggest that you start from my first post Deep Learning from first principles in Python, R and Octave-Part 1 and work your way up. Feel free to take the code apart and try out things. That is the only way you will learn.

Hope you had as much fun as I had. Stay tuned. I will be back!!!

Also see
1. My book ‘Practical Machine Learning with R and Python’ on Amazon
2. Revisiting crimes against women in India
3. Literacy in India – A deepR dive
4. Sixer – R package cricketr’s new Shiny avatar
5. Bend it like Bluemix, MongoDB using Auto-scale – Part 1!
6. Computer Vision: Ramblings on derivatives, histograms and contours
7. Introducing QCSimulator: A 5-qubit quantum computing simulator in R
8. A closer look at “Robot Horse on a Trot” in Android

To see all post click Index of Posts

Deep Learning from first principles in Python, R and Octave – Part 7

Artificial Intelligence is the new electricity. – Prof Andrew Ng

Most of human and animal learning is unsupervised learning. If intelligence was a cake, unsupervised learning would be the cake, supervised learning would be the icing on the cake, and reinforcement learning would be the cherry on the cake. We know how to make the icing and the cherry, but we don’t know how to make the cake. We need to solve the unsupervised learning problem before we can even think of getting to true AI.  – Yann LeCun, March 14, 2016 (Facebook)

Introduction

In this post ‘Deep Learning from first principles with Python, R and Octave-Part 7’, I implement optimization methods used in Stochastic Gradient Descent (SGD) to speed up the convergence. Specifically I discuss and implement the following gradient descent optimization techniques

a.Vanilla Stochastic Gradient Descent
b.Learning rate decay
c. Momentum method
d. RMSProp
e. Adaptive Moment Estimation (Adam)

This post, further enhances my generic  L-Layer Deep Learning Network implementations in  vectorized Python, R and Octave to also include the Stochastic Gradient Descent optimization techniques. You can clone/download the code from Github at DeepLearning-Part7

You can view my video  presentation on Gradient Descent Optimization in Neural Networks 7

Incidentally, a good discussion of the various optimizations methods used in Stochastic Gradient Optimization techniques can be seen at Sebastian Ruder’s blog

Note: In the vectorized Python, R and Octave implementations below only a  1024 random training samples were used. This was to reduce the computation time. You are free to use the entire data set (60000 training data) for the computation.

This post is largely based of on Prof Andrew Ng’s Deep Learning Specialization.  All the above optimization techniques for Stochastic Gradient Descent are based on the technique of exponentially weighted average method. So for example if we had some time series data \theta_{1},\theta_{2},\theta_{3}... \theta_{t} then we we can represent the exponentially average value at time ‘t’ as a sequence of the the previous value v_{t-1} and \theta_{t} as shown below
v_{t} = \beta v_{t-1} + (1-\beta)\theta_{t}

Here v_{t} represent the average of the data set over \frac {1}{1-\beta}  By choosing different values of \beta, we can average over a larger or smaller number of the data points.
We can write the equations as follows
v_{t} = \beta v_{t-1} + (1-\beta)\theta_{t}
v_{t-1} = \beta v_{t-2} + (1-\beta)\theta_{t-1}
v_{t-2} = \beta v_{t-3} + (1-\beta)\theta_{t-2}
and
v_{t-k} = \beta v_{t-(k+1))} + (1-\beta)\theta_{t-k}
By substitution we have
v_{t} = (1-\beta)\theta_{t} + \beta v_{t-1}
v_{t} = (1-\beta)\theta_{t} + \beta ((1-\beta)\theta_{t-1}) + \beta v_{t-2}
v_{t} = (1-\beta)\theta_{t} + \beta ((1-\beta)\theta_{t-1}) + \beta ((1-\beta)\theta_{t-2}+ \beta v_{t-3} )

Hence it can be seen that the v_{t} is the weighted sum over the previous values \theta_{k}, which is an exponentially decaying function.

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- Machine Learning in stereo” available in Amazon in paperback($9.99) and Kindle($6.99) 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.

1.1a. Stochastic Gradient Descent (Vanilla) – Python

import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
exec(open("DLfunctions7.py").read())
exec(open("load_mnist.py").read())

# Read the training data
training=list(read(dataset='training',path=".\\mnist"))
test=list(read(dataset='testing',path=".\\mnist"))
lbls=[]
pxls=[]
for i in range(60000):
       l,p=training[i]
       lbls.append(l)
       pxls.append(p)
labels= np.array(lbls)
pixels=np.array(pxls)       
y=labels.reshape(-1,1)
X=pixels.reshape(pixels.shape[0],-1)
X1=X.T
Y1=y.T

# Create  a list of 1024 random numbers.
permutation = list(np.random.permutation(2**10))
# Subset 16384 from the data
X2 = X1[:, permutation]
Y2 = Y1[:, permutation].reshape((1,2**10))
# Set the layer dimensions  
layersDimensions=[784, 15,9,10] 
# Perform SGD with regular gradient descent
parameters = L_Layer_DeepModel_SGD(X2, Y2, layersDimensions, hiddenActivationFunc='relu', 
                                   outputActivationFunc="softmax",learningRate = 0.01 ,
                                   optimizer="gd",
                                   mini_batch_size =512, num_epochs = 1000, print_cost = True,figure="fig1.png")

1.1b. Stochastic Gradient Descent (Vanilla) – R

source("mnist.R")
source("DLfunctions7.R")
#Load and read MNIST data
load_mnist() 
x <- t(train$x)
X <- x[,1:60000]
y <-train$y
y1 <- y[1:60000]
y2 <- as.matrix(y1)
Y=t(y2)

# Subset 1024 random samples from MNIST 
permutation = c(sample(2^10))
# Randomly shuffle the training data
X1 = X[, permutation]
y1 = Y[1, permutation]
y2 <- as.matrix(y1)
Y1=t(y2)
# Set layer dimensions
layersDimensions=c(784, 15,9, 10) 
# Perform SGD with regular gradient descent
retvalsSGD= L_Layer_DeepModel_SGD(X1, Y1, layersDimensions,
                            hiddenActivationFunc='tanh',
                            outputActivationFunc="softmax",
                            learningRate = 0.05,
                            optimizer="gd",
                            mini_batch_size = 512, 
                            num_epochs = 5000, 
                            print_cost = True)
#Plot the cost vs iterations
iterations <- seq(0,5000,1000)
costs=retvalsSGD$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
 ggtitle("Costs vs no of epochs") + xlab("No of epochss") + ylab("Cost")

1.1c. Stochastic Gradient Descent (Vanilla) – Octave

source("DL7functions.m")
#Load and read MNIST
load('./mnist/mnist.txt.gz'); 
#Create a random permutatation from 1024
permutation = randperm(1024);
disp(length(permutation));

# Use this 1024 as the batch
X=trainX(permutation,:);
Y=trainY(permutation,:);

# Set layer dimensions
layersDimensions=[784, 15, 9, 10];
# Perform SGD with regular gradient descent
[weights biases costs]=L_Layer_DeepModel_SGD(X', Y', layersDimensions,
 hiddenActivationFunc='relu', 
 outputActivationFunc="softmax",
 learningRate = 0.005,
 lrDecay=true, 
 decayRate=1,
 lambd=0,
 keep_prob=1,
 optimizer="gd",
 beta=0.9,
 beta1=0.9,
 beta2=0.999,
 epsilon=10^-8,
 mini_batch_size = 512, 
 num_epochs = 5000);

plotCostVsEpochs(5000,costs);

2.1. Stochastic Gradient Descent with Learning rate decay

Since in Stochastic Gradient Descent,with  each epoch, we use slight different samples, the gradient descent algorithm, oscillates across the ravines and wanders around the minima, when a fixed learning rate is used. In this technique of ‘learning rate decay’ the learning rate is slowly decreased with the number of epochs and becomes smaller and smaller, so that gradient descent can take smaller steps towards the minima.

There are several techniques employed in learning rate decay

a) Exponential decay: \alpha = decayRate^{epochNum} *\alpha_{0}
b) 1/t decay : \alpha = \frac{\alpha_{0}}{1 + decayRate*epochNum}
c) \alpha = \frac {decayRate}{\sqrt(epochNum)}*\alpha_{0}

In my implementation I have used the ‘exponential decay’. The code snippet for Python is shown below

if lrDecay == True:
   learningRate = np.power(decayRate,(num_epochs/1000)) * learningRate

2.1a. Stochastic Gradient Descent with Learning rate decay – Python

import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
exec(open("DLfunctions7.py").read())
exec(open("load_mnist.py").read())

# Read the MNIST data
training=list(read(dataset='training',path=".\\mnist"))
test=list(read(dataset='testing',path=".\\mnist"))
lbls=[]
pxls=[]
for i in range(60000):
       l,p=training[i]
       lbls.append(l)
       pxls.append(p)
labels= np.array(lbls)
pixels=np.array(pxls)       
y=labels.reshape(-1,1)
X=pixels.reshape(pixels.shape[0],-1)
X1=X.T
Y1=y.T

# Create  a list of random numbers of 1024
permutation = list(np.random.permutation(2**10))
# Subset 16384 from the data
X2 = X1[:, permutation]
Y2 = Y1[:, permutation].reshape((1,2**10))
# Set layer dimensions
layersDimensions=[784, 15,9,10] 
# Perform SGD with learning rate decay
parameters = L_Layer_DeepModel_SGD(X2, Y2, layersDimensions, hiddenActivationFunc='relu', 
                                   outputActivationFunc="softmax",
                                   learningRate = 0.01 , lrDecay=True, decayRate=0.9999,
                                   optimizer="gd",
                                   mini_batch_size =512, num_epochs = 1000, print_cost = True,figure="fig2.png")

2.1b. Stochastic Gradient Descent with Learning rate decay – R

source("mnist.R")
source("DLfunctions7.R")
# Read and load MNIST
load_mnist()
x <- t(train$x)
X <- x[,1:60000]
y <-train$y
y1 <- y[1:60000]
y2 <- as.matrix(y1)
Y=t(y2)

# Subset 1024 random samples from MNIST 
permutation = c(sample(2^10))
# Randomly shuffle the training data
X1 = X[, permutation]
y1 = Y[1, permutation]
y2 <- as.matrix(y1)
Y1=t(y2)
# Set layer dimensions
layersDimensions=c(784, 15,9, 10) 
# Perform SGD with Learning rate decay
retvalsSGD= L_Layer_DeepModel_SGD(X1, Y1, layersDimensions,
                                  hiddenActivationFunc='tanh',
                                  outputActivationFunc="softmax",
                                  learningRate = 0.05,
                                  lrDecay=TRUE,
                                  decayRate=0.9999,
                                  optimizer="gd",
                                  mini_batch_size = 512, 
                                  num_epochs = 5000, 
                                  print_cost = True)
#Plot the cost vs iterations
iterations <- seq(0,5000,1000)
costs=retvalsSGD$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
 ggtitle("Costs vs number of epochs") + xlab("No of epochs") + ylab("Cost")

2.1c. Stochastic Gradient Descent with Learning rate decay – Octave

source("DL7functions.m")
#Load and read MNIST
load('./mnist/mnist.txt.gz'); 
#Create a random permutatation from 1024
permutation = randperm(1024);
disp(length(permutation));

# Use this 1024 as the batch
X=trainX(permutation,:);
Y=trainY(permutation,:);

# Set layer dimensions
layersDimensions=[784, 15, 9, 10];
# Perform SGD with regular Learning rate decay
[weights biases costs]=L_Layer_DeepModel_SGD(X', Y', layersDimensions,
 hiddenActivationFunc='relu', 
 outputActivationFunc="softmax",
 learningRate = 0.01,
 lrDecay=true, 
 decayRate=0.999,
 lambd=0,
 keep_prob=1,
 optimizer="gd",
 beta=0.9,
 beta1=0.9,
 beta2=0.999,
 epsilon=10^-8,
 mini_batch_size = 512, 
 num_epochs = 5000);
plotCostVsEpochs(5000,costs)

3.1. Stochastic Gradient Descent with Momentum

Stochastic Gradient Descent with Momentum uses the exponentially weighted average method discusses above and more generally moves faster into the ravine than across it. The equations are
v_{dW}^l = \beta v_{dW}^l + (1-\beta)dW^{l}
v_{db}^l = \beta v_{db}^l + (1-\beta)db^{l}
W^{l} = W^{l} - \alpha v_{dW}^l
b^{l} = b^{l} - \alpha v_{db}^l where
v_{dW} and v_{db} are the momentum terms which are exponentially weighted with the corresponding gradients ‘dW’ and ‘db’ at the corresponding layer ‘l’ The code snippet for Stochastic Gradient Descent with momentum in R is shown below

# Perform Gradient Descent with momentum
# Input : Weights and biases
#       : beta
#       : gradients
#       : learning rate
#       : outputActivationFunc - Activation function at hidden layer sigmoid/softmax
#output : Updated weights after 1 iteration
gradientDescentWithMomentum  <- function(parameters, gradients,v, beta, learningRate,outputActivationFunc="sigmoid"){

    L = length(parameters)/2 # number of layers in the neural network    
    # Update rule for each parameter. Use a for loop.
    for(l in 1:(L-1)){
        # Compute velocities
        # v['dWk'] = beta *v['dWk'] + (1-beta)*dWk
        v[[paste("dW",l, sep="")]] = beta*v[[paste("dW",l, sep="")]] + 
                   (1-beta) * gradients[[paste('dW',l,sep="")]]
        v[[paste("db",l, sep="")]] = beta*v[[paste("db",l, sep="")]] + 
            (1-beta) * gradients[[paste('db',l,sep="")]]
        
        parameters[[paste("W",l,sep="")]] = parameters[[paste("W",l,sep="")]] -
            learningRate* v[[paste("dW",l, sep="")]] 
        parameters[[paste("b",l,sep="")]] = parameters[[paste("b",l,sep="")]] -
            learningRate* v[[paste("db",l, sep="")]] 
    }    
    # Compute for the Lth layer
    if(outputActivationFunc=="sigmoid"){
        v[[paste("dW",L, sep="")]] = beta*v[[paste("dW",L, sep="")]] + 
            (1-beta) * gradients[[paste('dW',L,sep="")]]
        v[[paste("db",L, sep="")]] = beta*v[[paste("db",L, sep="")]] + 
            (1-beta) * gradients[[paste('db',L,sep="")]]
        
        parameters[[paste("W",L,sep="")]] = parameters[[paste("W",L,sep="")]] -
            learningRate* v[[paste("dW",l, sep="")]]  
        parameters[[paste("b",L,sep="")]] = parameters[[paste("b",L,sep="")]] -
            learningRate* v[[paste("db",l, sep="")]]
        
    }else if (outputActivationFunc=="softmax"){
        v[[paste("dW",L, sep="")]] = beta*v[[paste("dW",L, sep="")]] + 
            (1-beta) * t(gradients[[paste('dW',L,sep="")]])
        v[[paste("db",L, sep="")]] = beta*v[[paste("db",L, sep="")]] + 
            (1-beta) * t(gradients[[paste('db',L,sep="")]])       
        parameters[[paste("W",L,sep="")]] = parameters[[paste("W",L,sep="")]] -
            learningRate* t(gradients[[paste("dW",L,sep="")]])
        parameters[[paste("b",L,sep="")]] = parameters[[paste("b",L,sep="")]] -
            learningRate* t(gradients[[paste("db",L,sep="")]])
    }
    return(parameters)
}

3.1a. Stochastic Gradient Descent with Momentum- Python

import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
# Read and load data
exec(open("DLfunctions7.py").read())
exec(open("load_mnist.py").read())
training=list(read(dataset='training',path=".\\mnist"))
test=list(read(dataset='testing',path=".\\mnist"))
lbls=[]
pxls=[]
for i in range(60000):
       l,p=training[i]
       lbls.append(l)
       pxls.append(p)
labels= np.array(lbls)
pixels=np.array(pxls)       
y=labels.reshape(-1,1)
X=pixels.reshape(pixels.shape[0],-1)
X1=X.T
Y1=y.T

# Create  a list of random numbers of 1024
permutation = list(np.random.permutation(2**10))
# Subset 16384 from the data
X2 = X1[:, permutation]
Y2 = Y1[:, permutation].reshape((1,2**10))
layersDimensions=[784, 15,9,10] 
# Perform SGD with momentum
parameters = L_Layer_DeepModel_SGD(X2, Y2, layersDimensions, hiddenActivationFunc='relu', 
                                   outputActivationFunc="softmax",learningRate = 0.01 ,
                                   optimizer="momentum", beta=0.9,
                                   mini_batch_size =512, num_epochs = 1000, print_cost = True,figure="fig3.png")

3.1b. Stochastic Gradient Descent with Momentum- R

source("mnist.R")
source("DLfunctions7.R")
load_mnist()
x <- t(train$x)
X <- x[,1:60000]
y <-train$y
y1 <- y[1:60000]
y2 <- as.matrix(y1)
Y=t(y2)

# Subset 1024 random samples from MNIST 
permutation = c(sample(2^10))
# Randomly shuffle the training data
X1 = X[, permutation]
y1 = Y[1, permutation]
y2 <- as.matrix(y1)
Y1=t(y2)
layersDimensions=c(784, 15,9, 10) 
# Perform SGD with momentum
retvalsSGD= L_Layer_DeepModel_SGD(X1, Y1, layersDimensions,
                                  hiddenActivationFunc='tanh',
                                  outputActivationFunc="softmax",
                                  learningRate = 0.05,
                                  optimizer="momentum",
                                  beta=0.9,
                                  mini_batch_size = 512, 
                                  num_epochs = 5000, 
                                  print_cost = True)


#Plot the cost vs iterations
iterations <- seq(0,5000,1000)
costs=retvalsSGD$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
 ggtitle("Costs vs number of epochs") + xlab("No of epochs") + ylab("Cost")

3.1c. Stochastic Gradient Descent with Momentum- Octave

source("DL7functions.m")
#Load and read MNIST
load('./mnist/mnist.txt.gz'); 
#Create a random permutatation from 60K
permutation = randperm(1024);
disp(length(permutation));

# Use this 1024 as the batch
X=trainX(permutation,:);
Y=trainY(permutation,:);

# Set layer dimensions
layersDimensions=[784, 15, 9, 10];
# Perform SGD with Momentum
[weights biases costs]=L_Layer_DeepModel_SGD(X', Y', layersDimensions,
 hiddenActivationFunc='relu', 
 outputActivationFunc="softmax",
 learningRate = 0.01,
 lrDecay=false, 
 decayRate=1,
 lambd=0,
 keep_prob=1,
 optimizer="momentum",
 beta=0.9,
 beta1=0.9,
 beta2=0.999,
 epsilon=10^-8,
 mini_batch_size = 512, 
 num_epochs = 5000);

plotCostVsEpochs(5000,costs)

4.1. Stochastic Gradient Descent with RMSProp

Stochastic Gradient Descent with RMSProp tries to move faster towards the minima while dampening the oscillations across the ravine.
The equations are

s_{dW}^l = \beta_{1} s_{dW}^l + (1-\beta_{1})(dW^{l})^{2}
s_{db}^l = \beta_{1} s_{db}^l + (1-\beta_{1})(db^{l})^2
W^{l} = W^{l} - \frac {\alpha s_{dW}^l}{\sqrt (s_{dW}^l + \epsilon) }
b^{l} = b^{l} - \frac {\alpha s_{db}^l}{\sqrt (s_{db}^l + \epsilon) }
where s_{dW} and s_{db} are the RMSProp terms which are exponentially weighted with the corresponding gradients ‘dW’ and ‘db’ at the corresponding layer ‘l’

The code snippet in Octave is shown below

# Update parameters with RMSProp
# Input : parameters
#       : gradients
#       : s
#       : beta
#       : learningRate
#       : 
#output : Updated parameters RMSProp
function [weights biases] = gradientDescentWithRMSProp(weights, biases,gradsDW,gradsDB, sdW, sdB, beta1, epsilon, learningRate,outputActivationFunc="sigmoid")
    L = size(weights)(2); # number of layers in the neural network
    # Update rule for each parameter. 
    for l=1:(L-1)
        sdW{l} =  beta1*sdW{l} + (1 -beta1) * gradsDW{l} .* gradsDW{l};
        sdB{l} =  beta1*sdB{l} + (1 -beta1) * gradsDB{l} .* gradsDB{l};
        weights{l} = weights{l} - learningRate* gradsDW{l} ./ sqrt(sdW{l} + epsilon); 
        biases{l} = biases{l} -  learningRate* gradsDB{l} ./ sqrt(sdB{l} + epsilon);
    endfor
  
    if (strcmp(outputActivationFunc,"sigmoid"))
        sdW{L} =  beta1*sdW{L} + (1 -beta1) * gradsDW{L} .* gradsDW{L};
        sdB{L} =  beta1*sdB{L} + (1 -beta1) * gradsDB{L} .* gradsDB{L};
        weights{L} = weights{L} -learningRate* gradsDW{L} ./ sqrt(sdW{L} +epsilon); 
        biases{L} = biases{L} -learningRate* gradsDB{L} ./ sqrt(sdB{L} + epsilon);
     elseif (strcmp(outputActivationFunc,"softmax"))
        sdW{L} =  beta1*sdW{L} + (1 -beta1) * gradsDW{L}' .* gradsDW{L}';
        sdB{L} =  beta1*sdB{L} + (1 -beta1) * gradsDB{L}' .* gradsDB{L}';
        weights{L} = weights{L} -learningRate* gradsDW{L}' ./ sqrt(sdW{L} +epsilon); 
        biases{L} = biases{L} -learningRate* gradsDB{L}' ./ sqrt(sdB{L} + epsilon);
     endif   
end

4.1a. Stochastic Gradient Descent with RMSProp – Python

import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
exec(open("DLfunctions7.py").read())
exec(open("load_mnist.py").read())

# Read and load MNIST
training=list(read(dataset='training',path=".\\mnist"))
test=list(read(dataset='testing',path=".\\mnist"))
lbls=[]
pxls=[]
for i in range(60000):
       l,p=training[i]
       lbls.append(l)
       pxls.append(p)
labels= np.array(lbls)
pixels=np.array(pxls)       
y=labels.reshape(-1,1)
X=pixels.reshape(pixels.shape[0],-1)
X1=X.T
Y1=y.T

print("X1=",X1.shape)
print("y1=",Y1.shape)

# Create  a list of random numbers of 1024
permutation = list(np.random.permutation(2**10))
# Subset 16384 from the data
X2 = X1[:, permutation]
Y2 = Y1[:, permutation].reshape((1,2**10))
  
layersDimensions=[784, 15,9,10] 
# Use SGD with RMSProp
parameters = L_Layer_DeepModel_SGD(X2, Y2, layersDimensions, hiddenActivationFunc='relu', 
                                   outputActivationFunc="softmax",learningRate = 0.01 ,
                                   optimizer="rmsprop", beta1=0.7, epsilon=1e-8,
                                   mini_batch_size =512, num_epochs = 1000, print_cost = True,figure="fig4.png")

4.1b. Stochastic Gradient Descent with RMSProp – R

source("mnist.R")
source("DLfunctions7.R")
load_mnist()
x <- t(train$x)
X <- x[,1:60000]
y <-train$y
y1 <- y[1:60000]
y2 <- as.matrix(y1)
Y=t(y2)

# Subset 1024 random samples from MNIST 
permutation = c(sample(2^10))
# Randomly shuffle the training data
X1 = X[, permutation]
y1 = Y[1, permutation]
y2 <- as.matrix(y1)
Y1=t(y2)
layersDimensions=c(784, 15,9, 10) 
#Perform SGD with RMSProp
retvalsSGD= L_Layer_DeepModel_SGD(X1, Y1, layersDimensions,
                                  hiddenActivationFunc='tanh',
                                  outputActivationFunc="softmax",
                                  learningRate = 0.001,
                                  optimizer="rmsprop",
                                  beta1=0.9,
                                  epsilon=10^-8,
                                  mini_batch_size = 512, 
                                  num_epochs = 5000 , 
                                  print_cost = True)
#Plot the cost vs iterations
iterations <- seq(0,5000,1000)
costs=retvalsSGD$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
 ggtitle("Costs vs number of epochs") + xlab("No of epochs") + ylab("Cost")

4.1c. Stochastic Gradient Descent with RMSProp – Octave

source("DL7functions.m")
load('./mnist/mnist.txt.gz'); 
#Create a random permutatation from 1024
permutation = randperm(1024);

# Use this 1024 as the batch
X=trainX(permutation,:);
Y=trainY(permutation,:);

# Set layer dimensions
layersDimensions=[784, 15, 9, 10];
#Perform SGD with RMSProp
[weights biases costs]=L_Layer_DeepModel_SGD(X', Y', layersDimensions,
 hiddenActivationFunc='relu', 
 outputActivationFunc="softmax",
 learningRate = 0.005,
 lrDecay=false, 
 decayRate=1,
 lambd=0,
 keep_prob=1,
 optimizer="rmsprop",
 beta=0.9,
 beta1=0.9,
 beta2=0.999,
 epsilon=1,
 mini_batch_size = 512, 
 num_epochs = 5000);
plotCostVsEpochs(5000,costs)

5.1. Stochastic Gradient Descent with Adam

Adaptive Moment Estimate is a combination of the momentum (1st moment) and RMSProp(2nd moment). The equations for Adam are below
v_{dW}^l = \beta_{1} v_{dW}^l + (1-\beta_{1})dW^{l}
v_{db}^l = \beta_{1} v_{db}^l + (1-\beta_{1})db^{l}
The bias corrections for the 1st moment
vCorrected_{dW}^l= \frac {v_{dW}^l}{1 - \beta_{1}^{t}}
vCorrected_{db}^l= \frac {v_{db}^l}{1 - \beta_{1}^{t}}

Similarly the moving average for the 2nd moment- RMSProp
s_{dW}^l = \beta_{2} s_{dW}^l + (1-\beta_{2})(dW^{l})^2
s_{db}^l = \beta_{2} s_{db}^l + (1-\beta_{2})(db^{l})^2
The bias corrections for the 2nd moment
sCorrected_{dW}^l= \frac {s_{dW}^l}{1 - \beta_{2}^{t}}
sCorrected_{db}^l= \frac {s_{db}^l}{1 - \beta_{2}^{t}}

The Adam Gradient Descent is given by
W^{l} = W^{l} - \frac {\alpha vCorrected_{dW}^l}{\sqrt (s_{dW}^l + \epsilon) }
b^{l} = b^{l} - \frac {\alpha vCorrected_{db}^l}{\sqrt (s_{db}^l + \epsilon) }
The code snippet of Adam in R is included below

# Perform Gradient Descent with Adam
# Input : Weights and biases
#       : beta1
#       : epsilon
#       : gradients
#       : learning rate
#       : outputActivationFunc - Activation function at hidden layer sigmoid/softmax
#output : Updated weights after 1 iteration
gradientDescentWithAdam  <- function(parameters, gradients,v, s, t, 
                        beta1=0.9, beta2=0.999, epsilon=10^-8, learningRate=0.1,outputActivationFunc="sigmoid"){
    
    L = length(parameters)/2 # number of layers in the neural network
    v_corrected <- list()
    s_corrected <- list()
    # Update rule for each parameter. Use a for loop.
    for(l in 1:(L-1)){
        # v['dWk'] = beta *v['dWk'] + (1-beta)*dWk
        v[[paste("dW",l, sep="")]] = beta1*v[[paste("dW",l, sep="")]] + 
            (1-beta1) * gradients[[paste('dW',l,sep="")]]
        v[[paste("db",l, sep="")]] = beta1*v[[paste("db",l, sep="")]] + 
            (1-beta1) * gradients[[paste('db',l,sep="")]]
        
        
        # Compute bias-corrected first moment estimate. 
        v_corrected[[paste("dW",l, sep="")]] = v[[paste("dW",l, sep="")]]/(1-beta1^t)
        v_corrected[[paste("db",l, sep="")]] = v[[paste("db",l, sep="")]]/(1-beta1^t)
               
        # Element wise multiply of gradients
        s[[paste("dW",l, sep="")]] = beta2*s[[paste("dW",l, sep="")]] + 
            (1-beta2) * gradients[[paste('dW',l,sep="")]] * gradients[[paste('dW',l,sep="")]]
        s[[paste("db",l, sep="")]] = beta2*s[[paste("db",l, sep="")]] + 
            (1-beta2) * gradients[[paste('db',l,sep="")]] * gradients[[paste('db',l,sep="")]]
        
        # Compute bias-corrected second moment estimate. 
        s_corrected[[paste("dW",l, sep="")]] = s[[paste("dW",l, sep="")]]/(1-beta2^t)
        s_corrected[[paste("db",l, sep="")]] = s[[paste("db",l, sep="")]]/(1-beta2^t)
        
        # Update parameters. 
        d1=sqrt(s_corrected[[paste("dW",l, sep="")]]+epsilon)
        d2=sqrt(s_corrected[[paste("db",l, sep="")]]+epsilon)        
                
        parameters[[paste("W",l,sep="")]] = parameters[[paste("W",l,sep="")]] -
            learningRate * v_corrected[[paste("dW",l, sep="")]]/d1
        parameters[[paste("b",l,sep="")]] = parameters[[paste("b",l,sep="")]] -
            learningRate*v_corrected[[paste("db",l, sep="")]]/d2
    }    
    # Compute for the Lth layer
    if(outputActivationFunc=="sigmoid"){
        v[[paste("dW",L, sep="")]] = beta1*v[[paste("dW",L, sep="")]] + 
            (1-beta1) * gradients[[paste('dW',L,sep="")]]
        v[[paste("db",L, sep="")]] = beta1*v[[paste("db",L, sep="")]] + 
            (1-beta1) * gradients[[paste('db',L,sep="")]]
                
        # Compute bias-corrected first moment estimate. 
        v_corrected[[paste("dW",L, sep="")]] = v[[paste("dW",L, sep="")]]/(1-beta1^t)
        v_corrected[[paste("db",L, sep="")]] = v[[paste("db",L, sep="")]]/(1-beta1^t)
                
        # Element wise multiply of gradients
        s[[paste("dW",L, sep="")]] = beta2*s[[paste("dW",L, sep="")]] + 
            (1-beta2) * gradients[[paste('dW',L,sep="")]] * gradients[[paste('dW',L,sep="")]]
        s[[paste("db",L, sep="")]] = beta2*s[[paste("db",L, sep="")]] + 
            (1-beta2) * gradients[[paste('db',L,sep="")]] * gradients[[paste('db',L,sep="")]]
        
        # Compute bias-corrected second moment estimate. 
        s_corrected[[paste("dW",L, sep="")]] = s[[paste("dW",L, sep="")]]/(1-beta2^t)
        s_corrected[[paste("db",L, sep="")]] = s[[paste("db",L, sep="")]]/(1-beta2^t)
        
        # Update parameters. 
        d1=sqrt(s_corrected[[paste("dW",L, sep="")]]+epsilon)
        d2=sqrt(s_corrected[[paste("db",L, sep="")]]+epsilon)  
        
        parameters[[paste("W",L,sep="")]] = parameters[[paste("W",L,sep="")]] -
            learningRate * v_corrected[[paste("dW",L, sep="")]]/d1
        parameters[[paste("b",L,sep="")]] = parameters[[paste("b",L,sep="")]] -
            learningRate*v_corrected[[paste("db",L, sep="")]]/d2
        
    }else if (outputActivationFunc=="softmax"){
        v[[paste("dW",L, sep="")]] = beta1*v[[paste("dW",L, sep="")]] + 
            (1-beta1) * t(gradients[[paste('dW',L,sep="")]])
        v[[paste("db",L, sep="")]] = beta1*v[[paste("db",L, sep="")]] + 
            (1-beta1) * t(gradients[[paste('db',L,sep="")]])
                
        # Compute bias-corrected first moment estimate. 
        v_corrected[[paste("dW",L, sep="")]] = v[[paste("dW",L, sep="")]]/(1-beta1^t)
        v_corrected[[paste("db",L, sep="")]] = v[[paste("db",L, sep="")]]/(1-beta1^t)        
        
        # Element wise multiply of gradients
        s[[paste("dW",L, sep="")]] = beta2*s[[paste("dW",L, sep="")]] + 
            (1-beta2) * t(gradients[[paste('dW',L,sep="")]]) * t(gradients[[paste('dW',L,sep="")]])
        s[[paste("db",L, sep="")]] = beta2*s[[paste("db",L, sep="")]] + 
            (1-beta2) * t(gradients[[paste('db',L,sep="")]]) * t(gradients[[paste('db',L,sep="")]])
        
        # Compute bias-corrected second moment estimate. 
        s_corrected[[paste("dW",L, sep="")]] = s[[paste("dW",L, sep="")]]/(1-beta2^t)
        s_corrected[[paste("db",L, sep="")]] = s[[paste("db",L, sep="")]]/(1-beta2^t)
        
        # Update parameters. 
        d1=sqrt(s_corrected[[paste("dW",L, sep="")]]+epsilon)
        d2=sqrt(s_corrected[[paste("db",L, sep="")]]+epsilon) 
        
        parameters[[paste("W",L,sep="")]] = parameters[[paste("W",L,sep="")]] -
            learningRate * v_corrected[[paste("dW",L, sep="")]]/d1
        parameters[[paste("b",L,sep="")]] = parameters[[paste("b",L,sep="")]] -
            learningRate*v_corrected[[paste("db",L, sep="")]]/d2
    }
    return(parameters)
}

5.1a. Stochastic Gradient Descent with Adam – Python

import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
import sklearn
import sklearn.datasets
exec(open("DLfunctions7.py").read())
exec(open("load_mnist.py").read())
training=list(read(dataset='training',path=".\\mnist"))
test=list(read(dataset='testing',path=".\\mnist"))
lbls=[]
pxls=[]
print(len(training))
#for i in range(len(training)):
for i in range(60000):
       l,p=training[i]
       lbls.append(l)
       pxls.append(p)
labels= np.array(lbls)
pixels=np.array(pxls)       
y=labels.reshape(-1,1)
X=pixels.reshape(pixels.shape[0],-1)
X1=X.T
Y1=y.T


# Create  a list of random numbers of 1024
permutation = list(np.random.permutation(2**10))
# Subset 16384 from the data
X2 = X1[:, permutation]
Y2 = Y1[:, permutation].reshape((1,2**10))
layersDimensions=[784, 15,9,10] 
#Perform SGD with Adam optimization
parameters = L_Layer_DeepModel_SGD(X2, Y2, layersDimensions, hiddenActivationFunc='relu', 
                                   outputActivationFunc="softmax",learningRate = 0.01 ,
                                   optimizer="adam", beta1=0.9, beta2=0.9, epsilon = 1e-8,
                                   mini_batch_size =512, num_epochs = 1000, print_cost = True, figure="fig5.png")

5.1b. Stochastic Gradient Descent with Adam – R

source("mnist.R")
source("DLfunctions7.R")
load_mnist()
x <- t(train$x)
X <- x[,1:60000]
y <-train$y
y1 <- y[1:60000]
y2 <- as.matrix(y1)
Y=t(y2)

# Subset 1024 random samples from MNIST 
permutation = c(sample(2^10))
# Randomly shuffle the training data
X1 = X[, permutation]
y1 = Y[1, permutation]
y2 <- as.matrix(y1)
Y1=t(y2)
layersDimensions=c(784, 15,9, 10) 
#Perform SGD with Adam
retvalsSGD= L_Layer_DeepModel_SGD(X1, Y1, layersDimensions,
                                  hiddenActivationFunc='tanh',
                                  outputActivationFunc="softmax",
                                  learningRate = 0.005,
                                  optimizer="adam",
                                  beta1=0.7,
                                  beta2=0.9,
                                  epsilon=10^-8,
                                  mini_batch_size = 512, 
                                  num_epochs = 5000 , 
                                  print_cost = True)
#Plot the cost vs iterations
iterations <- seq(0,5000,1000)
costs=retvalsSGD$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
 ggtitle("Costs vs number of epochs") + xlab("No of epochs") + ylab("Cost")

5.1c. Stochastic Gradient Descent with Adam – Octave

source("DL7functions.m")
load('./mnist/mnist.txt.gz'); 
#Create a random permutatation from 1024
permutation = randperm(1024);
disp(length(permutation));

# Use this 1024 as the batch
X=trainX(permutation,:);
Y=trainY(permutation,:);
# Set layer dimensions
layersDimensions=[784, 15, 9, 10];

# Note the high value for epsilon. 
#Otherwise GD with Adam does not seem to converge   
# Perform SGD with Adam         
[weights biases costs]=L_Layer_DeepModel_SGD(X', Y', layersDimensions,
                       hiddenActivationFunc='relu', 
                       outputActivationFunc="softmax",
                       learningRate = 0.1,
                       lrDecay=false, 
                       decayRate=1,
                       lambd=0,
                       keep_prob=1,
                       optimizer="adam",
                       beta=0.9,
                       beta1=0.9,
                       beta2=0.9,
                       epsilon=100,
                       mini_batch_size = 512, 
                       num_epochs = 5000);
plotCostVsEpochs(5000,costs)

Conclusion: In this post I discuss and implement several Stochastic Gradient Descent optimization methods. The implementation of these methods enhance my already existing generic L-Layer Deep Learning Network implementation in vectorized Python, R and Octave, which I had discussed in the previous post in this series on Deep Learning from first principles in Python, R and Octave. Check it out, if you haven’t already. As already mentioned the code for this post can be cloned/forked from Github at DeepLearning-Part7

Watch this space! I’ll be back!

Also see
1.My book ‘Practical Machine Learning with R and Python’ on Amazon
2. Deep Learning from first principles in Python, R and Octave – Part 3
3. Experiments with deblurring using OpenCV
3. Design Principles of Scalable, Distributed Systems
4. Natural language processing: What would Shakespeare say?
5. yorkr crashes the IPL party! – Part 3!
6. cricketr flexes new muscles: The final analysis

To see all post click Index of posts

Deep Learning from first principles in Python, R and Octave – Part 5

Introduction

a. A robot may not injure a human being or, through inaction, allow a human being to come to harm.
b. A robot must obey orders given it by human beings except where such orders would conflict with the First Law.
c. A robot must protect its own existence as long as such protection does not conflict with the First or Second Law.

      Isaac Asimov's Three Laws of Robotics

Any sufficiently advanced technology is indistinguishable from magic.

      Arthur C Clarke.

In this 5th part on Deep Learning from first Principles in Python, R and Octave, I solve the MNIST data set of handwritten digits (shown below), from the basics. To do this, I construct a L-Layer, vectorized Deep Learning implementation in Python, R and Octave from scratch and classify the  MNIST data set. The MNIST training data set  contains 60000 handwritten digits from 0-9, and a test set of 10000 digits. MNIST, is a popular dataset for running Deep Learning tests, and has been rightfully termed as the ‘drosophila’ of Deep Learning, by none other than the venerable Prof Geoffrey Hinton.

The ‘Deep Learning from first principles in Python, R and Octave’ series, so far included  Part 1 , where I had implemented logistic regression as a simple Neural Network. Part 2 implemented the most elementary neural network with 1 hidden layer, but  with any number of activation units in that layer, and a sigmoid activation at the output layer.

This post, ‘Deep Learning from first principles in Python, R and Octave – Part 5’ largely builds upon Part3. in which I implemented a multi-layer Deep Learning network, with an arbitrary number of hidden layers and activation units per hidden layer and with the output layer was based on the sigmoid unit, for binary classification. In Part 4, I derive the Jacobian of a Softmax, the Cross entropy loss and the gradient equations for a multi-class Softmax classifier. I also  implement a simple Neural Network using Softmax classifications in Python, R and Octave.

In this post I combine Part 3 and Part 4 to to build a L-layer Deep Learning network, with arbitrary number of hidden layers and hidden units, which can do both binary (sigmoid) and multi-class (softmax) classification.

Note: A detailed discussion of the derivation for multi-class clasification can be seen in my video presentation Neural Networks 5

The generic, vectorized L-Layer Deep Learning Network implementations in Python, R and Octave can be cloned/downloaded from GitHub at DeepLearning-Part5. This implementation allows for arbitrary number of hidden layers and hidden layer units. The activation function at the hidden layers can be one of sigmoid, relu and tanh (will be adding leaky relu soon). The output activation can be used for binary classification with the ‘sigmoid’, or multi-class classification with ‘softmax’. Feel free to download and play around with the code!

I thought the exercise of combining the two parts(Part 3, & Part 4)  would be a breeze. But it was anything but. Incorporating a Softmax classifier into the generic L-Layer Deep Learning model was a challenge. Moreover I found that I could not use the gradient descent on 60,000 training samples as my laptop ran out of memory. So I had to implement Stochastic Gradient Descent (SGD) for Python, R and Octave. In addition, I had to also implement the numerically stable version of Softmax, as the softmax and its derivative would result in NaNs.

Numerically stable Softmax

The Softmax function S_{j} =\frac{e^{Z_{j}}}{\sum_{i}^{k}e^{Z_{i}}} can be numerically unstable because of the division of large exponentials.  To handle this problem we have to implement stable Softmax function as below

S_{j} =\frac{e^{Z_{j}}}{\sum_{i}^{k}e^{Z_{i}}}
S_{j} =\frac{e^{Z_{j}}}{\sum_{i}^{k}e^{Z_{i}}} = \frac{Ce^{Z_{j}}}{C\sum_{i}^{k}e^{Z_{i}}} = \frac{e^{Z_{j}+log(C)}}{\sum_{i}^{k}e^{Z_{i}+log(C)}}
Therefore S_{j}  = \frac{e^{Z_{j}+ D}}{\sum_{i}^{k}e^{Z_{i}+ D}}
Here ‘D’ can be anything. A common choice is
D=-max(Z_{1},Z_{2},... Z_{k})

Here is the stable Softmax implementation in Python

# A numerically stable Softmax implementation
def stableSoftmax(Z):  
    #Compute the softmax of vector x in a numerically stable way.
    shiftZ = Z.T - np.max(Z.T,axis=1).reshape(-1,1)
    exp_scores = np.exp(shiftZ)
    # normalize them for each example
    A = exp_scores / np.sum(exp_scores, axis=1, keepdims=True) 
    cache=Z
    return A,cache

While trying to create a L-Layer generic Deep Learning network in the 3 languages, I found it useful to ensure that the model executed correctly on smaller datasets.  You can run into numerous problems while setting up the matrices, which becomes extremely difficult to debug. So in this post, I run the model on 2 smaller data for sets used in my earlier posts(Part 3 & Part4) , in each of the languages, before running the generic model on MNIST.

Here is a fair warning. if you think you can dive directly into Deep Learning, with just some basic knowledge of Machine Learning, you are bound to run into serious issues. Moreover, your knowledge will be incomplete. It is essential that you have a good grasp of Machine and Statistical Learning, the different algorithms, the measures and metrics for selecting the models etc.It would help to be conversant with all the ML models, ML concepts, validation techniques, classification measures  etc. Check out the internet/books for background.

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.

1. Random dataset with Sigmoid activation – Python

This random data with 9 clusters, was used in my post Deep Learning from first principles in Python, R and Octave – Part 3 , and was used to test the complete L-layer Deep Learning network with Sigmoid activation.

import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import pandas as pd
from sklearn.datasets import make_classification, make_blobs
exec(open("DLfunctions51.py").read()) # Cannot import in Rmd.
# Create a random data set with 9 centeres
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 L -layer DL network
layersDimensions = [2, 9, 9,1] #  4-layer model
# Execute DL network with hidden activation=relu and sigmoid output function
parameters = L_Layer_DeepModel(X2, Y2, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid",learningRate = 0.3,num_iterations = 2500, print_cost = True)

2. Spiral dataset with Softmax activation – Python

The Spiral data was used in my post Deep Learning from first principles in Python, R and Octave – Part 4 and was used to test the complete L-layer Deep Learning network with multi-class Softmax activation at the output layer

import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import pandas as pd
from sklearn.datasets import make_classification, make_blobs
exec(open("DLfunctions51.py").read())

# Create an input data set - Taken from CS231n Convolutional Neural networks
# http://cs231n.github.io/neural-networks-case-study/
N = 100 # number of points per class
D = 2 # dimensionality
K = 3 # number of classes
X = np.zeros((N*K,D)) # data matrix (each row = single example)
y = np.zeros(N*K, dtype='uint8') # class labels
for j in range(K):
  ix = range(N*j,N*(j+1))
  r = np.linspace(0.0,1,N) # radius
  t = np.linspace(j*4,(j+1)*4,N) + np.random.randn(N)*0.2 # theta
  X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]
  y[ix] = j

X1=X.T
Y1=y.reshape(-1,1).T
numHidden=100 # No of hidden units in hidden layer
numFeats= 2 # dimensionality
numOutput = 3 # number of classes
# Set the dimensions of the layers
layersDimensions=[numFeats,numHidden,numOutput]
parameters = L_Layer_DeepModel(X1, Y1, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax",learningRate = 0.6,num_iterations = 9000, print_cost = True)
## Cost after iteration 0: 1.098759
## Cost after iteration 1000: 0.112666
## Cost after iteration 2000: 0.044351
## Cost after iteration 3000: 0.027491
## Cost after iteration 4000: 0.021898
## Cost after iteration 5000: 0.019181
## Cost after iteration 6000: 0.017832
## Cost after iteration 7000: 0.017452
## Cost after iteration 8000: 0.017161

3. MNIST dataset with Softmax activation – Python

In the code below, I execute Stochastic Gradient Descent on the MNIST training data of 60000. I used a mini-batch size of 1000. Python takes about 40 minutes to crunch the data. In addition I also compute the Confusion Matrix and other metrics like Accuracy, Precision and Recall for the MNIST data set. I get an accuracy of 0.93 on the MNIST test set. This accuracy can be improved by choosing more hidden layers or more hidden units and possibly also tweaking the learning rate and the number of epochs.

import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import pandas as pd
import math
from sklearn.datasets import make_classification, make_blobs
from sklearn.metrics import confusion_matrix
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score
exec(open("DLfunctions51.py").read())
exec(open("load_mnist.py").read())
# Read the MNIST training and test sets
training=list(read(dataset='training',path=".\\mnist"))
test=list(read(dataset='testing',path=".\\mnist"))
# Create labels and pixel arrays
lbls=[]
pxls=[]
print(len(training))
#for i in range(len(training)):
for i in range(60000):
       l,p=training[i]
       lbls.append(l)
       pxls.append(p)
labels= np.array(lbls)
pixels=np.array(pxls)       
y=labels.reshape(-1,1)
X=pixels.reshape(pixels.shape[0],-1)
X1=X.T
Y1=y.T
# Set the dimensions of the layers. The MNIST data is 28x28 pixels= 784
# Hence input layer is 784. For the 10 digits the Softmax classifier
# has to handle 10 outputs
layersDimensions=[784, 15,9,10] # Works very well,lr=0.01,mini_batch =1000, total=20000
np.random.seed(1)
costs = []  
# Run Stochastic Gradient Descent with Learning Rate=0.01, mini batch size=1000
# number of epochs=3000
parameters = L_Layer_DeepModel_SGD(X1, Y1, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax",learningRate = 0.01 ,mini_batch_size =1000, num_epochs = 3000, print_cost = True)

# Compute the Confusion Matrix on Training set
# Compute the training accuracy, precision and recall
proba=predict_proba(parameters, X1,outputActivationFunc="softmax")
#A2, cache = forwardPropagationDeep(X1, parameters)
#proba=np.argmax(A2, axis=0).reshape(-1,1)
a=confusion_matrix(Y1.T,proba)
print(a)
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score
print('Accuracy: {:.2f}'.format(accuracy_score(Y1.T, proba)))
print('Precision: {:.2f}'.format(precision_score(Y1.T, proba,average="micro")))
print('Recall: {:.2f}'.format(recall_score(Y1.T, proba,average="micro")))

# Read the test data
lbls=[]
pxls=[]
print(len(test))
for i in range(10000):
       l,p=test[i]
       lbls.append(l)
       pxls.append(p)
testLabels= np.array(lbls)
testPixels=np.array(pxls)       
ytest=testLabels.reshape(-1,1)
Xtest=testPixels.reshape(testPixels.shape[0],-1)
X1test=Xtest.T
Y1test=ytest.T

# Compute the Confusion Matrix on Test set
# Compute the test accuracy, precision and recall
probaTest=predict_proba(parameters, X1test,outputActivationFunc="softmax")
#A2, cache = forwardPropagationDeep(X1, parameters)
#proba=np.argmax(A2, axis=0).reshape(-1,1)
a=confusion_matrix(Y1test.T,probaTest)
print(a)
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score
print('Accuracy: {:.2f}'.format(accuracy_score(Y1test.T, probaTest)))
print('Precision: {:.2f}'.format(precision_score(Y1test.T, probaTest,average="micro")))
print('Recall: {:.2f}'.format(recall_score(Y1test.T, probaTest,average="micro")))

##1.  Confusion Matrix of Training set
       0     1    2    3    4    5    6    7    8    9
## [[5854    0   19    2   10    7    0    1   24    6]
##  [   1 6659   30   10    5    3    0   14   20    0]
##  [  20   24 5805   18    6   11    2   32   37    3]
##  [   5    4  175 5783    1   27    1   58   60   17]
##  [   1   21    9    0 5780    0    5    2   12   12]
##  [  29    9   21  224    6 4824   18   17  245   28]
##  [   5    4   22    1   32   12 5799    0   43    0]
##  [   3   13  148  154   18    3    0 5883    4   39]
##  [  11   34   30   21   13   16    4    7 5703   12]
##  [  10    4    1   32  135   14    1   92  134 5526]]

##2. Accuracy, Precision, Recall of  Training set
## Accuracy: 0.96
## Precision: 0.96
## Recall: 0.96

##3. Confusion Matrix of Test set
       0     1    2    3    4    5    6    7    8    9
## [[ 954    1    8    0    3    3    2    4    4    1]
##  [   0 1107    6    5    0    0    1    2   14    0]
##  [  11    7  957   10    5    0    5   20   16    1]
##  [   2    3   37  925    3   13    0    8   18    1]
##  [   2    6    1    1  944    0    7    3    4   14]
##  [  12    5    4   45    2  740   24    8   42   10]
##  [   8    4    4    2   16    9  903    0   12    0]
##  [   4   10   27   18    5    1    0  940    1   22]
##  [  11   13    6   13    9   10    7    2  900    3]
##  [   8    5    1    7   50    7    0   20   29  882]]
##4. Accuracy, Precision, Recall of  Training set
## Accuracy: 0.93
## Precision: 0.93
## Recall: 0.93

4. Random dataset with Sigmoid activation – R code

This is the random data set used in the Python code above which was saved as a CSV. The code is used to test a L -Layer DL network with Sigmoid Activation in R.

source("DLfunctions5.R")
# Read the random data set
z <- as.matrix(read.csv("data.csv",header=FALSE)) 
x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
# Set the dimensions of the  layer
layersDimensions = c(2, 9, 9,1)

# Run Gradient Descent on the data set with relu hidden unit activation 
# sigmoid activation unit in the output layer
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
                            hiddenActivationFunc='relu',
                            outputActivationFunc="sigmoid",
                            learningRate = 0.3,
                            numIterations = 5000, 
                            print_cost = True)
#Plot the cost vs iterations
iterations <- seq(0,5000,1000)
costs=retvals$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
 ggtitle("Costs vs iterations") + xlab("Iterations") + ylab("Loss")

5. Spiral dataset with Softmax activation – R

The spiral data set used in the Python code above, is reused to test  multi-class classification with Softmax.

source("DLfunctions5.R")
Z <- as.matrix(read.csv("spiral.csv",header=FALSE)) 

# Setup the data
X <- Z[,1:2]
y <- Z[,3]
X <- t(X)
Y <- t(y)

# Initialize number of features, number of hidden units in hidden layer and
# number of classes
numFeats<-2 # No features
numHidden<-100 # No of hidden units
numOutput<-3 # No of classes

# Set the layer dimensions
layersDimensions = c(numFeats,numHidden,numOutput)

# Perform gradient descent with relu activation unit for hidden layer
# and softmax activation in the output
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
                            hiddenActivationFunc='relu',
                            outputActivationFunc="softmax",
                            learningRate = 0.5,
                            numIterations = 9000, 
                            print_cost = True)
#Plot cost vs iterations
iterations <- seq(0,9000,1000)
costs=retvals$costs
df=data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() + geom_line(color="blue") +
 ggtitle("Costs vs iterations") + xlab("Iterations") + ylab("Costs")

6. MNIST dataset with Softmax activation – R

The code below executes a L – Layer Deep Learning network with Softmax output activation, to classify the 10 handwritten digits from MNIST with Stochastic Gradient Descent. The entire 60000 data set was used to train the data. R takes almost 8 hours to process this data set with a mini-batch size of 1000.  The use of ‘for’ loops is limited to iterating through epochs, mini batches and for creating the mini batches itself. All other code is vectorized. Yet, it seems to crawl. Most likely the use of ‘lists’ in R, to return multiple values is performance intensive. Some day, I will try to profile the code, and see where the issue is. However the code works!

Having said that, the Confusion Matrix in R dumps a lot of interesting statistics! There is a bunch of statistical measures for each class. For e.g. the Balanced Accuracy for the digits ‘6’ and ‘9’ is around 50%. Looks like, the classifier is confused by the fact that 6 is inverted 9 and vice-versa. The accuracy on the Test data set is just around 75%. I could have played around with the number of layers, number of hidden units, learning rates, epochs etc to get a much higher accuracy. But since each test took about 8+ hours, I may work on this, some other day!

source("DLfunctions5.R")
source("mnist.R")
#Load the mnist data
load_mnist()
show_digit(train$x[2,])
#Set the layer dimensions
layersDimensions=c(784, 15,9, 10) # Works at 1500
x <- t(train$x)
X <- x[,1:60000]
y <-train$y
y1 <- y[1:60000]
y2 <- as.matrix(y1)
Y=t(y2)

# Subset 32768 random samples from MNIST 
permutation = c(sample(2^15))
# Randomly shuffle the training data
X1 = X[, permutation]
y1 = Y[1, permutation]
y2 <- as.matrix(y1)
Y1=t(y2)

# Execute Stochastic Gradient Descent on the entire training set
# with Softmax activation
retvalsSGD= L_Layer_DeepModel_SGD(X1, Y1, layersDimensions,
                            hiddenActivationFunc='relu',
                            outputActivationFunc="softmax",
                            learningRate = 0.05,
                            mini_batch_size = 512, 
                            num_epochs = 1, 
                            print_cost = True)


# Compute the Confusion Matrix
library(caret)
library(e1071)
predictions=predictProba(retvalsSGD[['parameters']], X,hiddenActivationFunc='relu',
                   outputActivationFunc="softmax")
confusionMatrix(predictions,Y)
# Confusion Matrix on the Training set
> confusionMatrix(predictions,Y)
Confusion Matrix and Statistics

          Reference
Prediction    0    1    2    3    4    5    6    7    8    9
         0 5738    1   21    5   16   17    7   15    9   43
         1    5 6632   21   24   25    3    2   33   13  392
         2   12   32 5747  106   25   28    3   27   44 4779
         3    0   27   12 5715    1   21    1   20    1   13
         4   10    5   21   18 5677    9   17   30   15  166
         5  142   21   96  136   93 5306 5884   43   60  413
         6    0    0    0    0    0    0    0    0    0    0
         7    6    9   13   13    3    4    0 6085    0   55
         8    8   12    7   43    1   32    2    7 5703   69
         9    2    3   20   71    1    1    2    5    6   19

Overall Statistics
                                          
               Accuracy : 0.777           
                 95% CI : (0.7737, 0.7804)
    No Information Rate : 0.1124          
    P-Value [Acc > NIR] : < 2.2e-16       
                                          
                  Kappa : 0.7524          
 Mcnemar's Test P-Value : NA              

Statistics by Class:

                     Class: 0 Class: 1 Class: 2 Class: 3 Class: 4 Class: 5 Class: 6
Sensitivity           0.96877   0.9837  0.96459  0.93215  0.97176  0.97879  0.00000
Specificity           0.99752   0.9903  0.90644  0.99822  0.99463  0.87380  1.00000
Pos Pred Value        0.97718   0.9276  0.53198  0.98348  0.95124  0.43513      NaN
Neg Pred Value        0.99658   0.9979  0.99571  0.99232  0.99695  0.99759  0.90137
Prevalence            0.09872   0.1124  0.09930  0.10218  0.09737  0.09035  0.09863
Detection Rate        0.09563   0.1105  0.09578  0.09525  0.09462  0.08843  0.00000
Detection Prevalence  0.09787   0.1192  0.18005  0.09685  0.09947  0.20323  0.00000
Balanced Accuracy     0.98314   0.9870  0.93551  0.96518  0.98319  0.92629  0.50000
                     Class: 7 Class: 8  Class: 9
Sensitivity            0.9713  0.97471 0.0031938
Specificity            0.9981  0.99666 0.9979464
Pos Pred Value         0.9834  0.96924 0.1461538
Neg Pred Value         0.9967  0.99727 0.9009521
Prevalence             0.1044  0.09752 0.0991500
Detection Rate         0.1014  0.09505 0.0003167
Detection Prevalence   0.1031  0.09807 0.0021667
Balanced Accuracy      0.9847  0.98568 0.5005701

# Confusion Matrix on the Training set xtest <- t(test$x) Xtest <- xtest[,1:10000] ytest <-test$y ytest1 <- ytest[1:10000] ytest2 <- as.matrix(ytest1) Ytest=t(ytest2)

Confusion Matrix and Statistics

          Reference
Prediction    0    1    2    3    4    5    6    7    8    9
         0  950    2    2    3    0    6    9    4    7    6
         1    3 1110    4    2    9    0    3   12    5   74
         2    2    6  965   21    9   14    5   16   12  789
         3    1    2    9  908    2   16    0   21    2    6
         4    0    1    9    5  938    1    8    6    8   39
         5   19    5   25   35   20  835  929    8   54   67
         6    0    0    0    0    0    0    0    0    0    0
         7    4    4    7   10    2    4    0  952    5    6
         8    1    5    8   14    2   16    2    3  876   21
         9    0    0    3   12    0    0    2    6    5    1

Overall Statistics
                                          
               Accuracy : 0.7535          
                 95% CI : (0.7449, 0.7619)
    No Information Rate : 0.1135          
    P-Value [Acc > NIR] : < 2.2e-16       
                                          
                  Kappa : 0.7262          
 Mcnemar's Test P-Value : NA              

Statistics by Class:

                     Class: 0 Class: 1 Class: 2 Class: 3 Class: 4 Class: 5 Class: 6
Sensitivity            0.9694   0.9780   0.9351   0.8990   0.9552   0.9361   0.0000
Specificity            0.9957   0.9874   0.9025   0.9934   0.9915   0.8724   1.0000
Pos Pred Value         0.9606   0.9083   0.5247   0.9390   0.9241   0.4181      NaN
Neg Pred Value         0.9967   0.9972   0.9918   0.9887   0.9951   0.9929   0.9042
Prevalence             0.0980   0.1135   0.1032   0.1010   0.0982   0.0892   0.0958
Detection Rate         0.0950   0.1110   0.0965   0.0908   0.0938   0.0835   0.0000
Detection Prevalence   0.0989   0.1222   0.1839   0.0967   0.1015   0.1997   0.0000
Balanced Accuracy      0.9825   0.9827   0.9188   0.9462   0.9733   0.9043   0.5000
                     Class: 7 Class: 8  Class: 9
Sensitivity            0.9261   0.8994 0.0009911
Specificity            0.9953   0.9920 0.9968858
Pos Pred Value         0.9577   0.9241 0.0344828
Neg Pred Value         0.9916   0.9892 0.8989068
Prevalence             0.1028   0.0974 0.1009000
Detection Rate         0.0952   0.0876 0.0001000
Detection Prevalence   0.0994   0.0948 0.0029000
Balanced Accuracy      0.9607   0.9457 0.4989384

7. Random dataset with Sigmoid activation – Octave

The Octave code below uses the random data set used by Python. The code below implements a L-Layer Deep Learning with Sigmoid Activation.


source("DL5functions.m")
# Read the data
data=csvread("data.csv");

X=data(:,1:2);
Y=data(:,3);
#Set the layer dimensions
layersDimensions = [2 9 7  1]; #tanh=-0.5(ok), #relu=0.1 best!
# Perform gradient descent 
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
                               hiddenActivationFunc='relu', 
                               outputActivationFunc="sigmoid",
                               learningRate = 0.1,
                               numIterations = 10000);
# Plot cost vs iterations
plotCostVsIterations(10000,costs);

8. Spiral dataset with Softmax activation – Octave

The  code below uses the spiral data set used by Python above. The code below implements a L-Layer Deep Learning with Softmax Activation.

# Read the data
data=csvread("spiral.csv");

# Setup the data
X=data(:,1:2);
Y=data(:,3);

# Set the number of features, number of hidden units in hidden layer and number of classess
numFeats=2; #No features
numHidden=100; # No of hidden units
numOutput=3; # No of  classes
# Set the layer dimensions
layersDimensions = [numFeats numHidden  numOutput];  
#Perform gradient descent with softmax activation unit
[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,
                               hiddenActivationFunc='relu', 
                               outputActivationFunc="softmax",
                               learningRate = 0.1,
                               numIterations = 10000);

9. MNIST dataset with Softmax activation – Octave

The code below implements a L-Layer Deep Learning Network in Octave with Softmax output activation unit, for classifying the 10 handwritten digits in the MNIST dataset. Unfortunately, Octave can only index to around 10000 training at a time,  and I was getting an error ‘error: out of memory or dimension too large for Octave’s index type error: called from…’, when I tried to create a batch size of 20000.  So I had to come with a work around to create a batch size of 10000 (randomly) and then use a mini-batch of 1000 samples and execute Stochastic Gradient Descent. The performance was good. Octave takes about 15 minutes, on a batch size of 10000 and a mini batch of 1000.

I thought if the performance was not good, I could iterate through these random batches and refining the gradients as follows

# Pseudo code that could be used since Octave only allows 10K batches
# at a time
# Randomly create weights
[weights biases] = initialize_weights()
for i=1:k
    # Create a random permutation and create a random batch
    permutation = randperm(10000);
    X=trainX(permutation,:);
    Y=trainY(permutation,:);
    # Compute weights from SGD and update weights in the next batch update
    [weights biases costs]=L_Layer_DeepModel_SGD(X,Y,mini_bactch=1000,weights, biases,...);
    ...
endfor
# Load the MNIST data
load('./mnist/mnist.txt.gz'); 
#Create a random permutatation from 60K
permutation = randperm(10000);
disp(length(permutation));

# Use this 10K as the batch
X=trainX(permutation,:);
Y=trainY(permutation,:);

# Set layer dimensions
layersDimensions=[784, 15, 9, 10];

# Run Stochastic Gradient descent with batch size=10K and mini_batch_size=1000
[weights biases costs]=L_Layer_DeepModel_SGD(X', Y', layersDimensions,
                       hiddenActivationFunc='relu', 
                       outputActivationFunc="softmax",
                       learningRate = 0.01,
                       mini_batch_size = 2000, num_epochs = 5000);

9. Final thoughts

Here are some of my final thoughts after working on Python, R and Octave in this series and in other projects
1. Python, with its highly optimized numpy library, is ideally suited for creating Deep Learning Models, which have a lot of matrix manipulations. Python is a real workhorse when it comes to Deep Learning computations.
2. R is somewhat clunky in comparison to its cousin Python in handling matrices or in returning multiple values. But R’s statistical libraries, dplyr, and ggplot are really superior to the Python peers. Also, I find R handles  dataframes,  much better than Python.
3. Octave is a no-nonsense,minimalist language which is very efficient in handling matrices. It is ideally suited for implementing Machine Learning and Deep Learning from scratch. But Octave has its problems and cannot handle large matrix sizes, and also lacks the statistical libaries of R and Python. They possibly exist in its sibling, Matlab

Feel free to clone/download the code from  GitHub at DeepLearning-Part5.

Conclusion

Building a Deep Learning Network from scratch is quite challenging, time-consuming but nevertheless an exciting task.  While the statements in the different languages for manipulating matrices, summing up columns, finding columns which have ones don’t take more than a single statement, extreme care has to be taken to ensure that the statements work well for any dimension.  The lessons learnt from creating L -Layer Deep Learning network  are many and well worth it. Give it a try!

Hasta la vista! I’ll be back, so stick around!
Watch this space!

References
1. Deep Learning Specialization
2. Neural Networks for Machine Learning
3. CS231 Convolutional Neural Networks for Visual Recognition
4. Eli Bendersky’s Website – The Softmax function and its derivative

Also see
1. My book ‘Practical Machine Learning with R and Python’ on Amazon
2. Presentation on Wireless Technologies – Part 1
3. Exploring Quantum Gate operations with QCSimulator
4. What’s up Watson? Using IBM Watson’s QAAPI with Bluemix, NodeExpress – Part 1
5. TWS-4: Gossip protocol: Epidemics and rumors to the rescue
6. cricketr plays the ODIs!
7. “Is it an animal? Is it an insect?” in Android
8. The 3rd paperback & kindle editions of my books on Cricket, now on Amazon
9. Deblurring with OpenCV: Weiner filter reloaded
10. GooglyPlus: yorkr analyzes IPL players, teams, matches with plots and tables

To see all posts click Index of Posts