Dr. Seuss

*“Explanations exist; they have existed for all time; there is always a well-known solution to every human problem — neat, plausible, and wrong.”*

H L Mencken

In this 6th instalment of ‘Deep Learning from first principles in Python, R and Octave-Part6’, I look at a couple of different initialization techniques used in Deep Learning, L2 regularization and the ‘dropout’ method. Specifically, I implement “He initialization” & “Xavier Initialization”. My earlier posts in this series of Deep Learning included

1. Part 1 – In the 1st part, I implemented logistic regression as a simple 2 layer Neural Network

2. Part 2 – In part 2, implemented the most basic of Neural Networks, with just 1 hidden layer, and any number of activation units in that hidden layer. The implementation was in vectorized Python, R and Octave

3. Part 3 -In part 3, I derive the equations and also implement a L-Layer Deep Learning network with either the relu, tanh or sigmoid activation function in Python, R and Octave. The output activation unit was a sigmoid function for logistic classification

4. Part 4 – This part looks at multi-class classification, and I derive the Jacobian of a Softmax function and implement a simple problem to perform multi-class classification.

5. Part 5 – In the 5th part, I extend the L-Layer Deep Learning network implemented in Part 3, to include the Softmax classification. I also use this L-layer implementation to classify MNIST handwritten digits with Python, R and Octave.

The code in Python, R and Octave are identical, and just take into account some of the minor idiosyncrasies of the individual language. In this post, I implement different initialization techniques (random, He, Xavier), L2 regularization and finally dropout. Hence my generic L-Layer Deep Learning network includes these additional enhancements for enabling/disabling initialization methods, regularization or dropout in the algorithm. It already included sigmoid & softmax output activation for binary and multi-class classification, besides allowing relu, tanh and sigmoid activation for hidden units.

This R Markdown file and the code for Python, R and Octave can be cloned/downloaded from **Github** at DeepLearning-Part6

The usual initialization technique is to generate Gaussian or uniform random numbers and multiply it by a small value like 0.01. Two techniques which are used to speed up convergence is the He initialization or Xavier. These initialization techniques enable gradient descent to converge faster.

This technique just initializes the weights to small random values based on Gaussian or uniform distribution

```
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("DLfunctions61.py").read())
#Load the data
train_X, train_Y, test_X, test_Y = load_dataset()
# Set the layers dimensions
layersDimensions = [2,7,1]
# Train a deep learning network with random initialization
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid",learningRate = 0.6, num_iterations = 9000, initType="default", print_cost = True,figure="fig1.png")
# Clear the plot
plt.clf()
plt.close()
# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), train_X, train_Y,str(0.6),figure1="fig2.png")
```

‘He’ initialization attributed to He et al, multiplies the random weights by

```
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("DLfunctions61.py").read())
#Load the data
train_X, train_Y, test_X, test_Y = load_dataset()
# Set the layers dimensions
layersDimensions = [2,7,1]
# Train a deep learning network with He initialization
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid", learningRate =0.6, num_iterations = 10000,initType="He",print_cost = True, figure="fig3.png")
plt.clf()
plt.close()
# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), train_X, train_Y,str(0.6),figure1="fig4.png")
```

Xavier initialization multiply the random weights by

```
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("DLfunctions61.py").read())
#Load the data
train_X, train_Y, test_X, test_Y = load_dataset()
# Set the layers dimensions
layersDimensions = [2,7,1]
# Train a L layer Deep Learning network
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid",
learningRate = 0.6,num_iterations = 10000, initType="Xavier",print_cost = True,
figure="fig5.png")
# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), train_X, train_Y,str(0.6),figure1="fig6.png")
```

```
source("DLfunctions61.R")
#Load the data
z <- as.matrix(read.csv("circles.csv",header=FALSE))
x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
#Set the layer dimensions
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
numIterations = 9000,
initType="default",
print_cost = True)
```

```
#Plot the 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("No of iterations") + ylab("Cost")
```

```
# Plot the decision boundary
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",lr=0.5)
```

```
source("DLfunctions61.R")
# Load the data
z <- as.matrix(read.csv("circles.csv",header=FALSE))
x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
# Set the layer dimensions
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
numIterations = 9000,
initType="He",
print_cost = True)
```

```
#Plot the 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("No of iterations") + ylab("Cost")
```

```
# Plot the decision boundary
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",0.5,lr=0.5)
```

```
## Xav initialization
# Set the layer dimensions
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
numIterations = 9000,
initType="Xav",
print_cost = True)
```

```
#Plot the 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("No of iterations") + ylab("Cost")
```

```
# Plot the decision boundary
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",0.5)
```

source("DL61functions.m") # Read the data data=csvread("circles.csv"); X=data(:,1:2); Y=data(:,3); # Set the layer dimensions layersDimensions = [2 11 1]; #tanh=-0.5(ok), #relu=0.1 best! # Train a deep learning network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid", learningRate = 0.5, lambd=0, keep_prob=1, numIterations = 10000, initType="default"); # Plot cost vs iterations plotCostVsIterations(10000,costs) #Plot decision boundary plotDecisionBoundary(data,weights, biases,keep_prob=1, hiddenActivationFunc="relu")

source("DL61functions.m") #Load data data=csvread("circles.csv"); X=data(:,1:2); Y=data(:,3); # Set the layer dimensions layersDimensions = [2 11 1]; #tanh=-0.5(ok), #relu=0.1 best! # Train a deep learning network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid", learningRate = 0.5, lambd=0, keep_prob=1, numIterations = 8000, initType="He"); plotCostVsIterations(8000,costs) #Plot decision boundary plotDecisionBoundary(data,weights, biases,keep_prob=1,hiddenActivationFunc="relu")

source("DL61functions.m") #Load data data=csvread("circles.csv"); X=data(:,1:2); Y=data(:,3); #Set layer dimensions layersDimensions = [2 11 1]; #tanh=-0.5(ok), #relu=0.1 best! # Train a deep learning network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid", learningRate = 0.5, lambd=0, keep_prob=1, numIterations = 8000, initType="Xav"); plotCostVsIterations(8000,costs) plotDecisionBoundary(data,weights, biases,keep_prob=1,hiddenActivationFunc="relu")

The cross entropy cost for Logistic classification is given as

The regularized L2 cost is given by

Incidentally, a detailed implementation of L1 and L2 regularization in Machine Learning, besides other feature selection methods can be found in my book ‘Practical Machine Learning with R and Python: Machine Learning in stereo‘ available in Amazon. My book is compact and can be used as a quick reference for ML algorithms and associated metrics

```
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("DLfunctions61.py").read())
#Load the data
train_X, train_Y, test_X, test_Y = load_dataset()
# Set the layers dimensions
layersDimensions = [2,7,1]
# Train a deep learning network
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",learningRate = 0.6, lambd=0.1, num_iterations = 9000,
initType="default", print_cost = True,figure="fig7.png")
# Clear the plot
plt.clf()
plt.close()
# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), train_X, train_Y,str(0.6),figure1="fig8.png")
plt.clf()
plt.close()
#Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T,keep_prob=0.9), train_X, train_Y,str(2.2),"fig8.png",)
```

```
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("DLfunctions61.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)
plt.clf()
plt.close()
#Set layer dimensions
layersDimensions = [2,100,3]
y1=y.reshape(-1,1).T
# Train a deep learning network
parameters = L_Layer_DeepModel(X.T, y1, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax",
learningRate = 0.6,lambd=0.2, num_iterations = 5000, print_cost = True,figure="fig9.png")
plt.clf()
plt.close()
W1=parameters['W1']
b1=parameters['b1']
W2=parameters['W2']
b2=parameters['b2']
plot_decision_boundary1(X, y1,W1,b1,W2,b2,figure2="fig10.png")
```

```
source("DLfunctions61.R")
#Load data
df=read.csv("circles.csv",header=FALSE)
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(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
lambd=0.1,
numIterations = 9000,
initType="default",
print_cost = True)
```

```
```

```
#Plot the 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("No of iterations") + ylab("Cost")
```

```
# Plot the decision boundary
plotDecisionBoundary(z,retvals,hiddenActivationFunc="relu",0.5)
```

```
# Read the spiral dataset
#Load the data
source("DLfunctions61.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, 15, 6, 3)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="softmax",
learningRate = 5.1,
lambd=0.01,
numIterations = 9000,
print_cost = True)
```

```
parameters<-retvals$parameters
plotDecisionBoundary1(Z,parameters)
```

2.3a Regularization: Circles data – Octave

source("DL61functions.m") #Load data data=csvread("circles.csv"); X=data(:,1:2); Y=data(:,3); layersDimensions = [2 11 1]; #tanh=-0.5(ok), #relu=0.1 best! # Train a deep learning network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid", learningRate = 0.5, lambd=0.2, keep_prob=1, numIterations = 8000, initType="default"); plotCostVsIterations(8000,costs) #Plot decision boundary plotDecisionBoundary(data,weights, biases,keep_prob=1,hiddenActivationFunc="relu")

source("DL61functions.m") data=csvread("spiral.csv"); # Setup the data X=data(:,1:2); Y=data(:,3); layersDimensions = [2 100 3] # Train a deep learning network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax", learningRate = 0.6, lambd=0.2, keep_prob=1, numIterations = 10000); plotCostVsIterations(10000,costs) #Plot decision boundary plotDecisionBoundary1(data,weights, biases,keep_prob=1,hiddenActivationFunc="relu")

The ‘dropout’ regularization technique was used with great effectiveness, to prevent overfitting by Alex Krizhevsky, Ilya Sutskever and Prof Geoffrey E. Hinton in the Imagenet classification with Deep Convolutional Neural Networks

The technique of dropout works by dropping a random set of activation units in each hidden layer, based on a ‘keep_prob’ criteria in the forward propagation cycle. Here is the code for Octave. A ‘dropoutMat’ is created for each layer which specifies which units to drop **Note**: The same ‘dropoutMat has to be used which computing the gradients in the backward propagation cycle. Hence the dropout matrices are stored in a cell array.

for l =1:L-1 ... D=rand(size(A)(1),size(A)(2)); D = (D < keep_prob) ; # Zero out some hidden units A= A .* D; # Divide by keep_prob to keep the expected value of A the same A = A ./ keep_prob; # Store D in a dropoutMat cell array dropoutMat{l}=D; ... endfor

In the backward propagation cycle we have

for l =(L-1):-1:1 ... D = dropoutMat{l}; # Zero out the dAl based on same dropout matrix dAl= dAl .* D; # Divide by keep_prob to maintain the expected value dAl = dAl ./ keep_prob; ... endfor

```
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("DLfunctions61.py").read())
#Load the data
train_X, train_Y, test_X, test_Y = load_dataset()
# Set the layers dimensions
layersDimensions = [2,7,1]
# Train a deep learning network
parameters = L_Layer_DeepModel(train_X, train_Y, layersDimensions, hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",learningRate = 0.6, keep_prob=0.7, num_iterations = 9000,
initType="default", print_cost = True,figure="fig11.png")
# Clear the plot
plt.clf()
plt.close()
# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T,keep_prob=0.7), train_X, train_Y,str(0.6),figure1="fig12.png")
```

```
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("DLfunctions61.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
# Plot the data
plt.scatter(X[:, 0], X[:, 1], c=y, s=40, cmap=plt.cm.Spectral)
plt.clf()
plt.close()
layersDimensions = [2,100,3]
y1=y.reshape(-1,1).T
# Train a deep learning network
parameters = L_Layer_DeepModel(X.T, y1, layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax",
learningRate = 0.6,keep_prob=0.5, num_iterations = 5000, print_cost = True,figure="fig13.png")
plt.clf()
plt.close()
W1=parameters['W1']
b1=parameters['b1']
W2=parameters['W2']
b2=parameters['b2']
#Plot decision boundary
plot_decision_boundary1(X, y1,W1,b1,W2,b2,figure2="fig14.png")
```

```
source("DLfunctions61.R")
#Load data
df=read.csv("circles.csv",header=FALSE)
z <- as.matrix(read.csv("circles.csv",header=FALSE))
x <- z[,1:2]
y <- z[,3]
X <- t(x)
Y <- t(y)
layersDimensions = c(2,11,1)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="sigmoid",
learningRate = 0.5,
keep_prob=0.8,
numIterations = 9000,
initType="default",
print_cost = True)
```

```
# Plot the decision boundary
plotDecisionBoundary(z,retvals,keep_prob=0.6, hiddenActivationFunc="relu",0.5)
```

```
# Read the spiral dataset
source("DLfunctions61.R")
# Load data
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, 15, 6, 3)
# Train a deep learning network
retvals = L_Layer_DeepModel(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="softmax",
learningRate = 5.1,
keep_prob=0.9,
numIterations = 9000,
print_cost = True)
```

```
parameters<-retvals$parameters
#Plot decision boundary
plotDecisionBoundary1(Z,parameters)
```

data=csvread("circles.csv"); X=data(:,1:2); Y=data(:,3); layersDimensions = [2 11 1]; #tanh=-0.5(ok), #relu=0.1 best! # Train a deep learning network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="sigmoid", learningRate = 0.5, lambd=0, keep_prob=0.8, numIterations = 10000, initType="default"); plotCostVsIterations(10000,costs) #Plot decision boundary plotDecisionBoundary1(data,weights, biases,keep_prob=1, hiddenActivationFunc="relu")

source("DL61functions.m") data=csvread("spiral.csv"); # Setup the data X=data(:,1:2); Y=data(:,3); layersDimensions = [numFeats numHidden numOutput]; # Train a deep learning network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', outputActivationFunc="softmax", learningRate = 0.1, lambd=0, keep_prob=0.8, numIterations = 10000); plotCostVsIterations(10000,costs) #Plot decision boundary plotDecisionBoundary1(data,weights, biases,keep_prob=1, hiddenActivationFunc="relu")

**Note**: The Python, R and Octave code can be cloned/downloaded from Github at DeepLearning-Part6

**Conclusion**

This post further enhances my earlier L-Layer generic implementation of a Deep Learning network to include options for initialization techniques, L2 regularization or dropout regularization

**References**

1. Deep Learning Specialization

2. Neural Networks for Machine Learning

Also see

1. Architecting a cloud based IP Multimedia System (IMS)

2. Using Linear Programming (LP) for optimizing bowling change or batting lineup in T20 cricket

3. My book ‘Practical Machine Learning with R and Python’ on Amazon

4. Simulating a Web Joint in Android

5. Inswinger: yorkr swings into International T20s

6. Introducing QCSimulator: A 5-qubit quantum computing simulator in R

7. Computer Vision: Ramblings on derivatives, histograms and contours

8. Bend it like Bluemix, MongoDB using Auto-scale – Part 1!

9. The 3rd paperback & kindle editions of my books on Cricket, now on Amazon

To see all posts click Index of posts

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.

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.

The Softmax function can be numerically unstable because of the division of large exponentials. To handle this problem we have to implement stable Softmax function as below

Therefore

Here ‘D’ can be anything. A common choice is

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.

Incidentally you could also check out my book, Practical Machine Learning in R and Python – Machine Learning in Stereo, available on Amazon. In this book, I implement different ML algorithms, regularization techniques, model selection, ensemble methods, key measures and metrics. My book is compact, minimalist and includes implementations of many ML algorithms in R and Python, and can be used as a handy reference.

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)
```

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
```

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
```

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")
```

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")
```

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)
# Execute Stochastic Gradient Descent on the entire training set
# with Softmax activation
retvalsSGD= L_Layer_DeepModel_SGD(X, Y, layersDimensions,
hiddenActivationFunc='relu',
outputActivationFunc="softmax",
learningRate = 0.01,
mini_batch_size = 1000,
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
```

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);
```

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);
```

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);
```

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.

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

]]>

The implementations of all the discussed algorithm are are available in my book which is available on Amazon My book ‘Practical Machine Learning with R and Python’ on Amazon

You may also like

1. My TEDx talk on the “Internet of Things”

2. Deep Learning from first principles in Python, R and Octave – Part 2

3. De-blurring revisited with Wiener filter using OpenCV

4. Architecting a cloud based IP Multimedia System (IMS)

5.The 3rd paperback & kindle editions of my books on Cricket, now on Amazon

To see all posts click Index of posts

]]>If you would like to see the implementations of the discussed algorithms, in this presentation, do check out my book My book ‘Practical Machine Learning with R and Python’ on Amazon

You may also like

1.A primer on Qubits, Quantum gates and Quantum Operations

2.Deep Learning from first principles in Python, R and Octave – Part 4

3.Latency, throughput implications for the Cloud

4.Brewing a potion with Bluemix, PostgreSQL, Node.js in the cloud

5.Practical Machine Learning with R and Python – Part 6

6. Introducing cricketr! : An R package to analyze performances of cricketers

To see all post click Index of posts

]]>It is hoped that these presentations will trigger sufficient interest in you, to explore this fascinating field further

To see actual implementations of the most widely used Machine Learning algorithms in R and Python, check out My book ‘Practical Machine Learning with R and Python’ on Amazon

Also see

1. Practical Machine Learning with R and Python – Part 3

2.R vs Python: Different similarities and similar differences

3. Perils and pitfalls of Big Data

4. Deep Learning from first principles in Python, R and Octave – Part 2

5. Getting started with memcached-libmemcached

To see all post see “Index of posts“

]]>1. Deep Learning from first principles in Python, R and Octave – Part 1. In this post I implemented logistic regression as a simple Neural Network in vectorized Python, R and Octave

2. Deep Learning from first principles in Python, R and Octave – Part 2. This 2nd part implemented the most elementary neural network with 1 hidden layer and any number of activation units in the hidden layer with sigmoid activation at the output layer

3. Deep Learning from first principles in Python, R and Octave – Part 3. The 3rd implemented a multi-layer Deep Learning network with an arbitrary number if hidden layers and activation units per hidden layer. The output layer was for binary classification which was based on the sigmoid unit. This multi-layer deep network was implemented in vectorized Python, R and Octave

This 4th part takes a swing at multi-class classification and uses the Softmax as the activation unit in the output layer. Inclusion of the Softmax activation unit in the activation layer requires us to compute the derivative of Softmax, or rather the “Jacobian” of the Softmax function, besides also computing the log loss for this Softmax activation during back propagation. Since the derivation of the Jacobian of a Softmax and the computation of the Cross Entropy/log loss is very involved, I have implemented a basic neural network with just 1 hidden layer with the Softmax activation at the output layer. I also perform multi-class classification based on the ‘spiral’ data set from CS231n Convolutional Neural Networks Stanford course, to test the performance and correctness of the implementations in Python, R and Octave. You can clone download the code for the Python, R and Octave implementations from Github at Deep Learning – Part 4

The Softmax function takes an N dimensional vector as input and generates a N dimensional vector as output.

The Softmax function is given by

There is a probabilistic interpretation of the Softmax, since the sum of the Softmax values of a set of vectors will always add up to 1, given that each Softmax value is divided by the total of all values.

As mentioned earlier, the Softmax takes a vector input and returns a vector of outputs. For e.g. the Softmax of a vector a=[1, 3, 6] is another vector S=[0.0063,0.0471,0.9464]. Notice that vector output is proportional to the input vector. Also, taking the derivative of a vector by another vector, is known as the Jacobian. By the way, The Matrix Calculus You Need For Deep Learning by Terence Parr and Jeremy Howard, is very good paper that distills all the main mathematical concepts for Deep Learning in one place.

Let us take a simple 2 layered neural network with just 2 activation units in the hidden layer is shown below

and

where g'() is the activation unit in the hidden layer which can be a relu, sigmoid or a

tanh function

**Note**: The superscript denotes the layer. The above denotes the equation for layer 1

of the neural network. For layer 2 with the Softmax activation, the equations are

and

where S() is the Softmax activation function

The Jacobian of the softmax ‘S’ is given by

– (A)

Now the ‘division-rule’ of derivatives is as follows. If u and v are functions of x, then

Using this to compute each element of the above Jacobian matrix, we see that

when i=j we have

and when

This is of the general form

when i=j

and

when

**Note**: Since the Softmax essentially gives the probability the following

notation is also used

when i=j

and

If you throw the “Kronecker delta” into the equation, then the above equations can be expressed even more concisely as

where when i=j and 0 when

This reduces the Jacobian of the simple 2 output softmax vectors equation (A) as

The loss of Softmax is given by

For the 2 valued Softmax output this is

Using the chain rule we can write

(1)

and

(2)

In expanded form this is

Also

Therefore

Since

when i=j

and

when

which simplifies to

Since

Similarly

In general this is of the form

For e.g if the probabilities computed were p=[0.1, 0.7, 0.2] then this implies that the class with probability 0.7 is the likely class. This would imply that the ‘One hot encoding’ for yi would be yi=[0,1,0] therefore the gradient pi-yi = [0.1,-0.3,0.2]

<strong>Note: Further, we could extend this derivation for a Softmax activation output that outputs 3 classes

We could derive

which similarly reduces to

Interestingly, despite the lengthy derivations the final result is simple and intuitive!

As seen in my post ‘Deep Learning from first principles with Python, R and Octave – Part 3 the key equations for forward and backward propagation are

**Forward propagation equations layer 1**

and

**Forward propagation equations layer 1**

and

**Backward propagation equations layer 2**

**Backward propagation equations layer 1**

As I mentioned earlier, I will be using the ‘spiral’ data from CS231n Convolutional Neural Networks to ensure that my vectorized implementations in Python, R and Octave are correct. Here is the ‘spiral’ data set.

```
import numpy as np
import matplotlib.pyplot as plt
import os
os.chdir("C:/junk/dl-4/dl-4")
exec(open("././DLfunctions41.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
# Plot the data
plt.scatter(X[:, 0], X[:, 1], c=y, s=40, cmap=plt.cm.Spectral)
plt.savefig("fig1.png", bbox_inches='tight')
```

The implementations of the vectorized Python, R and Octave code are shown diagrammatically below

A simple 2 layer Neural network with a single hidden layer , with 100 Relu activation units in the hidden layer and the Softmax activation unit in the output layer is used for multi-class classification. This Deep Learning Network, plots the non-linear boundary of the 3 classes as shown below

```
import numpy as np
import matplotlib.pyplot as plt
import os
os.chdir("C:/junk/dl-4/dl-4")
exec(open("././DLfunctions41.py").read())
# Read the input data
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
# Set the number of features, hidden units in hidden layer and number of classess
numHidden=100 # No of hidden units in hidden layer
numFeats= 2 # dimensionality
numOutput = 3 # number of classes
# Initialize the model
parameters=initializeModel(numFeats,numHidden,numOutput)
W1= parameters['W1']
b1= parameters['b1']
W2= parameters['W2']
b2= parameters['b2']
# Set the learning rate
learningRate=0.6
# Initialize losses
losses=[]
# Perform Gradient descent
for i in range(10000):
# Forward propagation through hidden layer with Relu units
A1,cache1= layerActivationForward(X.T,W1,b1,'relu')
# Forward propagation through output layer with Softmax
A2,cache2 = layerActivationForward(A1,W2,b2,'softmax')
# No of training examples
numTraining = X.shape[0]
# Compute log probs. Take the log prob of correct class based on output y
correct_logprobs = -np.log(A2[range(numTraining),y])
# Conpute loss
loss = np.sum(correct_logprobs)/numTraining
# Print the loss
if i % 1000 == 0:
print("iteration %d: loss %f" % (i, loss))
losses.append(loss)
dA=0
# Backward propagation through output layer with Softmax
dA1,dW2,db2 = layerActivationBackward(dA, cache2, y, activationFunc='softmax')
# Backward propagation through hidden layer with Relu unit
dA0,dW1,db1 = layerActivationBackward(dA1.T, cache1, y, activationFunc='relu')
#Update paramaters with the learning rate
W1 += -learningRate * dW1
b1 += -learningRate * db1
W2 += -learningRate * dW2.T
b2 += -learningRate * db2.T
#Plot losses vs iterations
i=np.arange(0,10000,1000)
plt.plot(i,losses)
plt.xlabel('Iterations')
plt.ylabel('Loss')
plt.title('Losses vs Iterations')
plt.savefig("fig2.png", bbox="tight")
#Compute the multi-class Confusion Matrix
from sklearn.metrics import confusion_matrix
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score
# We need to determine the predicted values from the learnt data
# Forward propagation through hidden layer with Relu units
A1,cache1= layerActivationForward(X.T,W1,b1,'relu')
# Forward propagation through output layer with Softmax
A2,cache2 = layerActivationForward(A1,W2,b2,'softmax')
#Compute predicted values from weights and biases
yhat=np.argmax(A2, axis=1)
a=confusion_matrix(y.T,yhat.T)
print("Multi-class Confusion Matrix")
print(a)
```

```
## iteration 0: loss 1.098507
## iteration 1000: loss 0.214611
## iteration 2000: loss 0.043622
## iteration 3000: loss 0.032525
## iteration 4000: loss 0.025108
## iteration 5000: loss 0.021365
## iteration 6000: loss 0.019046
## iteration 7000: loss 0.017475
## iteration 8000: loss 0.016359
## iteration 9000: loss 0.015703
## Multi-class Confusion Matrix
## [[ 99 1 0]
## [ 0 100 0]
## [ 0 1 99]]
```

To know more details about Confusion Matrix and other related measures check out my book My book ‘Practical Machine Learning with R and Python’ on Amazon

The spiral data set created with Python was saved, and is used as the input with R code. The R Neural Network seems to perform much,much slower than both Python and Octave. Not sure why! Incidentally the computation of loss and the softmax derivative are identical for both R and Octave. yet R is much slower. To compute the softmax derivative I create matrices for the One Hot Encoded yi and then stack them before subtracting pi-yi. I am sure there is a more elegant and more efficient way to do this, much like Python. Any suggestions?

```
library(ggplot2)
library(dplyr)
library(RColorBrewer)
```

```
source("DLfunctions41.R")
# Read the spiral dataset
Z <- as.matrix(read.csv("spiral.csv",header=FALSE))
Z1=data.frame(Z)
#Plot the dataset
ggplot(Z1,aes(x=V1,y=V2,col=V3)) +geom_point() +
scale_colour_gradientn(colours = brewer.pal(10, "Spectral"))
```

```
# Setup the data
X <- Z[,1:2]
y <- Z[,3]
X1 <- t(X)
Y1 <- 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
# Initialize model
parameters <-initializeModel(numFeats, numHidden,numOutput)
W1 <-parameters[['W1']]
b1 <-parameters[['b1']]
W2 <-parameters[['W2']]
b2 <-parameters[['b2']]
# Set the learning rate
learningRate <- 0.5
# Initialize losses
losses <- NULL
# Perform gradient descent
for(i in 0:9000){
# Forward propagation through hidden layer with Relu units
retvals <- layerActivationForward(X1,W1,b1,'relu')
A1 <- retvals[['A']]
cache1 <- retvals[['cache']]
forward_cache1 <- cache1[['forward_cache1']]
activation_cache <- cache1[['activation_cache']]
# Forward propagation through output layer with Softmax units
retvals = layerActivationForward(A1,W2,b2,'softmax')
A2 <- retvals[['A']]
cache2 <- retvals[['cache']]
forward_cache2 <- cache2[['forward_cache1']]
activation_cache2 <- cache2[['activation_cache']]
# No oftraining examples
numTraining <- dim(X)[1]
dA <-0
# Select the elements where the y values are 0, 1 or 2 and make a vector
a=c(A2[y==0,1],A2[y==1,2],A2[y==2,3])
# Take log
correct_probs = -log(a)
# Compute loss
loss= sum(correct_probs)/numTraining
if(i %% 1000 == 0){
sprintf("iteration %d: loss %f",i, loss)
print(loss)
}
# Backward propagation through output layer with Softmax units
retvals = layerActivationBackward(dA, cache2, y, activationFunc='softmax')
dA1 = retvals[['dA_prev']]
dW2= retvals[['dW']]
db2= retvals[['db']]
# Backward propagation through hidden layer with Relu units
retvals = layerActivationBackward(t(dA1), cache1, y, activationFunc='relu')
dA0 = retvals[['dA_prev']]
dW1= retvals[['dW']]
db1= retvals[['db']]
# Update parameters
W1 <- W1 - learningRate * dW1
b1 <- b1 - learningRate * db1
W2 <- W2 - learningRate * t(dW2)
b2 <- b2 - learningRate * t(db2)
}
```

```
## [1] 1.212487
## [1] 0.5740867
## [1] 0.4048824
## [1] 0.3561941
## [1] 0.2509576
## [1] 0.7351063
## [1] 0.2066114
## [1] 0.2065875
## [1] 0.2151943
## [1] 0.1318807
```

```
#Create iterations
iterations <- seq(0,10)
#df=data.frame(iterations,losses)
ggplot(df,aes(x=iterations,y=losses)) + geom_point() + geom_line(color="blue") +
ggtitle("Losses vs iterations") + xlab("Iterations") + ylab("Loss")
plotDecisionBoundary(Z,W1,b1,W2,b2)
```

**Multi-class Confusion Matrix**

```
library(caret)
library(e1071)
# Forward propagation through hidden layer with Relu units
retvals <- layerActivationForward(X1,W1,b1,'relu')
A1 <- retvals[['A']]
# Forward propagation through output layer with Softmax units
retvals = layerActivationForward(A1,W2,b2,'softmax')
A2 <- retvals[['A']]
yhat <- apply(A2, 1,which.max) -1
Confusion Matrix and Statistics
Reference
Prediction 0 1 2
0 97 0 1
1 2 96 4
2 1 4 95
Overall Statistics
Accuracy : 0.96
95% CI : (0.9312, 0.9792)
No Information Rate : 0.3333
P-Value [Acc > NIR] : <2e-16
Kappa : 0.94
Mcnemar's Test P-Value : 0.5724
Statistics by Class:
Class: 0 Class: 1 Class: 2
Sensitivity 0.9700 0.9600 0.9500
Specificity 0.9950 0.9700 0.9750
Pos Pred Value 0.9898 0.9412 0.9500
Neg Pred Value 0.9851 0.9798 0.9750
Prevalence 0.3333 0.3333 0.3333
Detection Rate 0.3233 0.3200 0.3167
Detection Prevalence 0.3267 0.3400 0.3333
Balanced Accuracy 0.9825 0.9650 0.9625
```

My book “Practical Machine Learning with R and Python” includes the implementation for many Machine Learning algorithms and associated metrics. Pick up your copy today!

A 2 layer Neural network with the Softmax activation unit in the output layer is constructed in Octave. The same spiral data set is used for Octave also

source("DL41functions.m")

# Read the spiral 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 classes

numFeats=2; #No features

numHidden=100; # No of hidden units

numOutput=3; # No of classes

# Initialize model

[W1 b1 W2 b2] = initializeModel(numFeats,numHidden,numOutput);

# Initialize losses

losses=[]

#Initialize learningRate

learningRate=0.5;

for k =1:10000

# Forward propagation through hidden layer with Relu units

[A1,cache1 activation_cache1]= layerActivationForward(X',W1,b1,activationFunc ='relu');

# Forward propagation through output layer with Softmax units

[A2,cache2 activation_cache2] =

layerActivationForward(A1,W2,b2,activationFunc='softmax');

# No of training examples

numTraining = size(X)(1);

# Select rows where Y=0,1,and 2 and concatenate to a long vector

a=[A2(Y==0,1) ;A2(Y==1,2) ;A2(Y==2,3)];

#Select the correct column for log prob

correct_probs = -log(a);

#Compute log loss

loss= sum(correct_probs)/numTraining;

if(mod(k,1000) == 0)

disp(loss);

losses=[losses loss];

endif

dA=0;

# Backward propagation through output layer with Softmax units

[dA1 dW2 db2] = layerActivationBackward(dA, cache2, activation_cache2,Y,activationFunc='softmax');

# Backward propagation through hidden layer with Relu units

[dA0,dW1,db1] = layerActivationBackward(dA1', cache1, activation_cache1, Y, activationFunc='relu');

#Update parameters

W1 += -learningRate * dW1;

b1 += -learningRate * db1;

W2 += -learningRate * dW2';

b2 += -learningRate * db2';

endfor

# Plot Losses vs Iterations

iterations=0:1000:9000

plotCostVsIterations(iterations,losses)

# Plot the decision boundary

plotDecisionBoundary( X,Y,W1,b1,W2,b2)

The code for the Python, R and Octave implementations can be downloaded from Github at Deep Learning – Part 4

In this post I have implemented a 2 layer Neural Network with the Softmax classifier. In Part 3, I implemented a multi-layer Deep Learning Network. I intend to include the Softmax activation unit into the generalized multi-layer Deep Network along with the other activation units of sigmoid,tanh and relu.

Stick around, I’ll be back!!

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

5. Cross Validated – Backpropagation with Softmax / Cross Entropy

6. Stackoverflow – CS231n: How to calculate gradient for Softmax loss function?

7. Math Stack Exchange – Derivative of Softmax

8. The Matrix Calculus for Deep Learning

**You may like**

1.My book ‘Practical Machine Learning with R and Python’ on Amazon

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

3. Deblurring with OpenCV: Weiner filter reloaded

4. A method to crowd source pothole marking on (Indian) roads

5. Rock N’ Roll with Bluemix, Cloudant & NodeExpress

6. Sea shells on the seashore

7. Design Principles of Scalable, Distributed Systems

To see all post click Index of posts

]]>from The Brain: The Story of you – David Eagleman

*“Thought is a great big vector of neural activity”*

Prof Geoffrey Hinton

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

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

1. Python and Octave allow multiple return values to be unpacked in a single statement. With R, unpacking multiple return values from a list, requires the list returned, to be unpacked separately. I did see that there is a package gsubfn, which does this. I hope this feature becomes a base R feature.

2. Python and R allow dissimilar elements to be saved and returned from functions using dictionaries or lists respectively. However there is no real equivalent in Octave. The closest I got to this functionality in Octave, was the ‘cell array’. But the cell array can be accessed only by the index, and not with the key as in a Python dictionary or R list. This makes things just a bit more difficult in Octave.

3. Python and Octave include implicit broadcasting. In R, broadcasting is not implicit, but R has a nifty function, the sweep(), with which we can broadcast either by columns or by rows

4. The closest equivalent of Python’s dictionary, or R’s list, in Octave is the cell array. However I had to manage separate cell arrays for weights and biases and during gradient descent and separate gradients dW and dB

5. In Python the rank-1 numpy arrays can be annoying at times. This issue is not present in R and Octave.

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

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

Lets take a simple 3 layer Neural network with 3 hidden layers and an output layer

In the forward propagation cycle the equations are

and

and

and

The loss function is given by

and

For a binary classification the output activation function is the sigmoid function given by

. It can be shown that

see equation 2 in Part 1

see equation (f) in Part 1

and since

because -(1a)

and -(1b)

-(1c)

since

and

-(1d)

because

Also

– (2a)

– (2b)

– (2c)

– (2d)

Inspecting the above equations (1a – 1d & 2a-2d), our ‘Uber deep, bottomless’ brain can easily discern the pattern in these equations. The equation for any layer ‘l’ is of the form

and

The equation for the backward propagation have the general form

Some other important results The derivatives of the activation functions in the implemented Deep Learning network

g(z) = sigmoid(z) = = a g’(z) = a(1-a) – See Part 1

g(z) = tanh(z) = a g’(z) =

g(z) = relu(z) = z when z>0 and 0 when z 0 and 0 when z <= 0

While it appears that there is a discontinuity for the derivative at 0 the small value at the discontinuity does not present a problem

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

V (in Python)= , …

V (in R)= , …

V (in Octave)= [ …

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

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

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

`import os import numpy as np import matplotlib.pyplot as plt import matplotlib.colors import sklearn.linear_model from sklearn.model_selection import train_test_split from sklearn.datasets import make_classification, make_blobs from matplotlib.colors import ListedColormap import sklearn import sklearn.datasets #from DLfunctions import plot_decision_boundary execfile("./DLfunctions34.py") # os.chdir("C:\\software\\DeepLearning-Posts\\part3") # Create clusters of 2 classes X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9, cluster_std = 1.3, random_state = 4) #Create 2 classes Y1=Y1.reshape(400,1) Y1 = Y1 % 2 X2=X1.T Y2=Y1.T # Set the dimensions of DL Network # Below we have # 2 - 2 input features # 9,9 - 2 hidden layers with 9 activation units per layer and # 1 - 1 sigmoid activation unit in the output layer as this is a binary classification # The activation in the hidden layer is the 'relu' specified in L_Layer_DeepModel layersDimensions = [2, 9, 9,1] # 4-layer model parameters = L_Layer_DeepModel(X2, Y2, layersDimensions,hiddenActivationFunc='relu', learning_rate = 0.3,num_iterations = 2500, fig="fig1.png") #Plot the decision boundary plot_decision_boundary(lambda x: predict(parameters, x.T), X2,Y2,str(0.3),"fig2.png") # Compute the confusion matrix yhat = predict(parameters,X2) from sklearn.metrics import confusion_matrix a=confusion_matrix(Y2.T,yhat.T) from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score print('Accuracy: {:.2f}'.format(accuracy_score(Y2.T, yhat.T))) print('Precision: {:.2f}'.format(precision_score(Y2.T, yhat.T))) print('Recall: {:.2f}'.format(recall_score(Y2.T, yhat.T))) print('F1: {:.2f}'.format(f1_score(Y2.T, yhat.T)))`

`## Accuracy: 0.90 ## Precision: 0.91 ## Recall: 0.87 ## F1: 0.89`

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

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

```
library(ggplot2)
# Read the data
z <- as.matrix(read.csv("data.csv",header=FALSE))
x <- z[,1:2]
y <- z[,3]
X1 <- t(x)
Y1 <- t(y)
# Set the dimensions of the Deep Learning network
# No of input features =2, 2 hidden layers with 9 activation units and 1 output layer
layersDimensions = c(2, 9, 9,1)
# Execute the Deep Learning Neural Network
retvals = L_Layer_DeepModel(X1, Y1, layersDimensions,
hiddenActivationFunc='relu',
learningRate = 0.3,
numIterations = 5000,
print_cost = True)
```

```
library(ggplot2)
source("DLfunctions33.R")
# Get the computed costs
costs <- retvals[['costs']]
# Create a sequence of iterations
numIterations=5000
iterations <- seq(0,numIterations,by=1000)
df <-data.frame(iterations,costs)
# Plot the Costs vs number of iterations
ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") +
xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations")
```

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

`library(caret)`

```
# Predict the output for the data values
yhat <-predict(retvals$parameters,X1,hiddenActivationFunc="relu")
yhat[yhat==FALSE]=0
yhat[yhat==TRUE]=1
# Compute the confusion matrix
confusionMatrix(yhat,Y1)
```

```
## Confusion Matrix and Statistics
##
## Reference
## Prediction 0 1
## 0 201 10
## 1 21 168
##
## Accuracy : 0.9225
## 95% CI : (0.8918, 0.9467)
## No Information Rate : 0.555
## P-Value [Acc > NIR] : < 2e-16
##
## Kappa : 0.8441
## Mcnemar's Test P-Value : 0.07249
##
## Sensitivity : 0.9054
## Specificity : 0.9438
## Pos Pred Value : 0.9526
## Neg Pred Value : 0.8889
## Prevalence : 0.5550
## Detection Rate : 0.5025
## Detection Prevalence : 0.5275
## Balanced Accuracy : 0.9246
##
## 'Positive' Class : 0
##
```

Included below is the code for performing classification. Incidentally Octave does not seem to have implemented the confusion matrix, but confusionmat is available in Matlab.

# Read the data

data=csvread("data.csv");

X=data(:,1:2);

Y=data(:,3);

# Set layer dimensions

layersDimensions = [2 9 7 1] #tanh=-0.5(ok), #relu=0.1 best!

# Execute Deep Network

[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,

hiddenActivationFunc='relu',

learningRate = 0.1,

numIterations = 10000);

plotCostVsIterations(10000,costs);

plotDecisionBoundary(data,weights, biases,hiddenActivationFunc="tanh")

Below the Tanh activation function is used to perform the same classification. I found the Tanh activation required a simpler Neural Network of 3 layers.

```
# Tanh activation
import os
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.colors
import sklearn.linear_model
from sklearn.model_selection import train_test_split
from sklearn.datasets import make_classification, make_blobs
from matplotlib.colors import ListedColormap
import sklearn
import sklearn.datasets
#from DLfunctions import plot_decision_boundary
os.chdir("C:\\software\\DeepLearning-Posts\\part3")
execfile("./DLfunctions34.py")
# Create the dataset
X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9,
cluster_std = 1.3, random_state = 4)
#Create 2 classes
Y1=Y1.reshape(400,1)
Y1 = Y1 % 2
X2=X1.T
Y2=Y1.T
# Set the dimensions of the Neural Network
layersDimensions = [2, 4, 1] # 3-layer model
# Compute the DL network
parameters = L_Layer_DeepModel(X2, Y2, layersDimensions, hiddenActivationFunc='tanh', learning_rate = .5,num_iterations = 2500,fig="fig3.png")
#Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), X2,Y2,str(0.5),"fig4.png")
```

R performs better with a Tanh activation than the Relu as can be seen below

```
#Set the dimensions of the Neural Network
layersDimensions = c(2, 9, 9,1)
library(ggplot2)
# Read the data
z <- as.matrix(read.csv("data.csv",header=FALSE))
x <- z[,1:2]
y <- z[,3]
X1 <- t(x)
Y1 <- t(y)
# Execute the Deep Model
retvals = L_Layer_DeepModel(X1, Y1, layersDimensions,
hiddenActivationFunc='tanh',
learningRate = 0.3,
numIterations = 5000,
print_cost = True)
```

```
# Get the costs
costs <- retvals[['costs']]
iterations <- seq(0,numIterations,by=1000)
df <-data.frame(iterations,costs)
# Plot Cost vs number of iterations
ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") +
xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations")
```

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

The code below uses the Tanh activation in the hidden layers for Octave

# Read the data

data=csvread("data.csv");

X=data(:,1:2);

Y=data(:,3);

# Set layer dimensions

layersDimensions = [2 9 7 1] #tanh=-0.5(ok), #relu=0.1 best!

# Execute Deep Network

[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,

hiddenActivationFunc='tanh',

learningRate = 0.1,

numIterations = 10000);

plotCostVsIterations(10000,costs);

plotDecisionBoundary(data,weights, biases,hiddenActivationFunc="tanh")

To make things more interesting, I create a 2D figure of the Bernoulli’s lemniscate to perform non-linear classification. The Lemniscate is given by the equation

=

`import os import numpy as np import matplotlib.pyplot as plt os.chdir("C:\\software\\DeepLearning-Posts\\part3") execfile("./DLfunctions33.py") x1=np.random.uniform(0,10,2000).reshape(2000,1) x2=np.random.uniform(0,10,2000).reshape(2000,1) X=np.append(x1,x2,axis=1) X.shape # Create a subset of values where squared is <0,4. Perform ravel() to flatten this vector # Create the equation # (x^{2} + y^{2})^2 - 2a^2*(x^{2}-y^{2}) <= 0 a=np.power(np.power(X[:,0]-5,2) + np.power(X[:,1]-5,2),2) b=np.power(X[:,0]-5,2) - np.power(X[:,1]-5,2) c= a - (b*np.power(4,2)) <=0 Y=c.reshape(2000,1) # Create a scatter plot of the lemniscate plt.scatter(X[:,0], X[:,1], c=Y, marker= 'o', s=15,cmap="viridis") Z=np.append(X,Y,axis=1) plt.savefig("fig50.png",bbox_inches='tight') plt.clf() # Set the data for classification X2=X.T Y2=Y.T # These settings work the best # Set the Deep Learning layer dimensions for a Relu activation layersDimensions = [2,7,4,1] #Execute the DL network parameters = L_Layer_DeepModel(X2, Y2, layersDimensions, hiddenActivationFunc='relu', learning_rate = 0.5,num_iterations = 10000, fig="fig5.png") #Plot the decision boundary plot_decision_boundary(lambda x: predict(parameters, x.T), X2, Y2,str(2.2),"fig6.png") # Compute the Confusion matrix yhat = predict(parameters,X2) from sklearn.metrics import confusion_matrix a=confusion_matrix(Y2.T,yhat.T) from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score print('Accuracy: {:.2f}'.format(accuracy_score(Y2.T, yhat.T))) print('Precision: {:.2f}'.format(precision_score(Y2.T, yhat.T))) print('Recall: {:.2f}'.format(recall_score(Y2.T, yhat.T))) print('F1: {:.2f}'.format(f1_score(Y2.T, yhat.T)))`

`## Accuracy: 0.93 ## Precision: 0.77 ## Recall: 0.76 ## F1: 0.76`

We could get better performance by tuning further. Do play around if you fork the code.

**Note:**: The lemniscate data is saved as a CSV and then read in R and also in Octave. I do this instead of recreating the lemniscate shape

The R decision boundary for the Bernoulli’s lemniscate is shown below

```
Z <- as.matrix(read.csv("lemniscate.csv",header=FALSE))
Z1=data.frame(Z)
# Create a scatter plot of the lemniscate
ggplot(Z1,aes(x=V1,y=V2,col=V3)) +geom_point()
```

```
#Set the data for the DL network
X=Z[,1:2]
Y=Z[,3]
X1=t(X)
Y1=t(Y)
# Set the layer dimensions for the tanh activation function
layersDimensions = c(2,5,4,1)
# Execute the Deep Learning network with Tanh activation
retvals = L_Layer_DeepModel(X1, Y1, layersDimensions,
hiddenActivationFunc='tanh',
learningRate = 0.3,
numIterations = 20000, print_cost = True)
```

```
# Plot cost vs iteration
costs <- retvals[['costs']]
numIterations = 20000
iterations <- seq(0,numIterations,by=1000)
df <-data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") +
xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations")
```

```
#Plot the decision boundary
plotDecisionBoundary(Z,retvals,hiddenActivationFunc="tanh",0.3)
```

Octave is used to generate the non-linear lemniscate boundary.

# Read the data

data=csvread("lemniscate.csv");

X=data(:,1:2);

Y=data(:,3);

# Set the dimensions of the layers

layersDimensions = [2 9 7 1]

# Compute the DL network

[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,

hiddenActivationFunc='relu',

learningRate = 0.20,

numIterations = 10000);

plotCostVsIterations(10000,costs);

plotDecisionBoundary(data,weights, biases,hiddenActivationFunc="relu")

Finally I perform a simple classification using the MNIST handwritten digits, which according to Prof Geoffrey Hinton is “the Drosophila of Deep Learning”.

The Python code for reading the MNIST data is taken from Alex Kesling’s github link MNIST.

In the Python code below, I perform a simple binary classification between the handwritten digit ‘5’ and ‘not 5’ which is all other digits. I will perform the proper classification of all digits using the Softmax classifier some time later.

```
import os
import numpy as np
import matplotlib.pyplot as plt
os.chdir("C:\\software\\DeepLearning-Posts\\part3")
execfile("./DLfunctions34.py")
execfile("./load_mnist.py")
training=list(read(dataset='training',path="./mnist"))
test=list(read(dataset='testing',path="./mnist"))
lbls=[]
pxls=[]
print(len(training))
# Select the first 10000 training data and the labels
for i in range(10000):
l,p=training[i]
lbls.append(l)
pxls.append(p)
labels= np.array(lbls)
pixels=np.array(pxls)
# Sey y=1 when labels == 5 and 0 otherwise
y=(labels==5).reshape(-1,1)
X=pixels.reshape(pixels.shape[0],-1)
# Create the necessary feature and target variable
X1=X.T
Y1=y.T
# Create the layer dimensions. The number of features are 28 x 28 = 784 since the 28 x 28
# pixels is flattened to single vector of length 784.
layersDimensions=[784, 15,9,7,1] # Works very well
parameters = L_Layer_DeepModel(X1, Y1, layersDimensions, hiddenActivationFunc='relu', learning_rate = 0.1,num_iterations = 1000, fig="fig7.png")
# Test data
lbls1=[]
pxls1=[]
for i in range(800):
l,p=test[i]
lbls1.append(l)
pxls1.append(p)
testLabels=np.array(lbls1)
testData=np.array(pxls1)
ytest=(testLabels==5).reshape(-1,1)
Xtest=testData.reshape(testData.shape[0],-1)
Xtest1=Xtest.T
Ytest1=ytest.T
yhat = predict(parameters,Xtest1)
from sklearn.metrics import confusion_matrix
a=confusion_matrix(Ytest1.T,yhat.T)
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score
print('Accuracy: {:.2f}'.format(accuracy_score(Ytest1.T, yhat.T)))
print('Precision: {:.2f}'.format(precision_score(Ytest1.T, yhat.T)))
print('Recall: {:.2f}'.format(recall_score(Ytest1.T, yhat.T)))
print('F1: {:.2f}'.format(f1_score(Ytest1.T, yhat.T)))
probs=predict_proba(parameters,Xtest1)
from sklearn.metrics import precision_recall_curve
precision, recall, thresholds = precision_recall_curve(Ytest1.T, probs.T)
closest_zero = np.argmin(np.abs(thresholds))
closest_zero_p = precision[closest_zero]
closest_zero_r = recall[closest_zero]
plt.xlim([0.0, 1.01])
plt.ylim([0.0, 1.01])
plt.plot(precision, recall, label='Precision-Recall Curve')
plt.plot(closest_zero_p, closest_zero_r, 'o', markersize = 12, fillstyle = 'none', c='r', mew=3)
plt.xlabel('Precision', fontsize=16)
plt.ylabel('Recall', fontsize=16)
plt.savefig("fig8.png",bbox_inches='tight')
```

```
## Accuracy: 0.99
## Precision: 0.96
## Recall: 0.89
## F1: 0.92
```

In addition to plotting the Cost vs Iterations, I also plot the Precision-Recall curve to show how the Precision and Recall, which are complementary to each other vary with respect to the other. To know more about Precision-Recall, please check my post Practical Machine Learning with R and Python – Part 4. You could also check out my book My book ‘Practical Machine Learning with R and Python’ on Amazon for details on the key metrics and algorithms for classification and regression problems. A physical copy of the book is much better than scrolling down a webpage. Personally, I tend to use my own book quite frequently to refer to R, Python constructs, subsetting, machine Learning function calls and the necessary parameters etc. It is useless to commit any of this to memory, and a physical copy of a book is much easier to thumb through for the relevant code snippet. Pick up your copy today!

In the R code below the same binary classification of the digit ‘5’ and the ‘not 5’ is performed. The code to read and display the MNIST data is taken from Brendan O’ Connor’s github link at MNIST

```
source("mnist.R")
load_mnist()
#show_digit(train$x[2,]
layersDimensions=c(784, 7,7,3,1) # Works at 1500
x <- t(train$x)
# Choose only 5000 training data
x2 <- x[,1:5000]
y <-train$y
# Set labels for all digits that are 'not 5' to 0
y[y!=5] <- 0
# Set labels of digit 5 as 1
y[y==5] <- 1
# Set the data
y1 <- as.matrix(y)
y2 <- t(y1)
# Choose the 1st 5000 data
y3 <- y2[,1:5000]
#Execute the Deep Learning Model
retvals = L_Layer_DeepModel(x2, y3, layersDimensions,
hiddenActivationFunc='tanh',
learningRate = 0.3,
numIterations = 3000, print_cost = True)
```

```
# Plot cost vs iteration
costs <- retvals[['costs']]
numIterations = 3000
iterations <- seq(0,numIterations,by=1000)
df <-data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") +
xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations")
```

```
# Compute probability scores
scores <- computeScores(retvals$parameters, x2,hiddenActivationFunc='relu')
a=y3==1
b=y3==0
# Compute probabilities of class 0 and class 1
class1=scores[a]
class0=scores[b]
# Plot ROC curve
pr <-pr.curve(scores.class0=class1,
scores.class1=class0,
curve=T)
plot(pr)
```

The AUC curve hugs the top left corner and hence the performance of the classifier is quite good.

This code to load MNIST data was taken from Daniel E blog.

Precision recall curves are available in Matlab but are yet to be implemented in Octave’s statistics package.

load('./mnist/mnist.txt.gz'); % load the dataset

# Subset the 'not 5' digits

a=(trainY != 5);

# Subset '5'

b=(trainY == 5);

#make a copy of trainY

#Set 'not 5' as 0 and '5' as 1

y=trainY;

y(a)=0;

y(b)=1;

X=trainX(1:5000,:);

Y=y(1:5000);

# Set the dimensions of layer

layersDimensions=[784, 7,7,3,1];

# Compute the DL network

[weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions,

hiddenActivationFunc='relu',

learningRate = 0.1,

numIterations = 5000);

It was quite a challenge coding a Deep Learning Network in Python, R and Octave. The Deep Learning network implementation, in this post,is the base Deep Learning network, without any of the regularization methods included. Here are some key learning that I got while playing with different multi-layer networks on different problems

a. Deep Learning Networks come with many levers, the hyper-parameters,

– learning rate

– activation unit

– number of hidden layers

– number of units per hidden layer

– number of iterations while performing gradient descent

b. Deep Networks are very sensitive. A change in any of the hyper-parameter makes it perform very differently

c. Initially I thought adding more hidden layers, or more units per hidden layer will make the DL network better at learning. On the contrary, there is a performance degradation after the optimal DL configuration

d. At a sub-optimal number of hidden layers or number of hidden units, gradient descent seems to get stuck at a local minima

e. There were occasions when the cost came down, only to increase slowly as the number of iterations were increased. Probably early stopping would have helped.

f. I also did come across situations of ‘exploding/vanishing gradient’, cost went to Inf/-Inf. Here I would think inclusion of ‘momentum method’ would have helped

I intend to add the additional hyper-parameters of L1, L2 regularization, momentum method, early stopping etc. into the code in my future posts.

Feel free to fork/clone the code from Github Deep Learning – Part 3, and take the DL network apart and play around with it.

I will be continuing this series with more hyper-parameters to handle vanishing and exploding gradients, early stopping and regularization in the weeks to come. I also intend to add some more activation functions to this basic Multi-Layer Network.

Hang around, there are more exciting things to come.

Watch this space!

References

1. Deep Learning Specialization

2. Neural Networks for Machine Learning

3. Deep Learning, Ian Goodfellow, Yoshua Bengio and Aaron Courville

4. Neural Networks: The mechanics of backpropagation

5. Machine Learning

Also see

1.My book ‘Practical Machine Learning with R and Python’ on Amazon

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

3. Designing a Social Web Portal

4. GooglyPlus: yorkr analyzes IPL players, teams, matches with plots and tables

4. Introducing QCSimulator: A 5-qubit quantum computing simulator in R

6. Presentation on “Intelligent Networks, CAMEL protocol, services & applications

7. Design Principles of Scalable, Distributed Systems

To see all posts see Index of posts

]]>The Brain: The Story of You” by David Eagleman

“*The world is Maya, illusory. The ultimate reality, the Brahman, is all-pervading and all-permeating, which is colourless, odourless, tasteless, nameless and formless*“

Bhagavad Gita

This post is a follow-up post to my earlier post Deep Learning from first principles in Python, R and Octave-Part 1. In the first part, I implemented Logistic Regression, in vectorized Python,R and Octave, with a wannabe Neural Network (a Neural Network with no hidden layers). In this second part, I implement a regular, but somewhat primitive Neural Network (a Neural Network with just 1 hidden layer). The 2nd part implements classification of manually created datasets, where the different clusters of the 2 classes are not linearly separable.

Neural Network perform really well in learning all sorts of non-linear boundaries between classes. Initially logistic regression is used perform the classification and the decision boundary is plotted. Vanilla logistic regression performs quite poorly. Using SVMs with a radial basis kernel would have performed much better in creating non-linear boundaries. To see R and Python implementations of SVMs take a look at my post Practical Machine Learning with R and Python – Part 4.

You could also check out my book on Amazon Practical Machine Learning with R and Python – Machine Learning in Stereo, in which I implement several Machine Learning algorithms on regression and classification, along with other necessary metrics that are used in Machine Learning.

You can clone and fork this R Markdown file along with the vectorized implementations of the 3 layer Neural Network for Python, R and Octave from Github DeepLearning-Part2

A simple representation of a 3 layer Neural Network (NN) with 1 hidden layer is shown below.

In the above Neural Network, there are 2 input features at the input layer, 3 hidden units at the hidden layer and 1 output layer as it deals with binary classification. The activation unit at the hidden layer can be a tanh, sigmoid, relu etc. At the output layer the activation is a sigmoid to handle binary classification

# Superscript indicates layer 1

Also

# Superscript indicates layer 2

Hence

And

Similarly

and

These equations can be written as

**I) Some important results** (a memory refresher!)

and -(a) and

and

Using (a) we can shown that and (b)

Now -(c)

Since and using (b) we get

Using the values of the derivatives of sinhx and coshx from (b) above we get

Since

-(d)

**II) Derivatives**

Since therefore see Part1

and

**III) Back propagation**

Using the derivatives from II) we can derive the following results using Chain Rule

-(A)

-(B)

-(C)

-(D)

**IV) Gradient Descent**

The key computations in the backward cycle are

– From (C)

– From (D)

– From (A)

– From (B)

The weights and biases (W1,b1,W2,b2) are updated for each iteration thus minimizing the loss/cost.

These derivations can be represented pictorially using the computation graph (from the book Deep Learning by Ian Goodfellow, Joshua Bengio and Aaron Courville)

Initially I create a dataset with 2 classes which has around 9 clusters that cannot be separated by linear boundaries. **Note**: *This data set is saved as data.csv and is used for the R and Octave Neural networks to see how they perform on the same dataset.*

```
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.colors
import sklearn.linear_model
from sklearn.model_selection import train_test_split
from sklearn.datasets import make_classification, make_blobs
from matplotlib.colors import ListedColormap
import sklearn
import sklearn.datasets
colors=['black','gold']
cmap = matplotlib.colors.ListedColormap(colors)
X, y = make_blobs(n_samples = 400, n_features = 2, centers = 7,
cluster_std = 1.3, random_state = 4)
#Create 2 classes
y=y.reshape(400,1)
y = y % 2
#Plot the figure
plt.figure()
plt.title('Non-linearly separable classes')
plt.scatter(X[:,0], X[:,1], c=y,
marker= 'o', s=50,cmap=cmap)
plt.savefig('fig1.png', bbox_inches='tight')
```

On the above created dataset, classification with logistic regression is performed, and the decision boundary is plotted. It can be seen that logistic regression performs quite poorly

`import numpy as np import matplotlib.pyplot as plt import matplotlib.colors import sklearn.linear_model from sklearn.model_selection import train_test_split from sklearn.datasets import make_classification, make_blobs from matplotlib.colors import ListedColormap import sklearn import sklearn.datasets #from DLfunctions import plot_decision_boundary execfile("./DLfunctions.py") # Since import does not work in Rmd!!! colors=['black','gold'] cmap = matplotlib.colors.ListedColormap(colors) X, y = make_blobs(n_samples = 400, n_features = 2, centers = 7, cluster_std = 1.3, random_state = 4) #Create 2 classes y=y.reshape(400,1) y = y % 2 # Train the logistic regression classifier clf = sklearn.linear_model.LogisticRegressionCV(); clf.fit(X, y); # Plot the decision boundary for logistic regression plot_decision_boundary_n(lambda x: clf.predict(x), X.T, y.T,"fig2.png")`

The vectorized implementation is included below. Note that in the case of Python a learning rate of 0.5 and 3 hidden units performs very well.

```
## Random data set with 9 clusters
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
from sklearn.datasets import make_classification, make_blobs
execfile("./DLfunctions.py") # Since import does not work in Rmd!!!
X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9,
cluster_std = 1.3, random_state = 4)
#Create 2 classes
Y1=Y1.reshape(400,1)
Y1 = Y1 % 2
X2=X1.T
Y2=Y1.T
#Perform gradient descent
parameters,costs = computeNN(X2, Y2, numHidden = 4, learningRate=0.5, numIterations = 10000)
plot_decision_boundary(lambda x: predict(parameters, x.T), X2, Y2,str(4),str(0.5),"fig3.png")
```

```
## Cost after iteration 0: 0.692669
## Cost after iteration 1000: 0.246650
## Cost after iteration 2000: 0.227801
## Cost after iteration 3000: 0.226809
## Cost after iteration 4000: 0.226518
## Cost after iteration 5000: 0.226331
## Cost after iteration 6000: 0.226194
## Cost after iteration 7000: 0.226085
## Cost after iteration 8000: 0.225994
## Cost after iteration 9000: 0.225915
```

For this the dataset created by Python is saved to see how R performs on the same dataset. The vectorized implementation of a Neural Network was just a little more interesting as R does not have a similar package like ‘numpy’. While numpy handles broadcasting implicitly, in R I had to use the ‘sweep’ command to broadcast. The implementaion is included below. Note that since the initialization with random weights is slightly different, R performs best with a learning rate of 0.1 and with 6 hidden units

`source("DLfunctions2_1.R")`

```
z <- as.matrix(read.csv("data.csv",header=FALSE)) #
x <- z[,1:2]
y <- z[,3]
x1 <- t(x)
y1 <- t(y)
#Perform gradient descent
nn <-computeNN(x1, y1, 6, learningRate=0.1,numIterations=10000) # Good
```

```
## [1] 0.7075341
## [1] 0.2606695
## [1] 0.2198039
## [1] 0.2091238
## [1] 0.211146
## [1] 0.2108461
## [1] 0.2105351
## [1] 0.210211
## [1] 0.2099104
## [1] 0.2096437
## [1] 0.209409
```

`plotDecisionBoundary(z,nn,6,0.1)`

This uses the same dataset that was generated using Python code.

`source("DL-function2.m")`

data=csvread("data.csv");

X=data(:,1:2);

Y=data(:,3);

# Make sure that the model parameters are correct. Take the transpose of X & Y

`#Perform gradient descent`

[W1,b1,W2,b2,costs]= computeNN(X', Y',4, learningRate=0.5, numIterations = 10000);

```
import numpy as np
import matplotlib
import matplotlib.pyplot as plt
import sklearn.linear_model
import pandas as pd
from sklearn.datasets import make_classification, make_blobs
execfile("./DLfunctions.py") # Since import does not work in Rmd!!!
# Create data
X1, Y1 = make_blobs(n_samples = 400, n_features = 2, centers = 9,
cluster_std = 1.3, random_state = 4)
#Create 2 classes
Y1=Y1.reshape(400,1)
Y1 = Y1 % 2
X2=X1.T
Y2=Y1.T
# Create a list of learning rates
learningRate=[0.5,1.2,3.0]
df=pd.DataFrame()
#Compute costs for each learning rate
for lr in learningRate:
parameters,costs = computeNN(X2, Y2, numHidden = 4, learningRate=lr, numIterations = 10000)
print(costs)
df1=pd.DataFrame(costs)
df=pd.concat([df,df1],axis=1)
#Set the iterations
iterations=[0,1000,2000,3000,4000,5000,6000,7000,8000,9000]
#Create data frame
#Set index
df1=df.set_index([iterations])
df1.columns=[0.5,1.2,3.0]
fig=df1.plot()
fig=plt.title("Cost vs No of Iterations for different learning rates")
plt.savefig('fig4.png', bbox_inches='tight')
```

```
source("DLfunctions2_1.R")
# Read data
z <- as.matrix(read.csv("data.csv",header=FALSE)) #
x <- z[,1:2]
y <- z[,3]
x1 <- t(x)
y1 <- t(y)
#Loop through learning rates and compute costs
learningRate <-c(0.1,1.2,3.0)
df <- NULL
for(i in seq_along(learningRate)){
nn <- computeNN(x1, y1, 6, learningRate=learningRate[i],numIterations=10000)
cost <- nn$costs
df <- cbind(df,cost)
}
```

```
#Create dataframe
df <- data.frame(df)
iterations=seq(0,10000,by=1000)
df <- cbind(iterations,df)
names(df) <- c("iterations","0.5","1.2","3.0")
library(reshape2)
```

```
df1 <- melt(df,id="iterations") # Melt the data
#Plot
ggplot(df1) + geom_line(aes(x=iterations,y=value,colour=variable),size=1) +
xlab("Iterations") +
ylab('Cost') + ggtitle("Cost vs No iterations for different learning rates")
```

`source("DL-function2.m")`

plotLRCostVsIterations()

print -djph figa.jpg

In this 2nd part I create a a central region of positives and and the outside region as negatives. The points are generated using the equation of a circle (x – a)^{2} + (y -b) ^{2} = R^{2} . How does the 3 layer Neural Network perform on this? Here’s a look! **Note**: *The same dataset is also used for R and Octave Neural Network constructions*

```
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.colors
import sklearn.linear_model
from sklearn.model_selection import train_test_split
from sklearn.datasets import make_classification, make_blobs
from matplotlib.colors import ListedColormap
import sklearn
import sklearn.datasets
colors=['black','gold']
cmap = matplotlib.colors.ListedColormap(colors)
x1=np.random.uniform(0,10,800).reshape(800,1)
x2=np.random.uniform(0,10,800).reshape(800,1)
X=np.append(x1,x2,axis=1)
X.shape
# Create (x-a)^2 + (y-b)^2 = R^2
# Create a subset of values where squared is <0,4. Perform ravel() to flatten this vector
a=(np.power(X[:,0]-5,2) + np.power(X[:,1]-5,2) <= 6).ravel()
Y=a.reshape(800,1)
cmap = matplotlib.colors.ListedColormap(colors)
plt.figure()
plt.title('Non-linearly separable classes')
plt.scatter(X[:,0], X[:,1], c=Y,
marker= 'o', s=15,cmap=cmap)
plt.savefig('fig6.png', bbox_inches='tight')
```

` David Eagleman - The Brain: The Story of You`

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

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

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

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

```
## Accuracy of Logistic regression classifier on training set: 0.96
## Accuracy of Logistic regression classifier on test set: 0.96
```

To check on other classification algorithms, check my post Practical Machine Learning with R and Python – Part 2. You could also take a look at my book, available on Amazon , in which I implement the most popular Machine Learning algorithms in both R and Python – My book ‘Practical Machine Learning with R and Python’ on Amazon

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

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

This simple 2 layer Neural Network is shown below

At the input layer there are 30 features and the corresponding weights of these inputs which are initialized to small random values.

where ‘b’ is the bias term

The Activation function is the sigmoid function which is

The Loss, when the sigmoid function is used in the output layer, is given by

(1)

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

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

**Chain rule of differentiation**

Let y=f(u)

and u=g(x) then

**Derivative of sigmoid**

Let then

Using the chain rule of differentiation we get

Therefore -(2)

The 3 equations for the 2 layer Neural Network representation of Logistic Regression are

-(a)

-(b)

-(c)

The back propagation step requires the computation of and . In the case of regression it would be and where dE is the Mean Squared Error function.

Computing the derivatives for back propagation we have

-(d)

because

Also from equation (2) we get

– (e)

By chain rule

therefore substituting the results of (d) & (e) we get

(f)

Finally

-(g)

– (h)

and from (f) we have

Therefore (g) reduces to

-(i)

Also

-(j)

Since

The gradient computes the weights at the input layer and the corresponding bias by using the values

of and

I found the computation graph representation in the book Deep Learning: Ian Goodfellow, Yoshua Bengio, Aaron Courville, very useful to visualize and also compute the backward propagation. For the 2 layer Neural Network of Logistic Regression the computation graph is shown below

```
import numpy as np
import pandas as pd
import os
import matplotlib.pyplot as plt
from sklearn.model_selection import train_test_split
# Define the sigmoid function
def sigmoid(z):
a=1/(1+np.exp(-z))
return a
# Initialize
def initialize(dim):
w = np.zeros(dim).reshape(dim,1)
b = 0
return w
# Compute the loss
def computeLoss(numTraining,Y,A):
loss=-1/numTraining *np.sum(Y*np.log(A) + (1-Y)*(np.log(1-A)))
return(loss)
# Execute the forward propagation
def forwardPropagation(w,b,X,Y):
# Compute Z
Z=np.dot(w.T,X)+b
# Determine the number of training samples
numTraining=float(len(X))
# Compute the output of the sigmoid activation function
A=sigmoid(Z)
#Compute the loss
loss = computeLoss(numTraining,Y,A)
# Compute the gradients dZ, dw and db
dZ=A-Y
dw=1/numTraining*np.dot(X,dZ.T)
db=1/numTraining*np.sum(dZ)
# Return the results as a dictionary
gradients = {"dw": dw,
"db": db}
loss = np.squeeze(loss)
return gradients,loss
# Compute Gradient Descent
def gradientDescent(w, b, X, Y, numIerations, learningRate):
losses=[]
idx =[]
# Iterate
for i in range(numIerations):
gradients,loss=forwardPropagation(w,b,X,Y)
#Get the derivates
dw = gradients["dw"]
db = gradients["db"]
w = w-learningRate*dw
b = b-learningRate*db
# Store the loss
if i % 100 == 0:
idx.append(i)
losses.append(loss)
# Set params and grads
params = {"w": w,
"b": b}
grads = {"dw": dw,
"db": db}
return params, grads, losses,idx
# Predict the output for a training set
def predict(w,b,X):
size=X.shape[1]
yPredicted=np.zeros((1,size))
Z=np.dot(w.T,X)
# Compute the sigmoid
A=sigmoid(Z)
for i in range(A.shape[1]):
#If the value is > 0.5 then set as 1
if(A[0][i] > 0.5):
yPredicted[0][i]=1
else:
# Else set as 0
yPredicted[0][i]=0
return yPredicted
#Normalize the data
def normalize(x):
x_norm = None
x_norm = np.linalg.norm(x,axis=1,keepdims=True)
x= x/x_norm
return x
# Run the 2 layer Neural Network on the cancer data set
from sklearn.datasets import load_breast_cancer
# Load the cancer data
(X_cancer, y_cancer) = load_breast_cancer(return_X_y = True)
# Create train and test sets
X_train, X_test, y_train, y_test = train_test_split(X_cancer, y_cancer,
random_state = 0)
# Normalize the data for better performance
X_train1=normalize(X_train)
# Create weight vectors of zeros. The size is the number of features in the data set=30
w=np.zeros((X_train.shape[1],1))
#w=np.zeros((30,1))
b=0
#Normalize the training data so that gradient descent performs better
X_train1=normalize(X_train)
#Transpose X_train so that we have a matrix as (features, numSamples)
X_train2=X_train1.T
# Reshape to remove the rank 1 array and then transpose
y_train1=y_train.reshape(len(y_train),1)
y_train2=y_train1.T
# Run gradient descent for 4000 times and compute the weights
parameters, grads, costs,idx = gradientDescent(w, b, X_train2, y_train2, numIerations=4000, learningRate=0.75)
w = parameters["w"]
b = parameters["b"]
# Normalize X_test
X_test1=normalize(X_test)
#Transpose X_train so that we have a matrix as (features, numSamples)
X_test2=X_test1.T
#Reshape y_test
y_test1=y_test.reshape(len(y_test),1)
y_test2=y_test1.T
# Predict the values for
yPredictionTest = predict(w, b, X_test2)
yPredictionTrain = predict(w, b, X_train2)
# Print the accuracy
print("train accuracy: {} %".format(100 - np.mean(np.abs(yPredictionTrain - y_train2)) * 100))
print("test accuracy: {} %".format(100 - np.mean(np.abs(yPredictionTest - y_test)) * 100))
# Plot the Costs vs the number of iterations
fig1=plt.plot(idx,costs)
fig1=plt.title("Gradient descent-Cost vs No of iterations")
fig1=plt.xlabel("No of iterations")
fig1=plt.ylabel("Cost")
fig1.figure.savefig("fig1", bbox_inches='tight')
```

```
## train accuracy: 90.3755868545 %
## test accuracy: 89.5104895105 %
```

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

`source("RFunctions-1.R")`

```
# Define the sigmoid function
sigmoid <- function(z){
a <- 1/(1+ exp(-z))
a
}
# Compute the loss
computeLoss <- function(numTraining,Y,A){
loss <- -1/numTraining* sum(Y*log(A) + (1-Y)*log(1-A))
return(loss)
}
# Compute forward propagation
forwardPropagation <- function(w,b,X,Y){
# Compute Z
Z <- t(w) %*% X +b
#Set the number of samples
numTraining <- ncol(X)
# Compute the activation function
A=sigmoid(Z)
#Compute the loss
loss <- computeLoss(numTraining,Y,A)
# Compute the gradients dZ, dw and db
dZ<-A-Y
dw<-1/numTraining * X %*% t(dZ)
db<-1/numTraining*sum(dZ)
fwdProp <- list("loss" = loss, "dw" = dw, "db" = db)
return(fwdProp)
}
# Perform one cycle of Gradient descent
gradientDescent <- function(w, b, X, Y, numIerations, learningRate){
losses <- NULL
idx <- NULL
# Loop through the number of iterations
for(i in 1:numIerations){
fwdProp <-forwardPropagation(w,b,X,Y)
#Get the derivatives
dw <- fwdProp$dw
db <- fwdProp$db
#Perform gradient descent
w = w-learningRate*dw
b = b-learningRate*db
l <- fwdProp$loss
# Stoe the loss
if(i %% 100 == 0){
idx <- c(idx,i)
losses <- c(losses,l)
}
}
# Return the weights and losses
gradDescnt <- list("w"=w,"b"=b,"dw"=dw,"db"=db,"losses"=losses,"idx"=idx)
return(gradDescnt)
}
# Compute the predicted value for input
predict <- function(w,b,X){
m=dim(X)[2]
# Create a ector of 0's
yPredicted=matrix(rep(0,m),nrow=1,ncol=m)
Z <- t(w) %*% X +b
# Compute sigmoid
A=sigmoid(Z)
for(i in 1:dim(A)[2]){
# If A > 0.5 set value as 1
if(A[1,i] > 0.5)
yPredicted[1,i]=1
else
# Else set as 0
yPredicted[1,i]=0
}
return(yPredicted)
}
# Normalize the matrix
normalize <- function(x){
#Create the norm of the matrix.Perform the Frobenius norm of the matrix
n<-as.matrix(sqrt(rowSums(x^2)))
#Sweep by rows by norm. Note '1' in the function which performing on every row
normalized<-sweep(x, 1, n, FUN="/")
return(normalized)
}
# Run the 2 layer Neural Network on the cancer data set
# Read the data (from sklearn)
cancer <- read.csv("cancer.csv")
# Rename the target variable
names(cancer) <- c(seq(1,30),"output")
# Split as training and test sets
train_idx <- trainTestSplit(cancer,trainPercent=75,seed=5)
train <- cancer[train_idx, ]
test <- cancer[-train_idx, ]
# Set the features
X_train <-train[,1:30]
y_train <- train[,31]
X_test <- test[,1:30]
y_test <- test[,31]
# Create a matrix of 0's with the number of features
w <-matrix(rep(0,dim(X_train)[2]))
b <-0
X_train1 <- normalize(X_train)
X_train2=t(X_train1)
# Reshape then transpose
y_train1=as.matrix(y_train)
y_train2=t(y_train1)
# Perform gradient descent
gradDescent= gradientDescent(w, b, X_train2, y_train2, numIerations=3000, learningRate=0.77)
# Normalize X_test
X_test1=normalize(X_test)
#Transpose X_train so that we have a matrix as (features, numSamples)
X_test2=t(X_test1)
#Reshape y_test and take transpose
y_test1=as.matrix(y_test)
y_test2=t(y_test1)
# Use the values of the weights generated from Gradient Descent
yPredictionTest = predict(gradDescent$w, gradDescent$b, X_test2)
yPredictionTrain = predict(gradDescent$w, gradDescent$b, X_train2)
sprintf("Train accuracy: %f",(100 - mean(abs(yPredictionTrain - y_train2)) * 100))
```

`## [1] "Train accuracy: 90.845070"`

`sprintf("test accuracy: %f",(100 - mean(abs(yPredictionTest - y_test)) * 100))`

`## [1] "test accuracy: 87.323944"`

```
df <-data.frame(gradDescent$idx, gradDescent$losses)
names(df) <- c("iterations","losses")
ggplot(df,aes(x=iterations,y=losses)) + geom_point() + geom_line(col="blue") +
ggtitle("Gradient Descent - Losses vs No of Iterations") +
xlab("No of iterations") + ylab("Losses")
```

1;

# Define sigmoid function

function a = sigmoid(z)

a = 1 ./ (1+ exp(-z));

end

# Compute the loss

function loss=computeLoss(numtraining,Y,A)

loss = -1/numtraining * sum((Y .* log(A)) + (1-Y) .* log(1-A));

end

# Perform forward propagation

function [loss,dw,db,dZ] = forwardPropagation(w,b,X,Y)

% Compute Z

Z = w' * X + b;

numtraining = size(X)(1,2);

# Compute sigmoid

A = sigmoid(Z);

#Compute loss. Note this is element wise product

loss =computeLoss(numtraining,Y,A);

# Compute the gradients dZ, dw and db

dZ = A-Y;

dw = 1/numtraining* X * dZ';

db =1/numtraining*sum(dZ);

end

# Compute Gradient Descent

function [w,b,dw,db,losses,index]=gradientDescent(w, b, X, Y, numIerations, learningRate)

#Initialize losses and idx

losses=[];

index=[];

# Loop through the number of iterations

for i=1:numIerations,

[loss,dw,db,dZ] = forwardPropagation(w,b,X,Y);

# Perform Gradient descent

w = w - learningRate*dw;

b = b - learningRate*db;

if(mod(i,100) ==0)

# Append index and loss

index = [index i];

losses = [losses loss];

endif

end

end

# Determine the predicted value for dataset

function yPredicted = predict(w,b,X)

m = size(X)(1,2);

yPredicted=zeros(1,m);

# Compute Z

Z = w' * X + b;

# Compute sigmoid

A = sigmoid(Z);

for i=1:size(X)(1,2),

# Set predicted as 1 if A > 0,5

if(A(1,i) >= 0.5)

yPredicted(1,i)=1;

else

yPredicted(1,i)=0;

endif

end

end

# Normalize by dividing each value by the sum of squares

function normalized = normalize(x)

# Compute Frobenius norm. Square the elements, sum rows and then find square root

a = sqrt(sum(x .^ 2,2));

# Perform element wise division

normalized = x ./ a;

end

# Split into train and test sets

function [X_train,y_train,X_test,y_test] = trainTestSplit(dataset,trainPercent)

# Create a random index

ix = randperm(length(dataset));

# Split into training

trainSize = floor(trainPercent/100 * length(dataset));

train=dataset(ix(1:trainSize),:);

# And test

test=dataset(ix(trainSize+1:length(dataset)),:);

X_train = train(:,1:30);

y_train = train(:,31);

X_test = test(:,1:30);

y_test = test(:,31);

end

cancer=csvread("cancer.csv");

[X_train,y_train,X_test,y_test] = trainTestSplit(cancer,75);

w=zeros(size(X_train)(1,2),1);

b=0;

X_train1=normalize(X_train);

X_train2=X_train1';

y_train1=y_train';

[w1,b1,dw,db,losses,idx]=gradientDescent(w, b, X_train2, y_train1, numIerations=3000, learningRate=0.75);

# Normalize X_test

X_test1=normalize(X_test);

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

X_test2=X_test1';

y_test1=y_test';

# Use the values of the weights generated from Gradient Descent

yPredictionTest = predict(w1, b1, X_test2);

yPredictionTrain = predict(w1, b1, X_train2);

trainAccuracy=100-mean(abs(yPredictionTrain - y_train1))*100

testAccuracy=100- mean(abs(yPredictionTest - y_test1))*100

trainAccuracy = 90.845

testAccuracy = 89.510

graphics_toolkit('gnuplot')

plot(idx,losses);

title ('Gradient descent- Cost vs No of iterations');

xlabel ("No of iterations");

ylabel ("Cost");

**Conclusion**

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

The Deep Learning journey has begun… Don’t miss the bus!

Stay tuned for more interesting posts in Deep Learning!!

References

1. Deep Learning Specialization

2. Neural Networks for Machine Learning

3. Deep Learning, Ian Goodfellow, Yoshua Bengio and Aaron Courville

4. Neural Networks: The mechanics of backpropagation

5. Machine Learning

Also see

1. My book ‘Practical Machine Learning with R and Python’ on Amazon

2. Simplifying Machine Learning: Bias, Variance, regularization and odd facts – Part 4

3. The 3rd paperback & kindle editions of my books on Cricket, now on Amazon

4. Practical Machine Learning with R and Python – Part 4

5. Introducing QCSimulator: A 5-qubit quantum computing simulator in R

6. A Bluemix recipe with MongoDB and Node.js

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

To see all posts check Index of posts

]]>a) Cricket analytics with cricketr, Third Edition. The paperback edition is $12.99 and the kindle edition is $4.99/Rs320. This book is based on my R package ‘cricketr‘, available on CRAN and uses ESPN Cricinfo Statsguru

b) Beaten by sheer pace! Cricket analytics with yorkr, 3rd edition . The paperback is $12.99 and the kindle version is $6.99/Rs448. This is based on my R package ‘yorkr‘ on CRAN and uses data from Cricsheet

Pick up your copies today!!

**Note**: In the 3rd edition of the paperback book, the charts will be in black and white. If you would like the charts to be in color, please check out the 2nd edition of these books see More book, more cricket! 2nd edition of my books now on Amazon

You may also like

1. My book ‘Practical Machine Learning with R and Python’ on Amazon

2. A crime map of India in R: Crimes against women

3. What’s up Watson? Using IBM Watson’s QAAPI with Bluemix, NodeExpress – Part 1

4. Bend it like Bluemix, MongoDB with autoscaling – Part 2

5. Informed choices through Machine Learning : Analyzing Kohli, Tendulkar and Dravid

6. Thinking Web Scale (TWS-3): Map-Reduce – Bring compute to data

To see all posts see Index of posts

]]>