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

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

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

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

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

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

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

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

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

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

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

library(ggplot2)
# 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               
## 

1c. Classification with Multi layer Deep Learning Network – Relu activation(Octave)

Included below is the code for performing classification. Incidentally Octave does not seem to have implemented the confusion matrix,  but confusionmat is available in Matlab. # Read the data data=csvread("data.csv"); X=data(:,1:2); Y=data(:,3); # Set layer dimensions layersDimensions = [2 9 7 1] #tanh=-0.5(ok), #relu=0.1 best! # Execute Deep Network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', learningRate = 0.1, numIterations = 10000); plotCostVsIterations(10000,costs); plotDecisionBoundary(data,weights, biases,hiddenActivationFunc="tanh")

2a. Classification with Multi layer Deep Learning Network – Tanh activation(Python)

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

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

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

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

2b. Classification with Multi layer Deep Learning Network – Tanh activation(R)

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

 #Set the dimensions of the Neural Network
layersDimensions = c(2, 9, 9,1)
library(ggplot2)
# Read the data
z <- as.matrix(read.csv("data.csv",header=FALSE)) 
x <- z[,1:2]
y <- z[,3]
X1 <- t(x)
Y1 <- t(y)
# Execute the Deep Model
retvals = L_Layer_DeepModel(X1, Y1, layersDimensions,
                            hiddenActivationFunc='tanh', 
                            learningRate = 0.3,
                            numIterations = 5000, 
                            print_cost = True)
# Get the costs
costs <- retvals[['costs']]
iterations <- seq(0,numIterations,by=1000)
df <-data.frame(iterations,costs)
# Plot Cost vs number of iterations
ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") +
    xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations")

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

2c. Classification with Multi layer Deep Learning Network – Tanh activation(Octave)

The code below uses the   Tanh activation in the hidden layers for Octave # Read the data data=csvread("data.csv"); X=data(:,1:2); Y=data(:,3); # Set layer dimensions layersDimensions = [2 9 7 1] #tanh=-0.5(ok), #relu=0.1 best! # Execute Deep Network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='tanh', learningRate = 0.1, numIterations = 10000); plotCostVsIterations(10000,costs); plotDecisionBoundary(data,weights, biases,hiddenActivationFunc="tanh")

3. Bernoulli’s Lemniscate

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

3a. Classifying a lemniscate with Deep Learning Network – Relu activation(Python)

import os
import numpy as np 
import matplotlib.pyplot as plt
os.chdir("C:\\software\\DeepLearning-Posts\\part3")
execfile("./DLfunctions33.py") 
x1=np.random.uniform(0,10,2000).reshape(2000,1)
x2=np.random.uniform(0,10,2000).reshape(2000,1)

X=np.append(x1,x2,axis=1)
X.shape

# Create a subset of values where squared is <0,4. Perform ravel() to flatten this vector
# Create the equation
# (x^{2} + y^{2})^2 - 2a^2*(x^{2}-y^{2}) <= 0
a=np.power(np.power(X[:,0]-5,2) + np.power(X[:,1]-5,2),2)
b=np.power(X[:,0]-5,2) - np.power(X[:,1]-5,2)
c= a - (b*np.power(4,2)) <=0
Y=c.reshape(2000,1)
# Create a scatter plot of the lemniscate
plt.scatter(X[:,0], X[:,1], c=Y, marker= 'o', s=15,cmap="viridis")
Z=np.append(X,Y,axis=1)
plt.savefig("fig50.png",bbox_inches='tight')
plt.clf()

# Set the data for classification
X2=X.T
Y2=Y.T
# These settings work the best
# Set the Deep Learning layer dimensions for a Relu activation
layersDimensions = [2,7,4,1]
#Execute the DL network
parameters = L_Layer_DeepModel(X2, Y2, layersDimensions, hiddenActivationFunc='relu', learning_rate = 0.5,num_iterations = 10000, fig="fig5.png")
#Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), X2, Y2,str(2.2),"fig6.png")

# Compute the Confusion matrix
yhat = predict(parameters,X2)
from sklearn.metrics import confusion_matrix
a=confusion_matrix(Y2.T,yhat.T)
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score
print('Accuracy: {:.2f}'.format(accuracy_score(Y2.T, yhat.T)))
print('Precision: {:.2f}'.format(precision_score(Y2.T, yhat.T)))
print('Recall: {:.2f}'.format(recall_score(Y2.T, yhat.T)))
print('F1: {:.2f}'.format(f1_score(Y2.T, yhat.T)))
## Accuracy: 0.93
## Precision: 0.77
## Recall: 0.76
## F1: 0.76

We could get better performance by tuning further. Do play around if you fork the code.
Note:: The lemniscate data is saved as a CSV and then read in R and also in Octave. I do this instead of recreating the lemniscate shape

3b. Classifying a lemniscate with Deep Learning Network – Relu activation(R code)

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

Z <- as.matrix(read.csv("lemniscate.csv",header=FALSE))
Z1=data.frame(Z)
# Create a scatter plot of the lemniscate
ggplot(Z1,aes(x=V1,y=V2,col=V3)) +geom_point()
#Set the data for the DL network
X=Z[,1:2]
Y=Z[,3]

X1=t(X)
Y1=t(Y)

# Set the layer dimensions for the tanh activation function
layersDimensions = c(2,5,4,1)
# Execute the Deep Learning network with Tanh activation
retvals = L_Layer_DeepModel(X1, Y1, layersDimensions, 
                               hiddenActivationFunc='tanh', 
                               learningRate = 0.3,
                               numIterations = 20000, print_cost = True)
# Plot cost vs iteration
costs <- retvals[['costs']]
numIterations = 20000
iterations <- seq(0,numIterations,by=1000)
df <-data.frame(iterations,costs)
ggplot(df,aes(x=iterations,y=costs)) + geom_point() +geom_line(color="blue") +
    xlab('No of iterations') + ylab('Cost') + ggtitle("Cost vs No of iterations")

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

3c. Classifying a lemniscate with Deep Learning Network – Relu activation(Octave code)

Octave is used to generate the non-linear lemniscate boundary.
# Read the data data=csvread("lemniscate.csv"); X=data(:,1:2); Y=data(:,3); # Set the dimensions of the layers layersDimensions = [2 9 7 1] # Compute the DL network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', learningRate = 0.20, numIterations = 10000); plotCostVsIterations(10000,costs); plotDecisionBoundary(data,weights, biases,hiddenActivationFunc="relu")

4a. Binary Classification using MNIST – Python code

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

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

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

import os
import numpy as np 
import matplotlib.pyplot as plt
os.chdir("C:\\software\\DeepLearning-Posts\\part3")
execfile("./DLfunctions34.py") 
execfile("./load_mnist.py")
training=list(read(dataset='training',path="./mnist"))
test=list(read(dataset='testing',path="./mnist"))
lbls=[]
pxls=[]
print(len(training))

# Select the first 10000 training data and the labels
for i in range(10000):
       l,p=training[i]
       lbls.append(l)
       pxls.append(p)
labels= np.array(lbls)
pixels=np.array(pxls)   

#  Sey y=1  when labels == 5 and 0 otherwise
y=(labels==5).reshape(-1,1)
X=pixels.reshape(pixels.shape[0],-1)

# Create the necessary feature and target variable
X1=X.T
Y1=y.T

# Create the layer dimensions. The number of features are 28 x 28 = 784 since the 28 x 28
# pixels is flattened to single vector of length 784.
layersDimensions=[784, 15,9,7,1] # Works very well
parameters = L_Layer_DeepModel(X1, Y1, layersDimensions, hiddenActivationFunc='relu', learning_rate = 0.1,num_iterations = 1000, fig="fig7.png")

# Test data
lbls1=[]
pxls1=[]
for i in range(800):
       l,p=test[i]
       lbls1.append(l)
       pxls1.append(p)
 
testLabels=np.array(lbls1)
testData=np.array(pxls1)

ytest=(testLabels==5).reshape(-1,1)
Xtest=testData.reshape(testData.shape[0],-1)
Xtest1=Xtest.T
Ytest1=ytest.T

yhat = predict(parameters,Xtest1)
from sklearn.metrics import confusion_matrix
a=confusion_matrix(Ytest1.T,yhat.T)
from sklearn.metrics import accuracy_score, precision_score, recall_score, f1_score
print('Accuracy: {:.2f}'.format(accuracy_score(Ytest1.T, yhat.T)))
print('Precision: {:.2f}'.format(precision_score(Ytest1.T, yhat.T)))
print('Recall: {:.2f}'.format(recall_score(Ytest1.T, yhat.T)))
print('F1: {:.2f}'.format(f1_score(Ytest1.T, yhat.T)))

probs=predict_proba(parameters,Xtest1)
from sklearn.metrics import precision_recall_curve

precision, recall, thresholds = precision_recall_curve(Ytest1.T, probs.T)
closest_zero = np.argmin(np.abs(thresholds))
closest_zero_p = precision[closest_zero]
closest_zero_r = recall[closest_zero]
plt.xlim([0.0, 1.01])
plt.ylim([0.0, 1.01])
plt.plot(precision, recall, label='Precision-Recall Curve')
plt.plot(closest_zero_p, closest_zero_r, 'o', markersize = 12, fillstyle = 'none', c='r', mew=3)
plt.xlabel('Precision', size=16)
plt.ylabel('Recall', size=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!

4b. Binary Classification using MNIST – R code

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

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

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

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

4c. Binary Classification using MNIST – Octave code

This code to load MNIST data was taken from Daniel E blog.
Precision recall curves are available in Matlab but are yet to be implemented in Octave’s statistics package.
load('./mnist/mnist.txt.gz'); % load the dataset # Subset the 'not 5' digits a=(trainY != 5); # Subset '5' b=(trainY == 5); #make a copy of trainY #Set 'not 5' as 0 and '5' as 1 y=trainY; y(a)=0; y(b)=1; X=trainX(1:5000,:); Y=y(1:5000); # Set the dimensions of layer layersDimensions=[784, 7,7,3,1]; # Compute the DL network [weights biases costs]=L_Layer_DeepModel(X', Y', layersDimensions, hiddenActivationFunc='relu', learningRate = 0.1, numIterations = 5000);

Conclusion

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

a. Deep Learning Networks come with many levers, the hyper-parameters,
– learning rate
– activation unit
– number of hidden layers
– number of units per hidden layer
– number of iterations while performing gradient descent
b. Deep Networks are very sensitive. A change in any of the hyper-parameter makes it perform very differently
c. Initially I thought adding more hidden layers, or more units per hidden layer will make the DL network better at learning. On the contrary, there is a performance degradation after the optimal DL configuration
d. At a sub-optimal number of hidden layers or number of hidden units, gradient descent seems to get stuck at a local minima
e. There were occasions when the cost came down, only to increase slowly as the number of iterations were increased. Probably early stopping would have helped.
f. I also did come across situations of ‘exploding/vanishing gradient’, cost went to Inf/-Inf. Here I would think inclusion of ‘momentum method’ would have helped

I intend to add the additional hyper-parameters of L1, L2 regularization, momentum method, early stopping etc. into the code in my future posts.
Feel free to fork/clone the code from Github Deep Learning – Part 3, and take the DL network apart and play around with it.

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

Watch this space!