Backpropagation Algorithm Machine Learning

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Backpropagation Algorithm – Machine Learning – Artificial Neural Network

In this tutorial i will discuss the Backpropagation Algorithm and its implementation in Python.

Video Tutorial on Backpropagation Algorithm

BACKPROPAGATION (training_example, ƞ, nin, nout, nhidden)

Each training example is a pair of the form (𝑥, 𝑡), where (𝑥) is the vector of network input values, and (𝑡) is the vector of target network output values.

ƞ is the learning rate (e.g., 0.05).

ni, is the number of network inputs,

nhidden the number of units in the hidden layer, and

nout the number of output units.

The input from unit i into unit j is denoted xji, and the weight from unit i to unit j is denoted wji

Steps in Backpropagation algorithm

1. Create a feed-forward network with ni inputs, nhidden hidden units, and nout output units.

2. Initialize all network weights to small random numbers

3. Until the termination condition is met, Do

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       For each (𝑥, t), in training examples, Do

           Propagate the input forward through the network:

               1. Input the instance 𝑥, to the network and compute the output ou of every unit u in the network.

           Propagate the errors backward through the network

               2. For each network unit k, calculate its error term δk

               3. For each network unit h, calculate its error term δh

               4. Update each network weight wji

Python Program to Implement and Demonstrate Backpropagation Algorithm Machine Learning

import numpy as np

X = np.array(([2, 9], [1, 5], [3, 6]), dtype=float)
y = np.array(([92], [86], [89]), dtype=float)
X = X/np.amax(X,axis=0) #maximum of X array longitudinally
y = y/100

#Sigmoid Function
def sigmoid (x):
    return 1/(1 + np.exp(-x))

#Derivative of Sigmoid Function
def derivatives_sigmoid(x):
    return x * (1 - x)

#Variable initialization
epoch=5 #Setting training iterations
lr=0.1 #Setting learning rate

inputlayer_neurons = 2 #number of features in data set
hiddenlayer_neurons = 3 #number of hidden layers neurons
output_neurons = 1 #number of neurons at output layer
#weight and bias initialization

wh=np.random.uniform(size=(inputlayer_neurons,hiddenlayer_neurons))
bh=np.random.uniform(size=(1,hiddenlayer_neurons))
wout=np.random.uniform(size=(hiddenlayer_neurons,output_neurons))
bout=np.random.uniform(size=(1,output_neurons))

#draws a random range of numbers uniformly of dim x*y
for i in range(epoch):
    #Forward Propogation
    hinp1=np.dot(X,wh)
    hinp=hinp1 + bh
    hlayer_act = sigmoid(hinp)
    outinp1=np.dot(hlayer_act,wout)
    outinp= outinp1+bout
    output = sigmoid(outinp)
    
    #Backpropagation
    EO = y-output
    outgrad = derivatives_sigmoid(output)
    d_output = EO * outgrad
    EH = d_output.dot(wout.T)
    hiddengrad = derivatives_sigmoid(hlayer_act)#how much hidden layer wts contributed to error
    d_hiddenlayer = EH * hiddengrad
    
    wout += hlayer_act.T.dot(d_output) *lr   # dotproduct of nextlayererror and currentlayerop
    wh += X.T.dot(d_hiddenlayer) *lr
    
    print ("-----------Epoch-", i+1, "Starts----------")
    print("Input: \n" + str(X)) 
    print("Actual Output: \n" + str(y))
    print("Predicted Output: \n" ,output)
    print ("-----------Epoch-", i+1, "Ends----------\n")
        
print("Input: \n" + str(X)) 
print("Actual Output: \n" + str(y))
print("Predicted Output: \n" ,output)

Training Examples:

ExampleSleepStudyExpected % in Exams
12992
21586
33689

Normalize the input

ExampleSleepStudyExpected % in Exams
12/3 = 0.666666679/9 = 10.92
21/3 = 0.333333335/9 = 0.555555560.86
33/3 = 16/9 = 0.666666670.89

Output

———–Epoch- 1 Starts———-
Input:
[[0.66666667 1. ]
[0.33333333 0.55555556]
[1. 0.66666667]]
Actual Output:
[[0.92]
[0.86]
[0.89]]
Predicted Output:
[[0.81951208]
[0.8007242 ]
[0.82485744]]
———–Epoch- 1 Ends———-

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———–Epoch- 2 Starts———-
Input:
[[0.66666667 1. ]
[0.33333333 0.55555556]
[1. 0.66666667]]
Actual Output:
[[0.92]
[0.86]
[0.89]]
Predicted Output:
[[0.82033938]
[0.80153634]
[0.82568134]]
———–Epoch- 2 Ends———-

———–Epoch- 3 Starts———-
Input:
[[0.66666667 1. ]
[0.33333333 0.55555556]
[1. 0.66666667]]
Actual Output:
[[0.92]
[0.86]
[0.89]]
Predicted Output:
[[0.82115226]
[0.80233463]
[0.82649072]]
———–Epoch- 3 Ends———-

———–Epoch- 4 Starts———-
Input:
[[0.66666667 1. ]
[0.33333333 0.55555556]
[1. 0.66666667]]
Actual Output:
[[0.92]
[0.86]
[0.89]]
Predicted Output:
[[0.82195108]
[0.80311943]
[0.82728598]]
———–Epoch- 4 Ends———-

———–Epoch- 5 Starts———-
Input:
[[0.66666667 1. ]
[0.33333333 0.55555556]
[1. 0.66666667]]
Actual Output:
[[0.92]
[0.86]
[0.89]]
Predicted Output:
[[0.8227362 ]
[0.80389106]
[0.82806747]]
———–Epoch- 5 Ends———-

Input:
[[0.66666667 1. ]
[0.33333333 0.55555556]
[1. 0.66666667]]
Actual Output:
[[0.92]
[0.86]
[0.89]]
Predicted Output:
[[0.8227362 ]
[0.80389106]
[0.82806747]]

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Summary

This tutorial discusses Backpropagation Algorithm in Machine Learning and how to Implement and demonstrate the Backpropagation Algorithm in Python. If you like the tutorial share it with your friends. Like the Facebook page for regular updates and YouTube channel for video tutorials.

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