11. Implement AND function using perceptron networks for bipolar inputs and targets by Mahesh Huddar

Summary

This video tutorial offers a clear explanation of the perceptron rule, guiding viewers through the process of designing and functioning a perceptron with a numerical example. It covers the fundamental concept of the perceptron, the training rule, and weight updates based on the error between calculated and target outputs. The video demonstrates the perceptron's learning process with an AND function example, showing how weights are adjusted through iterations until the network correctly classifies all inputs. The final weights are presented, illustrating the perceptron's ability to model the AND function.

Takeaways

• 🧠 The video explains the Perceptron Rule, a fundamental concept in machine learning for binary classification.
• 📚 The Perceptron training rule involves adjusting weights based on the error, which is the difference between the actual and calculated output.
• 🔢 The net input to the neuron is calculated using an equation that includes the bias and the weighted sum of inputs.
• 📈 The output of the neuron is determined by comparing the net input to a threshold, outputting 1, 0, or -1 based on whether the net input is greater than, equal to, or less than the threshold.
• 🔄 Weights are updated using the formula new weight = old weight + alpha * (target - output) * input, where alpha is the learning rate.
• 🔧 The AND function is used as an example to demonstrate how a perceptron can be trained to perform logical operations.
• 🤖 The perceptron learns by iteratively adjusting weights until it correctly classifies all given inputs, as shown in the video's example.
• 🔄 The process involves multiple epochs, or iterations, where weights are updated until the perceptron correctly learns the AND function.
• 📉 If the calculated output matches the target, no weight update is necessary; otherwise, the weights are adjusted accordingly.
• 📊 The final weights represent the learned model, capable of correctly classifying inputs for the AND function as demonstrated.
• 👍 The video encourages viewers to like, share, subscribe, and turn on notifications for more content on similar topics.

Q & A

What is the perceptron rule discussed in the video?

The perceptron rule is a training algorithm used to train a single-layer perceptron, which is a type of neural network. It involves adjusting the weights and bias of the perceptron based on the error between the calculated output and the target output.

How does the perceptron training rule determine if the network has learned the correct output?

The perceptron training rule determines if the network has learned correctly by comparing the calculated output of the neuron with the actual target output. If there is no difference, the network has learned; if there is a difference, the network needs further learning.

What is the formula used to calculate the net input to the output unit in a perceptron?

The net input to the output unit is calculated using the formula: net input = B + Σ (x_i * w_i), where B is the bias, x_i is the input, and w_i is the weight of the input.

What is the role of the threshold in the perceptron's activation function?

The threshold in the perceptron's activation function determines the output of the perceptron. If the net input is greater than the threshold, the output is 1; if it is equal to the threshold, the output is 0; and if it is less than the threshold, the output is -1.

How are the weights updated in the perceptron training rule?

The weights are updated based on the difference between the calculated output and the target output. If they are not the same, the new weight is calculated as the old weight plus alpha * T * x, where alpha is the learning rate, T is the target, and x is the input.

What is the AND function in the context of the video?

In the context of the video, the AND function is a logical operation where the output is high (1) only when both inputs (X1 and X2) are high (1). In all other cases, the target output is low (-1 or 0).

How is the learning rate (alpha) used in the weight update process?

The learning rate (alpha) is a scaling factor that determines the step size during the weight update process. It is multiplied by the difference between the target and the calculated output, and then this product is added to the old weight to get the new weight.

What is the purpose of initializing weights and bias to zero at the beginning of the perceptron training?

Initializing weights and bias to zero is a starting point for the training process. It ensures that the perceptron begins learning from a neutral state without any preconceived notions or biases.

Can you explain the process of calculating the net input and output for the first epoch with the given inputs and weights?

In the first epoch, the input (X1, X2) is presented, and the net input is calculated using the initialized weights and bias (all set to zero). The activation function then determines the output based on whether the net input is greater than, equal to, or less than the threshold. If the calculated output does not match the target, the weights and bias are updated accordingly.

What does it mean when the calculated output matches the target output during the training process?

When the calculated output matches the target output, it means that the perceptron has correctly classified the input for that particular training example, and there is no need to update the weights for that instance.

How many iterations or epochs are typically required for a perceptron to learn correctly?

The number of iterations or epochs required for a perceptron to learn correctly depends on the complexity of the problem and the initial weights. In some cases, it may take only a few iterations, while in others, it may require many iterations or might not converge at all if the problem is not linearly separable.

Outlines

00:00

🤖 Introduction to Perceptron Rule

This paragraph introduces the topic of the video, which is the perceptron rule. It explains the perceptron as a fundamental unit of neural networks capable of linear classification. The speaker outlines the plan to first define the perceptron rule and then demonstrate its application through a training rule with a simple numerical example. The training rule involves calculating the output of a neuron, comparing it with the target output, and updating weights based on the error (difference between the calculated and target outputs). The process of calculating the net input, setting a threshold, and updating weights with the learning rate and input values is also discussed.

05:01

🔍 Detailed Explanation of Perceptron Training

The speaker delves deeper into the perceptron training process, explaining how to calculate the net input using the bias and weighted sum of inputs. The activation function is introduced, which determines the output based on whether the net input is above or below a threshold. The paragraph illustrates the weight update mechanism with a learning rate, target, and input values. An example is given where the initial weights are incorrect, and the perceptron rule iteratively adjusts the weights to minimize the error between the calculated and target outputs. The process is shown in action with specific input values, targets, and the resulting weight updates.

10:02

📈 Iterative Weight Update and Convergence

This paragraph continues the discussion on the perceptron training process, focusing on the iterative nature of weight updates. The speaker provides a step-by-step account of how the perceptron rule is applied to a series of inputs and their corresponding targets. The narrative includes the calculation of net inputs, the determination of outputs, and the comparison with targets to decide whether weight updates are necessary. The example demonstrates the perceptron's learning process, where weights are adjusted until the perceptron can correctly classify all given examples. The final weights are presented, and the paragraph concludes with the speaker summarizing the perceptron's functionality and inviting viewers to engage with the content.

Keywords

💡Perceptron Rule

The Perceptron Rule is a fundamental algorithm used in machine learning for training a perceptron, which is a type of linear classifier. It is central to the video's theme as it explains how to adjust the weights of a perceptron to minimize errors. In the script, the Perceptron Rule is used to update weights based on the difference between the calculated output and the target output.

💡Learning Signal

In the context of the video, the learning signal refers to the error signal or the difference between the actual output and the calculated output of the perceptron. It is crucial for the training process as it guides the weight updates. The script mentions that if there is no difference, the network has learned the pattern; otherwise, weight updates are necessary.

💡Calculated Output

The calculated output is the result produced by the perceptron for a given input, which is compared against the target output to determine the error. In the script, the process of calculating the output involves using an equation that includes the net input to the perceptron and a threshold to decide the final output value.

💡Target Output

The target output is the desired result that the perceptron should produce for a given input. It serves as a benchmark to evaluate the accuracy of the perceptron's calculated output. The script describes how the difference between the target and calculated output is used to update the weights of the perceptron.

💡Net Input

Net input is the sum of the product of the input values and their corresponding weights, plus a bias term. It is a key concept in the script as it represents the initial computation that leads to the activation of the perceptron. The formula for net input is given as B + Σ(xi * wi), where B is the bias, xi is the input, and wi is the weight.

💡Activation Function

The activation function determines the output of the perceptron based on the net input. It acts as a threshold that decides whether the perceptron should 'fire' or not. In the script, the activation function is described with a threshold, where the output is 1 if the net input is greater than the threshold, 0 if equal, and -1 if less than the threshold.

💡Weight Update

Weight update is the process of adjusting the weights of the perceptron in response to the learning signal. It is essential for the learning process as it allows the perceptron to improve its predictions over time. The script explains that if the calculated output does not match the target, the weights are updated using the formula: new weight = old weight + alpha * (T - Yn) * Xi.

💡Learning Rate (Alpha)

The learning rate, denoted as Alpha in the script, is a hyperparameter that controls the step size at each iteration while moving toward a minimum of a loss function. It is used in the weight update formula to determine how much the weights should be adjusted. The script sets the learning rate to 1 for simplicity.

💡AND Function

The AND function is a logical operation that outputs true or high only when all inputs are true or high. In the script, the perceptron is trained to mimic the AND function, where the perceptron's output is high only when both inputs X1 and X2 are high, and the target output is low in all other cases.

💡Epoch

An epoch in the context of machine learning refers to one complete pass through the entire training dataset. The script describes the process of training the perceptron over multiple epochs, where the weights are updated after each presentation of the inputs until the perceptron correctly classifies all the targets.

💡Bias

Bias is a parameter in the perceptron model that allows the decision boundary to shift. It is added to the net input before the activation function is applied. In the script, the bias is initialized to zero and then updated along with the weights to improve the perceptron's performance.

Highlights

Introduction to the Perceptron Rule and its application in designing and functioning a perceptron.

Explanation of the learning signal in perceptron training as the error between calculated and actual output.

Process of calculating the output at the output unit using a specific equation and net input calculation.

Description of the net input formula involving bias, input, and weight.

Threshold setting for the perceptron's output based on the net input.

Weight update mechanism when there is a discrepancy between calculated and target outputs.

The role of the learning rate (alpha), target (T), and input (X) in updating weights.

Designing the AND function using the perceptron rule with high output only when both inputs are high.

Initial weight and bias settings for the AND function perceptron.

Iterative weight update process during the perceptron training epochs.

Calculation of the change in weights (Delta W1, Delta W2, and Delta B) for weight update.

First epoch demonstration with input, target, and weight update steps.

Final weights determination after successful classification of all targets.

Visualization of the final AND function with perceptron rule showing bias and weights.

Conclusion emphasizing the clarity of the concept and call to action for likes, shares, and subscriptions.

Transcripts

welcome back in this video I will

discuss how to design and function using

perceptron rule with a very simple

numerical example

first we will try to understand briefly

what is perceptron Rule and then we will

try to understand how to design and

function with the help of perceptron

training rule

in in case of perceptron training rule

the learning signal that is the error

you can say is the difference between

the calculated output so we need to

calculate the output of the output

neuron and then we need to compare that

particular thing with the actual output

or the target output so if we see the

difference between these two things if

there is no difference the meaning is

the network has learned it otherwise if

there is a difference the meaning is the

network has not learned so we need to go

back and then do some weight updation in

this case

so how to calculate this particular

output at the output unit so that's the

next question comes in front of us to

calculate that particular thing the very

first thing what we need to do is we

need to use this particular equation and

then calculate the net input to that

particular node

next we need to use this particular

equation to calculate the output at the

output unit so how this particular net

input is calculated that is B plus

summation of I is equal to 1 to n x i w

i where B is the bias x i is the input

and a w i is the weight of I T net in

this case

now once you get this particular wine we

need to set some threshold initially if

the wine is greater than threshold the

output is one if it is in the range of

minus Theta 2 plus Theta that is Theta

as a threshold in this case it is zero

if it is less than minus Theta it will

be minus 1 in this case now we have to

update the weights based on what we have

calculated here and what is the actual

Target if they are not same we need to

update the weights otherwise we should

not update the weights if the calculated

output is not equivalent to Target new

weight is equivalent to Old weight plus

Alpha TX so Alpha TX is added to

old weight what is alpha alpha is a

learning rate T is the target of that

input and X is the input over here now

once you calculate this you will get the

updated weight here if both of them are

same there is no need to do the weight

updation or new weight is equivalent to

Old weight here

so now we try to understand how and

function is designed with the help of

this particular perceptron rule this is

how actually the uh and function looks

like the output of ion function is high

whenever both the inputs that is X1 and

X2 are high in all other cases the

target is low in this case

this is how actually the sample and

function looks like we have two inputs

X1 and X2 uh this is the bias the value

of B we need to set W 1 and W2 we need

to set initially and then we need to go

on calculating the output at this

particular output in it the calculated

output and this target is same uh the

meaning is we did we don't need to do

the weight updation otherwise we need to

do the weight updation over here so that

is uh what is written at this particular

state

now uh how to calculate the y-in I think

I have already told you that is a

summation function we need to use bias

plus X1 W1 this is uh W1 this is X2 W2

in this case

and once you calculate this particular

line based on this particular activation

function we will get the output here and

then

how to update the weights uh because

once uh you find that the target what

you is not equivalent to what you have

calculated we need to do the weight

updation so if you want to do the weight

updation we need to calculate the change

in weights that is nothing but the Delta

W1 Delta W 2 and then Delta B Delta W 1

is not something great it is nothing but

alpha tx1 alpha is the learning rate T

is the target X1 is the input here

Delta W 2 is nothing but what Alpha t x

2 Alpha is the learning rate T is the

Target and x 2 is again input here Delta

B is equivalent to Alpha T where Alpha

is the learning rate and T is the target

here again

now I will show you the first Epoch that

is the first run in this case

uh this particular F1 this is the input

x 1 and X2 this is the target uh this is

fixed we need we we need to we should

not change this particular thing what we

need to do is we need to calculate the

net input so first we will present this

particular input once you present this

particular input this is what the

equation here that is B plus x 1 W 1

plus x 2 W 2 we have to initialize the

uh W 1 W 2 and B we have initialized

them to zero zero zero here so once you

set 0 0 0 B 0

and then X1 what is the X1 here 1 1 into

0 that is what the thing plus uh I will

make it as multiplication symbol plus x

2 x 2 is what one again here and W 2 is

again 0 we have set so once you

calculate this particular thing we will

get zero so that is what I have written

here now if you see this particular zero

uh this is the Y in so 0 y n is

equivalent to 0 in this case if it is

greater than 0 1 if it is equivalent to

0 it is 0 if it is less than 0 it will

be minus 1 so that is what the

activation function I have considered

here so because y n is equivalent to 0

we will set the output as 0 in this case

now once you set this particular output

we need to compare this particular

output this is y and this is the target

both of them are not same so what we

need to do we need to update this

particular previous weights to update

the weights we need to know uh that's a

change in weight that is Delta W1 Delta

W1 is equivalent to what here this is

the Delta W1 column Alpha t x 1 Alpha is

the learning rate that is set to 1 here

and then T is the target what is the

target here 1 and then X1 what is the X1

here 1 1 into 1 into 1 which is

equivalent to 1 here and then Delta W 2

that is Alpha t x 2 Alpha is 1 T is one

here Target again x two is one

everything is equal to 1 here Alpha into

T is nothing but B here Alpha is one t

is Target is equivalent to 1 B is

equivalent to 1 over here so these are

the change in weights add this

particular change in weights to the

previous weights to you will get the new

weight that is Delta W 1 is added to all

the W1 so 0 plus 1 is equivalent to 1 0

plus 1 is equal to 1 and B plus this

change in B is equivalent to 1 here so

we got the new weights here that is one

one one

now we will present this particular new

input once you project this particular

new input the Target in this case is -1

so we have to calculate what is that

called as again Y in so how to calculate

Y in here b b what is the value we have

1 so that is what I will write here Plus

uh x 1 W 1 x 1 is what here 1 W 1 is

what 1 here so 1 to 1 is equivalent to 1

plus x 2 W 2 x 2 is what minus 1 W 2 is

equal to 1 so this will become minus 1

here so 1 plus 1 minus 1 is equivalent

to 1 so that is nothing but the net

input over here we have calculated a net

input now this net input is greater than

0 because of that the output is

equivalent to 1 here

so once you calculate the output we need

to compare this output with the target

here so Target is minus 1 you have

calculated one both of them are not same

the meaning is what we need to update

the weights again if you want to update

the weights again we have to calculate

Delta W1 Delta W 2 and Delta Delta B

here what is Delta W1 alpha tx1 alpha is

known to us that is 1 T is Target is

minus 1 x 1 is equivalent to 1 here so 1

into minus 1 into 1 that is minus 1 here

again alpha tx2 alpha is 1 and then T is

minus 1 and x 2 is minus 1 minus 1 into

minus 1 into 1 that will become 1 here

Alpha into T is nothing but B Alpha is 1

T is minus 1 this will become minus 1

over here

now we need to add this thing to the

previous weights so minus 1 1 will

become 0 1 plus 1 will become 2 minus 1

1 will become 0 here so these are the

new modified weights here because

weights were modified uh we have to

repeat this particular thing for the

next input so next input is equal to

this one minus 1 1 is the input Target

is minus 1 we will calculate the net

input again with the same equation we

will get 2 here once you get this

particular 2 2 is greater than 0 so the

output is equivalent to 1 here the

calculated output is 1 Target is minus

one both of them are not same here

because they are not same we have to

update the weights again to update the

weights we need to calculate Delta W 1

Delta W 2 and Delta B with the same

equations here we got plus 1 minus 1

minus 1 here and then we need to add

these two these things to the previous

weights that is 0 plus 1 will become 1

here 2 plus minus 1 will become one here

and then 0 plus minus 1 will become

minus 1 here so these are the

uh the modified bits what we got again

we will present the next input here once

you present the next input YN is minus 3

because it is minus 3 minus 3 is less

than 0 because minus 3 is less than 0

the output is equal to minus 1 so minus

1 is equal to minus 1 we should not do

any weight modification so Delta W 1

Delta W 2 and Delta B are 0 and weights

are remains same over here

now what has happened in this case is so

only the this particular thing was

classified correctly the previous one

were not classified correctly in the

first iteration or the epoch so we will

try to repeat the same thing again here

now what are the modified weights 1 1

minus one so with the help of this

particular 1 1 minus 1 we will try to

classify the same examples again we will

try to consider the first input 1 1

Target is 1 here net input we will

calculate here what is the net input we

got one because we have to use the same

equation we will get this particular

thing as one because it is greater than

0 the output is equivalent to what uh

one in this case okay now this is one

and the target is also one both of them

are same so no need to do the weight

updation so Delta terms are zero and the

weights remains same over here

now we will present the next input this

is the next input the Target in this

case is minus 1 the net input that is y

in we have to calculate with the help of

this equation that is minus 1 minus 1 is

less than 0 because of that it'll be

minus 1 here so that is what I have

written here now the calculated output

is minus 1 and the actual Target is also

minus 1 because they are same we should

not do the weight updation Delta terms

remains 0 and the uh the weights will

remains same here

we will present the next input the

target is -1 Y in calculated is minus 1

here and the calculated output is equal

to because y minus 1 is less than 0 the

calculator output is minus 1 minus 1 and

minus 1 that's Target and the calculated

outputs are matching here so no need to

do the weight updation so Delta terms

remains same weights remains same here

now we will try to present the next

input once you present the next input

the target is minus 1 y n is minus 3

minus 3 is less than 0 because of that

we got the output as minus 1 so minus 1

is equal to minus 1 then there is no

need to do the weight updation so these

are the final weights after the weight

of the second iteration you can say now

after second iteration we were able to

classify all the targets correctly this

one was classified correctly second one

was correctly classified third one was

classified correctly fourth one was

classified correctly so you can say that

these are the final weights of this

particular Network or you can say that

the and function in this case

for Alpha is equivalent to 1 uh these

are the final weights so that can be

drawn something like this the bias is

equivalent to -1

the weight one that is W1 and W two are

equivalent to 1 1 in this case so this

is how actually the final and function

looks like with the help of perceptron

rule

I hope the concept is clear if you like

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