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‫In this lecture, we are going to understand the concept behind how neural networks actually learn.

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‫So tell this lecture, we have covered what that neural network is.

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‫Now we are starting with how does a neural network work?

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‫Here's a quick recap.

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‫A neural network is a network of cells in our network.

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‫We are going to use sigmoid neurons because these learn in a more controllable manner in a sigmoid neuron.

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‫Two things happen.

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‫We first multiply the input features with the weights that are represented by W and then add a bias.

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‫Term b, this value

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‫We name as Z or Z.

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‫The second step is the application of sigmoid or the resistory function.

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‫That is, the cell calculates one upon one

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‫Plus e raise to power minus e

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‫This value is the output of the cell.

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‫And is always between zero and one.

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‫This output becomes the new input for the next layer.

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‫This continues till the last layer, till we get the final output as per our network.

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‫Now the problem for which we are solving is this.

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‫We want to find out the weights and biases of all the cells in the system so that the final output of this

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‫network is as close to the actual value of the variable to be predicted.

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‫For better understanding, let us just calculate the number of variables we need to calculate.

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‫For this small network here, we have two neurons in the hidden layer and one neuron in the output layer.

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‫And there are two input features, x1 and X2.

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‫So for the first neuron we are trying to calculate W1 into X1 plus W2 into X2 plus B1 is equal

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‫to Z.

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‫This z will be put into the activation function.

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‫That is the sigmoid function.

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‫And that will be the output of this neuron.

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‫Let us say that the output of this neuron is represented by a1 for the second neuron.

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‫We have two new weights, W 3 and W 4.

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‫We calculate this value w3 x1 plus W4 x2 plus B2.

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‫This B2 is the bias of this neuron is equal to Z2.

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‫We apply activation function on the Z2 to get A2.  These A1 and A2

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‫Are the inputs to this final output neuron for these two inputs.

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‫We need two new weights, W5 and W6.

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‫So the equation at this output neuron is W5 into a1 plus W6 into A2 plus B3 gives Z3.

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‫When we apply activation function on this Z3, we get the predicted output from this output neuron.

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‫So if you look at the variables that we need to estimate for weights, we have W1, W2, W3, W4 W5

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‫and W6.

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‫So we are estimating six weights. For biases

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‫We have B1, B2 and B3 three Bias's.

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‫So for this small network we need to establish the values of nine variables to make this neural network

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‫ready for predictions.

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‫Now how do we find out the values of Ws and Bs the technique followed for

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‫This is called gradient descent.

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‫Gradient descent is just another optimization technique to find minimum of a function.

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‫There are other optimization techniques also, such as ordinarily squared, which is used in linear

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‫regression.

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‫But for a large number of features and complex relationships, gradient descent shows much better computational

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‫performance than any other technique.

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‫This means that if you have a large number of input variables and a very complex relationship between

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‫input on output, gradient descent will train the model in a much faster way as compared to other optimization.

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‫So let's first discuss the process followed and gradient descent in a stepwise manner.

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‫We start by assigning a random weights and bias values to all the cells in our network.

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‫Since all the weights and biased values are available, that is, we have randomly assigned all weight

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‫and biases.

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‫Our model is ready to give out output.

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‫The second step is we input one training example.

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‫We use the X values of the training example and calculate the final output of the network using these

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‫weights and bias values.

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‫Third step is that we compare the predicted values vs. the actual values and we note the difference

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‫between these two using some error function.

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‫E will come back to this error function later.

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‫Remember that we have the actual Y value because this was a training observation.

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‫So these actual values are being used to give feedback to our network that how bad it is performing.

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‫The fourth step is we try to find out those weights and biases changing, which we can reduce this error.

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‫Lastly, we update the values of the weights and bias.

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‫And repeat this process from step two.

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‫This loop goes on till no

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‫Further reduction in error function can be achieved.

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‫And these steps, the first step is called initialization. Here

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‫We just gave some random initial values to weights and bias.

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‫The second step is called forward propagation.

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‫This is because in this step, we start with input values.

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‫Process them in layer one.

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‫Then we take the output of layer one and process it.

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‫In layer two.

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‫And so on.

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‫Till we get one final predicted output.

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‫We are simply moving forward in terms of the layers of the network.

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‫So this is forward propagation.

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‫The third and fourth step are called backward propagation in these steps.

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‫We already have the final error function and we look backwards in our network to find out which weights

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‫and biases have maximum impact on this error function.

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‫Once we establish which weights and biases have maximum impact.

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‫We update these weights biases slightly to reduce the error.

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‫So this is the process we follow to implement gradient descent in neural networks.

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‫But we are still not discussed.

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‫The concept behind gradient descent

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‫gradient descent is a mathematical technique which is used to find out B minimum of A function.

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‫Let's see this example in the graph on the left on the x axis.

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‫I have this variable on the Y axis.

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‫I have a function applied on this variable.

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‫This is the plot of this function.

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‫Now, if you want to find out the value of X at which the function has minimum value, there are two

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‫ways to it.

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‫One is if you know the exact relationship between X and the function, you can use calculus to find

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‫the minimum of this function.

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‫But as you know, in our machine learning problems, we do not have this exact relationship.

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‫So we use a second technique, which is an iterative technique.

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‫In this technique.

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‫We start at a random point on this plot.

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‫Say we have this value of X and fx. Now instead of focusing on the whole graph, we focus only

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‫on this small part of the graph and try to find out what happens if we slightly increase the value of

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‫X or decrease the value of X.

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‫In other words, we are trying to find out which way is the slope.

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‫If the slope is negative, that is like this, we increase the value of X a little bit.

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‫And then we will see that fx will also decrease.

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‫Similarly, if the slope is positive, we decrease the value of X, which will slightly decrease the

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‫value of fx.

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‫We continue taking these small, small steps.

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‫Till we reach the final minimum point when we are at this point moving either side only increases

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‫the value of the function.

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‫So we stop our process here.

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‫This iterative technique of finding instantaneous slope, also known as gradient and slightly moving

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‫down that slope that is descent is called gradient descent.

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‫If you want to picturise this, you can think of yourself being on top of a hill.

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‫You cannot see anything around you because it is dark and foggy.

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‫Now, you want to come down the hill as fast as possible.

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‫What do you do?

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‫Ideally, if you could see, you would spot the closest downhill point and run to it.

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‫But since you cannot see and you know, the gradient descent technique, you take a step in each direction,

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‫see which direction is more down, and then you move from your current position to that new position.

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‫Then you again take out your right foot, Jack, which is the direction of steep slope and move.

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‫Did you keep doing this?

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‫And eventually you will come downhill.

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‫This is the concept behind gradient descent.

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‫In the next lecture we will merge the two ideas.

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‫The first is the process that neural networks use to implement gradient descent.

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‫And the second idea was what gradient descent is.

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‫Mathematically, we will merge these two and understand how gradient descent is helping us achieve the

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‫minima in neural networks.