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We've just looked at error sanctioning, and that has permitted us to see how this model A um, with

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parameters M and C different from this other model B with its own parameters m and C.

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Let's say, for example, given that we have um, this line y so m is say for example zero and c three.

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And then this other m could be two and then c could be let's say four.

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With the model B having a higher loss or having a higher error as compared to the model A.

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Now the whole idea of um training optimization is simple.

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What we want to do is take a model which has been initialized.

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Let's take a model whose parameters have been initialized to some random values, like for example,

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zero three or even zero, zero or whatever random values, and then update those parameters such that

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we get a better performing model like this one here with parameters two four.

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So let's suppose that we randomly initialize our model, um, for to have m equals zero and c equal

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three.

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Then we want to update this such that zero tends to something close to two, and three turns to something

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close to four.

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Now, to carry out this transformation, we shall make use of um, an algorithm known as the stochastic

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gradient descent.

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This algorithm is quite simple.

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Suppose we have a weight.

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This weight could be m or c.

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Thus we have the a parameter.

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Uh, let's call this parameter.

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Or let's say let's say this initial value.

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We are going to subtract.

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We're going to take this weight minus, uh, learning rate.

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Times the partial derivative of the loss.

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This loss is the same as the error we had computed, uh, with respect to that particular weight.

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So let's have this.

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This is minus all this here.

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And when you have this, I.

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So this initial width or the initial weight value um is here.

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So we have w I minus learning rate times the partial derivative of l the loss with respect to I.

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And then this now gives us w f.

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That's this final value.

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So we could go from um if we pick out some let's pick out some values.

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Let's suppose that uh our w initially is zero minus, um, let's say a learning rate of um, 0.1.

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Times.

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Um, a partial derivative of, let's say -20 will now give us a w f, which is two.

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And that's how we could move from, um, this initial w to this two.

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But it should be noted that in practice we don't generally just move from, uh, very poorly performing

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model to one that is very well performing in a single step.

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So we are going to go through this same algorithm in many steps.

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So we could start with let's get back.

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We could start with um, this partial derivative of um let's say even five.

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So here, here we have 0.1 times ten.

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Uh, times one times ten is actually uh, one divided by two.

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That's 0.5.

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So we could go from 0 to 0.5.

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You see we go from 0 to 0.5.

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And then in our next step, because now our W has turned to 0.5.

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That's from zero.

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We went from 0 to 0.5.

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We will replace this zero here with 0.5.

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So instead of having zero we would have now 0.5.

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The learning rate is still the same.

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Um, although it could change as we keep training.

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And then this partial derivative, we could say for example that is ten.

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So let's um, this is 0.5.

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Let's take this off here and then replace this with a negative ten.

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So here we have negative ten.

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Now you see we have one.

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So we have 0.5 minus or rather 0.5 plus one.

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So 0.5 plus one gives us um 1.5.

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So let's take this off again.

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So instead of 0.5 now we have 1.5.

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Take this off.

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And we have now 1.5.

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So you see how we go from zero.

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We go from 0 to 0.5.

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And then 21.5.

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And the reason why I would expect to have uh, better values of w better in this case is values of w

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such that the, the straight line would be close to enough to this point as compared to this other poorly

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performing model.

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And so though, as we're saying, the reason why we are sure that we are going to keep getting towards

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this, um, better performing values of W is because we are including the loss in this, um, algorithm.

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So when we calculate our loss, we look for this partial derivative with respect to the weights.

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And is this partial derivative that permits us to adjust the weights.

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Now this learning rate helps us adjust by how much we are going to, um, modify the weights.

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So if we had a large learning rate here, or let's say if we had a larger learning rate, not large

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because learning rates are relative.

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So if we had a larger learning rate of um instead of 0.1 if we had one.

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So let's take this off.

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If we had one, you would find that right from the first step we would have um, five.

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So we would, we would just jump from 0 to 5.

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So this controls how much we, we improve our model.

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Now, if you have no background in calculus, you shouldn't bother.

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As with TensorFlow, all we need to do to make use of this algorithm is by just specifying that our

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optimizer is SGD.

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So you just need to put in the string and you're good to go.

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You don't really need to write all this code.

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So getting back um here again we have let's let's plot the loss against the weights.

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So we have a plot of loss against weights.

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Now our loss here is actually something like y minus y prime.

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All of that square is true.

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We have a sum but we're going to ignore that.

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So it's going to look like it's going to look something like this.

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You see this quadratic function.

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And even if you have no background in calculus, you should just understand that when we talk about,

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uh, partial derivative or derivative in general, uh, and we have a curve that has to do with the

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slope of that curve.

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So if you, if you have at this point, if at this point you have w and you have the loss at this point

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the, the slope, you have a slope which is, which looks like this.

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If you draw a tangent and you calculate a slope, you try to calculate the slope.

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You find that here the slope is going to be higher than if we draw a tangent at this point.

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If you look at this point here and you draw this tangent well let's get back and try to draw a tangent.

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And we have a slope.

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The slope is smaller as compared to this.

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So if this is let's call this slope a and let's call this slope b.

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So slope a is greater than slope b.

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And so what this means is the partial derivative of the loss with respect to the weight at this point

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is greater than the partial derivative of the loss with respect to the weight at this other point.

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Now let's see how this, uh, plays when we when it comes to updating the weights, remember that the

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ideal situation is that our loss should be zero.

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So what we want to do here is minimize the loss.

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So, um, if we get to a point like this, then that's fine because we've gotten to the smallest possible

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value of our loss.

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And what this means is you may even have two and four and, uh, as weight as, um, M and C as the

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parameters, but the, the loss will be maybe somewhere about, uh, around here.

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And so there might still be room for improvement.

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And so, uh, what happens is we are going to take this weight.

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You see, we have the weight, subtract the take the subtract the learning rate times this slope actually.

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So when we say partial derivative of the loss with respect to the weight, what we actually are doing

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here is we're subtracting that slope.

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And so if we are having, um, this weight, we imagine we have this weight here and that this, this

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weight links leads us to this point here.

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There we go.

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We have this point here where our loss is, is quite high, which is not what we desire.

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Then you would find that the slope, the slope is much larger.

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Just like with this slope here, the slope is much larger as compared to if we have a weight value around

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here where we have a smaller slope, and because the slope is much larger, you would find that, um,

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multiplying this larger slope by the learning rate would lead to a larger change in the weight.

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So, um, this larger change in the weight will tend to make the weight, um, or this weight value

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move to the left.

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Because now we're taking all the weight minus a large value.

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And so we'll tend to move now to the left.

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Now if we're in this other direction, if we come in this other direction, that if we have a weight

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value this other direction, then, um, taking w minus this learning rate times the slope, because

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the slope is now negative, we have a negative slope.

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We'll have w minus a negative.

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That's w plus the learning rate times the absolute value of the slope.

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And so now we'll be moving towards the right.

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And so you see how graphically the stochastic gradient descent algorithm permits us to move the weight

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towards values which will permit us have, um, a smaller loss.

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Another interesting point to note is the fact that as we as our weight approaches this values where

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the corresponding loss is very small, the slope also becomes very small.

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And so the change in weight isn't that large.

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So so if we have if we get to a point like this here, if we get to this point, let's say we get to

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this point here, which is very close to the optimal weight value, where we're going to have the smallest

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possible loss, then this slope here is going to be very small.

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And so when you do w minus learning rate times that very small slope, you will find that the value

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of w will change just a little bit.

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And so what this tells us is when we are far away from that um optimal weight value, the slope is large.

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And then when we approach that, uh, optimal weight value, the slope is small.

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And so what happens generally during training is as we start training, the loss value, the loss the

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loss value drops much faster.

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And as we keep getting much better weight values, this loss value continues dropping but much more

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slowly.

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And this is what we call um, model convergence.

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Now the choice of this learning rate is very important because if our learning rate is too big, then,

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um, multiplying it by a large slope will take this weight to, uh, this other side, which is not

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what we want.

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And if it's too small, then we'll even if we have a large slope, the change in the weight is going

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to be also too small.

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And so the learning process is going to be too slow.

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So we want to choose, uh, an adequate value for the learning rate so that our training goes fast and

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smooth.

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Diving back to the code, you see all we need to do, um, is just specify SGD and we're going to have

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the the w minus learning rate times the partial derivative with respect to the learning with respect

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to the weights, um, which are going to be calculated um, under the hood.

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So that's it.

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Let's get back and print out the shape that's y and x and y.

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And then we could start with the training.

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So here we just need to do model dot fit.

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And then we specify x.

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We specify y we give number of epochs.

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Remember we said we are not going to start updating these parameters and then dive.

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Or after one step just get straight to the optimal values.

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Generally we go through several steps and each one of these step is called an epoch.

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So we'll specify a number of epochs to say um ten.

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And then we'll variables we say one.

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So let's run that and see what we get.

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Um has no this should be epochs on that.

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And as you could see, the model, um, has trained for ten epochs.

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Now you'll notice how this loss value keeps dropping.

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See, we go from 308 or 308,519.

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We drop um to 308,516.

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If you increase the number of epochs, you would find that these values are going to drop even much

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more, um, as compared to what we've seen already.

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So you see, keeps dropping.

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Now we get to 308, um, 485.

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Now, although we specify this optimizer to be SGD, that's stochastic gradient descent.

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Um, we could also have other optimizers.

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So you could check out here.

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But most of these are just flavors of the SGD, like the Adam and Adam W, which are the one of the

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most popular optimizers we have today.

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So click on this.

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Adam.

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You see we have Adam Optimizer, which as we said is a flavor of the SGD and it has now many more parameters.

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You could feel free to check out all this in the documentation.

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So let's just let's just copy this out or let's just get back to the code and import.

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Um, Adam.

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So we have get back to the top we have from TensorFlow.

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TensorFlow Keras, Keras optimizers.

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We're going to import um Adam.

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So that's it.

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You could import Adam.

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You can import Adam um w or any other optimizer available in TensorFlow.

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So that's it.

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Once you import that you could now replace this SGD we had right here.

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So instead of this now we just have Adam.

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And then you could specify it's learning rate.

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So we have learning rate by default it was given to be 0.001.

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Yeah that was it.

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Um take that off okay.

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So by default we have this.

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We also have the values for beta one, beta two and so on and so forth.

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But let's just focus on the learning rate.

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So by default we have this.

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We could change this to say um one e negative five or some other value.

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So let's just maintain the one e negative three.

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And then just as before we compile the model.

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And then we go ahead and train.

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Now while training we did not keep the history of all these losses.

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So let's say history.

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Um there we go.

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So we're still going to have the same output.

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But now after the training is done, you could do history dot history, and you would see that you will

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get all these different loss values.

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So you get all the different loss values.

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See we have this dictionary.

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We have loss um, and we have all these values right up to the end.

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So let's, let's now do history dot history and then we get the loss.

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See we have directly all the different loss values from the epoch zero right up to the last epoch.

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We can now plot out this loss values um using matplotlib.

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So let's simply do, let's first of all import matplotlib.

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We have from here um import matplotlib.pyplot as plt on that.

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Well think of this.

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Oh and that should be fine.

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So we run that again.

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That's fine.

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We get back and then we, um, try to plot all this.

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Uh, I'll try to plot out this loss.

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So we see that, um, graphically.

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There we go.

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We have our matplotlib plot, um, history, history loss.

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There we go.

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We also want to have a title.

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So here we have model model loss.

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Then we want to have a y label.

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Y label is simply the loss loss.

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And then we have the X label.

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The X label is the epoch um epoch.

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And then we have um legend.

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Legend.

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We have train.

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Well let's see train.

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Okay, let's put on the list and we have train.

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That's fine.

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And now we do plot show.

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There we go.

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You can see our loss and you see that, um, it's actually dropping.

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And if we compare this, uh, plot to what we had already in the board, we see that we're still very

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far from the model convergence.

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So we could still, uh, do better with this model, or we could still get, uh, much smaller loss

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values.