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In this lecture, we are going to answer the question, how does a model learn?

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Let's start with linear regression and its most basic form.

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Linear regression just means line of best fit.

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As you may recall from your high school math studies, a line has the equation, Y equals M, X plus

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B here.

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X is the input variable, M is the slope and B is the Y intercept.

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When we put these together, we get the equation for a line.

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And our job, of course, is to find a line that best fits our input data.

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So you can imagine that our input data is a bunch of data points that approximately create the shape

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of a line.

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This plot is called a scatterplot.

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So for each point X and Y in our data set, we're going to draw a dot on this chart.

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Our goal is to find a line that best passes through all these data points.

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In practice, this means finding the slope and y intercept.

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One method you may have used in high school is to take a ruler and to try and draw the best line on

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paper using visual inspection.

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Of course, since this is data science, we no longer use such methods.

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We have to be more systematic.

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The line that you draw might not be the same as the line that I draw.

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And furthermore, if we are in multiple dimensions, this won't be a line at all.

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So it's clear that we need something better and more quantitative.

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The idea is we're going to define an era function for each data point, which is a pair made up of the

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input EXI and the target.

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Why we're going to calculate a prediction why Hatti equals to Mexi plus B, then we are going to take

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the squared difference between why and why haddi.

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We're going to do this for each of our data points.

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For I equals one up to end.

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Once we've done this, we can add them all up and divide by N.

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This is called the mean squared error.

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It's the average squared deviation between our predictions and our targets.

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Basically, this is going to tell us how accurate our model is.

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If our predictions are equal to the targets, then this error will be zero because why I would be equal

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to I had I then the more wrong we are, the larger this error becomes.

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So that's generally how an error function should work.

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As a side note, since some students get confused by this, at first we have multiple names for this

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error and we usually use them interchangeably.

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Sometimes we call it a loss, sometimes we call it an objective, or sometimes we call it a cost.

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However, these all mean the same thing.

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I like the term cost because it makes intuitive sense for business people as any good businessmen would

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do.

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Your job is to minimize your cost.

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And in fact, this statement alone tells us how to find the ways W.

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So how do we minimize this cost?

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In other words, make it as small as possible?

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Well, it's time to turn it to our old friend calculus.

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If you recall, we can find the minimum or a maximum of a function by finding where its derivative is

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equal to zero.

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If you're not convinced by this, try to draw a curve which has a minimum or a maximum and draw the

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slope at the minimum or maximum point.

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If you drew your line correctly, it should be a horizontal line, meaning that the slope at this point

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is zero.

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And remember, the slope at each point is just the derivative.

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So that's why we want to find the derivative set at zero and then solve for the parameter in question.

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Now, usually we have more than one parameter, even in one dimensional linear regression.

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This is the case because we have both the slope and the Y intercept.

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So what happens when we have a function of more than one variable?

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In this case, the derivative is actually called the gradient.

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If you recall from your calculus studies, the gradient is just a vector or tensor of partial derivatives

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for each of the variables we're taking the gradient with respect to.

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So, for example, if you take the square function in one dimension, that's called a parabola.

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But if you have two input dimensions, it's called a parabola, Lloyd.

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Beyond that, we can't picture it.

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But the basic idea is still the same.

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Find the gradient set at zero, solve for the parameters.

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By the way, the term parameter is just another name for the weights.

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So that's the WNBA we've been referring to throughout this section.

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Luckily, in this course, we won't be calculating any derivatives manually, since I have many other

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courses which do that a lot and by a lot I really mean a lot.

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So in this course, we're kind of excused from doing that because to flow already does that for us using

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a process called automatic differentiation.

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Luckily, you don't have to know how that works either, because as its name suggests, it's automatic.

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So the only thing you really do have to know is that a tensor flow is going to automatically find the

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gradient of all your weights and be essential.

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Flow uses these gradients to train your model.

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At this point, you might be wondering if all I have to do is find the gradient and set it to zero,

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then why in our previous code did it involve an iterative training process?

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Recall that we had to specify the number of epochs to train for, and this resulted in a plot of lost

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per iteration, which we could check to confirm that the training algorithm converges nicely.

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Well, in actuality, it's not possible to actually solve the equation that you get when you set the

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gradient to zero most of the time.

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The one exception to this, in this course at least, is linear regression where we can solve it.

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We call the solution an analytical solution or a closed form solution.

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Basically, this means that we can express the optimal value of W and be using an equation for logistic

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regression and the rest of the models will be discussing it's not possible to do this.

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Therefore, we need another approach.

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That approach is called gradient descent.

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Now, there's a lot more to gradient descent than what's in this lecture, and you can check out the

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in-depth section of this course for that if you're interested.

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For now, let's just talk about the basics.

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Essentially, what we do is we start at a randomly initialized point for both WMP.

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Since this is random, these probably don't lead to a small cost.

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So then we find the gradient of our loss with respect to W and B, we then take small steps in this

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direction to update W and B on each iteration.

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Remember, these iterations are called epochs.

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Mathematically, you can prove that this leads to a decrease in cost, although we won't discuss that

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in this lecture.

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You can imagine that this is exactly what goes inside the CARUS fit function.

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This is really all that's happening for epoch in range epochs and then set W equal to W minus eight

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times the gradient of G with respect to W and set B to B minus eight times the gradient of G with respect

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to be.

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So, again, all we're doing is taking small steps in the direction of the gradient with respect W and

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B, and one obvious question that arises from this is how small should these small steps actually be?

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Well, the step size is specified by this Greek letter, ADA, which we call the learning rate that

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specifies how fast or slow we want to train our model.

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It's important to set this value right, because if you don't, then your model won't get good results,

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even if your model is actually a good model.

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Unfortunately, there's no direct method of choosing a good learning rate.

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Generally speaking, the learning rate is something we call a hyper parameter.

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This is to differentiate it from WSB, which are just regular old parameters.

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It's called a hyper parameter because it's still a parameter, but it's not a perimeter of the model

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itself.

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It's sort of like a meta parameter.

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And in fact, no hyper parameters are really chosen directly.

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It's more of a process of trial and error, along with intuition that you gain from practicing a lot

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and seeing a lot of different examples.

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As mentioned before, one way to know if your learning rate is too high or too low is the check the

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loss per iteration after training.

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This is somewhat unfortunate because training can take a long time and you only know the result once

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it's done.

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If you use a bad learning rate and got bad results while you still had to wait for them.

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I really like this visualization because it encapsulates how to choose the learning rate quite well.

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If your lending rate is too high, then your loss will just shoot off to infinity.

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Intuitively, this is because you're overshooting the minimum and just ending up on the other side of

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the canyon.

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That means your steps are too large and you need to make them smaller, sometimes your lending rate

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might be a bit too high.

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So you'll see some convergence, but it'll converge to a suboptimal value.

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If your lending rate is too low, then you'll see a really shallow curve.

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This is also not good because as I mentioned, training takes time.

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So if you're lending rate is too low, you'll have to wait longer.

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But not only that, sometimes you can just get stuck at a suboptimal point.

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That's also not good.

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A good learning rate is in between these two extremes.

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And so what you'll probably end up doing is finding the limits of your particular data set, what's

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too high, what's too low, and then trying different numbers in between until you find something good.

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Normally we try numbers that have powers of 10, for example, zero point one zero point zero one zero

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point zero zero one and so forth.