Original subtitles

Welcome back. In the last video,

we saw visualizations of

the cost function j and how you can try

different choices of the parameters w and

b and see what cost value they get you.

It would be nice if we had

a more systematic way to find the values of w and b,

that results in the smallest possible cost,

j of w, b.

It turns out there's an algorithm called

gradient descent that you can use to do that.

Gradient descent is used

all over the place in machine learning,

not just for linear regression,

but for training for example some of

the most advanced neural network models,

also called deep learning models.

Deep learning models are something you

learned about in the second course.

Learning these two of gradient descent will set you

up with one of the most

important building blocks in machine learning.

Here's an overview of what

we'll do with gradient descent.

You have the cost function j of w,

b right here that you want to minimize.

In the example we've seen so far,

this is a cost function for linear regression,

but it turns out that gradient descent

is an algorithm that you

can use to try to minimize any function,

not just a cost function for linear regression.

Just to make this discussion on

gradient descent more general,

it turns out that gradient descent

applies to more general functions,

including other cost functions that work

with models that have more than two parameters.

For instance, if you have a cost function

J as a function of w_1,

w_2 up to w_n and b,

your objective is to minimize j

over the parameters w_1 to w_n and b.

In other words, you want to pick values

for w_1 through w_n and b,

that gives you the smallest possible value of j.

It turns out that gradient descent

is an algorithm that you can apply

to try to minimize this cost function j as well.

What you're going to do is just to start off

with some initial guesses for w and b.

In linear regression, it won't matter too

much what the initial value are,

so a common choice is to set them both to 0.

For example, you can set w to 0

and b to 0 as the initial guess.

With the gradient descent algorithm,

what you're going to do is,

you'll keep on changing the parameters w and b a bit

every time to try to reduce the cost j of w,

b until hopefully j settles at or near a minimum.

One thing I should note is that for some functions

j that may not be a bow shape or a hammock shape,

it is possible for there to be

more than one possible minimum.

Let's take a look at an example of

a more complex surface plot j

to see what gradient is doing.

This function is not a squared error cost function.

For linear regression with

the squared error cost function,

you always end up with a bow shape or a hammock shape.

But this is a type of cost function you

might get if you're training a neural network model.

Notice the axes, that is w and b on the bottom axis.

For different values of w and b,

you get different points on this surface,

j of w, b,

where the height of the surface at

some point is the value of the cost function.

Now, let's imagine that

this surface plot is actually a view of

a slightly hilly outdoor park or

a golf course where the high points are

hills and the low points are valleys like so.

I'd like you to imagine if you will,

that you are physically standing

at this point on the hill.

If it helps you to relax,

imagine that there's lots of

really nice green grass and

butterflies and flowers is a really nice hill.

Your goal is to start up here and get

to the bottom of one of

these valleys as efficiently as possible.

What the gradient descent algorithm does is,

you're going to spin around 360 degrees

and look around and ask yourself,

if I were to take

a tiny little baby step in one direction,

and I want to go downhill as quickly

as possible to or one of these valleys.

What direction do I choose to take that baby step?

Well, if you want to walk down

this hill as efficiently as possible,

it turns out that if you're standing

at this point in the hill and you look around,

you will notice that the best direction to take

your next step downhill is roughly that direction.

Mathematically, this is

the direction of steepest descent.

It means that when you take a tiny baby little step,

this takes you downhill faster than

a tiny little baby step you could

have taken in any other direction.

After taking this first step,

you're now at this point on the hill over here.

Now let's repeat the process.

Standing at this new point,

you're going to again spin

around 360 degrees and ask yourself,

in what direction will I take

the next little baby step in order to move downhill?

If you do that and take another step,

you end up moving a bit in

that direction and you can keep going.

From this new point,

you can again look around and decide

what direction would take you downhill most quickly.

Take another step, another step, and so on,

until you find yourself at the bottom of this valley,

at this local minimum, right here.

What you just did was go through

multiple steps of gradient descent.

It turns out, gradient descent

has an interesting property.

Remember that you can choose a starting point at

the surface by choosing

starting values for the parameters w and b.

When you perform gradient descent a moment ago,

you had started at this point over here.

Now, imagine if you try gradient descent again,

but this time you choose

a different starting point by choosing

parameters that place your starting point

just a couple of steps to the right over here.

If you then repeat the gradient descent process,

which means you look around,

take a little step in the direction of

steepest ascent so you end up here.

Then you again look around,

take another step, and so on.

If you were to run gradient descent this second time,

starting just a couple steps in

the right of where we did it the first time,

then you end up in a totally different valley.

This different minimum over here on the right.

The bottoms of both the first and

the second valleys are called local minima.

Because if you start going down the first valley,

gradient descent won't lead you to the second valley,

and the same is true if you

started going down the second valley,

you stay in that second minimum and

not find your way into the first local minimum.

This is an interesting property

of the gradient descent algorithm,

and you see more about this later.

In this video, you saw how

gradient descent helps you go downhill.

In the next video,

let's look at the mathematical expressions that you

can implement to make gradient descent work.

Let's go on to the next video.