All language subtitles for 05_visualizing-the-cost-function.en

In the last video, you
saw one visualization of

the cost function J
of w or J of w, b.

Let's look at some further

richer visualizations
so that you can get

an even better intuition

about what the cost
function is doing.

Here is what we've seen so far.

There's the model, the
model's parameters w and b,

the cost function J of w and b,

as well as the goal
of linear regression,

which is to minimize the
cost function J of w

and b over parameters w and b.

In the last video,

we had temporarily set b to

zero in order to simplify
the visualizations.

Now, let's go back to
the original model with

both parameters w and b

without setting b
to be equal to 0.

Same as last time,

we want to get a
visual understanding

of the model function, f of x,

shown here on the left,

and how it relates to the
cost function J of w,

b, shown here on the right.

Here's a training set of
house sizes and prices.

Let's say you pick
one possible function

of x, like this one.

Here, I've set w
to 0.06 and b to

50. f of x is 0.06
times x plus 50.

Note that this is not

a particularly good model
for this training set,

is actually a pretty bad model.

It seems to consistently
underestimate housing prices.

Given these values for w
and b let's look at what

the cost function J of
w and b may look like.

Recall what we saw last time
was when you had only w,

because we temporarily set b
to zero to simplify things,

but then we had come up with
a plot of the cost function

that look like this as
a function of w only.

When we had only
one parameter, w,

the cost function had
this U-shaped curve,

shaped a bit like a soup bowl.

That sounds delicious.

Now, in this housing
price example

that we have on this slide,

we have two parameters, w and b.

The plots becomes a
little more complex.

It turns out that
the cost function

also has a similar
shape like a soup bowl,

except in three dimensions
instead of two.

In fact, depending on
your training set,

the cost function will
look something like this.

To me, this looks
like a soup bowl,

maybe because I'm a
little bit hungry,

or maybe to you it looks like

a curved dinner
plate or a hammock.

Actually that sounds
relaxing too,

and there's your coconut drink.

Maybe when you're done
with this course,

you should treat yourself to

vacation and relax in
a hammock like this.

What you see here is
a 3D-surface plot

where the axes are
labeled w and b.

As you vary w and b,

which are the two
parameters of the model,

you get different values for

the cost function J of w, and b.

This is a lot like
the U-shaped curve

you saw in the last video,

except instead of having

one parameter w as
input for the j,

you now have two parameters,

w and b as inputs into

this soup bowl or this
hammock-shaped function

J. I just want to point out
that any single point on

this surface represents
some particular choice

of w and b.

For example, if w was minus
10 and b was minus 15,

then the height of the surface

above this point is the value of

j when w is minus 10
and b is minus 15.

Now, in order to look even more

closely at specific points,

there's another way of plotting

the cost function J

that would be useful
for visualization,

which is, rather than using
these 3D-surface plots,

I like to take this
exact same function

J. I'm not changing

the function J at all and

plot it using something
called a contour plot.

If you've ever seen
a topographical map

showing how high
different mountains are,

the contours in a topographical
map are basically

horizontal slices of the
landscape of say, a mountain.

This image is of
Mount Fuji in Japan.

I still remember
my family visiting

Mount Fuji when I
was a teenager.

It's beautiful sights.

If you fly directly
above the mountain,

that's what this
contour map looks like.

It shows all the points,

they're at the same height
for different heights.

At the bottom of this slide
is a 3D-surface plot of

the cost function J.

I know it doesn't look
very bowl-shaped,

but it is actually a bowl
just very stretched out,

which is why it looks like that.

In an optional lab,

that is shortly to follow,

you will be able to see
this in 3D and spin around

the surface yourself
and it'll look

more obviously
bowl-shaped there.

Next, here on the upper
right is a contour plot of

this exact same cost function

as that shown at the bottom.

The two axes on this
contour plots are b,

on the vertical axis,

and w on the horizontal axis.

What each of these ovals,

also called ellipses,

shows, is the center points on

the 3D surface which are
at the exact same height.

In other words, the set
of points which have

the same value for
the cost function J.

To get the contour plots,

you take the 3D surface
at the bottom and you

use a knife to slice
it horizontally.

You take horizontal slices of

that 3D surface and
get all the points,

they're at the same height.

Therefore, each horizontal
slice ends up being shown

as one of these ellipses
or one of these ovals.

Concretely, if you
take that point,

and that point, and that point,

all of these three points have

the same value for
the cost function J,

even though they have
different values for w and b.

In the figure on the upper left,

you see also that
these three points

correspond to
different functions,

f, all three of which
are actually pretty

bad for predicting housing
prices in this case.

Now, the bottom of the bowl,

where the cost function
J is at a minimum,

is this point right here,

at the center of this
concentric ovals.

If you haven't seen
contour plots much before,

I'd like you to
imagine, if you will,

that you are flying
high up above

the bowl in an airplane
or in a rocket ship,

and you're looking
straight down at it.

That is as if you set

your computer monitor
flat on your desk

facing up and the bowl shape

is coming directly
out of your screen,

rising above you desk.

Imagine that the bowl
shape grows out of

your computer screen
lying flat like that,

so that each of these ovals have

the same height above
your screen and

the minimum of the bowl is right

down there in the center
of the smallest oval.

It turns out that the
contour plots are

a convenient way to visualize
the 3D cost function J,

but in a way, there's
plotted in just 2D.

In this video, you saw how

the 3D bowl-shaped surface plot

can also be visualized
as a contour plot.

Using this visualization too,

in the next video,

let's visualize some
specific choices

of w and b in the
linear regression model

so that you can see how
these different choices

affect the straight line
you're fitting to the data.

Let's go on to the next video.