Original subtitles

In this video, we'll figure out

how to get gradient descent to

work with regularized linear regression. Let's jump in.

Here Is a cost function we've come up with in

the last video for regularized linear regression.

The first part is the usual squared error cost function,

and now you have this additional regularization term,

where Lambda is the regularization parameter,

and you'd like to find parameters w and

b that minimize the regularized cost function.

Previously we were using

gradient descent for the original cost function,

just the first term before we

added that second regularization term,

and previously, we had

the following gradient descent algorithm,

which is that we repeatedly update the parameters w, j,

and b for j equals 1 through n

according to this formula

and b is also updated similarly.

Again, Alpha is

a very small positive number called the learning rate.

In fact, the updates for

a regularized linear regression look exactly the same,

except that now the cost,

J, is defined a bit differently.

Previously the derivative of J with respect

to w_j was given by this expression over here,

and the derivative respect to b was

given by this expression over here.

Now that we've added this additional regularization term,

the only thing that changes is that the expression for

the derivative with respect to

w_j ends up with one additional term,

this plus Lambda over m times w_j.

And in particular for

the new definition of the cost function j,

these two expressions over here,

these are the new derivatives of J with respect

to w_j and the derivative of J with respect to b.

Recall that we don't regularize b,

so we're not trying to shrink B.

That's why the updated B remains the same as before,

whereas the updated w changes because

the regularization term causes us to try to shrink w_j.

Let's take these definitions for

the derivatives and put them back into

the expression on the left to write out

the gradient descent algorithm

for regularized linear regression.

To implement gradient descent

for regularized linear regression,

this is what you would have your code do.

Here is the update for w_j,

for j equals 1 through n,

and here's the update for b.

As usual, please remember to carry out

simultaneous updates for all of these parameters.

Now, in order for you to get this algorithm to work,

this is all you need to know.

But what I like to do in

the remainder of this video is to go over

some optional material to convey

a slightly deeper intuition about

what this formula is actually doing,

as well as chat briefly about how

these derivatives are derived.

The rest of this video is completely optional.

It's completely okay if you skip the rest of this video,

but if you have a strong interests

in math, then stick with me.

It is always nice to hang out with you here,

and through these equations,

perhaps you can build a deeper intuition

about what the math and what

the derivatives are doing as well.

Let's take a look. Let's look at

the update rule for w_j and rewrite it in another way.

We're updating w_j as 1 times

w_j minus Alpha times Lambda over m times w_j.

I've moved the term from the end to the front here.

Then minus Alpha times 1 over m,

and then the rest of that term over there.

We just rearranged the terms a little bit.

If we simplify, then we're saying that w_j is updated

as w_j times 1 minus Alpha times Lambda over m,

minus Alpha times this other term over here.

You might recognize the second term as

the usual gradient descent update

for unregularized linear regression.

This is the update for

linear regression before we had regularization,

and this is the term we saw in Week 2 of this course.

The only change we add

regularization is that instead of w_j being set

to be equal to w_j minus Alpha times

this term is now

w times this number minus the usual update.

This is what we had in Week 1 of this course.

What is this first term over here?

Well, Alpha is a very small positive number, say 0.01.

Lambda is usually a small number,

say 1 or maybe 10.

Let's say lambda is 1 for this example

and m is the training set size, say 50.

When you multiply Alpha Lambda over m,

say 0.01 times 1 divided by 50,

this term ends up being a small positive number

, say 0.0002,

and thus, 1 minus Alpha Lambda

over m is going to be

a number just slightly less than 1,

in this case, 0.9998.

The effect of this term is that

on every single iteration of gradient descent,

you're taking w_j and multiplying it by 0.9998,

that is by some numbers slightly less than

one and for carrying out the usual update.

What regularization is doing on every single iteration

is you're multiplying w

by a number slightly less than 1,

and that has effect of shrinking

the value of w_j just a little bit.

This gives us another view on why

regularization has the effect of

shrinking the parameters w_j

a little bit on every iteration,

and so that's how regularization works.

If you're curious about how

these derivative terms were computed,

I've just one last optional slide that goes through

just a little bit of

a calculation of the derivative term.

Again, this slide and the rest of

this video are completely optional,

meaning you won't need any of this to

do the practice labs and the quizzes.

Let's step through quickly to derivative calculation.

The derivative of J with respect to w_j looks like this.

Recall that f of x for linear regression is defined as

w dot x plus b or w dot product x plus b.

It turns out that by the rules of calculus,

the derivatives look like this,

is 1 over 2m times the sum i equals 1 through m of

w dot x plus b minus y

times 2x_j plus the derivative

of the regularization term,

which is Lambda over 2m times 2 w_j.

Notice that the second term does not have

the summation term from j equals 1 through n anymore.

The 2's cancel out here and here,

and also here and here.

It simplifies to this expression over here.

Finally, remember that wx plus b is f of x,

and so you can rewrite it as this expression down here.

This is why this expression is used to

compute the gradient in regularized linear regression.

You now know how to

implement regularized linear regression.

Using this, you really reduce overfitting when you

have a lot of features and

a relatively small training set.

This should let you get linear regression to

work much better on many problems.

In the next video,

we'll take this regularization idea and apply it to

logistic regression to avoid

overfitting for logistic regression as well.

Let's take a look at that in the next video.