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Cost Function
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From the course by Stanford University
Machine Learning
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From the lesson
Linear Regression with One Variable
Linear regression predicts a real-valued output based on an input value. We discuss the application of linear regression to housing price prediction, present the notion of a cost function, and introduce the gradient descent method for learning.
Meet the Instructors
Andrew NgCEO/Founder Landing AI; Co-founder, Coursera; Adjunct Professor, Stanford University; formerly Chief Scientist,Baidu and founding lead of Google Brain
In this video we'll define something called the cost function,
this will let us figure out how to fit the best possible straight line to our data.
In linear progression, we have a training set that I showed here remember on
notation M was the number of training examples, so maybe m equals 47.
And the form of our hypothesis,
which we use to make predictions is this linear function.
To introduce a little bit more terminology, these theta zero and
theta one, they stabilize what I call the parameters of the model.
And what we're going to do in this video is talk about how to
go about choosing these two parameter values, theta 0 and theta 1.
With different choices of the parameter's theta 0 and theta 1,
we get different hypothesis, different hypothesis functions.
I know some of you will probably be already familiar with
what I am going to do on the slide, but just for review, here are a few examples.
If theta 0 is 1.5 and theta 1 is 0,
then the hypothesis function will look like this.
Because your hypothesis function will be h of x equals 1.5 plus 0 times
x which is this constant value function which is phat at 1.5.
If theta0 = 0, theta1 = 0.5, then the hypothesis will look like this, and
it should pass through this point 2,1 so
that you now have h(x).
Or really h of theta(x), but sometimes I'll just omit theta for brevity.
So h(x) will be equal to just 0.5 times x, which looks like that.
And finally, if theta zero equals one, and theta one equals 0.5,
then we end up with a hypothesis that looks like this.
Let's see, it should pass through the two-two point.
Like so, and this is my new vector of x, or my new h subscript theta of x.
Whatever way you remember, I said that this is h subscript theta of x, but
that's a shorthand, sometimes I'll just write this as h of x.
In linear regression, we have a training set, like maybe the one I've plotted here.
What we want to do, is come up with values for the parameters theta zero and
theta one so that the straight line we get out of this, corresponds to
a straight line that somehow fits the data well, like maybe that line over there.
So, how do we come up with values,
theta zero, theta one, that corresponds to a good fit to the data?
The idea is we get to choose our parameters theta 0, theta 1 so
that h of x, meaning the value we predict on input x,
that this is at least close to the values y for
the examples in our training set, for our training examples.
So in our training set, we've given a number of examples where we know X
decides the wholes and we know the actual price is was sold for.
So, let's try to choose values for the parameters so that,
at least in the training set, given the X
in the training set we make reason of the active predictions for the Y values.
Let's formalize this.
So linear regression, what we're going to do is,
I'm going to want to solve a minimization problem.
So I'll write minimize over theta0 theta1.
And I want this to be small, right?
I want the difference between h(x) and y to be small.
And one thing I might do is try to minimize the square difference
between the output of the hypothesis and the actual price of a house.
Okay. So lets find some details.
You remember that I was using the notation (x(i),y(i))
to represent the ith training example.
So what I want really is to sum over my training set,
something i = 1 to m,
of the square difference between, this is the prediction
of my hypothesis when it is input to size of house number i.
Right? Minus the actual price that house number
I was sold for, and I want to minimize the sum of my training set,
sum from I equals one through M, of the difference of this squared error,
the square difference between the predicted price of a house, and
the price that it was actually sold for.
And just remind you of notation, m here was the size of my training set right?
So my m there is my number of training examples.
Right that hash sign is the abbreviation for number of training examples, okay?
And to make some of our, make the math a little bit easier,
I'm going to actually look at we are 1 over m times that so
let's try to minimize my average minimize one over 2m.
Putting the 2 at the constant one half in front, it may just sound
the math probably easier so minimizing one-half of something, right, should give
you the same values of the process, theta 0 theta 1, as minimizing that function.
And just to be sure, this equation is clear, right?
This expression in here, h subscript
theta(x), this is our usual, right?
That is equal to this plus theta one xi.
And this notation, minimize over theta 0 theta 1, this means
you'll find me the values of theta 0 and theta 1 that causes this expression
to be minimized and this expression depends on theta 0 and theta 1, okay?
So just a recap.
We're closing this problem as, find me the values of theta zero and theta one so
that the average, the 1 over the 2m,
times the sum of square errors between my predictions on the training set
minus the actual values of the houses on the training set is minimized.
So this is going to be my overall objective function for linear regression.
And just to rewrite this out a little bit more cleanly, what I'm going to do is,
by convention we usually define a cost function,
which is going to be exactly this, that formula I have up here.
And what I want to do is minimize over theta0 and theta1.
My function j(theta0, theta1).
Just write this out.
This is my cost function.
So, this cost function is also called the squared error function.
When sometimes called the squared error cost function and
it turns out that why do we take the squares of the erros.
It turns out that these squared error cost function is a reasonable choice and
works well for problems for most regression programs.
There are other cost functions that will work pretty well.
But the square cost function is probably the most commonly used one for
regression problems.
Later in this class we'll talk about alternative cost functions as well,
but this choice that we just had should be a pretty reasonable thing to try for
most linear regression problems.
Okay.
So that's the cost function.
So far we've just seen a mathematical definition of this cost function.
In case this function j of theta zero, theta one.
In case this function seems a little bit abstract,
and you still don't have a good sense of what it's doing,
in the next video, in the next couple videos, I'm actually going
to go a little bit deeper into what the cause function "J" is doing and try to
give you better intuition about what is computing and why we want to use it...
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