1.1 Machine learning lecture 2  (Page 9/15)

In particular, it turns out for ordinary release squares, the function J of theta is – it's just a quadratic function. And so we'll always have a nice bow shape, like what you see up here, and only have one global minimum with no other local optima.

So when you run gradient descent, here are actually the contours of the function J. So the contours of a bow shaped function like that are going to be ellipses, and if you run gradient descent on this algorithm, here's what you might get. Let's see, so I initialize the parameters. So let's say randomly at the position of that cross over there, right, that cross on the upper right.

And so after one iteration of gradient descent, as you change the space of parameters, so if that's the result of one step of gradient descent, two steps, three steps, four steps, five steps, and so on, and it, you know, converges easily, rapidly to the global minimum of this function J of theta.

Okay, and this is a property of [inaudible] regression with a linear hypothesis cost. The function, J of theta has no local optima. Yes, question?

Student: Is the alpha changing every time? Because the step is not [inaudible].

Instructor (Andrew Ng) :So it turns out that – yes, so it turns out – this was done with a – this is with a fake value of alpha, and one of the properties of gradient descent is that as you approach the local minimum, it actually takes smaller and smaller steps so they'll converge. And the reason is, the update is – you update theta by subtracting from alpha times the gradient. And so as you approach the local minimum, the gradient also goes to zero.

As you approach the local minimum, at the local minimum the gradient is zero, and as you approach the local minimum, the gradient also gets smaller and smaller. And so gradient descent will automatically take smaller and smaller steps as you approach the local minimum. Make sense?

And here's the same plot – here's actually a plot of the housing prices data. So here, lets you initialize the parameters to the vector of all zeros, and so this blue line at the bottom shows the hypothesis with the parameters of initialization. So initially theta zero and theta one are both zero, and so your hypothesis predicts that all prices are equal to zero.

After one iteration of gradient descent, that's the blue line you get. After two iterations, three, four, five, and after a few more iterations, excuse me, it converges, and you've now found the least square fit for the data. Okay, let's switch back to the chalkboard. Are there questions about this? Yeah?

Student: [Inaudible] iteration, do we mean that we run each sample – all the sample cases [inaudible]the new values?

Instructor (Andrew Ng) :Yes, right.

Student: And converged means that the value will be the same [inaudible] roughly the same?

Instructor (Andrew Ng) :Yeah, so this is sort of a question of how do you test the convergence. And there's different ways of testing for convergence. One is you can look at two different iterations and see if theta has changed a lot, and if theta hasn't changed much within two iterations, you may say it's sort of more or less converged.