1.1 Machine learning lecture 2  (Page 6/15)

And so what we'll do, let's say, is minimize as a function of the parameters of theta, this quantity J of theta. I should say, to those of you who have taken sort of linear algebra classes, or maybe basic statistics classes, some of you may have seen things like these before and seen least [inaudible] regression or [inaudible]squares.

Many of you will not have seen this before. I think some of you may have seen it before, but either way, regardless of whether you've seen it before, let's keep going. Just for those of you that have seen it before, I should say eventually, we'll actually show that this algorithm is a special case of a much broader class of algorithms. But let's keep going. We'll get there eventually.

So I'm going to talk about a couple of different algorithms for performing that minimization over theta of J of theta. The first algorithm I'm going to talk about is a search algorithm, where the basic idea is we'll start with some value of my parameter vector theta. Maybe initialize my parameter vector theta to be the vector of all zeros, and excuse me, have to correct that. I sort of write zero with an arrow on top to denote the vector of all zeros.

And then I'm going to keep changing my parameter vector theta to reduce J of theta a little bit, until we hopefully end up at the minimum with respect to theta of J of theta. So switch the laptops please, and lower the big screen. So let me go ahead and show you an animation of this first algorithm for minimizing J of theta, which is an algorithm called grading and descent.

So here's the idea. You see on the display a plot and the axes, the horizontal axes are theta zero and theta one. That's usually – minimize J of theta, which is represented by the height of this plot. So the surface represents the function J of theta and the axes of this function, or the inputs of this function are the parameters theta zero and theta one, written down here below.

So here's the gradient descent algorithm. I'm going to choose some initial point. It could be vector of all zeros or some randomly chosen point. Let's say we start from that point denoted by the star, by the cross, and now I want you to imagine that this display actually shows a 3D landscape. Imagine you're all in a hilly park or something, and this is the 3D shape of, like, a hill in some park.

So imagine you're actually standing physically at the position of that star, of that cross, and imagine you can stand on that hill, right, and look all 360 degrees around you and ask, if I were to take a small step, what would allow me to go downhill the most? Okay, just imagine that this is physically a hill and you're standing there, and would look around ask, "If I take a small step, what is the direction of steepest descent, that would take me downhill as quickly as possible?"

So the gradient descent algorithm does exactly that. I'm going to take a small step in this direction of steepest descent, or the direction that the gradient turns out to be. And then you take a small step and you end up at a new point shown there, and it would keep going. You're now at a new point on this hill, and again you're going to look around you, look all 360 degrees around you, and ask, "What is the direction that would take me downhill as quickly as possible?"