1.1 Machine learning lecture 2  (Page 10/15)

Something that's done maybe slightly more often is look at the value of J of theta, and if J of theta – if the quantity you're trying to minimize is not changing much anymore, then you might be inclined to believe it's converged. So these are sort of standard heuristics, or standard rules of thumb that are often used to decide if gradient descent has converged.

Yeah?

Student: I may have missed something, but especially in [inaudible] descent. So one feature [inaudible]curve and can either go this way or that way. But the math at incline [inaudible] where that comes in. When do you choose whether you go left, whether you're going this way or that way?

Instructor (Andrew Ng) :I see. It just turns out that – so the question is, how is gradient descent looking 360 around you and choosing the direction of steepest descent. So it actually turns out – I'm not sure I'll answer the second part, but it turns out that if you stand on the hill and if you – it turns out that when you compute the gradient of the function, when you compute the derivative of the function, then it just turns out that that is indeed the direction of steepest descent.

By the way, I just want to point out, you would never want to go in the opposite direction because the opposite direction would actually be the direction of steepest ascent, right. So as it turns out – maybe the TAs can talk a bit more about this at the section if there's interest. It turns out, when you take the derivative of a function, the derivative of a function sort of turns out to just give you the direction of steepest descent.

And so you don't explicitly look all 360 degrees around you. You sort of just compute the derivative and that turns out to be the direction of steepest descent. Yeah, maybe the TAs can talk a bit more about this on Friday.

Okay, let's see, so let me go ahead and give this algorithm a specific name. So this algorithm here is actually called batch gradient descent, and the term batch isn't a great term, but the term batch refers to the fact that on every step of gradient descent you're going to look at your entire training set. You're going to perform a sum over your M training examples.

So [inaudible] descent often works very well. I use it very often, and it turns out that sometimes if you have a really, really large training set, imagine that instead of having 47 houses from Portland, Oregon in our training set, if you had, say, the U.S. Census Database or something, with U.S. census size databases you often have hundreds of thousands or millions of training examples.

So if M is a few million then if you're running batch rate and descent, this means that to perform every step of gradient descent you need to perform a sum from J equals one to a million. That's sort of a lot of training examples where your computer programs have to look at, before you can even take one step downhill on the function J of theta.

So it turns out that when you have very large training sets, you should write down an alternative algorithm that is called [inaudible] gradient descent. Sometimes I'll also call it incremental gradient descent, but the algorithm is as follows. Again, it will repeat until convergence and will iterate for J equals one to M, and will perform one of these sort of gradient descent updates using just the J training example.