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Well, we can use the same type of algorithm to find the minimum of
a function.
So, here, our interest is min over all w g of w.
And on this picture here, for this convex function, that's this point right here.
And, but let's think a little bit about what happens in this case.
So let's say we're starting at some w value here,
and I'd like to know whether I should move again to the left or to the right.
So increase or decrease w?
Well let's look at the derivative of the function.
And what I see is that the derivative is negative.
The derivative is negative, and yet in this case,
I want to be moving to the right and increasing w.
Now, let's look at a point on the other side of the optimum, so some point w here.
Look at the derivative.
In this case the derivative is positive and
when I ask whether I want to move to the left or to the right,
the answer in this case is, I want to decrease the value of w.
I want to move to the left.
So what we're saying is that,
when the derivative is
positive we want to decrease w and
when the derivative is negative,
you wanna increase w.
So again, in this picture I have that the derivative of this
function g everywhere on the left-hand side of the optimum,
in this case, is negative, everywhere on the right-hand side is positive.
So when I go to do what I'm gonna call a hill descent algorithm to contrast
with the hill climbing algorithm,
the update is gonna look almost exactly the same as the hill climbing.
Except because of what we just discussed,
instead of having a plus sign here, and moving in the same direction,
meaning the same sign of the derivative, we're going to move in the opposite.
Okay, so when the derivative, just to be very clear,
when the derivative is positive, what's going to happen?
Well this term is going to be negative, we're going to decrease w.
When the derivative is negative, this term,
this joint term here is going to be positive, we're going to increase w.
So that satisfies exactly what we stated here.
Okay.
So that is finding the minimum of a convex function.
And I wanna emphasize this slide right here because we're gonna be
looking at a lot of convex functions in this course, and in this module.
So this is really the picture that I want you to have in mind.
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