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Well, we can use the same type of
algorithm to find the minimum of

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a function.

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So, here,
our interest is min over all w g of w.

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And on this picture here, for this convex
function, that's this point right here.

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And, but let's think a little bit
about what happens in this case.

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So let's say we're starting
at some w value here,

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and I'd like to know whether I should
move again to the left or to the right.

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So increase or decrease w?

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Well let's look at
the derivative of the function.

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And what I see is that
the derivative is negative.

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The derivative is negative,
and yet in this case,

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I want to be moving to the right and
increasing w.

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Now, let's look at a point on the other
side of the optimum, so some point w here.

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Look at the derivative.

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In this case the derivative
is positive and

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when I ask whether I want to move
to the left or to the right,

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the answer in this case is,
I want to decrease the value of w.

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I want to move to the left.

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So what we're saying is that,

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when the derivative is

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positive we want to decrease w and

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when the derivative is negative,

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you wanna increase w.

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So again, in this picture I have
that the derivative of this

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function g everywhere on
the left-hand side of the optimum,

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in this case, is negative, everywhere
on the right-hand side is positive.

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So when I go to do what I'm gonna call
a hill descent algorithm to contrast

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with the hill climbing algorithm,

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the update is gonna look almost
exactly the same as the hill climbing.

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Except because of what we just discussed,

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instead of having a plus sign here,
and moving in the same direction,

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meaning the same sign of the derivative,
we're going to move in the opposite.

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Okay, so when the derivative,
just to be very clear,

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when the derivative is positive,
what's going to happen?

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Well this term is going to be negative,
we're going to decrease w.

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When the derivative is negative,
this term,

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this joint term here is going to be
positive, we're going to increase w.

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So that satisfies exactly
what we stated here.

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Okay.

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So that is finding the minimum
of a convex function.

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And I wanna emphasize this slide
right here because we're gonna be

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looking at a lot of convex functions
in this course, and in this module.

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So this is really the picture
that I want you to have in mind.

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