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Now that we have an understanding of what
the fitted line is, and how we can use it,

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let's talk about an algorithm, or really
algorithms for searching out the space of

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all possible lines that we might use, and
finding the one that best fits the data.

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So in particular what we're going to
be doing is focusing in on this machine

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learning algorithm, which is this dark
gray square shown in this flow chart.

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Okay, so recall that our cost was to
find us this residual sum of squares,

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and for any given line,
we can compute the cost of that line.

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So, for example,
we showed three different lines and

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three different residual
sum of squares here, but

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our goal was to minimize over all
possible W0 and W1 slopes and

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intercepts, but a question is,
how are we going to do this?

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So that's the key question that
we're looking to address in this

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part of the module.

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Let's formalize this
idea a little bit more.

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So here, what we're showing is
our residual sum of squares and

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what we see is it's a function
of two variables, w0 and w1.

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So we can write it generically,

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let's just write it as some function
g of a variable w0 and a variable w1.

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And what I've done is I've gone ahead and

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plotted the residual sum of squares
versus w zero and w one for

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the data set you guys played within
the first course of this specialization.

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So here along this axis is w zero and
along this axis is w one.

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And then we are plotting here
our residual sum of squares.

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And that is this blue mesh surface here,
that's

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our residual sum of squares for
any given w zero, w one pair.

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And our objective here is to minimize
over all possible combinations of w zero,

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and w one, so mathematically we write
that, great, I wrote right over what I

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wanted to show you guys, so
let me erase what I wrote before, and

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rewrite it so
that it's a little bit more intelligible.

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But, as I was saying, our mathematical
notation for this minimization over

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all possible W0,
W1 is this notation right here.

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Okay, so in terms of the picture
what we want to do is

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over this entire space of w zero and
w one, we want to find the place,

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let me find the place so that this is
a little easier to see on this slide,

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so we want to find
the specific value of W0.

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So, we'll call that W0 hat.

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And W1 hat, that minimize
this residual summit squares.

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So, this is our objective.

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And switching back to our blue color here,
this is an optimization problem,

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where specifically the optimization
objective is to minimize a function,

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in this case, of two parameters,
two different variables.

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[MUSIC]