Now that we have an understanding of what the fitted line is, and how we can use it,
let's talk about an algorithm, or really algorithms for searching out the space of
all possible lines that we might use, and finding the one that best fits the data.
So in particular what we're going to be doing is focusing in on this machine
learning algorithm, which is this dark gray square shown in this flow chart.
Okay, so recall that our cost was to find us this residual sum of squares,
and for any given line, we can compute the cost of that line.
So, for example, we showed three different lines and
three different residual sum of squares here, but
our goal was to minimize over all possible W0 and W1 slopes and
intercepts, but a question is, how are we going to do this?
So that's the key question that we're looking to address in this
part of the module.
Let's formalize this idea a little bit more.
So here, what we're showing is our residual sum of squares and
what we see is it's a function of two variables, w0 and w1.
So we can write it generically,
let's just write it as some function g of a variable w0 and a variable w1.
And what I've done is I've gone ahead and
plotted the residual sum of squares versus w zero and w one for
the data set you guys played within the first course of this specialization.
So here along this axis is w zero and along this axis is w one.
And then we are plotting here our residual sum of squares.
And that is this blue mesh surface here, that's
our residual sum of squares for any given w zero, w one pair.
And our objective here is to minimize over all possible combinations of w zero,
and w one, so mathematically we write that, great, I wrote right over what I
wanted to show you guys, so let me erase what I wrote before, and
rewrite it so that it's a little bit more intelligible.
But, as I was saying, our mathematical notation for this minimization over
all possible W0, W1 is this notation right here.
Okay, so in terms of the picture what we want to do is
over this entire space of w zero and w one, we want to find the place,
let me find the place so that this is a little easier to see on this slide,
so we want to find the specific value of W0.
So, we'll call that W0 hat.
And W1 hat, that minimize this residual summit squares.
So, this is our objective.
And switching back to our blue color here, this is an optimization problem,
where specifically the optimization objective is to minimize a function,
in this case, of two parameters, two different variables.
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