[MUSIC] Well, we've motivated analytically how
the coefficients that we get when solving this ridge regression problem are gonna
change for different settings of lambda. Specifically, we saw that when lambda
was 0, we get our least square solution. When lambda goes to infinity, we get very,
very small coefficients approaching 0. And in between,
we get some other set of coefficients and then we explore this experimentally
in this polynomial regression demo. But one thing that's
interesting to draw is what's called the coefficient path for
ridge regression. Which shows as you vary lambda,
all the way from 0 up towards infinity,
how do the coefficients change? So how does my solution change
as a function of lambda? And what we're doing in this plot
here is we're drawing this for our housing example,
where we have eight different features. Number of bedrooms, bathrooms,
square feet of the living space, number of square feet of the lot size. Number of floors, the year the house was
built, the year the house was renovated, and whether or
not the property is waterfront. And for each one of these different
inputs to our model are different, and these we're just gonna use
as different features, we're drawing what the coefficients,
so this would be, Coefficient value for square feet living. For some specific choice of lambda and
how that coefficient varies as I increase lambda and I'm showing this for each one
of the eight different coefficients. And I just want to briefly mention that in
this figure, we've rescaled the features so that they all have unit norm so
each one of these different inputs. That's why all of these coefficients
are roughly on the same scale. They're roughly the same
order of magnitude. Okay, and so what we see in this plot is,
as lambda goes towards 0, or when it's specifically at 0,
our solution here. The value of each of these coefficients,
so each of these circles touching this line, this is gonna
be my w hat least squares solution. And as I increase lambda
out towards infinity, I see that my solution,
w hat, approaches 0. There's a vector of
coefficients is going to 0. And we haven't made lambda large enough
in this plot to see them actually really, really, really, really close to 0,
but you see the trend happening here. And then there's some sweet spot in
this model, sorry not in this model, in this plot. Which we're gonna talk
about later in this module. Whoops, I should draw it actually
hitting some of these circles. One of these considered points. So this is gonna represent, erase this, this is gonna represent some lambda star. Which will be the value of lambda that
we wanna use when we're selecting our specific regularized model to use for
forming predictions. And we're gonna discuss how we choose
which lambda to use later in the module. But for now, the main point of
this plot is to realize that for every value of lambda,
every slice of this plot, we get a different solution,
a different w hat vector. [MUSIC]