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Collaborative Filtering Algorithm
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Do curso por Stanford University
Aprendizagem Automática
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Na lição
Recommender Systems
When you buy a product online, most websites automatically recommend other products that you may like. Recommender systems look at patterns of activities between different users and different products to produce these recommendations. In this module, we introduce recommender algorithms such as the collaborative filtering algorithm and low-rank matrix factorization.
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Andrew NgCo-founder, Coursera; Adjunct Professor, Stanford University; formerly head of Baidu AI Group/Google Brain
In the last couple videos, we
talked about the ideas of
how, first, if you're
given features for movies, you
can use that to learn parameters data for users.
And second, if you're given parameters for the users,
you can use that to learn features for the movies.
In this video we're going
to take those ideas and put
them together to come up
with a collaborative filtering algorithm.
So one of the things we worked
out earlier is that if
you have features for the
movies then you can solve
this minimization problem to find
the parameters theta for your users.
And then we also
worked that out, if you
are given the parameters theta,
you can also use that to
estimate the features x, and
you can do that by solving this minimization problem.
So one thing you
could do is actually go back and forth.
Maybe randomly initialize the parameters
and then solve for theta,
solve for x, solve for theta,
solve for x. But, it
turns out that there is a
more efficient algorithm that doesn't
need to go back and forth
between the x's and the
thetas, but that can solve
for theta and x simultaneously.
And here it is. What we are going to do, is basically take
both of these optimization objectives, and
put them into the same objective.
So I'm going to define the
new optimization objective j, which
is a cost function, that
is a function of my features
x and a function
of my parameters theta.
And, it's basically the two optimization objectives
I had on top, but I put together.
So, in order to
explain this, first, I want to point out that this
term over here, this squared
error term, is the same
as this squared error term and the
summations look a little bit
different, but let's see what the summations are really doing.
The first summation is sum
over all users J and
then sum over all movies rated by that user.
So, this is really summing over all
pairs IJ, that correspond
to a movie that was rated by a user.
Sum over J says, for every
user, the sum of
all the movies rated by that user.
This summation down here, just does things in the opposite order.
This says for every movie
I, sum over all
the users J that have
rated that movie and so, you
know these summations, both of these
are just summations over all pairs
ij for which
r of i J is equal to 1.
It's just something over all the
user movie pairs for which you have a rating.
and so those two terms
up there is just
exactly this first term, and
I've just written the summation here explicitly,
where I'm just saying the sum
of all pairs IJ, such that
RIJ is equal to 1.
So what we're going
to do is define a
combined optimization objective that
we want to minimize in order
to solve simultaneously for x and theta.
And then the other terms in
the optimization objective are this,
which is a regularization in terms of theta.
So that came down here and
the final piece is this
term which is
my optimization objective for
the x's and that became this.
And this optimization objective
j actually has an interesting property
that if you were to hold
the x's constant and just
minimize with respect to the thetas then
you'd be solving exactly this problem,
whereas if you were to do
the opposite, if you were to
hold the thetas constant, and minimize
j only with respect to
the x's, then it becomes equivalent to this.
Because either this term
or this term is constant if
you're minimizing only the respective x's or only respective thetas.
So here's an optimization
objective that puts together my
cost functions in terms of x and in terms of theta.
And in order to
come up with just one
optimization problem, what we're going
to do, is treat this
cost function, as a
function of my features
x and of my user
pro user parameters data and
just minimize this whole thing, as
a function of both the
Xs and a function of the thetas.
And really the only difference
between this and the older
algorithm is that, instead
of going back and forth, previously
we talked about minimizing with respect
to theta then minimizing with respect to x,
whereas minimizing with respect to theta,
minimizing with respect to x and so on.
In this new version instead of
sequentially going between the
2 sets of parameters x and theta,
what we are going to do
is just minimize with respect
to both sets of parameters simultaneously.
Finally one last detail
is that when we're learning the features this way.
Previously we have been using
this convention that
we have a feature x0 equals
one that corresponds to an interceptor.
When we are using this
sort of formalism where we're are actually learning the features,
we are actually going to do away with this convention.
And so the features we are going to learn x, will be in Rn.
Whereas previously we had
features x and Rn + 1 including the intercept term.
By getting rid of x0 we now just have x in Rn.
And so similarly, because the
parameters theta is in
the same dimension, we now
also have theta in RN
because if there's no
x0, then there's no need
parameter theta 0 as well.
And the reason we do away
with this convention is because
we're now learning all the features, right?
So there is no need
to hard code the feature that is always equal to one.
Because if the algorithm really wants
a feature that is always equal
to 1, it can choose to learn one for itself.
So if the algorithm chooses,
it can set the feature X1 equals 1.
So there's no need
to hard code the feature of
001, the algorithm now has
the flexibility to just learn it by itself. So, putting
everything together, here is
our collaborative filtering algorithm.
first we are going to
initialize x and
theta to small random values.
And this is a little bit
like neural network training, where there
we were also initializing all the parameters of a neural network to small random values.
Next we're then going
to minimize the cost function using
great intercepts or one of the advance optimization algorithms.
So, if you take derivatives you
find that the great intercept
like these and so this
term here is the
partial derivative of the cost function,
I'm not going to write that out,
with respect to the feature
value Xik and similarly
this term here is also
a partial derivative value of
the cost function with respect to the parameter
theta that we're minimizing.
And just as a reminder, in
this formula that we no
longer have this X0 equals
1 and so we have
that x is in Rn and theta is a Rn.
In this new formalism, we're regularizing
every one of our perimeters theta, you know, every one of our parameters Xn.
There's no longer the special
case theta zero, which was
regularized differently, or which
was not regularized compared to
the parameters theta 1 down to theta.
So there is now no longer a
theta 0, which is why
in these updates, I did not
break out a special case for k equals 0.
So we then use gradient descent
to minimize the cost function
j with respect to the
features x and with respect to the parameters theta.
And finally, given a
user, if a user
has some parameters, theta, and
if there's a movie with
some sort of learned features x,
we would then predict that that
movie would be given a
star rating by that user
of theta transpose j. Or
just to fill those in,
then we're saying that if user
J has not yet
rated movie I, then
what we do is predict that
user J is going to
rate movie I according to
theta J transpose Xi.
So that's the collaborative
filtering algorithm and if
you implement this algorithm you actually get a pretty
decent algorithm that will simultaneously
learn good features for hopefully
all the movies as well as
learn parameters for all the
users and hopefully give
pretty good predictions for how
different users will rate different movies that they have not yet rated
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