In the previous video, we talked
about how to use back propagation
to compute the derivatives of your cost function.
In this video, I want
to quickly tell you about one implementational detail of
unrolling your parameters from
matrices into vectors, which we
need in order to use the advanced optimization routines.
Concretely, let's say
you've implemented a cost function
that takes this input, you know, parameters
theta and returns the cost function and returns derivatives.
Then you can pass this to
an advanced authorization algorithm by fminunc
and fminunc
isn't the only one by the way.
There are also other advanced authorization algorithms.
But what all of them
do is take those input
pointedly the cost function,
and some initial value of theta.
And both, and these
routines assume that theta and
the initial value of theta, that
these are parameter vectors, maybe
Rn or Rn plus 1.
But these are vectors and it
also assumes that, you know, your cost
function will return as
a second return value this
gradient which is also Rn
and Rn plus 1. So also a vector.
This worked fine when we
were using logistic progression but
now that we're using a neural
network our parameters are
no longer vectors, but instead
they are these matrices where for
a full neural network we would
have parameter matrices theta 1, theta 2, theta 3
that we might represent in Octave
as these matrices theta 1, theta 2, theta 3.
And similarly these gradient
terms that were expected to return.
Well, in the previous video we
showed how to compute these
gradient matrices, which was
capital D1, capital D2,
capital D3, which we
might represent an octave as matrices D1, D2, D3.
In this video I want
to quickly tell you about the
idea of how to take
these matrices and unroll them into vectors.
So that they end up
being in a format suitable for
passing into as theta here off for getting
out for a gradient there.
Concretely, let's say we
have a neural network with one
input layer with ten units,
hidden layer with ten units
and one output layer with
just one unit, so s1
is the number of units in layer one
and s2 is the
number of units in layer two, and s3 is a number
of units in layer three.
In this case, the dimension of
your matrices theta and
D are going to be
given by these expressions.
For example, theta one
is going to a 10 by 11 matrix and so on.
So in if you want
to convert between these matrices.
vectors.
What you can do is take
your theta 1, theta
2, theta 3, and write this
piece of code and this will
take all the elements of
your three theta matrices and
take all the elements
of theta one, all the
elements of theta 2, all the
elements of theta 3,
and unroll them and put
all the elements into a big long vector.
Which is thetaVec and similarly
the second command would take
all of your D matrices and
unroll them into a big
long vector and call them
DVec.
And finally
if you want to go back from
the vector representations to the matrix representations.
What you do to get back
to theta one say is take
thetaVec and pull
out the first 110 elements.
So theta 1 has 110
elements because it's a
10 by 11 matrix so that
pulls out the first 110 elements
and then you can
use the reshape command to reshape those back into theta 1.
And similarly, to get
back theta 2 you pull
out the next 110 elements and reshape it.
And for theta 3, you pull out
the final eleven elements and run
reshape to get back the theta 3.
Here's a quick Octave demo of that process.
So for this example
let's set theta 1 equal
to be ones of 10 by
11, so it's a matrix of all ones. And
just to make this easier seen,
let's set that to be 2
times ones, 10 by
11 and let's
set theta 3 equals 3
times 1's of 1 by 11.
So this is 3
separate matrices: theta 1, theta 2, theta 3.
We want to put all of these as a vector.
ThetaVec equals theta
1; theta 2
theta 3.
Right, that's a colon
in the middle and like so
and now thetavec is
going to be a very long vector.
That's 231 elements.
If I display it, I find
that this very long vector with
all the elements of the first
matrix, all the elements of
the second matrix, then all the elements of the third matrix.
And if I want to get back
my original matrices, I can
do reshape thetaVec.
Let's pull out the first 110
elements and reshape them to a 10 by 11 matrix.
This gives me back theta 1.
And if I then pull
out the next 110 elements.
So that's indices 111 to 220.
I get back all of my 2's.
And if I go
from 221 up to
the last element, which is
element 231, and reshape to
1 by 11, I get back theta 3.
To make this process really concrete,
here's how we use the unrolling
idea to implement our learning algorithm.
Let's say that you have some
initial value of the parameters
theta 1, theta 2, theta 3.
What we're going to do
is take these and unroll
them into a long vector
we're gonna call initial theta to
pass in to fminunc
as this initial setting of the parameters theta.
The other thing we need to do is implement the cost function.
Here's my implementation of the cost function.
The cost function is going to
give us input, thetaVec,
which is going to be all
of my parameters vectors that in
the form that's been unrolled into a vector.
So the first thing I'm going to
do is I'm going to use
thetaVec and I'm going to use the reshape functions.
So I'll pull out elements from
thetaVec and use reshape
to get back my
original parameter matrices, theta 1, theta 2, theta 3.
So these are going to be matrices that I'm going to get.
So that gives me a
more convenient form in which
to use these matrices so that I
can run forward propagation and
back propagation to compute my
derivatives, and to compute my cost function j of theta.
And finally, I can then
take my derivatives and unroll
them, to keeping the elements
in the same ordering as I did when I unroll my thetas.
But I'm gonna unroll D1, D2,
D3, to get gradientVec
which is now what my cost function can return.
It can return a vector of these derivatives.
So, hopefully, you now have
a good sense of how to
convert back and forth between
the matrix representation of the
parameters versus the vector representation of the parameters.
The advantage of the matrix
representation is that when
your parameters are stored as
matrices it's more convenient when
you're doing forward propagation and
back propagation and it's easier
when your parameters are stored as
matrices to take advantage
of the, sort of, vectorized implementations.
Whereas in contrast the advantage of
the vector representation, when you
have like thetaVec or DVec is that
when you are using the advanced optimization algorithms.
Those algorithms tend to
assume that you have
all of your parameters unrolled into a big long vector.
And so with what we just
went through, hopefully you can now quickly
convert between the two as needed.