In the previous video, we talked about
a cost function for the neural network.
In this video, let's start to talk about an algorithm,
for trying to minimize the cost function.
In particular, we'll talk about the back propagation algorithm.
Here's the cost function that
we wrote down in the previous video.
What we'd like to do is
try to find parameters theta
to try to minimize j of theta.
In order to use either gradient descent
or one of the advance optimization algorithms.
What we need to do therefore is
to write code that takes
this input the parameters theta
and computes j of theta
and these partial derivative terms.
Remember, that the parameters
in the the neural network of these things,
theta superscript l subscript ij,
that's the real number
and so, these are the partial derivative terms
we need to compute.
In order to compute the
cost function j of theta,
we just use this formula up here
and so, what I want to do
for the most of this video is
focus on talking about
how we can compute these
partial derivative terms.
Let's start by talking about
the case of when we have only
one training example,
so imagine, if you will that our entire
training set comprises only one
training example which is a pair xy.
I'm not going to write x1y1
just write this.
Write a one training example
as xy and let's tap through
the sequence of calculations
we would do with this one training example.
The first thing we do is
we apply forward propagation in
order to compute whether a hypotheses
actually outputs given the input.
Concretely, the called the
a(1) is the activation values
of this first layer that was the input there.
So, I'm going to set that to x
and then we're going to compute
z(2) equals theta(1) a(1)
and a(2) equals g, the sigmoid
activation function applied to z(2)
and this would give us our
activations for the first middle layer.
That is for layer two of the network
and we also add those bias terms.
Next we apply 2 more steps
of this four and propagation
to compute a(3) and a(4)
which is also the upwards
of a hypotheses h of x.
So this is our vectorized implementation of
forward propagation
and it allows us to compute
the activation values
for all of the neurons
in our neural network.
Next, in order to compute
the derivatives, we're going to use
an algorithm called back propagation.
The intuition of the back propagation algorithm
is that for each note
we're going to compute the term
delta superscript l subscript j
that's going to somehow
represent the error
of note j in the layer l.
So, recall that
a superscript l subscript j
that does the activation of
the j of unit in layer l
and so, this delta term
is in some sense
going to capture our error
in the activation of that neural duo.
So, how we might wish the activation
of that note is slightly different.
Concretely, taking the example
neural network that we have
on the right which has four layers.
And so capital L is equal to 4.
For each output unit, we're going to compute this delta term.
So, delta for the j of unit in the fourth layer is equal to
just the activation of that
unit minus what was
the actual value of 0 in our training example.
So, this term here can
also be written h of
x subscript j, right.
So this delta term is just
the difference between when a
hypotheses output and what
was the value of y
in our training set whereas
y subscript j is
the j of element of the
vector value y in our labeled training set.
And by the way, if you
think of delta a and
y as vectors then you can
also take those and come
up with a vectorized implementation of
it, which is just
delta 4 gets set as
a4 minus y. Where
here, each of these delta
4 a4 and y, each of
these is a vector whose
dimension is equal to
the number of output units in our network.
So we've now computed the
era term's delta
4 for our network.
What we do next is compute
the delta terms for the earlier layers in our network.
Here's a formula for computing delta
3 is delta 3 is equal
to theta 3 transpose times delta 4.
And this dot times, this
is the element y's multiplication operation
that we know from MATLAB.
So delta 3 transpose delta
4, that's a vector; g prime
z3 that's also a vector
and so dot times is
in element y's multiplication between these two vectors.
This term g prime of
z3, that formally is actually
the derivative of the activation
function g evaluated at
the input values given by z3.
If you know calculus, you
can try to work it out yourself
and see that you can simplify it to the same answer that I get.
But I'll just tell you pragmatically what that means.
What you do to compute this g
prime, these derivative terms is
just a3 dot times1
minus A3 where A3
is the vector of activations.
1 is the vector of
ones and A3 is
again the activation
the vector of activation values for that layer.
Next you apply a similar
formula to compute delta 2
where again that can be
computed using a similar formula.
Only now it is a2
like so and I
then prove it here but you
can actually, it's possible to
prove it if you know calculus
that this expression is equal
to mathematically, the derivative of
the g function of the activation
function, which I'm denoting
by g prime. And finally,
that's it and there is
no delta1 term, because the
first layer corresponds to the
input layer and that's just the
feature we observed in our
training sets, so that doesn't have any error associated with that.
It's not like, you know,
we don't really want to try to change those values.
And so we have delta
terms only for layers 2, 3 and for this example.
The name back propagation comes from
the fact that we start by
computing the delta term for
the output layer and then
we go back a layer and
compute the delta terms for the
third hidden layer and then we
go back another step to compute
delta 2 and so, we're sort of
back propagating the errors from
the output layer to layer 3
to their to hence the name back complication.
Finally, the derivation is
surprisingly complicated, surprisingly involved but
if you just do this few steps
steps of computation it is possible
to prove viral frankly some
what complicated mathematical proof.
It's possible to prove that if
you ignore authorization then the
partial derivative terms you want
are exactly given by the
activations and these delta terms.
This is ignoring lambda or
alternatively the regularization
term lambda will
equal to 0.
We'll fix this detail later
about the regularization term, but
so by performing back propagation
and computing these delta terms,
you can, you know, pretty
quickly compute these partial
derivative terms for all of your parameters.
So this is a lot of detail.
Let's take everything and put
it all together to talk about
how to implement back propagation
to compute derivatives with respect to your parameters.
And for the case of when
we have a large training
set, not just a training
set of one example, here's what we do.
Suppose we have a training
set of m examples like
that shown here.
The first thing we're going to do is
we're going to set these delta
l subscript i j. So this triangular symbol?
That's actually the capital Greek
alphabet delta .
The symbol we had on the previous slide was the lower case delta.
So the triangle is capital delta.
We're gonna set this equal to zero
for all values of l i j.
Eventually, this capital delta
l i j will be used
to compute the partial
derivative term, partial derivative
respect to theta l i j of
J of theta.
So as we'll see in
a second, these deltas are going
to be used as accumulators that
will slowly add things in
order to compute these partial derivatives.
Next, we're going to loop through our training set.
So, we'll say for i equals
1 through m and so
for the i iteration, we're
going to working with the training example xi, yi.
So
the first thing we're going to do
is set a1 which is the
activations of the input layer,
set that to be equal to
xi is the inputs for our
i training example, and then
we're going to perform forward propagation to
compute the activations for
layer two, layer three and so
on up to the final
layer, layer capital L. Next,
we're going to use the output
label yi from this
specific example we're looking
at to compute the error
term for delta L for the output there.
So delta L is what
a hypotheses output minus what
the target label was?
And then we're going to use
the back propagation algorithm to
compute delta L minus 1,
delta L minus 2, and
so on down to delta 2 and once again
there is now delta 1 because
we don't associate an error term with the input layer.
And finally, we're going to
use these capital delta terms
to accumulate these partial derivative
terms that we wrote down on the previous line.
And by the way, if you
look at this expression, it's possible to vectorize this too.
Concretely, if you think
of delta ij as
a matrix, indexed by subscript ij.
Then, if delta L is
a matrix we can rewrite
this as delta L, gets
updated as delta L plus
lower case delta L plus
one times aL transpose.
So that's a vectorized implementation of
this that automatically does
an update for all values of
i and j.
Finally, after
executing the body of
the four-loop we then go outside the four-loop
and we compute the following.
We compute capital D as
follows and we have
two separate cases for j
equals zero and j not equals zero.
The case of j equals zero
corresponds to the bias
term so when j equals
zero that's why we're missing
is an extra regularization term.
Finally, while the formal proof
is pretty complicated what you
can show is that once
you've computed these D terms,
that is exactly the partial
derivative of the cost
function with respect to each
of your perimeters and so you
can use those in either gradient
descent or in one of the advanced authorization
algorithms.
So that's the back propagation
algorithm and how you compute
derivatives of your cost
function for a neural network.
I know this looks like this
was a lot of details and this was a lot of steps strung together.
But both in the programming
assignments write out and later
in this video, we'll give
you a summary of this so
we can have all the pieces
of the algorithm together so that
you know exactly what you need
to implement if you want
to implement back propagation to compute
the derivatives of your neural network's
cost function with respect to those parameters.