If you run a learning algorithm and it
doesn't do as well as you are hoping,
almost all the time it'll be because you
have either a high finance problem or a
high variance problem. In other words,
either an under fitting problem or an over
fitting problem. And in this case it's
very important to figure out which of
these two problems, is it high or variance
or a bit of both that you actually have
because knowing which of these two things
is happening will give a strong indicator
for whether their useful and how much they
[inaudible] in trying to improve your
algorithm. In this video, I'd like to
delve more deeply into this bias. Various
issues and understand them better. Let's
what take around how to knock the learning
algorithm and evaluate or diagnose whether
we might have a bias problem or variance
problem since this will be critical to
figuring out how to improve the
performance of the learning algorithm that
you may implement. So you've already seen
this figure a few times where if you fit
too simple hypothesis that gives a
straight line that under-fits the data
complex. If you fit a too complex
hypothesis then that might fit the
training set perfectly but over-fit the
data and this may be. Hypothesis of some
intermediate level of complexities of some
two polynomials of not too low and not
high degree that's just right that gives
you the best generalization for these
options. Now that we're arms with the
notion of train, training and validation
the test says we can understand the
concepts that bios and theories a little
bit better. Concretely lets, let our
training error and cross validation error
be defined as in the previous videos just
say the square error the average square
error as mentioned. On the training sets
all has measured on the cross validation
set. Now let's plot the following figure.
On the horizontal axis, I'm going to plot
the degree of polynomial. So as it goes to
the right, I'm going to, I'm going to be
fitting higher and higher order
polynomials. So will the left of this
figure, where maybe D equals one, we're
going to be fitting very simple figures.
Whereas way here on the right of the
horizontal axis, have much larger values
of D. So a much higher degree of
polynomial. And so here that's going to
correspond to fitting. Much more complex
functions to your training set. Let's look
at the training error and the cross
validation error and plot them on this
figure. Let's start with the training
error. As we increase the degree of the
polynomial, we're going to be able to fit
our training set better and better. And
so, if D=1 then it's a relatively high
training error. If we have a very high
degree polynomial our training error is
going to be really low maybe even zero
because we'll fit the training set really
well. And so as we increase the degree of
polynomial, we find typically that the
training error decreases. So, I'm going to
write J. Subscript three of data there.
Because our training error tends to
decrease with the degree of polynomial
that we fit to the data. Next let's look
at the cross validation error. Or for that
matter, if we look at the test set error
we'll get a pretty similar result as if we
were to plot the cross validation error.
So, we know that if D=1. We're fitting a
very simple function. And so we may be
under fitting the training set. And so
we're going to have a very high cross
validation error. If we fit, you know, an
intermediate degree polynomial, there's,
we have a D equals two in our example on
the previous slide, we're going to have a
much lower cross validation error, because
we're just fitting, finding a much better
fit to the data. And conversely, if D were
too high, so if D took on, say, a value of
four, then we're getting over fitting, and
so we ended with a high value for cross
validation error. So, if you were to. Very
[inaudible] and plot the curve. You might
end up with a curve like that. Where,
that's J.C.V. Of staza. Indicating that
you plot J. Tesla's data, you get
something very similar. And so, this sort
of plot also helps us to better understand
the notions of bias and variance.
Concretely, suppose you've applied a
learning algorithm, and it's not
performing as well as you were hoping. So,
so if your cross-validation set error, or
your test set error is high. How can we
figure out if the learning algorithm is
suffering from high bias or if it's suffer
from high variance? So the setting of the
cause validation error being high,
corresponds to either this regime or this
regime. So this regime on the left
corresponds to a high bias problem. That
is, if you're fitting a overly low order
polynomial, such as a D=1. When we really
needed a higher order polynomial to fit
the data. Whereas in contrast, this regime
corresponds to a high variance problem.
That is, if D, the degree of polynomial
was too large for the data set that we
have. And this figure just has a clue for
how to distinguish between these two
cases. Concretely for the high bias case.
That is the case of [inaudible]. What we
find is that both the cross validation
error and the trading error are going to
be high. So if your algorithm is suffering
from a bias problem. The training set
error, will be high. And you might find
that the cross validation error will also
be high. It might be a close. Maybe just
slightly higher than a training error. And
so, if you see this combination that's a
sign your algorithm may be suffering from
high bias. In contrast if your algorithm
is suffering from high variance then if
you look here. We'll notice that J-train
that is the training error is going to be
low. That is your fitting the training set
very well. Where as your, cross validation
error. Assuming that this is say the
squared era. Which we're trying to
minimize [inaudible]. Where as in
contrast, your arrow on the cross
validation set or your cos function in the
cross validation set will be much bigger.
Then your training set error. So, there's
a double greater than sign. That's the
math symbol for much greater than, denoted
by two greater than signs. And so, if you
see this combination of values then that
might give you, that's a clue that your
learning algorithm maybe suffering from
high variance. And might be over
emphasizing. And the key that
distinguishes these two cases is if you
have a high bias problem your training set
error will also be high. Your hypothesis
is just not fitting the training set well.
And if you have a high variance problem.
Your training set error will usually be
low. That is much lower than your cross
allegation error. So hopefully that gives
you a somewhat better understanding of the
two problems of bias and variants. I still
have a lot more to say about bias and
variants in the next few videos. But what
we'll see later is that by diagnosing
whether a learning algorithm may be
suffering from high bias or high variance,
we'll show you even more details of how to
do that in later videos. We'll see that by
figuring out whether a learning algorithm
may be suffering from high bias or high
variance, or a combination of both, that,
that would give us much better guidance
for what might be [inaudible]. Things to
try in order to improve the performance of
a learning algorithm.