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If you run a learning algorithm and it
doesn't do as well as you are hoping,

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almost all the time it'll be because you
have either a high finance problem or a

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high variance problem. In other words,
either an under fitting problem or an over

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fitting problem. And in this case it's
very important to figure out which of

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these two problems, is it high or variance
or a bit of both that you actually have

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because knowing which of these two things
is happening will give a strong indicator

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for whether their useful and how much they
[inaudible] in trying to improve your

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algorithm. In this video, I'd like to
delve more deeply into this bias. Various

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issues and understand them better. Let's
what take around how to knock the learning

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algorithm and evaluate or diagnose whether
we might have a bias problem or variance

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problem since this will be critical to
figuring out how to improve the

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performance of the learning algorithm that
you may implement. So you've already seen

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this figure a few times where if you fit
too simple hypothesis that gives a

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straight line that under-fits the data
complex. If you fit a too complex

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hypothesis then that might fit the
training set perfectly but over-fit the

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data and this may be. Hypothesis of some
intermediate level of complexities of some

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two polynomials of not too low and not
high degree that's just right that gives

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you the best generalization for these
options. Now that we're arms with the

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notion of train, training and validation
the test says we can understand the

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concepts that bios and theories a little
bit better. Concretely lets, let our

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training error and cross validation error
be defined as in the previous videos just

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say the square error the average square
error as mentioned. On the training sets

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all has measured on the cross validation
set. Now let's plot the following figure.

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On the horizontal axis, I'm going to plot
the degree of polynomial. So as it goes to

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the right, I'm going to, I'm going to be
fitting higher and higher order

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polynomials. So will the left of this
figure, where maybe D equals one, we're

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going to be fitting very simple figures.
Whereas way here on the right of the

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horizontal axis, have much larger values
of D. So a much higher degree of

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polynomial. And so here that's going to
correspond to fitting. Much more complex

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functions to your training set. Let's look
at the training error and the cross

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validation error and plot them on this
figure. Let's start with the training

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error. As we increase the degree of the
polynomial, we're going to be able to fit

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our training set better and better. And
so, if D=1 then it's a relatively high

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training error. If we have a very high
degree polynomial our training error is

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going to be really low maybe even zero
because we'll fit the training set really

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well. And so as we increase the degree of
polynomial, we find typically that the

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training error decreases. So, I'm going to
write J. Subscript three of data there.

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Because our training error tends to
decrease with the degree of polynomial

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that we fit to the data. Next let's look
at the cross validation error. Or for that

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matter, if we look at the test set error
we'll get a pretty similar result as if we

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were to plot the cross validation error.
So, we know that if D=1. We're fitting a

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very simple function. And so we may be
under fitting the training set. And so

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we're going to have a very high cross
validation error. If we fit, you know, an

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intermediate degree polynomial, there's,
we have a D equals two in our example on

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the previous slide, we're going to have a
much lower cross validation error, because

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we're just fitting, finding a much better
fit to the data. And conversely, if D were

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too high, so if D took on, say, a value of
four, then we're getting over fitting, and

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so we ended with a high value for cross
validation error. So, if you were to. Very

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[inaudible] and plot the curve. You might
end up with a curve like that. Where,

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that's J.C.V. Of staza. Indicating that
you plot J. Tesla's data, you get

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something very similar. And so, this sort
of plot also helps us to better understand

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the notions of bias and variance.
Concretely, suppose you've applied a

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learning algorithm, and it's not
performing as well as you were hoping. So,

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so if your cross-validation set error, or
your test set error is high. How can we

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figure out if the learning algorithm is
suffering from high bias or if it's suffer

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from high variance? So the setting of the
cause validation error being high,

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corresponds to either this regime or this
regime. So this regime on the left

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corresponds to a high bias problem. That
is, if you're fitting a overly low order

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polynomial, such as a D=1. When we really
needed a higher order polynomial to fit

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the data. Whereas in contrast, this regime
corresponds to a high variance problem.

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That is, if D, the degree of polynomial
was too large for the data set that we

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have. And this figure just has a clue for
how to distinguish between these two

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cases. Concretely for the high bias case.
That is the case of [inaudible]. What we

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find is that both the cross validation
error and the trading error are going to

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be high. So if your algorithm is suffering
from a bias problem. The training set

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error, will be high. And you might find
that the cross validation error will also

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be high. It might be a close. Maybe just
slightly higher than a training error. And

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so, if you see this combination that's a
sign your algorithm may be suffering from

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high bias. In contrast if your algorithm
is suffering from high variance then if

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you look here. We'll notice that J-train
that is the training error is going to be

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low. That is your fitting the training set
very well. Where as your, cross validation

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error. Assuming that this is say the
squared era. Which we're trying to

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minimize [inaudible]. Where as in
contrast, your arrow on the cross

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validation set or your cos function in the
cross validation set will be much bigger.

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Then your training set error. So, there's
a double greater than sign. That's the

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math symbol for much greater than, denoted
by two greater than signs. And so, if you

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see this combination of values then that
might give you, that's a clue that your

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learning algorithm maybe suffering from
high variance. And might be over

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emphasizing. And the key that
distinguishes these two cases is if you

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have a high bias problem your training set
error will also be high. Your hypothesis

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is just not fitting the training set well.
And if you have a high variance problem.

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Your training set error will usually be
low. That is much lower than your cross

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allegation error. So hopefully that gives
you a somewhat better understanding of the

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two problems of bias and variants. I still
have a lot more to say about bias and

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variants in the next few videos. But what
we'll see later is that by diagnosing

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whether a learning algorithm may be
suffering from high bias or high variance,

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we'll show you even more details of how to
do that in later videos. We'll see that by

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figuring out whether a learning algorithm
may be suffering from high bias or high

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variance, or a combination of both, that,
that would give us much better guidance

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for what might be [inaudible]. Things to
try in order to improve the performance of

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a learning algorithm.