ANNOUNCER: The following program is brought to you by Caltech.
YASER ABU-MOSTAFA: Welcome back.
Last time, we introduced the learning problem.
And if you have an application in your domain that you wonder if machine
learning is the right technique for it, we found that there are three
criteria that you should check.
You should ask yourself: is there a pattern to begin
with that we can learn?
And we realize that this condition can be intuitively met in many applications,
even if we don't know mathematically what the pattern is.
The example we gave was the credit card approval.
There is clearly a pattern-- if someone has a particular salary, has
been in a residence for so long, has that much debt, and so on, that this
is somewhat correlated to their credit behavior.
And therefore, we know that the pattern exists in spite of the fact
that we don't know exactly what the pattern is.
The second item is that we cannot pin down the pattern mathematically, like
the example I just gave.
And this is why we resort to machine learning.
The third one is that we have data that represents that pattern.
In the case of the credit application, for example, there are historical
records of previous customers, and we have the data they wrote in their
application when they applied, and we have some years' worth of record of
their credit behavior.
So we have data that are going to enable us to correlate what they wrote in the
application to their eventual credit behavior, and that is what we are
going to learn from.
Now, if you look at the three criteria, basically there are two that
you can do without, and one that is absolutely essential.
What do I mean?
Let's say that you don't have a pattern.
Well, if you don't have a pattern, then you can try learning.
And the only problem is that you will fail.
That doesn't sound very encouraging.
But the idea here is that, when we develop the theory of learning, we will
realize that you can apply the technique regardless of whether there
is a pattern or not.
And you are going to determine whether there's a pattern or not.
So you are not going to be fooled and think, I learned, and then give the
system to your customer, and the customer will be disappointed.
There is something you can actually measure that will tell you whether you
learned or not.
So if there's no pattern, there is no harm done in trying machine learning.
The other one, also, you can do without.
Let's say that we can pin the thing down mathematically.
Well, in that case, machine learning is not the recommended technique.
It will still work.
It may not be the optimal technique.
If you can outright program it, and find the result perfectly, then why
bother generate examples, and try to learn, and go through all of that?
But machine learning is not going to refuse.
It is going to learn, and it is going to give you a system.
It may not be the best system in this case, but it's a system nonetheless.
The third one, I'm afraid you cannot do without.
You have to have data.
Machine learning is about learning from data.
And if you don't have data, there is absolutely nothing you can do.
So this is basically the picture about the context of machine learning.
Now, we went on to focus on one type, which is supervised learning.
And in the case of supervised learning, we have a target function.
The target function we are going to call f.
That is our standard notation.
And this corresponds, for example, to the credit application.
x is your application, and f of x is whether you are a good credit risk or
not, for the bank.
So if you look at the target function, the main criterion about the target
function is that it's unknown.
This is a property that we are going to insist on.
And obviously, unknown is a very generous assumption, which means that
you don't have to worry about what pattern you are trying to learn.
It could be anything, and you will learn it-- if we manage to do that.
There's still a question mark about that.
But it's a good assumption to have, or lack of assumption, if you will,
because then you know that you don't worry about the environment that
generated the examples.
You only worry about the system that you use to implement machine learning.
Now, you are going to be given data.
And the reason it's called supervised learning is that you are not only
given the input x's, as you can see here.
You're also given the output--
the target outputs.
So in spite of the fact that the target function is generally unknown,
it is known on the data that I give you.
This is the data that you are going to use as training examples, and that you
are going to use to figure out what the target function is.
So in the case of supervised learning, you have the targets
explicitly.
In the other cases, you have less information than the target, and we
talked about it-- like unsupervised learning, where you don't have
anything, and reinforcement learning, where you have partial information,
which is just a reward or punishment for a choice of a value of y that
may or may not be the target.
Finally, you have the solution tools.
These are the things that you're going to choose in order to solve the
problem, and they are called the learning model, as we discussed.
They are the learning algorithm and the hypothesis set.
And the learning algorithm will produce a hypothesis--
the final hypothesis, the one that you are going to give your customer, and
we give the symbol g for that.
And hopefully g approximates f, the actual target function,
which remains unknown.
And g is picked from a hypothesis set, and the general the symbol for
a member of the hypothesis set is h.
So h is a generic hypothesis.
The one you happen to pick, you are going to call g.
Now, we looked at an example of a learning algorithm.
First, the learning model-- the perceptron itself, which is a linear
function, thresholded.
That happens to be the hypothesis set.
And then, there is an algorithm that goes with it that chooses which
hypothesis to report based on the data.
And the hypothesis in this case is represented by the purple line.
Different hypotheses in the set H will result
in different lines.
Some of them are good and some of them are bad, in terms of separating
correctly the examples which are the pluses and minuses.
And we found that there's a very simple rule to adjust the current
hypothesis, while the algorithm is still running, in order to get a better
hypothesis.
And once you have all the points classified correctly, which is
guaranteed in the case of the perceptron learning algorithm if the
data was linearly separable in the first place,
then you will get there, and that will be the g that you are going to report.
Now, we ended the lecture on sort of a sad note, because after all of this
encouragement about learning, we asked ourselves: well,
can we actually learn?
So we said it's an unknown function.
Unknown function is an attractive assumption, as I said.
But can we learn an unknown function, really?
And then we realized that if you look at it, it's really impossible.
Why is it impossible?
Because I'm going to give you a finite data set, and I'm going to give you
the value of the function on this set.
Good.
Now, I'm going to ask you what is the function outside that set?
How in the world are you going to tell what the function is outside, if the
function is genuinely unknown?
Couldn't it assume any value it wants?
Yes, it can.
I can give you 1000 points, a million points, and on the million-and-first point,
still the function can behave any way it wants.
So it doesn't look like the statement we made is feasible in terms of
learning, and therefore we have to do something about it.
And what we are going to do about it is the subject of this lecture.
Now, the lecture is called Is Learning Feasible?
And I am going to address this question in extreme detail from
beginning to end.
This is the only topic for this lecture.
Now, if you want an outline--
it's really a logical flow.
But if you want to cluster it into points--
we are going to start with a probabilistic situation, that is a very
simple probabilistic situation.
It doesn't seem to relate to learning.
But it will capture the idea--
can we say something outside the sample data that we have?
So we're going to answer it in a way that is concrete, and where the
mathematics is very friendly.
And then after that, I'm going to be able to relate that probabilistic
situation to learning as we stated.
It will take two stages.
First, I will just translate the expressions into something that
relates to learning, and then we will move forward and make it correspond to
real learning.
That's the last one.
And then after we do that, and we think we are done, we find that there is
a serious dilemma that we have.
And we will find a solution to that dilemma, and then declare victory-- that
indeed, learning is feasible in a very particular sense.
So let's start with the experiment that I talked about.
Consider the following situation.
You have a bin, and the bin has marbles.
The marbles are either red or green.
That's what it looks like.
And we are going to do an experiment with this bin.
And the experiment is to pick a sample from the bin--
some marbles.
Let's formalize what the probability distribution is.
There is a probability of picking a red marble, and let's call it mu.
So now you think of mu as the probability of a red marble.
Now, the bin is really just a visual aid to make us relate to the
experiment.
You can think of this abstractly as a binary experiment--
two outcomes, red or green.
Probability of red is mu, independently from
one point to another.
If you want to stick to the bin, you can say the bin has an infinite number
of marbles and the fraction of red marbles is mu.
Or maybe it has a finite number of marbles, and you are going to pick the
marbles, but replace them.
But the idea now is that every time you reach in the bin, the probability
of picking a red marble is mu.
That's the rule.
Now, there's a probability of picking a green marble.
And what might that be?
That must be 1 minus mu.
So that's the setup.
Now, the value of mu is unknown to us.
So in spite of the fact that you can look at this particular bin and see
there's less red than green, so mu must be small.
and all of that.
You don't have that advantage in real.
The bin is opaque-- it's sitting there, and I reach for it like this.
So now that I declare mu is unknown, you probably see
where this is going.
Unknown is a famous word from last lecture, and that will be the link to
what we have.
Now, we pick N marbles independently.
Capital N. And I'm using the same notation for N, which is the
number of data points in learning, deliberately.
So the sample will look like this.
And it will have some red and some green.
It's a probabilistic situation.
And we are going to call the fraction of marbles in the sample--
this now is a probabilistic quantity.
mu is an unknown constant sitting there.
If you pick a sample, someone else picks a sample, you will have a different
frequency in sample from the other person.
And we are going to call it nu.
Now, interestingly enough, nu also should appear in the figure.
So it says nu equals fraction of red marbles.
So that's where it lies.
Here is nu!
For some reason that I don't understand, the app wouldn't show nu
in the figures.
So I decided maybe the app is actually a machine learning expert.
It doesn't like things in sample.
It only likes things that are real.
So it knows that nu is not important.
It's not an indication.
We are really interested in knowing what's outside.
So it kept the mu, but actually deleted the nu.
At least that's what we are going to believe for the rest of the lecture.
Now, this is the bin.
So now, the next step is to ask ourselves the question we asked in
machine learning.
Does nu, which is the sample frequency, tell us anything about mu,
which is the actual frequency in the bin that we are interested in knowing?
The short answer--
this is to remind you what it is.
The short answer is no.
Why?
Because the sample can be mostly green, while the bin is mostly red.
Anybody doubts that?
The thing could have 90% red, and I pick 100 marbles, and all
of them happen to be green.
This is possible, correct?
So if I ask you what is actually mu, you really don't know from the sample.
You don't know anything about the marbles you did not pick.
Well, that's the short answer.
The long answer is yes.
Not because no and yes, but this is more elaborate.
We have to really discuss a lot in order to get there.
So why is it yes?
Because if you know a little bit about probability, you realize that if the
sample is big enough, the sample frequency, which is nu-- the mysterious
disappearing quantity here-- that is likely to be close to mu.
Think of a presidential poll.
There are maybe 100 million or more voters in the US, and you make a poll
of 3000 people.
You have 3000 marbles, so to speak.
And you look at the result in the marbles, and you tell me how the 100
million will vote.
How the heck did you know that?
So now the statistics come in.
That's where the probability plays a role.
And the main distinction between the two answers is
possible versus probable.
In science and in engineering, you go a huge distance by settling for not
absolutely certain, but almost certain.
It opens a world of possibilities, and this is one of the
possibilities that it opens.
So now we know that, from a probabilistic point of view, nu does
tell me something about mu.
The sample frequency tells me something about the bin.
So what does it exactly say?
Now we go into a mathematical formulation.
In words, it says: in a big sample, nu, the sample frequency,
should be close to mu, the bin frequency.
So now, the symbols that go with that-- what is a big sample?
Large N, our parameter N.
And how do we say that nu is close to mu?
We say that they are within epsilon.
That is our criterion.
Now, with this in mind, we are going to formalize this.
The formula that I'm going to show you is a formula that is going to
stay with us for the rest of the course.
I would like you to pay attention.
And I'm going to build it gradually.
We are going to say that the probability of something is small.
So we're going to say that it's less than or equal to, and hopefully the
right-hand side will be a small quantity.
Now if I am claiming that the probability of something is small, it
must be that that thing is a bad event.
I don't want it to happen.
So we have a probability of something bad happening being small.
What is a bad event in the context we are talking about?
It is that nu does not approximate mu well.
They are not within epsilon of each other.
And if you look at it, here you have nu minus mu in absolute value, so
that's the difference in absolute value.
That happens to be bigger than epsilon.
So that's bad, because that tells us that they are further away from our
tolerance epsilon.
We don't want that to happen.
And we would like the probability of that happening to
be as small as possible.
Well, how small can we guarantee it?
Good news.
It's e to the minus N.
It's a negative exponential.
That is great, because negative exponentials tend to die very fast.
So if you get a bigger sample, this will be diminishingly small
probability.
So the probability of something bad happening will be very small, and we
can claims that, indeed, nu will be within epsilon from mu, and we will be
wrong for a very minute amount of the time.
But that's the good news.
Now the bad news--
ouch!
Epsilon is our tolerance.
If you're a very tolerant person, you say:
I just want nu and mu to be within, let's say, 0.1.
That's not very much to ask.
Now, the price you pay for that is that you plug in the exponent
not epsilon, but epsilon squared.
So that becomes 0.01.
0.01 will dampen N significantly, and you lose a lot of the benefit of the
negative exponential.
And if you are more stringent and you say, I really want nu
to be close to mu.
I am not fooling around here.
So I am going to pick epsilon to be 10 to the minus 6.
Good for you.
10 to the minus 6?
Pay the price for it.
You go here, and now that's 10 to the minus 12.
That will completely kill any N you will ever encounter.
So the exponent now will be around zero.
So this probability will be around 1, if that was the final answer.
That's not yet the final answer.
So now, you know that the probability is less than or equal to 1.
Congratulations!
You knew that already. [LAUGHTER]
Well, this is almost the formula, but it's not quite.
What we need is fairly trivial.
We just put 2 here, and 2 there.
Now, between you and me, I prefer the original formula
better, without the 2's.
However, the formula with the 2's has the distinct advantage of being: true. [LAUGHTER]
So we have to settle for that.
Now that inequality is called Hoeffding's Inequality.
It is the main inequality we are going to be using in the course.
You can look for the proof.
It's a basic proof in mathematics.
It's not that difficult, but definitely not trivial.
And we are going to use it all the way-- and this is the same formula
that will get us to prove something about the VC dimension.
If the buzzword 'VC dimension' means anything to you, it will come from
this after a lot of derivation.
So this is the building block that you have to really know cold.
Now, if you want to translate the Hoeffding Inequality into words, what
we have been talking about is that we would like to make the
statement: mu equals nu.
That would be the ultimate.
I look at the in-sample frequency, that's the out-of-sample frequency.
That's the real frequency out there.
But that's not the case.
We actually are making the statement mu equals nu, but we're not
making the statement--
we are making a PAC statement.
And that stands for: this statement is probably, approximately, correct.
Probably because of this.
This is small, so the probability of violation is small.
Approximately because of this.
We are not saying that mu equals nu.
We are saying that they are close to each other.
And that theme will remain with us in learning.
So we put the glorified Hoeffding's Inequality at the top, and we spend
a viewgraph analyzing what it means.
In case you forgot what nu and mu are, I put the figure.
So mu is the frequency within the bin.
This is the unknown quantity that we want to tell.
And nu is the disappearing quantity which happens to be the frequency in
the sample you have.
So what about the Hoeffding Inequality?
Well, one attraction of this inequality is that it is valid for
every N, positive integer, and every epsilon which is greater than zero.
Pick any tolerance you want, and for any number of examples you
want, this is true.
It's not an asymptotic result.
It's a result that holds for every N and epsilon.
That's a very attractive proposition for something that has
an exponential in it.
Now, Hoeffding Inequality belongs to a large class of mathematical laws,
which are called the Laws of Large Numbers.
So this is one law of large numbers, one form of it, and
there are tons of them.
This happens to be one of the friendliest, because it's not
asymptotic, and happens to have an exponential in it.
Now, one observation here is that if you look at the left-hand side, we are
computing this probability.
This probability patently depends on mu.
mu appears explicitly in it, and also mu affects the probability
distribution of nu.
Nu is the sample, in N marbles you picked.
That's a very simple binomial distribution.
You can find the probability that nu equals anything based on
the value of mu.
So the probability that this quantity, which depends on mu, exceeds epsilon--
the probability itself does depend on mu.
However, we are not interested in the exact probability.
We just want to bound it.
And in this case, we are bounding it uniformly.
As you see, the right-hand side does not have mu in it.
And that gives us a great tool, because now we don't use the quantity
that, we already declared, is unknown.
mu is unknown.
It would be a vicious cycle if I go and say that it depends on mu,
but I don't know what mu is.
Now you know uniformly, regardless of the value of mu-- mu could be anything
between 0 and 1, and this will still be bounding the deviation of the
sample frequency from the real frequency.
That's a good advantage.
Now, the other point is that there is a trade-off that you can read off the
inequality.
What is the trade-off?
The trade-off is between N and epsilon.
In a typical situation, if we think of N as the number of examples that are
given to you-- the amount of data-- in this case, the number of marbles out
of the bin,
N is usually dictated.
Someone comes and gives you a certain resource of examples.
Epsilon is your taste in tolerance.
You are very tolerant. You pick epsilon equals 0.5.
That will be very easy to satisfy.
And if you are very stringent, you can pick epsilon smaller and smaller.
Now, because they get multiplied here, the smaller the epsilon is, the bigger
than N you need in order to compensate for it and come up with the same level
of probability bound.
And that makes a lot of sense.
If you have more examples, you are more sure that nu and mu will be close
together, even closer and closer and closer,
as you get larger N.
So this makes sense.
Finally,
it's a subtle point, but it's worth saying.
We are making the statement that nu is approximately the same as mu.
And this implies that mu is approximately the same as nu.
What is this?
The logic here is a little bit subtle.
Obviously, the statement is a tautology, but I'm just making
a logical point, here.
When you run the experiment, you don't know what mu is.
mu is an unknown.
It's a constant.
The only random fellow in this entire operation is nu.
That is what the probability is with respect to.
You generate different samples, and you compute the probability.
This is the probabilistic thing.
This is a happy constant sitting there, albeit unknown.
Now, the way you are using the inequality is to infer mu, the sample
here, from nu.
That is not the cause and effect that actually takes place.
The cause and effect is that mu affects nu, not the other way around.
But we are using it the other way around.
Lucky for us, the form of the probability is symmetric.
Therefore, instead of saying that nu tends to be close to mu, which will
be the accurate logical statement-- mu is there, and nu has a tendency to be
close to it.
We, instead of that, say that I know already nu, and now mu tends to
be close to nu.
That's the logic we are using.
Now, I think we understand what the bin situation is, and we know what the
mathematical condition that corresponds to it is.
What I'd like to do,
I'd like to connect that to the learning problem we have.
In the case of a bin, the unknown quantity that we want to decipher is
a number, mu.
Just unknown.
What is the frequency inside the bin.
In the learning situation that we had, the unknown quantity we would like to
decipher is a full-fledged function.
It has a domain, X, that could be a 10th-order Euclidean space.
Y could be anything.
It could be binary, like the perceptron.
It could be something else.
That's a huge amount of information.
The bin has only one number.
This one, if you want to specify it, that's a lot of specification.
So how am I going to be able to relate the learning problem to something that
simplistic?
The way we are going to do it is the following.
Think of the bin as your input space in the learning problem.
That's the correspondence.
So every marble here is a point x.
That is a credit card applicant.
So if you look closely at the gray thing, you will read: salary, years in
residence, and whatnot.
You can't see it here because it's too small!
Now the bin has all the points in the space. Therefore, this
is really the space.
That's the correspondence in our mind.
Now we would like to give colors to the marbles.
So here are the colors.
There are green marbles, and they correspond to something in the
learning problem.
What do they correspond to?
They correspond to your hypothesis getting it right.
So what does that mean?
There is a target function sitting there, right?
You have a hypothesis.
The hypothesis is a full function, like the target function is.
You can compare the hypothesis to the target function on every point.
And they either agree or disagree.
If they agree, please color the corresponding point
in the input space--
Color it green.
Now, I'm not saying that you know which ones are green and which ones
are not, because you don't know the target function overall.
I'm just telling you the mapping that takes an unknown target function into
an unknown mu.
So both of them are unknown, admittedly, but that's the
correspondence that maps it.
And now you go, and there are some red ones.
And, you guessed it.
You color the thing red if your hypothesis got the answer wrong.
So now I am collapsing the entire thing into just agreement and
disagreement between your hypothesis and the target function, and that's
how you get to color the bin.
Because of that, you have a mapping for every point, whether it's green or
red, according to this rule.
Now, this will add a component to the learning problem that we
did not have before.
There is a probability associated with the bin.
There is a probability of picking a marble, and
independently, and all of that.
When we talked about the learning problem, there was no probability.
I will just give you a sample set, and that's what you work with.
So let's see what is the addition we need to do in order to adjust the
statement of the learning problem to accommodate the new ingredient.
And the new ingredient is important, because otherwise we cannot learn.
It's not like we have the luxury of doing without it.
So we go back to the learning diagram from last time.
Do you remember this one?
Let me remind you.
Here is your target function, and it's unknown.
And I promised you last time that it will remain unknown, and the promise
will be fulfilled.
We are not going to touch this box.
We're just going to add another box to accommodate the probability.
And the target function generates the training examples.
These are the only things that the learning algorithm sees.
It picks a hypothesis from the hypothesis set, and produces it as the
final hypothesis, which hopefully approximates f.
That's the game.
So what is the addition we are going to do?
In the bin analogy, this is the input space.
Now the input space has a probability.
So I need to apply this probability to the points from the input space that
are being generated.
I am going to introduce a probability distribution over the
input space.
Now the points in the input space-- let's say the d-dimensional
Euclidean space--
are not just generic points now.
There is a probability of picking one point versus the other.
And that is captured by the probability, which I'm going to call
capital P.
Now the interesting thing is that I'm making no assumptions about P. P can
be anything.
I just want a probability.
So invoke any probability you want, and I am ready with the machinery.
I am not going to restrict the probability distributions over X.
That's number one.
So this is not as bad as it looks.
Number two, I don't even need to know what P is.
Of course, the probability choice will affect the choice of the probability
of getting a green marble or a red marble, because now the probability of
different marbles changed, so it could change the value mu.
But the good news with the Hoeffding is that I could bound the performance
independently of mu.
So I can get away with not only any P, but with a P that I don't know, and
I'll still be able to make the mathematical statement.
So this is a very benign addition to the problem.
And it will give us very high dividends, which is the
feasibility of learning.
So what do you do with the probability?
You use the probability to generate the points x_1 up to x_N. So now
x_1 up to x_N are assumed to be generated by that probability
independently.
That's the only assumption that is made.
If you make that assumption, we are in business.
But the good news is, as I mentioned before,
we did not compromise about the target function.
You don't need to make assumptions about the function you don't know and
you want to learn, which is good news.
And the addition is almost technical.
That there is a probability somewhere, generating the points.
If I know that, then I can make a statement in probability.
Obviously, you can make that statement only to the extent that the assumption
is valid, and we can discuss that in later lectures when the
assumption is not valid.
So, OK.
Happy ending.
We are done, and we now have the correspondence.
Are we done?
Well, not quite.
Why are we not done?
Because the analogy I gave you requires a particular
hypothesis in mind.
I told you that the red and green marbles correspond to the agreement between h
and the target function.
So when you tell me what h is, you dictate the colors here.
All of these colors.
This is green not because it's inherently green, not because of
anything inherent about the target function.
It's because of the agreement between the target function and your
hypothesis, h.
That's fine, but what is the problem?
The problem is that I know that for this h, nu generalizes to mu.
You're probably saying, yeah, but h could be anything.
I don't see the problem yet.
Now here is the problem.
What we have actually discussed is not learning, it's verification.
The situation as I describe it--
you have a single bin and you have red and green marbles, and this and that,
corresponds to the following.
A bank comes to my office.
We would like a formula for credit approval.
And we have data.
So instead of actually taking the data, and searching hypotheses, and picking
one, like the perceptron learning algorithm, here is what I do that
corresponds to what I just described.
You guys want a linear formula?
OK.
I guess the salary should have a big weight.
Let's say 2.
The outstanding debt is negative, so that should be a weight minus 0.5.
And years in residence are important, but not that important.
So let's give them a 0.1.
And let's pick a threshold that is high, in order for
you not to lose money.
Let's pick a threshold of 0.5.
Sitting down, improvising an h.
Now, after I fix the h, I ask you for the data and just verify whether the h
I picked is good or bad.
That I can do with the bin, because I'm going to look at the data.
If I miraculously agree with everything in your data, I can
definitely declare victory by Hoeffding.
But what are the chances that this will happen in the first place?
I have no control over whether I will be good on the data or not.
The whole idea of learning is that I'm searching the space to deliberately
find a hypothesis that works well on the data.
In this case, I just dictated a hypothesis.
And I was able to tell you for sure what happens out-of-sample.
But I have no control of what news I'm going to tell you.
You can come to my office.
I improvise this.
I go to the data.
And I tell you, I have a fantastic system.
It generalizes perfectly, and it does a terrible job.
That's what I have, because when I tested it, nu was terrible.
So that's not what we are looking for.
What we are looking for is to make it learning.
So how do we do that?
No guarantee that nu will be small.
And we need to choose the hypothesis from multiple h's.
That's the game.
And in that case, you are going to go for the sample, so to speak, generated
by every hypothesis, and then you pick the hypothesis that is most favorable,
that gives you the least error.
So now, that doesn't look like a difficult thing.
It worked with one bin.
Maybe I can have more than one bin, to accommodate the situation where I have
more than one hypothesis.
It looks plausible.
So let's do that.
We will just take multiple bins.
So here is the first bin.
Now you can see that this is a bad bin.
So that hypothesis is terrible.
And the sample reflects that, to some extent.
But we are going to have other bins, so let's call this something.
So this bin corresponds to a particular h.
And since we are going to have other hypotheses, we are going to call this
h_1 in preparation for the next guy.
The next guy comes in, and you have h_2.
And you have another mu_2.
This one looks like a good hypothesis, and it's also reflected in the sample.
And it's important to look at the correspondence.
If you look at the top red point here and the top green point here, this is
the same point in the input space.
It just was colored red here and colored green here.
Why did that happen?
Because the target function disagrees with this h, and the target function
happens to agree with this h.
That's what got this the color green.
And when you pick a sample, the sample also will have different colors,
because the colors depend on which hypothesis.
And these are different hypotheses.
That looks simple enough.
So let's continue.
And we can have M of them.
I am going to consider a finite number of hypotheses, just to make the math
easy for this lecture.
And we're going to go more sophisticated when we get into the
theory of generalization.
So now I have this.
This is good.
I have samples, and the samples here are different.
And I can do the learning, and the learning now, abstractly, is to scan
these samples looking for a good sample.
And when you find a good sample, you declare victory, because of Hoeffding,
and you say that it must be that the corresponding bin is good, and the
corresponding bin happens to be the hypothesis you chose.
So that is an abstraction of learning.
That was easy enough.
Now, because this is going to stay with us, I am now going to introduce
the notation that will survive with us for the entire discussion of learning.
So here is the notation.
We realize that both mu, which happens to be inside the bin,
and nu, which happens to be the sample frequency--
in this case, the sample frequency of error-- both of them depend on h.
So I'd like to give a notation that makes that explicit.
The first thing,
I am going to call mu and nu with a descriptive name.
So nu, which is the frequency in the sample you have, is in-sample.
That is a standard definition for what happens in the data that I give you.
If you perform well in-sample, it means that your error in the sample
that I give you is small.
And because it is called in-sample, we are going to denote it by E_in.
I think this is worth blowing up, because it's an important one.
This is our standard notation for the error that you have in-sample.
Now, we go and get the other one, which happens to be mu.
And that is called out-of-sample.
So if you are in this field, I guess what matters is the out-of-sample
performance.
That's the lesson.
Out-of-sample means something that you haven't seen.
And if you perform out-of-sample, on something that you haven't seen, then
you must have really learned.
That's the standard for it, and the name for it is E_out.
With this in mind, we realize that we don't yet have the dependency on h
which we need.
So we are going to make the notation a little bit more elaborate, by calling
E_in and E_out--
calling them E_in of h, and E_out of h.
Why is that?
Well, the in-sample performance-- you are trying to see the error of
approximating the target function by your hypothesis.
That's what E_in is.
So obviously, it depends on your hypothesis.
So it's E_in of h.
Someone else picks another h, they will get another E_in of their h.
Similarly E_out, the corresponding one is E_out of h.
So now, what used to be nu is now E_in of h.
What used to be mu, inside the bin, is E_out of h.
Now, the Hoeffding Inequality, which we know all too well
by now, said that.
So all I'm going to do is just replace the notation.
And now it looks a little bit more crowded, but it's
exactly the same thing.
The probability that your in-sample performance deviates from your out-of-
sample performance by more than your prescribed tolerance is less than or
equal to a number that is hopefully small.
And you can go back and forth.
There's nu and mu, or you can go here and you get the new notation.
So we're settled on the notation now.
Now, let's go for the multiple bins and use this notation.
These are the multiple bins as we left them.
We have the hypotheses h_1 up to h_M, and we have the mu_1 and mu_M.
And if you see 1, 2, M, again, this is a disappearing nu--
the symbol that the app doesn't like.
But thank God we switched notations, so that
something will appear.
Yeah!
So right now, that's what we have.
Every bin has an out-of-sample performance, and out-of-
sample is: Out. Of. Sample.
So this is a sample.
What's in it is in-sample.
What is not in it is out-of-sample.
And the out-of-sample depends on h_1 here, h_2 here, and h_M here.
And obviously, these quantities will be different according to the sample, and
these quantities will be different according to the ultimate performance
of your hypothesis.
So we solved the problem.
It's not verification. It's not a single bin.
It's real learning.
I'm going to scan these.
So that's pretty good.
Are we done already?
Not so fast.
[LAUGHING]
What's wrong?
Let me tell you what's wrong.
The Hoeffding Inequality, that we have happily studied and declared important
and all of that, doesn't apply to multiple bins.
What?
You told us mathematics, and you go read the proof, and all of that.
Are you just pulling tricks on us?
What is the deal here?
And you even can complain.
We sat for 40 minutes now going from a single bin, mapping it to
the learning diagram, mapping it to multiple bins, and now you tell us
that the main tool we developed doesn't apply.
Why doesn't it apply, and what can we do about it?
Let me start by saying why it doesn't apply, and then we can go for what we
can do about it.
Now, everybody has a coin.
I hope the online audience have a coin ready.
I'd like to ask you to take the coin out and flip it,
let's say, five times.
And record what happens.
And when you at home flip the coin five times, please,
if you happen to get all five heads in your experiment, then text us that you
got all five heads.
If you get anything else, don't bother text us.
We just want to know if someone will get five heads.
Everybody is done flipping the coin.
Because you have been so generous and cooperative, you can keep the coin!
[LAUGHTER]
Now, did anybody get five heads?
All five heads?
Congratulations, sir.
You have a biased coin, right?
We just argued that in-sample corresponds to out-of-sample, and we
have this Hoeffding thing, and therefore if you get five heads, it
must be that this coin gives you heads.
We know better.
So in the online audience, what happened?
MODERATOR: Yeah, in the online audience, there's also five heads.
PROFESSOR: There are lots of biased coins out there.
Are they really biased coins?
No.
What is the deal here?
Let's look at it.
Here, with the audience here, I didn't want to push my luck with 10 flips,
because it's a live broadcast.
So I said five will work.
For the analytical example, let's take 10 flips.
Let's say you have a fair coin, which every coin is.
You have a fair coin.
And you toss it 10 times.
What is the probability that you will get all 10 heads?
Pretty easy.
One half, times one half, 10 times, and that will give
you about 1 in 1000.
No chance that you will get it--
not no chance, but very little chance.
Now, the second question is the one we actually ran the experiment for.
If you toss 1000 fair coins-- it wasn't 1000 here. It's how many there.
Maybe out there is 1000.
What is the probability that some coin will give you all 10 heads?
Not difficult at all to compute.
And when you get the answer, the answer will be it's actually more
likely than not.
So now it means that the 10 heads in this case are no indication at all of
the real probability.
That is the game we are playing.
Can I look at the sample and infer something about the real probability?
No.
In this case, you will get 10 heads, and the coin is fair.
Why did this happen?
This happened because you tried too hard.
Eventually what will happen is--
Hoeffding applies to any one of them.
But there is a probability, let's say half a percent, that you
will be off here.
Another half a percent that you will be off here.
If you do it often enough, and you are lucky enough that the half percents
are disjoint, you will end up with extremely high probability that
something bad will happen, somewhere.
That's the key.
So let's translate this into the learning situation.
Here are your coins.
And how do they correspond to the bins?
Well, it's a binary experiment, whether you are picking a red marble
or a green marble, or you are flipping a coin getting heads or tails.
It's a binary situation.
So there's a direct correspondence.
Just get the probability of heads being mu, which is the probability of
a red marble, corresponding to them.
So because the coins are fair,
actually all the bins in this case are half red, half green.
That's really bad news for a hypothesis.
The hypothesis is completely random.
Half the time it agrees with the target function.
Half the time it disagrees.
No information at all.
Now you apply the learning paradigm we mentioned, and you say: let me
generate a sample from the first hypothesis.
I get this, I look at it, and I don't like that.
It has some reds.
I want really a clean hypothesis that performs perfectly--
all green.
You move on.
And, OK.
This one--
even, I don't know.
This is even worse.
You go on and on and on.
And eventually, lo and behold, I have all greens.
Bingo.
I have the perfect hypothesis.
I am going to report this to my customer, and if my customer is in
financial forecasting, we are going to beat the stock market and
make a lot of money.
And you start thinking about the car you are going to buy, and all of that.
Well, is it bingo?
No, it isn't.
And that is the problem.
So now, we have to find something that makes us deal with
multiple bins properly.
Hoeffding Inequality-- if you have one experiment, it has a guarantee.
The guarantee gets terribly diluted as you go, and we want to know exactly
how the dilution goes.
So here is a simple solution.
This is a mathematical slide. I'll do it step-by-step.
There is absolutely nothing mysterious about it.
This is the quantity we've been talking about.
This is the probability of a bad event.
But in this case, you realize that I'm putting g.
Remember, g was our final hypothesis.
So this corresponds to a process where you had a bunch of h's, and you picked
one according to a criterion, that happens to be an in-sample criterion,
minimizing the error there, and then you report the g as the
one that you chose.
And you would like to make a statement that the probability for the g you
chose-- the in-sample error-- happens to be close to the out-of-sample error.
So you'd like the probability of the deviation being bigger than your
tolerance to be, again, small.
All we need to do is find a Hoeffding counterpart to this, because
now this fellow is loaded.
It's not just a fixed hypothesis and a fixed bin.
It actually corresponds to a large number of bins, and I am visiting the
random samples in order to pick one.
So clearly the assumptions of Hoeffding don't apply-- that correspond
to a single bin.
This probability is less than or equal to the
probability of the following.
I have M hypotheses--
capital M hypotheses.
h_1, h_2, h_3, h_M.
That's my entire learning model.
That's the hypothesis set that I have, finite as I said I would assume.
If you look at what is the probability that the hypothesis you
pick is bad? Well, this will be less than or equal to the probability that the
first hypothesis is bad, or the second hypothesis is bad, or, or, or the last
hypothesis is bad.
That is obvious.
g is one of them.
If it's bad, one of them is bad.
So less than or equal to that.
This is called the union bound in probability.
It's a very loose bound, in general, because it doesn't
consider the overlap.
Remember when I told you that the half a percent here, half a percent here,
half a percent here--
if you are very unlucky and these are non-overlapping, they add up.
The non-overlapping is the worst-case assumption, and it is the assumption
used by the union bound.
So you get this.
And the good news about this is that I have a handle on each term of them.
The union bound is coming up.
So I put the OR's.
And then I use the union bound to say that this is less than or equal to, and simply sum
the individual probabilities.
So the half a percent plus half a percent plus half a percent--
this will be an upper bound on all of them.
The probability that one of them goes wrong, the probability that someone
gets all heads, and I add the probability for all of you, and that
makes it a respectable probability.
So this event here is implied.
Therefore, I have the implication because of the OR, and this one
because of the union bound, where I have the pessimistic assumption that I
just need to add the probabilities.
Now, all of this-- again, we make simplistic assumptions, which is
really not simplistic as in trivially restricting, but rather the opposite.
We just don't want to make any assumptions that restrict the
applicability of our result.
So we took the worst case.
It cannot get worse than that.
If you look at this, now I have good news for you.
Because each term here is a fixed hypothesis.
I didn't choose anything.
Every one of them has a hypothesis that was declared ahead of time.
Every one of them is a bin.
So if I look at a term by itself, Hoeffding applies to this, exactly the
same way it applied before.
So this is a mathematical statement now.
I'm not looking at the bigger experiment.
I reduced the bigger experiment to a bunch of quantities.
Each of them corresponds to a simple experiment that we already solved.
So I can substitute for each of these by the bound that the
Hoeffding gives me.
So what is the bound that the Hoeffding gives me?
That's the one.
For every one of them, each of these guys was less than or
equal to this quantity.
One by one.
All of them are obviously the same.
So each of them is smaller than this quantity.
Each of them is smaller than this quantity.
Now I can be confident that the probability that I'm interested in,
which is the probability that the in-sample error
being close to the out-of-sample error-- the closeness of them is bigger
than my tolerance, the bad event.
Under the genuine learning scenario-- you generate marbles from every bin,
and you look deliberately for a sample that happens to be all green or as
green as possible, and you pick this one.
And you want an assurance that whatever that might be, the
corresponding bin will genuinely be good out-of-sample.
That is what is captured by this probability.
That is still bounded by something, which also has that exponential in it,
which is good.
But it has an added factor that will be a very bothersome factor, which is:
I have M of them.
Now, this is the bad event.
I'd like the probability to be small.
I don't like to magnify the right-hand side, because that is the probability
of something bad happening.
Now, with M, you realize that
if you use 10 hypotheses, this probability is probably tight.
If you use a million hypotheses, we probably are already in trouble.
There is no guarantee, because now the million gets multiplied by what used
to be a respectable probability, which is 1 in 100,000, and now you can make
the statement that the probability that something bad happens
is less than 10.
[LAUGHING]
Yeah, thank you very much.
We have to take a graduate course to learn that!
Now you see what the problem is.
And the problem is extremely intuitive.
In that Q&A session after the last lecture, we all got through the
discussion the assertion that if you have a more sophisticated model, the
chances are you will memorize in-sample, and you are not going to
really generalize well out-of-sample, because you have so many
parameters to work with.
There are so many ways to look at that intuitively, and this is one of them.
If you have a very sophisticated model-- M is huge, let alone infinite.
That's later to come.
That's what the theory of generalization is about.
But if you pick a very sophisticated example with a large M, you lose the
link between the in-sample and the out-of-sample.
So you look at here.
[LAUGHING], I didn't mean it this way, but let me go back just to show
you what it is.
At least you know it's over, so that's good.
So this fellow is supposed to track this fellow.
The in-sample is supposed to track the out-of-sample.
The more sophisticated the model you use, the looser that in-sample will
track the out-of-sample.
Because the probability of them deviating becomes bigger and bigger
and bigger.
And that is exactly the intuition we have.
Now, surprise.
The next one is for the Q&A. We will take a short break, and then we will
go to the questions and answers.
We are now in the Q&A session.
And if anybody wants to ask a question, they can go to the
microphone and ask, and we can start with the online audience questions, if
there are any.
MODERATOR: The first question is
what happens when the Hoeffding Inequality
gives you something trivial, like less than 2?
PROFESSOR: Well, it means that either the resources of the examples
you have, the amount of data you have, is not sufficient to guarantee any
generalization, or--
which is somewhat equivalent--
that your tolerance is too stringent.
The situation is not really mysterious.
Let's say that you'd like to take a poll for the president.
And let's say that you ask five people at random.
How can you interpret the result?
Nothing.
You need a certain amount of respondents in order for the
right-hand side to start becoming interesting.
Other than that, it's completely trivial.
It's very likely that what you have seen in-sample doesn't correspond to
anything out-of-sample.
MODERATOR: So in the case of the perceptron--
the question is would each set of w's be considered a new m?
PROFESSOR: The perceptron and, as
a matter of fact, every learning model of interest
that we're going to encounter, the number of hypotheses, M,
happens to be infinite.
We were just talking about the right-hand side not being meaningful
because it's bigger than 1. If you take an infinite hypothesis set and
verbatim apply what I said, then you find that the probability is actually
less than infinity.
That's very important.
However, this is our first step.
There will be another step, where we deal with infinite hypothesis sets.
And we are going to be able to describe them with an abstract quantity
that happens to be finite, and that abstract quantity will be the one we
are going to use in the counterpart for the Hoeffding Inequality.
That's why there is mathematics that needs to be done.
Obviously, the perceptron has an infinite number of hypotheses because
you have real space, and here is your hypothesis, and you can perturb this
continuously as you want.
Even just by doing this, you already have an infinite number of hypotheses
without even exploring further.
MODERATOR: OK, and this is a popular one.
Could you go over again in slide 6, of the implication of nu equals mu and
vice versa.
PROFESSOR: Six.
It's a subtle point, and it's common between machine learning and
statistics.
What do you do in statistics?
What is the cause and effect for a probability and a sample?
The probability results in a sample.
So if I know the probability, I can tell you exactly what is the
likelihood that you'll get one sample or another or another.
Now, what you do in statistics is the reverse of that.
You already have the sample, and you are trying to infer which probability
gave rise to it.
So you are using the effect to decide the cause rather than
the other way around.
So the same situation here.
The bin is the cause.
The frequency in the sample is the effect.
I can definitely tell you what the distribution is like in the sample,
based on the bin.
The utility, in terms of learning, is that I look at the sample
and infer the bin.
So I infer the cause based on the effect.
There's absolutely nothing terrible about that.
I just wanted to make the point clear, that when we write the Hoeffding
Inequality, which you can see here, we are talking about this event.
You should always remember that nu is the thing that plays around
and causes the probability to happen, and mu is a constant.
When we use it to predict that the out-of-sample will be the same as the in-
sample, we are really taking nu as fixed, because this is the in-
sample we've got.
And then we are trying to interpret what mu gave rise to it.
And I'm just saying that, in this case, since the statement is of the
form that the difference between them, which is symmetric, is greater than
epsilon, then if you look at this as saying mu is there and I know that nu
will be approximately the same, you can also flip that.
And you can say, nu is here, and I know that mu that gave rise to it must
be the same.
That's the whole idea.
It's a logical thing rather than a mathematical thing.
MODERATOR: OK.
Another conceptual question that is arising is that a more complicated
model corresponds to a larger number of h's.
And some people are asking--
they thought each h was a model.
PROFESSOR: OK.
Each h is a hypothesis.
A particular function, one of them you are going to pick, which is going to
be equal to g, and this is the g that you're going to report as your best
guess as an approximation for f.
The model is the hypotheses that you're allowed to visit in order to
choose one.
So that's the hypothesis set, which is H.
And again, but there is an interesting point.
I'm using the number of hypotheses as a measure for the complexity in the
intuitive argument that I gave you.
It's not clear at all that the pure number corresponds to the complexity.
It's not clear that anything that has to do with the size, really, is the
complexity.
Maybe the complexity has to do with the structure of individual
hypotheses.
And that's a very interesting point.
And that will be discussed at some point-- the complexity of individual
hypotheses versus the complexity of the model that captures all the
hypotheses.
This will be a topic that we will discuss much later in the course.
MODERATOR: Some people are getting ahead.
So how do you pick g?
PROFESSOR: OK.
We have one way of picking g-- that already was established last time--
which is the perceptron learning algorithm.
So your hypothesis set is H.
Script H.
It has a bunch of h's, which are the different lines in the plane.
And you pick g by applying the PLA, the perceptron learning algorithm,
playing around with this boundary, according to the update rule, until it
classifies the inputs correctly, assuming they are linearly separable,
and the one you end up with is what is declared g.
So g is just a matter of notation, a name for whichever one we settle on,
the final hypothesis.
How you pick g depends on what algorithm you use, and what
hypothesis set you use.
So it depends on the learning model, and obviously on the data.
MODERATOR: OK.
This is a popular question.
So it says: how would you extend the equation to support an output that
is a valid range of responses and not a binary response?
PROFESSOR: It can be done.
One of the things that I mentioned here is that this fellow, the
probability here, is uniform.
Now, let's say that you are not talking about a binary experiment.
Instead of taking the frequency of error versus the probability of error,
you take the expected value of something versus the
sample average of it.
And they will be close to each other, and some, obviously technical,
modification is needed to be here.
And basically, the set of laws of large numbers, from which this is one member,
has a bunch of members that actually have to do with expected value and
sample average, rather than just the specific case of probability and
sample average.
If you take your function as being 1, 0, and you take the expected value,
that will give you the sample as the sample average, and the probability as
the expected value.
So it's not a different animal.
It's just a special case that is easier to handle.
And in the other case, one of the things that matters is the variance of
your variable.
So it will affect the bounds.
Here, I'm choosing epsilon in general, because the variance of this variable
is very limited.
Let's say that the probability is mu, so the variance is mu
times 1 minus mu.
It goes from a certain value to a certain value.
So it can be absorbed.
It's bounded above and below.
And this is the reason why the right-hand side here can
be uniformly done.
If you have something that has variance that can be huge or small,
then that will play a role in your choice of epsilon, such that
this will be valid.
So the short answer is: it can be done.
There is a technical modification, and the main aspect of the technical
modification, that needs to be taken into consideration, is the variance of
the variable I'm talking about.
MODERATOR: OK.
There's also a common confusion.
Why are there are multiple bins?
PROFESSOR: OK.
The bin was only our conceptual tool to argue that learning is
feasible in a probabilistic sense.
When we used a single bin, we had a correspondence with a hypothesis, and
it looked like we actually captured the essence of learning, until we
looked closer and we realized that, if you restrict yourself to one bin and
apply the Hoeffding Inequality directly to it, what you are really
working with--
if you want to put it in terms of learning--
is that my hypothesis set has only one hypothesis.
And that corresponds to the bin.
So now I am picking it--
which is my only choice.
I don't have everything else.
And all I'm doing now is verifying that its in-sample performance will
correspond to the out-of-sample performance, and that is guaranteed by
the plain-vanilla Hoeffding.
Now, if you have actual learning, then you have more than one
hypothesis.
And we realize that the bin changes with the hypothesis, because whether
a marble is red or green depends on whether the hypothesis agrees or
disagrees with your target function.
Different hypotheses will lead to different colors.
Therefore, you need multiple bins to represent multiple hypotheses, which
is the only situation that admits learning as we know it--
that I'm going to explore the hypotheses, based on their performance in-sample,
and pick the one that performs best, perhaps, in-sample, and hope that it
will generalize well out-of-sample.
MODERATOR: OK.
Another confusion.
Can you resolve the relationship between the probability and the big H?
so I'm not clear exactly what--
PROFESSOR: We applied the--
there are a bunch of components in the learning
situation, so let me get the--
It's a big diagram, and it has lots of components.
So one big space or set is X, and another one is H. So if you
look at here.
This is hypothesis set H. It's a set.
OK, fine.
And also, if you look here, the target function is defined from X to Y, and
in this case, X is also a set.
The only invocation of probability that we needed to do, in order to get
the benefit of the probabilistic analysis in learning, was to put
a probability distribution on X.
H, which is down there, is left as a fixed hypothesis set.
There is no question of a probability on it.
When we talk about the Bayesian approach, in the last lecture in
fact, there will be a question of putting a probability distribution
here in order to make the whole situation probabilistic.
But that is not the approach that is followed for the entire course, until
we discuss that specific approach at the end.
Question.
STUDENT: What do we do when there are many possible hypotheses which
will satisfy my criteria?
Like, in perceptron, for example.
I could have several hyperplanes which could be separating the set.
So how do I pick the best--
PROFESSOR: Correct.
Usually, with a pre-specified algorithm,
you'll end up with something.
So the algorithm will choose it for you.
But your remark now is that,
given that there are many solutions that happen to have zero in-sample
error, there is really no distinction between them in terms of the out-of-
sample performance.
I'm using the same hypothesis set, so M is the same.
And the in-sample error is the same.
So my prediction for the out-of-sample error would be the same, as there's no
distinction between them.
The good news is that the learning algorithm will solve this for you, because
it will give you one specific, the one it ended with.
But even within the ones that achieve zero error, there is a method,
that we'll talk about later on when we talk about support vector machines,
that prefers one particular solution as having a better chance of
generalization.
Not clear at all given what I said so far, but I'm just telling you,
as an appetizer, there's something to be done in that regard.
MODERATOR: OK.
A question is does the inequality hold for any g,
even if g is not optimal?
PROFESSOR: What about the g?
MODERATOR: Does it hold for any g, no matter how you pick g?
PROFESSOR: Yeah.
So the whole idea--
once you write the symbol g, you already are talking about any
hypothesis.
Because by definition, g is the final hypothesis, and your algorithm is
allowed to pick any h from the hypothesis set and call it g.
Therefore, when I say g, don't look at a fixed hypothesis.
Look at the entire learning process that went through the H, the
set of hypotheses, according to the data and according to the learning
rule, and went through and ended up with one that is declared the right
one, and now we call this g.
So the answer is patently that g can be different.
Patently yes, just by the notation that I'm using.
MODERATOR: Also, some confusion.
With the perceptron algorithm or any linear algorithm--
there's a confusion that, at each step, there's a hypothesis, but--
PROFESSOR: Correct.
But these are hidden processes for us.
As far as analysis I mentioned, you get the data,
the algorithm does something magic, and ends up with a final hypothesis.
In the course of doing that, it will obviously be visiting lots of
hypotheses.
So the abstraction of having just the samples sitting there, and eyeballing
them and picking the one that happens to be green, is an abstraction.
In reality, these guys happen in a space, and you are moving from one
hypothesis to another by moving some parameters.
And in the course of doing that, including in the perceptron learning
algorithm, you are moving from one hypothesis to another.
But I'm not accounting for that, because I haven't found my final
hypothesis yet.
When you find the final hypothesis, you call it g.
On the other hand, because I use the union bound, I use the worst-case
scenario, the generalization bound applies to every single hypothesis you
visited or you didn't visit.
Because what I did to get the bound, of deviation between in-sample and out-of-
sample, is that I consider that all the hypotheses simultaneously behave from
in-sample to out-of-sample, closely according to your epsilon criterion.
And that obviously guarantees that whichever one you end up
with will be fine.
But obviously, it could be an overkill.
And among the positive side effects of that is that even the
intermediate values have good generalization--
not that we look at it or consider it, but just to answer the question.
MODERATOR: A question about the punchline.
They say that they don't understand exactly how the Hoeffding works--
shows that learning is feasible.
PROFESSOR: OK.
Hoeffding shows that verification is feasible.
The presidential poll makes sense.
That, if you have a sample and you have one question to ask, and you see
how the question is answered in the sample, then there is a reason to
believe that the answer in the general population, or in the big bin, will be
close to the answer you got in-sample.
So that's the verification.
In order to move from verification to learning, you need to be able to make
that statement, simultaneously on a number of these guys, and that's why
you had the modified Hoeffding Inequality at the end,
which is this one
that has the red M in it.
This is no longer the plain-vanilla Hoeffding Inequality.
We'll still call it Hoeffding.
But it basically deals with a situation where you have M of these
guys simultaneously, and you want to guarantee that all of them are
behaving well.
Under those conditions, this is the probability that the guarantee can
give, and the probability, obviously, is looser than it used to be.
So the probability that bad thing happens when you have many
possibilities is bigger than the probability that bad things happen when
you have one of them.
And this is the case where you added up as if they happen disjointly, as I
mentioned before.
MODERATOR: Can it be said that the bin corresponds to the entire
population in a--
PROFESSOR: The bin corresponds to the entire
population before coloring.
So remember the gray bin--
I have it somewhere.
We had a viewgraph where the bin had gray marbles.
So this is my way of saying this is a generic input, and we
call it X.
And this is indeed the input space in this case, or the general population.
Now, we start coloring it according to when you give me a hypothesis.
So now there's more in the process than just the input space.
But indeed, the bin can correspond to the general population, and the sample
will correspond to the people you polled over the phone, in the case of
the presidential thing.
MODERATOR: Is there a relation between the Hoeffding Inequality and the
p-values in statistics?
PROFESSOR: Yes.
The area where we are trying to say that if I have a sample and I get
an estimate on the sample, the estimate is reliable.
The estimate is close to the out-of-sample.
The probability that you will deviate-- is a huge body of work.
And the p-value in statistics is one approach.
And there are other laws of large numbers that come with it.
I don't want to venture too much into that.
I basically picked from that jungle of mathematics the single most useful
formula that will get me home when I talk about the theory of
generalization.
And I want to focus on it.
I want to understand it-- this specific formula-- perfectly, so when we
keep modifying it until we get to the VC dimension, things are clear.
And, obviously, if you get curious about the law of large numbers, and
different manifestations of in-sample being close to out-of-sample and
probabilities of error, that is a very fertile ground, and a very useful
ground to study.
But it is not a core subject of the course.
The subject is only borrowing one piece as a utility
to get what it wants.
So that ends the questions here?
Let's call it a day, and we will see you next week.