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Whatever our model,
One needs to minimize f of x minus y.

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Otherwise, the, the data, the value that
you want to predict you want to minimize

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the error between the, what you already
have.

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Real values and your predicted values for
those real values..

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Well, f is complicated, there's no formula
like the normal equations that we had for

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linear least squares.
So, all these methods, whether they are

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regression with logistic function, support
vector machines, neural networks, all

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learn the parameters through uniterative
process.

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We start with some f which is a
parameterization of the function f,

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It maybe not be linear like f transposed
x.

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It could be some other parameterization
so, you know, you could have squares, you

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could have exponentials, you can all kinds
of things.

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But, you have some parameters which tells
you what the function f for each choice of

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parameters is.
And, we need to refine it alliteratively

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to find the f which minimizes the error.
In the world of neural networks, the first

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search algorith was called back
propagation which has nothing going to do

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with the feed back, neural networks, is
just a new algorithms, it is just going to

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learn self.
While the other areas which were probably

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more mathematically founded, straight
forward optimization techniques were used

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to find the minimum of this error.
So, if we start off with a guess and we

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choose maybe a slightly different f as two
different guesses,

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One way to start minimizing this error or
moving towards a value of f which is

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likely to have lower error, is to use
something called gradient descent.

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This formula looks a bit daunting so let
me explain it in simple language.

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Epsilon of f is a function of the
parameters of f.

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So, one can measure how fast this function
changes if one changes each component of f

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separately, which is nothing but taking
the partial derivative of the function

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with respect to that component.
String all those partial derivatives

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together, and you'll get a vector which is
just this del sub f of epsilon.

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That vector tells you which direction to
move each component of f so as to reduce

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subset.
So, for example, the first component of f

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is such that if you increase it, then
epsilon decreases, then you would choose

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to increase that component of f.
If the second component is one which

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decreases epsilon, only if you decrease
that component, you will decrease f so

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then the, that particular element of del
would be negative.

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So, you'd get this vector del, computed
using your own value of f of r of f.

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That means, your previous iterations start
with f0, for example. And then, add some

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small multiple of this derivative to your
current guess to create a new guess.

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Now, you need to compute these derivatives
so you use the difference is between

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epsilon at f of i and eps, epsilon at f of
i minus one divided by, you know, f of i

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minus f of i minus one component-wise to
get estimates for this derivative and

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that's why you needed two guesses to start
with so, because you have to bootstrap

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this duration.
Now, this is a very popular and very

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simple method and you can have.
More complicated ways of finding the

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minimum value of, of epsilon by, by using
Newton's iteration where you use second

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derivatives in addition to first
derivatives and various different

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techniques in the mathematics literature
are available to you.

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This works fine if you have numbers, that
means if you have the x's are actual

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values as we've been assuming, but there
are, of course, some caveats.

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This iteration sort of works and all
situations work provided there is only,

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you know, one minima and you're actually
starting very close to that minima if, if

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there are multiple minimas.
And if there are many different minimas,

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you might get stuck in a local minima, and
not, and not really get to the real

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minimum value.
So, so you need better techniques.

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The other thing would be that the function
f might not necessarily be parametrized by

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a single set of values there may be some
constraints.

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You might have a function that says, that
you have five different lines which meet

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at, at the, the connecting points and
that's your function.

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So, you're going to try to fit five lines
to all your data.

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Then, you have constraints which will then
make this situation a little bit more

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complicated and, and, and, and, complex as
well.

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But, we won't go into those details right
now.

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The other problem is, is if x is not
numbers and if x is categorical, at least

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some of the x are categories like yes, no,
red, blue, or green, then you somehow have

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to convert them to a binary form.
So, if there are n categorical variables,

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you might want to convert, convert them
to, for example, zero or one.

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Of course, now, you're going to have many
more than n.

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So, you would say like words.
You, you have categorical variables which

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are words and you suddenly increase the
dimension of, you know, space from say,

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You're, you're talking about one word, but
words could be out of a set of millions.

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So now, you have a million dimensional
space and it's zero or one depending on

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whether that word is there.
And that tremendously complicates such

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methods and they often become unviable if
your categorical data is highly, very high

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dimension and you have to use different
techniques to learn parameters.

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You could also convert the these
techniques, these categorical variables to

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real numbers through a positive, through a
process of reversed classification, which

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we wont talk about, but essentially there,
you, you think about high, low, and medium

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as function which are not fixed as being
high, or low, or medium but could vary as

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something being 50% high, twenty percent
low, etc.

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And then, use such fuzzy membership to
convert your categorical variables to, to,

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into, into real numbers.
And then, once you have real numbers, you

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can use this technique or you could deal
with the space of categorical data itself.

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And then, there are techniques like
neighborhood search, heuristic AI search,

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genetic algorithms, and such techniques
which don't rely on converting your

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parameters into real space.
And then, you get the problem of a

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combinatorial explosion and you have to
use all kinds of techniques to figure out

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good ways of minimizing the error.
And lastly, as we have seen in previous

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lectures, probabilistic models can deal
with probabilities of variables taking on

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particular categorical values as opposed
to having to convert those to numbers or

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deal with them directly.
And these techniques have become

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increasingly powerful and, and popular,
precisely because you're able to deal

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with, with the numbers without having to
get into too much exponential

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combinatorial blowup.
Now, that's quite a mouthful and many of

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you might not have got it.
But those of you who are on the verge of

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graduate research, might want to listen to
that spill a couple of times cuz it does,

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sort of, unify different or sort of give a
bigger picture of how learning parameters

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in predictive models work and what are the
possible options.

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There are some more linkages which I'd
like to explore.

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For example,
Here we are trying to minimize the error

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in a predictive model.
In other situations, we might need to

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maximize a choice which is want to find
the best way to transport goods across the

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country,
Or the best way to allocate one's

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advertising funds across different
channels.

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So, there you're trying to maximize some
function and that's related closely to the

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problem of minimizing a prediction error.
Similar techniques, gradient-descent,

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Newton's method and various others apply
all the similar, similar problems in terms

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of categorical data,
Local minima, and constraints all apply as

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well.
Then, once one has decided where one wants

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to go, one needs to figure out how to get
there.

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So typically, if you're driving a car, for
example, you have a system and your angle

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of steering could be, say, theta.
And every time you, you, you change your

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theta, your car will move, move in a
certain direction so you'll have a, a new

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state of your system.
And the goal is to figure out which

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sequence of control actions will
ultimately minimize the difference between

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the state of your system and where you
want that state to be.

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And there could be other constraints that
you want that paths, paths to be fairly

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smooth, etc.
So, you get all those problems of

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constraints again.
Again, it's a minimization problem.

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And here, unlike in the first two cases of
minimizing the prediction error and, or

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optimizing a function each iteration is
actually executed and the response of a

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system, rather than a function kind of
abstractly is what you're dealing with.

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So, an actual control system will make a
change,

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See what the output is and make another
change again, techniques like

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gradient-descent,, Newton's method, and
all that start applying, but now you're

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controlling actions.
So, from predicting the future or

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predicting where things are going to go to
figuring out how to best utilize that

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information to optimize one's own goals.
And then, to decide how to change ones

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behavior or change ones system, so as to
get to the gold state, all these are

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tightly related.
And, I hope you've seen that the

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00:11:47,667 --> 00:11:53,940
mathematics is related, now lets talk
about some of the applications.

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Before we do that, of course, in practice,
whether you're dealing with support vector

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machines, logistic regression, neural
networks, or what have you, you won't ever

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really have to do these things.
There are packages available that let you

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choose let, let you apply any of these
techniques without having to get into the

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math, or get into the duration.
The important thing is, of course, trying

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to choose which package to use. And, as
soon as we do the applications,

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I'll try to summarize which package to
use, what kind of techniques to use from

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my experience.
And then, we move on to more issues of

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deep belief networks.