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In today lecture we start with the topic machine learning. We will have seven lectures of this
series. Today we will give introduction to this topic. The instructional objectives of
today's lectures are the following. We will first look at what we mean by machine learning
and several definitions of machine learning.
In this class the student will be introduced to different learning frameworks and then
we will introduce some of the basic notations. Also the student will be made familiar with
certain example applications of machine learning. Specifically we will give an introduction
to the type of learning which we call concept learning. In this context the student will
look at certain things; what are the concept of features used in a learning problem.
We will look at what we mean by a hypothesis base or a set of hypothesis and what we mean
by a hypothesis that we are trying to learn. We will introduce what we mean by a training
set and a desk set. We will also talk about the instance space. Also we will briefly introduce
the notion of inductive bias. On studying this lesson the student should be able to
formulate a given concept learning problem. Given a problem they should be identify possible
features that may be relevant to the problem and they should be able to get an idea of
the hypothesis base that they need to consider.
In subsequent lectures we will look at different algorithms and different types of representation
issues. Let us look at the definition of learning. Learning covers a broad range of processes.
Leaning means to gain knowledge or an understanding of skill in something. When we say we want
to learn then we want to gain knowledge or understanding or expertise in solving some
problems. And this expertise or this knowledge can come by looking at the examples or by
gaining example experience or this knowledge can come by studying the problem or by being
told by instructions.
There are several parallels between the process of human learning and artificial or machine
learning. Some techniques in machine learning derive from the efforts of psychologists to
derive theories of human or animal learning through computational models.
The field of cognitive psychology has dealt with in trying to understand how humans learn.
And in some cases they have also tried to identify certain computational models to model
the way humans think and the way humans learn. Machine learning on the other hand has dealt
with certain techniques some of which I inspired by human learning techniques as derived from
cognitive psychology, others by different symbolic techniques which are efficient to
do by machines. But it is conceivable that the concepts and techniques being explored
by researchers in machine learning may actually help understand certain biological processes.
So there is a lot of cross partilisation between the wheels of cognitive psychology which tries
to understand animal learning and the field of machine learning, their objective is to
solely learn by the help of machines. Before we formulate to define learning let us look
at few of the definitions that people have put forward on machine learning.
The first definition we will take up is the definition by Donald Michie in 1991. This
definition states that a learning system uses sample data to generate an updated basis for
improved performance on subsequent data from the same source and expresses the new basis
in intelligible symbolic form. So we have a learning system which uses sample data which
we call a training example. So there is sample data or there is some experience to go by
and on the basis of this experience the system tries to generate a new model so that it can
lead to improved performance on subsequent examples. So, the system uses input data to
get a model which can help it to improve its performance on new data. But data that is
coming from similar sort of examples data from the same source and this model that has
been learned is expressed in a symbolic form which can be understood and which can be manipulated.
This is the definition by Donald Michie.
Let us look at the definition by another pioneer field Herbert Simon. this definition says,
learning denotes changes in the system that are adaptive in the sense that they enable
the system to do the same task or tasks drawn from the same population more effectively
the next time. This is a similar definition as you can see. So this definition says, learning
means change in the system and this change enables the system to perform better on similar
tasks in future more effectively.
We see that they get several other definitions by several other practitioners in the field
but we see that this type of learning mainly talk about there being some input data or
experience in which some data system learns and the system tries to improve its performance.
And this improvement in performance must be measurable in some way so there must be performance
measure which must be improved due to learning. Therefore by learning from examples the system
is able to improve its performance.
Secondly, some of these definitions also emphasize the comprehensibility. The new thing the system
has learnt must be expressed in a form that will be understood.
Therefore now let us simply define a well posed learning problem. A computer program
is set to learn a task T, so T is the task the system is trying to learn. And what is
the basis by which the system is learning this task T?
The system is given some data which the experience e and the system is learning to improve its
performance in task t with respect to its performance metric p. That is, the system's
performance in the task T improves with experience e as measured by the performance metric p.
Therefore this is the definition of learning we will accept. To put it in a more natural
form, learning is the improvement of performance in some environment through the acquisition
of knowledge resulting from experience in that environment. To understand this definition
we need to go back to the framework of intelligent agent which we have been looking at in this
course.
We have this structure, we have this agent and we have this environment. The agent takes
action which changes the environment and the agent can sense the environment. The agent gets experience by interacting with
the environment. And this results in acquisition of some knowledge and using this knowledge
the agent can actually improve its performance in certain tasks. So this is the formal definition
of learning. Now let us look at few examples of learning problem to make this more concrete.
Let us say that we want to know whether a particular given patient has a possibility
of having brain tumor. What is the experience? We have the database of previous patients,
the ones who have been diagnosed with tumor and ones who have been certified as having
no tumor and we have accessed through the patient records which include different data
about the patient as well as some images like MRI or something. So we have some data about
past patients having tumor. And we wish to find out given a patient and the records we
want to know whether this patient is likely to have tumor.
The second example is our task is to recognize speech. So the machine should be able to hear
what we speak and it should able to recognize the speech and take dictations. So in this
case what can be used as an experience is a database of speech which has been already
recognized and their transcripts are available. So we have speech and their transcripts that
constitute the experience. And as a result of this experience the system should be able
to learn and how can we measure whether the system is successful? We measure the percentage
of correctly recognized words.
Therefore the success in the learning task can be measured by the accuracy or the precision.
That is, number of examples which have been correctly labeled as correctly recognized.
And the error can be measured by looking at the number of misclassifications, the number
of words which have been wrongly recognized. So we want to recognize all correct cases.
So, for tumor we want to label as positive all those patients who really have tumor and
none of the patients who do not have tumor. We can measure it by accuracy or you can measure
it by two terms procession and coverage. So procession accuracy is the number of examples
that we label as having tumor and we want to know the percentage of them who actually
have tumor and coverage means out of the patients who actually have tumor how many of them do
we correctly recognize as having tumor. These are the different measures we use for measuring
the accuracy or the correctness of a learning task.
So the learner is in an environment learner is tempting to learn something about environment
so that it can perform some of its tasks well. The learner is linked to some knowledge base
from which it can draw and which stores the acquired knowledge. Therefore the learner
has a knowledge base, there could be some prior knowledge in the knowledge base but
as a result of learning the learner is able to update its knowledge base. And this knowledge
is usually stored in some form of internal data structure.
You have studied logic and you have seen representational languages in logic, first order logic and
you have also looked at certain other representation schemes and frames etc. Knowledge is basically
stored in one of these representation schemes.
And what is experience? Experience is basically derived from the perceptual
input of the agent and the agent can take some action which is the output of the agent.
And the performance of the agent is measured quantitatively by several aspects and depending
on the task we decide what to use. Let us just review why study of machine learning
is so important. There are many tasks that require an adaptive system that require a
system which can learn.
For example, hand writing recognition, speech recognition are examples where an adaptive
system is required. Learning is also useful as an alternative to hard coding a program.
For example, suppose you want to develop a program which can play the game of chess.
Now you could write a program in which for every different possible situation you can
specify what move the agency. So you could hand code all the rules useful for playing
chess. Therefore an alternative of writing such a program would be to provide the system
with a database of chess, games and their outcomes and maybe the system can apply the
learning to learn to play a good game of chess without even instructing the system to make
the right move at the right situation. But you give a large database of games by which
the system can figure out. Providing database of games is usually easier than hand coding
the rules. So you can save a lot of manual effort if a system is able to learn.
Also, the study of machine learning gives us an insight into human learning. For example,
learning of language is a very non trivial task. So, trying to understand how we can
make a machine learn a language can give us a clue as to how humans acquire language.
Machine learning has been very useful in the curb of data mining which helps systems to
acquire hidden rules from data, which has opened a whole new area of applications. This
is a new kind of capability that our systems are provided with. There are many types of
learning and we can classify learning along different dimensions.
Supervised learning: In supervised learning the system is given labeled training examples.
In these labeled training examples we have a set of examples and we also have the labels.
So we have the inputs and the outputs of the examples. And unsupervised learning is the
learning where there are no labels given. We only have the examples which are not pre-classified.
So we have unclassified training example and there are situations where we like to learn
from them. For these lectures we will be mainly concerned with supervised machine learning
where we are given labeled examples. For the concept learning we have a set of labeled
examples during training using which we will learn to classify unknown example. There are
also other types of learning where we do not have labeled examples but we try to learn.
A third type of learning is reinforcement learning. In reinforcement learning we are
not given just examples and each one of them is labeled but we are given a sequence of
examples and at some points the system gets some reward or some punishments called reinforcement.
For example, when a system is playing a game of chess the system does not get to know whether
each move is good or bad each move is not rated. But at the end of the game the system
knows whether it has won or the game was a draw or it lost.
That is the reinforcement which is available at only certain points in the game. So, in
reinforcement learning the system is trying to learn but it gets reinforcements only at
a certain time. What are the different types of knowledge that can be acquired by learning?
The types of knowledge can be declarative knowledge. Declarative knowledge can be expressed
in terms of concepts, in terms of preferred value of parameters or in terms of grammar
or in terms of taxonomies. The knowledge acquired by learning can alternatively be expressed
as procedural knowledge. Procedural knowledge can be expressed in terms of rules, rule strengths,
graphs or networks, computer programs and plans.
Certain data structures can be used for storing knowledge. Certain data structures are decision
trees, logical expressions, neural networks, condition action rules, sets of rules, finite
state automata and programs. For example, concepts when we want to represent concepts
we can represent concepts by decision trees, or by logical expression or by neural networks.
Behavioral rules can be expressed as condition action rules.
Plans can be expressed as sets of rules or by finite set automata. Computer program can
be expressed as c code. And then there could be different strategies by which these learning
rules can be acquired. Inductive inference can be used to learn concepts and grammar.
Evolutionary learning or genetic algorithms can be used to learn certain preferred values
of parameters. In unsupervised learning clustering can be used for learning taxonomies.
Then we can also have analogy or induction to learn rules, we can have reinforcement
learning to learn plans or strategies or which you call policies, evolutionary learning,
stochastic learning etc. There is a wide choice of learning strategies and learning programs.
How do we evaluate learner? Once we have a learning strategy typically
what we do is we draw a learning curve. In a learning curve what we do is we plot the
accuracy of the learner along the y axis, this axis is the accuracy and along the x
axis we plot number for training example seen. So what we typically expect is that the accuracy
of the learner will increase as the number of training examples increases. These are
typical learning curves that we might get. So these learning curves plot the accuracy
or precision in terms of the number of training example seen. Next we will talk about inductive
learning for classification also known as concept learning. In fact in most of the lectures
on learning in this course we will mainly be looking at the idea of concept learning.
What is concept learning? Concept learning or classification means learning
at description of a class of objects. So we have some concept or some object whose description
we wish to learn and concept learning is learning this description.
Why do we want to learn this description? We want to use this description to predict
the class of a new object. When we have a new example we want to know which class this
new example belongs to so we try to learn the description of a class. So, if we are
given an animal we want to know whether this animal can be classified as a tiger or not.
And in the past we have seen several animals we are told they are tigers and we also have
seen several animals for which we were told they are not tigers and from this we will
form a model what is a tiger. And when given a new animal we will be able to know whether
this animal is a tiger or not a tiger, so this is the essence of concept learning.
There are other examples of concept learning tasks some of them are described here. Suppose
our objective is to classify parts as defective parts or ok parts. Second example is mammogram
analysis. We are given a mammogram and we want to classify whether the mammogram is
either normal or precancerous or cancerous.
Example 3; in document understanding we are given a rectangular region from a scanned
image and we should be able to say whether this is a text region or graphics region.
So we want to recognize a text region from a graphics region. These are certain examples
of concept learning tasks. Now we will be using inductive inference for concept learning.
And let us see what we mean by inductive inference.
Suppose H is a hypothesis and F is a set of facts, suppose we know that H implies F, if
H implies F is valid this rule implies f is valid and H is known to be true then by Modus
ponens F must be true. H implies F is valid, the antecedent H is known to be true then
logically by Modus ponens it follows that F must be true. This is an example of logical
deduction. So we can derive F from H. this process is truth preserving and this is called
deductive inference. Let us take an example. You know that all men are mortal and you know
that Socrates is a man and you can conclude that Socrates is mortal. This is an example
of deductive inference.
Next let us see what we mean by inductive inference. Suppose again let us assume that
h is a hypothesis and f is a set of facts and again let us say that H implies F is known
to be valid. Now suppose we know F, now knowing F in inductive inference we will try to derive
H. Now from F H does not follow by deductive inference. But if F is false then H must be
false, if F is true deductively we cannot say that H must be true but if F is false
we can say that H must be false. So deriving H from F is falsity preserving but not truth
preserving. So, if there are some facts which make f false then h must also be false. But
in inductive inference we will derive H from F. Suppose for 10 days I have woken up and
seen that in the morning the sky is blue and on the basis of this I form a hypothesis that
in the morning the sky is always blue. This is an example of an inductive inference. On
the basis of 10 data of the sky being blue in the morning I am inferring that the sky
is always blue. This is an example of inductive inference or induction.
Another example of induction is; suppose you see five white tigers, you see five tigers
and all of them are white and you conclude all tigers are white. This is an example of
inductive inference.
When can you be wrong? Suppose now you find a yellow tiger now this
falsifies your conjunction. So inductive inference is not a logical inference so it does not
preserve the truth but it is a useful leave to take, it gives you new knowledge, knowledge
that you have not already seen. An example of deduction which I already mentioned is
suppose you told that all men are mortal and you know that you are man and you conclude
that you are mortal. This is an example of deductive inference.
What we saw earlier was an example of inductive inference. Now let us look at concept learning
in more detail.
Suppose you have a goal concept that you are trying to learn so we will call this goal
concept as a target concept. A target concept is a concept you are trying to learn. For
example, you are trying to learn about a tiger so tiger is a target concept. And your guesses
of the target concept is the hypotheses. So you are trying to learn the concept and you
make hypothesis. You form a hypothesis about the description of the concept. So a hypothesis
is your guess or approximation of the target concept.
How do you form hypothesis? You form a hypothesis by looking at data,
by looking at many instances of data or examples. So, an object which is used to help you learn
the goal concept is called as instance or an example. An instance or an example is something
which helps you learn the goal concept. Therefore typically an instance is described by a vector
of features also called attributes.
Example of attributes; Suppose you are given a new animal and the attributes are color,
number of legs, whether the animal has viscous, length of the body, whether the animal has
fur and so on so these are the features. If you are given an animal whose color is white,
it is furry, it has a tail, it is 6 ft long, it has viscors then this is a description
of an instance of an animal. Therefore an instance is described by vector of features,
features are also called attributes. So, for example the instance is given by x1, x2, xn
and x1 is the value of the first feature, xn is the value of the nth feature. There
are different types of features that we can have, features can be nominal.
For example, color of the animal, color can be red, blue, yellow, green so these are some
specific values they are not directly linked to each other. Therefore color is a nominal
attribute similarly suppose you are trying to learn an object whether the object is a
table and one of the attributes it can have is from what material the object is made of?
And the material can be wood, steel, glass etc. So these are nominal attributes.
Secondly, you can have numeric attributes. Usually the attributes can be ordered the
other attributes are also in an order. Order means there is an ordering to the values.
For example, length of the animal is the order attribute; it is 5 ft, 6 ft, 4 ft, 8 ft so
they have some relationship within themselves. This is an order attribute. Temperature, weight,
length are examples of order attributes.
Thirdly attributes can be structured. There is some structure in the order of the values
but they are not fully ordered. For example, if the values can be put into some sort of
generalization hierarchy or some partial order. For example, if you consider the animal taxonomy,
this is an example of animal taxonomy.
Suppose you have the class vertebrates and you know that animals can be classified into
invertebrates and among vertebrates you can have mammals, reptiles etc and among mammals
birds, fish are vertebrates, so among mammals you can classify them as four legged mammals
and two legged mammal, under four legged mammal we have tiger, mouse, deer etc and under two
legged mammal we have humans, kangaroo etc. Therefore this is an example of classification
hierarchy and this is also an example of a structured attribute and in this direction
we have more general classes.
So the concept learning problem formally can be described as follows: We are given labeling
function f, f is a underlying function for describing the concept. This function f maps
feature vectors into some classes. Therefore we have a discrete set of k classes that are
k possible classes.
For example, we have two classes tiger or not tiger or we can have three classes whether
it is a tiger or a lion or a deer. So, in general we can have a finite set of classes
so we have k classes and we have an actual underlying function which maps to instance
to one of these classes. So the function f and maps an input instances to one of these
classes. In a special case we may have only two classes and in such a case we may say
that one of the classes is positive and the other one is negative. We are given some training
examples.
Each training example is a pair, the instance and its classification. So each training example
is given as a pair the instance and its class and we have a set of such training examples.
So using these training examples we want to learn an approximation of f. Therefore from
the set of x f(x) pair we have got we want to learn the target concept f. Therefore f
is what we wish to learn. So given a set of x of f(x) pairs we want to infer f we want
to inductively infer f.
Now, if you are given a finite sample and you are not shown all the instances it is
not really possible to guess the correct value of f with absolute certainty. So we will apply
inductive inference and we will try to find some pattern in the training examples and
we will assume that this pattern will hold for future examples also. This is an example
of training set, we have an instance x is equal to 1 then the classification is 1, x
is equal to 2 f(x) is equal to 3 f(x) is equal to 9 x is equal to 4, f(x) is equal to 16
and we want to know if x is 5 then what is f(x)? This is an example of a training set
that we are give. And if you want to learn the function f in this case one good guess
is that the function f is a square function that is f(x) is x square but this not really
a discrete learning problem but this is an example of learning a function.
Another example; suppose you want to learn the concept of an apple whether an object
is an apple or not, you are given some training examples. For training examples these are
the features that are given. For every object that you are given you know the color, shape,
diameter, whether it has a stem and then it is labeled whether the object is an apple
or not. You have got four examples.
The first example the color is red, shape is round, it has a diameter of four inch,
it has a stem and you know it is an apple.
The second example is, the color is yellow, shape is round, diameter is 4.3 inch. It does
not have a stem it is an apple.
The third example; the color is green, shape is square, diameter is 5 inch, no stem, not
an apple, this is a negative example of an apple.
Fourth example; the color is green, shape is round, diameter is 3 inch, it has a stem
and it is an apple. Now you could learn a rule or a set of rules to distinguish positive
examples of apple from negative examples. These rules are called classification rules.
For example, the rules could be, round means apple or stem implies apple or diameter less
than 5 inch means apple or round and diameter less than 5 inch means apple. So these are
some possible rules you can hypothesize about an apple.
In this case we are looking at rules expressed in a particular type of language. These rules
are given in terms of the features. But these features are expressed in some language. For
example, round and diameter is less than five inch so here we are expressing the antecedent
as a conjunction of constraints and features. Similarly we could express rule as a disjunction.
We could say round or diameter less than 5 inch implies apple. So this is the example
of disjunctive rule. Or we could express it other forms. We could decide a language to
express the rules. And this defines the set of hypothesis that we are considering.
What type of hypothesis should we consider? Suppose f is the set of rules then the space
of hypothesis is all rule sets. The space of hypothesis could be simple polynomials;
the space of hypothesis could be decision trees, the space of hypothesis we are considering
could be neural networks. Therefore we have a choice about what type of hypothesis we
are going to consider. We are trying to learn a function f which we do not know and we are
trying to find an approximation to f, we are trying to make a hypothesis. We will have
to choose this hypothesis from a set of hypothesis that we will consider. This is called our
hypothesis phase.
And what set of hypothesis we should consider? This is what we must decide. So we will express
by H the space of all possible hypothesis that a learning program consider. Then our
objective would be to find one hypothesis which is a member of this hypothesis space.
We want to find that hypothesis which is the best. We want to find the best hypothesis
of this hypothesis space.
What is the best hypothesis? It is that which fits the given data the best.
Therefore this is a process of search. In order to find its best hypothesis we will
search through this hypothesis space to find that hypothesis which fits the data in the
best possible manner.
This is an example of a hypothesis space. In this hypothesis space there is a large
number of hypothesis. For example h1 h2 h3 h4 h5 are some of the hypothesis in this hypothesis
space and we want to select the best of the hypothesis in this hypothesis space.
Suppose you want to learn the concept apple and this is an example of a hypothesis space,
you have a hypothesis here like; round and diameter less than 5 degree, red and round
has stem, not a square, round etc are some of the hypothesis that are present in the
hypothesis space and we want to select the best of these hypothesis.
Some definitions: A training set is the set of all examples that are given to the learner.
A testing set is a set of examples on which the learner tests his hypothesis. So, on the
basis of training set the learner will learn the hypothesis and the learner will try to
evaluate how good the hypothesis is by looking at another set of examples which is called
the test set. So the learner, on looking at the training set will find a hypothesis. The
hypothesis the learner has found will be set to be consistent if it is consistent with
all the training examples. That is, the hypothesis must predict the correct label of all the
training examples and such a hypothesis is called consistent hypothesis. We cannot guarantee
that a consistent hypothesis will necessarily always give the correct labeling for each
test example because we have not seen the text example but we can check whether the
hypothesis correctly labels the training example.
A testing set is the set of all examples that are given to the learner after it has learnt
the hypothesis on which its accuracy will be tested.
Suppose the examples are labeled as plus and minus then the consistent hypothesis is a
hypothesis that implies that labels are positive for all the plus examples and none of the
minus examples, that is a consistent hypothesis.
Now let us look at this diagram. Suppose this one is the instance space which consists of
all possible examples and suppose this is the true function f that our system is trying
to learn. So this f labels the instances inside the circle as positive and the instances outside
the circle as negative. Now this other circle is the hypothesis h that your learner finds.
In this region the h and f agree the label of instances. The h makes mistakes in this
blue region and in this black region. So these are the two regions in which h makes a mistake.
An example is called to be a false, negative if the hypothesis says it should be negative
but it is actually positive. In this blue region your hypothesis states that these examples
must be negative but these examples are actually positive according to f so these are falsely
negative by your hypothesis. So this is a zone of error of your hypothesis. In this
green region your hypothesis says that this instance should be positive but they are actually
negative according to f.
This is another error zone and this is said to be the false positive for your hypothesis.
So an example is a false negative for a hypothesis if the hypothesis says it would be negative
but it is actually positive. An example is false positive for hypothesis if the hypothesis
says it should be positive but it is negative. This big thing is the instance space, all
the instances are here and this is a particular concept, inside the circle we have the positive
example, outside the circle we have negative examples, and this other circle is your hypothesis.
Inductive bias is another definition.
When you are making an inductive inference on the basis of looking at some examples you
are making a hypothesis. That hypothesis may not be fully correct. For example, if you
are making a jump, you are making a conceptual jump, you are assuming something extreme so
when you see some data you could infer different possible hypothesis.
Which one of them would you select? You could try to find out that hypothesis
which is consistent but suppose you find several hypothesis which you are consistent with your
training set, like in the apple example we could find several hypothesis which are consistent
the four examples so which one of them would you select? In order to select one of them
we use the concept of inductive bias. Therefore inductive bias is a bias or a preference for
one hypothesis over another. And there are different types of inductive bias which are
used by different learners. For example, simplicity could be a bias or most general hypothesis
could be a bias or choose the most specific hypothesis could be a bias. Therefore among
the hypothesis which is both consistent we choose one of them according to some bias
we have.
For example, suppose you want to learn the concept apple then you could learn a set of
rules to distinguish positive from negative examples. These are called classification
rules and these are rules we have already seen. Then you could say in order to select
among rules you apply a bias this is called inductive bias.
How do you choose the best hypothesis? One way in which you could choose the best
hypothesis is, among two hypotheses which are equally consistent on the training examples
you should choose a hypothesis that is likely to agree with new examples. So our real goal
in machine learning is to find a hypothesis f' in h such that it will correctly classify
new examples. So the ideal thing would be to choose that f' so that probability that
f'x does not agree with f(x) is smallest.
We want to choose that f' so that it agrees with f most of the time. But we cannot compute
this probability because we do not before hand know all the instances all the labels
of instances etc.
How we will evaluate the performance of the learner?
We will evaluate the performance of the learner by seeing how well the learned hypothesis
can predict the classification of unseen examples, the examples that we have not yet seen.
If we use the training set for classification we will not really understand whether hypothesis
is performing well on unknown examples. So we should apply the hypothesis we learnt from
the training set and a new data. So, test set should be disjoint with the training set.
So, in a concept learning our problem is to learn the target concept f from a set of training
examples. We evaluate we measure the predictive performance of the learned hypothesis on a
set of test examples called the test set. And we require that the training and test
set are different from each other so that there is no bias.
Few questions to consider: 1) Consider the problem of trying to recognize
hand written digits. You have to formulate this as a concept learning problem. Specifically
you have three tasks; a) clearly specify what are the possible features?
b) How do you get the training set and the test set?
c) How will you measure the performance of your learning algorithm?
2) Consider the problem of trying to play a game of Ludo, formulate this as a learning
problem. Clearly specify what your system will try to learn.
b) How can you get the training examples for this system?
with this i stop todays lectures in the next class we will look further of certain definitions
of concept learning problem and then we will move on to some algorithms and frame works
fro learning thank you