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Data-dependencies and Learning in Artificial Systems
Palem GopalaKrishna krishna@cse.iitb.ac.in
Research Scholar, Computer Science & Engineering, Indian Institute of Technology - Bombay, Mumbai, India
Abstract
Data-dependencies play an important role in
the performance of learning algorithms. In
this paper we analyze the concepts of data
dependencies in the context of artificial sys-
tems. When a problem and its solution are
viewed as points in a system configuration,
variations in the problem configurations can
be used to study the variations in the so-
lution configurations and vice versa. These
variations could be used to infer solutions to
unknown instances of problems based on the
solutions to known instances, thus reducing
the problem of learning to that of identifying
the relations among problems and their solu-
tions. We use this concept in constructing a
formal framework for a learning mechanism
based on the relations among data attributes.
As part of the framework we provide metrics
–quality and quantity – for data samples and
establish a knowledge conservation theorem.
We explain how these concepts can be used in
practice by considering an example problem
and discuss the limitations.
1. Introduction
Two instances of a function can only differ in their
arguments, i.e. the input data. When a function is
sensitive to the data it is operating upon, even a slight
variation in the nature of data can cause large varia-
tions in the path of execution. This property of being
sensitive to data is termed as data-dependency which
poses critical restrictions on the applicability of algo-
rithms themselves.
The success of any data-dependent learning algorithm
highly depends on the nature of the data samples it
learns from. A well designed algorithm with mis-
matched data is unlikely to succeed in generalization.
Thus a careful analysis of the size and quality of the
input data samples is vital for the success of every
learning algorithm. While there exists sufficient num-
ber of metrics for learning in traditional systems in
this regard (Kearns, 1990; Angluin, 1992), there exists
almost none for learning in artificial systems, where
the typical requirements would be action selection and
planning implemented through agents (Wilson, 1994;
Bryson, 2003). These agents would act as determinis-
tic systems and thus demand non-probabilistic metrics
with data-independent algorithms.
Data-independence essentially means that the path
of execution (the series of instructions carried out)
is independent of the nature of the input data. In
other words, when an algorithm is said to be data-
independent, all instances of the algorithm would fol-
low the same execution path no matter what the input
data is. We can understand this with the following ex-
ample. Consider an algorithm to search a number in
a given array of numbers. Such an algorithm would
typically look like below.
int Search(int Array[], int ArrLen, int Number) {
for( int i=0; i < ArrLen; ++i )
if( Array[i] == Number)
return i;
return -1;
}
The above procedure sequentially scans a given array
of numbers to find if a given number is present in the
array. It returns the index of the number if it finds
a match and −1 otherwise. The time complexity of
this algorithm is O(1) in the best case and O(n) in the
average and worst cases. However, if we change the
iterator construct from for(i= 0; i <ArrLen; ++i) to
for(i=ArrLen-1; i≥0; − − i), then the performances
would vary from best to worst and vice versa.
On the other hand consider the following data-
independent version of the same code.
int Search1(int Array[], int ArrLen, int Number) {
int nIndex = -1;
for(int i=0; i < ArrLen; ++i) {
int bEqual = (Number == Array[i]);
nIndex = bEqual * i + !bEqual * nIndex;
}
return nIndex;
}
Data-dependencies and Learning in Artificial Systems
Search1 is same as Search with the mere exception
that we have replaced the non-deterministic if state-
ment with a series of deterministic arithmetic con-
structs that in the end produce same results. The
advantage with this replacement is that the path of
execution is deterministic and independent of the in-
put array values, thus facilitating us to reorder or even
parallelize the individual iterations. This is possible
because no (i+ 1)th iteration depends on the results
of ith iteration, unlike the case of search where the
(i+ 1)th iteration would be processed only if the ith
iteration fails to find a match.
Demanding a time complexity of O(n) in all cases,
it might appear that Search1 is inferior to Search in
performance. However, for this small cost of perfor-
mance we are gaining two invaluable properties that
are crucial for our present discussion: stability and
predictability.
It is a well-known phenomenon in the practice of
learning algorithms that the performance of learner is
highly affected by the order of the training data sam-
ples, making the learner unstable and at times unpre-
dictable. In this regard, what relation could one infer
between the stability of the learner and the depen-
dencies among data samples? How does such relation
affect the performance of learner? Can these depen-
dencies be analyzed in a formal framework to assist
the learning? These are some of the issues that we try
to address in the following.
2. Learning in Artificial Systems
By an artificial system we essentially mean a man-
made system that has a software module, commonly
known as agent, as one of its components. The artifi-
cial system itself could be a software program such as a
simulation program in a digital computer, or it could
be a hardware system such as an autonomous robot
in the real world. And there could be more than one
agent in an artificial system. The system can use the
agents in many ways as to steer the course of simula-
tion or to process the environmental inputs (or events)
and take the necessary action etc. . . . Additionally,
the functionality of agents could be static, i.e. does
not change with experience, or it could be dynamic,
varying with experience. The literature addressing
these can be broadly classified into two classes, namely
the theories that study the agents as pre-programmed
units (such as (Reynolds, 1987; Ray, 1991)), and the
theories that consider the agents as learning units
which can adjust their functionality based on their
experience (e.g. (Brooks, 1991; Ramamurthy et al.,
1998; Cliff & Grand, 1999)). The present discussion
Figure 1. Different paths indicate different algorithms to
solve a task instance in configuration space
falls into the second category. We discuss a learning
mechanism for agents based on the notion of data-
dependencies.
Consider an agent that is trying to accomplish a task,
such as solving a maze or sorting the events based on
priority etc..., in an artificial system.
Assume that the instantaneous configuration (the cur-
rent state) of artificial system is described by ngener-
alized coordinates q1, q2, . . . , qn,which corresponds to
a particular point in a Cartesian hyperspace, known
as the configuration space, where the q’s form the n
coordinate axes. As the state of the system changes
with time, the system point moves in the configuration
space tracing out a curve that represents ”the path of
motion of the system”.
In such a configuration space, a task is specified by a
set of system point pairs representing the initial and
final configurations for different instances of the task.
An instance of the task is said to be solvable if there
exists an algorithm that can compute the final con-
figuration from its initial configuration. The task is
said to be solvable if there exists an algorithm that
can solve all its instances.
Each instance of the algorithm solving an instance of
the task represents a path of the system in the con-
figuration space between the corresponding initial and
final system points. If there exists more than one al-
gorithm to solve the task then naturally there might
exist more than one path between the two points.
The goal of an agent that is trying to learn a task in
such a system is to observe the algorithm instances and
infer the algorithm. In this regard, all the information
that the agent would get from an algorithm instance
is just an initial-final configuration pair along with a
series of configuration changes that lead from initial
configuration to final configuration. The agent would
not be aware of the details of the underlying process
Data-dependencies and Learning in Artificial Systems
that is responsible for these changes, and has to infer
the process purely based on the observations.
The agent is said to have ”learned the task” if it can
perform the task on its own, by moving the system
from any given initial configuration to the correspond-
ing final configuration in the configuration space. It
should be noted that the procedure used (the algo-
rithm inferred) by the agent may not be the same
as the original algorithm from whose instances it has
learned.
It should also be noted that the notion of learning the
task, as described above, does not allow any proba-
bilistic or approximate solutions. The agent should be
able to perform the task correctly under all circum-
stances. An additional constraint that we put on the
agent is that it should infer the algorithm from as few
algorithmic instances as possible. This is important
for agents of both real world systems and simulation
systems alike, for in case of agents observing samples
from real world environment it may not be possible
to pickup as many samples as they want, and in case
of simulated environments each sample instance incurs
an execution cost in terms of time and other resources
and hence should be traded sparingly.
We formalize these concepts in the following.
2.1. A Formal Framework
Consider an agent that it trying to learn a task Tin a
system Swhose configuration space is given by
C(S) = {~s1, ~s2, . . . , ~sN},
where each ~siis a system point represented with
n−coordinates {qi1, qi2, . . . , qin}.
Let Abe an algorithm to solve the task T, and
A1, A2, . . . , Akbe the instances of Asolving the in-
stances T1, T2, . . . , Tkof Trespectively.
In the configuration space each Tiis represented by a
pair of system points (~si1, ~si2), and the corresponding
Aiby a path between those system points.
Let I, F be two operators that when applied to an
algorithm instance Ai,yield the corresponding initial
and final system points respectively, such as I(Ai) =
~si1and F(Ai) = ~si2. We also define the corresponding
set versions of these operators ~
Iand ~
Fas following.
For all A0⊆ {A1, A2, . . . , Ak},
~
I(A0) = {I(Ai)|Ai∈A0},
and
~
F(A0) = {F(Ai)|Ai∈A0}.
The goal of the agent is to perform T, by mimicking
or modeling A, inferring A’s details from a subset of
its instances.
By following the tradition of learning algorithms, let
us call Aas the target concept, and the subset of its in-
stances D={D1, D2, . . . , Dd} ⊆ {A1, A2, . . . , Ak}as
the training set or data samples, and the agent as the
learner. We use the symbol Lto denote the learner.
At any instance during the phase of learning the set
Dcan be partitioned into two subsets O, O0such that
(O∪O0=D)∧(O∩O0=∅).
The set O⊆Ddenotes the set of data samples that
the learner has already seen, and the set O0⊆Dde-
notes the set of data samples the learner has yet to see.
Learning progresses by presenting the learner with an
unseen sample Di∈O0,and marking it as seen, by
moving it to the set O. Starting from O=∅, O0=D,
this process of transition would be repeated till it be-
comes O=D, O0=∅.
In this process, each data sample Didecreases the ig-
norance of the learner Labout the target concept A,
and hence could be assigned some specific informa-
tion content value that indicates how much additional
information Lcan gain from Diabout A.
We can determine the information content values of
data samples by establishing the concept of a zone,
where we treat an ensemble of system points that share
a common relation as a single logical entity.
Definition. A set Z⊆ C(S) defines a zone if there
exists a function f:C(S)→ {0,1}such that for each
~si∈ C(S) :
f(si) = 1 if ~si∈Z,
0 if ~si/∈Z.
The function fis called the characteristic function of
Z.
For the configuration space C(S) = {~s1, ~s2, . . . , ~sm},
we can construct an equivalent zone-space Z(S) =
{Z1, Z2, . . . , Zr},such that the following holds.
∀~si∃Zi[~si∈Zi]∧ ∀r
i=1∀r
j=i+1 [Zi∩Zj=∅]∧
∀r
i=1 [Zi6=∅]∧ ∪r
i=1Zi=C(S).
The zone-space can be viewed as a m−dimensional
space with each Zibeing a point in it, where mis some
function of nwhose value depends upon and hence
would be decided by the nature of T. Let the range of
ith coordinate of this m−dimensional space be [0, ri].
Data-dependencies and Learning in Artificial Systems
If we use the notation |P|to indicate the size of any
set P, then we could represent the volume of the zone-
space as
vol(Z(S)) = |Z (S)|=
m
Y
i=1
ri.
Define an operator ∇:C(S)→ Z(S) that when ap-
plied to a system point in the configuration space
yields the corresponding zone in the zone-space. Sim-
ilarly, let ~
∇be the corresponding set version of this
operator defined as, for all S0⊆ C(S),
~
∇(S0) = {∇(~si)|~si∈S0}.
At any instance the knowledge of Labout Adepends
on the set of samples it has seen till then, and the infor-
mation content of a data sample depends on whether
the sample has already been seen by Lor not.
To define formally, the knowledge of the learner, after
having seen a set of samples O⊆D, is given by
KO(L) = X
Z∈~
∇(~
I(O))
|Z|.
The information content of any data sample Di∈D,
after the learner has seen a set of samples O⊆D, is
given by
ICO(Di) =
|∇(I(Di))|if ∀Dj∈O[∇(I(Di)) 6=
∇(I(Dj))];
0 otherwise;
and the information content of all data samples would
be given by
−→
ICO(D) = X
Z∈~
∇(~
I(D−O))
|Z|.
It should be noted that the above definitions measure
the information content of data samples relative to the
state of the learner and satisfy the limiting conditions
K∅(L) = 0,−→
ICD(D) = 0.
The process of learning is essentially a process of trans-
fer of information from data samples to the learner,
resulting in a change in the state of the learner. When
these changes are infinitesimal, spanning many steps,
the transformation process satisfies the condition that
the line integral
L=ZD
∅
E do, (2.1)
where E=−→
ICO(D)− KO(L),has a stationary value.
This is known as the Hamilton’s principle, which states
that out of all possible paths by which the learner
could move from K∅(L) to KD(L),it will actually
travel along that path for which the value of the line
integral (2.1) is stationary. The phrase ”stationary
value” for a line integral typically means that the in-
tegral along the given path has same value to within
first-order infinitesimals as that along all neighboring
paths (Goldstein, 1980; McCauley, 1997).
We can summarize this by saying that the process of
learning is such that the variation of the line integral
Lis zero.
δL=δZD
∅
E do = 0.
Thus we can formulate the following conservation the-
orem.
Theorem 1. The sum KO(L) + −→
ICO(D)is conserved
for all O⊆D.
Proof. We shall prove this by establishing that
KOi(L) + −→
ICOi(D) = KOj(L) + −→
ICOj(D) for all
Oi, Oj⊆D.
Consider O1, O2⊆Dsuch that |O2|−|O1|= 1.Let
O2−O1={Di}.To calculate the information content
value of Di,we need to consider two cases.
Case 1. ∇(I(Di)) = ∇(I(Dj)) for some Dj∈O1.
In such case, ~
∇(~
I(O2)) = ~
∇(~
I(O1)),and hence
ICO1(Di) = ICO2(Di) = 0.
KO2(L) = PZ∈~
∇(~
I(O2)) |Z|
=PZ∈~
∇(~
I(O1)) |Z|
=KO1(L).
−→
ICO2(D) = −→
ICO1(D)−I CO1(D)
=−→
ICO1(D).
KO2(L) + −→
ICO2(D) = KO1(L) + −→
ICO1(D).
Case 2. ∇(I(Di)) 6=∇(I(Dj)) for all Dj∈O1.In
such case, ICO1(Di) = |∇(I(Di))|.
−→
ICO2(D) = −→
ICO1(D)−I CO1(Di)
=−→
ICO1(D)− |∇(I(Di))|.
KO2(L) = PZ∈~
∇(~
I(O2)) |Z|
=PZ∈~
∇(~
I(O1+{Di})) |Z|
=X
Z∈~
∇(~
I(O1))
|Z|+X
Z∈{∇(I(Di))}
|Z|
=KO1(L) + |∇(I(Di))|.
KO2(L) + −→
ICO2(D) = KO1(L) + |∇(I(Di))|+
−→
ICO1(D)− |∇(I(Di))|
=KO1(L) + −→
ICO1(D).
Data-dependencies and Learning in Artificial Systems
Thus whenever |O2|−|O1|= 1,it holds that
KO2(L) + −→
ICO2(D) = KO1(L) + −→
ICO1(D).
Now consider two sets Oi, Oj⊆Dsuch that |Oj| −
|Oi|=l, l > 1.Let Oj−Oi={Dj1, . . . , Djl}.We can
construct sets P1, . . . , Pl−1such that
P1=Oi∪ {Dj1}, . . . , Pl−1=Oi∪ {Dj1, . . . , Djl−1}.
Then it holds that
|P1|−|Oi|=|P2|−|P1|=· · · =|Oj|−|Pl−1|= 1.
However, we have proved that
KO2(L) + −→
ICO2(D) = KO1(L) + −→
ICO1(D)
whenever |O2|−|O1|= 1,and hence it follows that
KOi(L) + −→
ICOi(D) = KP1(L) + −→
ICP1(D),
KP1(L) + −→
ICP1(D) = KP2(L) + −→
ICP2(D),
.
.
.
KPl−1(L) + −→
ICPl−1(D) = KOj(L) + −→
ICOj(D),
and thereby, KOi(L) + −→
ICOi(D) = KOj(L) +−→
ICOj(D).
Hence the sum KO(L) + −→
ICO(D) is conserved for all
O⊆D.
An important consequence of this theorem is that ir-
respective of the order of individual samples that L
chooses to learn from, the gain in its knowledge would
always be equal to the corresponding loss in the infor-
mation content of the data samples.
KOj(L)− KOi(L) = −→
ICOi(D)−−→
ICOj(D).
We now define two metrics – quality and quantity –
for the data samples to denote the notions of necessity
and sufficiency.
The metric quality measures the relative information
strength of individual samples, defined as
quality(D) =
|D| −
~
∇(~
I(D))
|D|×100%.
Ideally a data sample set should have this value to be
100%.Smaller values indicate the presence of unnec-
essary samples that do not contribute to learning.
Similarly we define the quantity of data samples as
quantity(D) =
~
∇(~
I(D))
|Z(S)|×100%.
This is a sufficiency measure and hence a value less
than 100% indicates the insufficiency of data samples
to complete the learning.
Theorem 2. The knowledge of the learner, after
completing the learning over data samples Dhaving
quantity(D) = 100%,would be equal to the volume of
the configuration space |C(S)|.
Proof. When the quantity(D) = 100%,
~
∇(~
I(D))
=|Z(S)|.
Since ~
∇(~
I(D)) ⊆ Z(S),
~
∇(~
I(D))
=|Z(S)| ⇒ ~
∇(~
I(D)) = Z(S).
From theorem 1 we have,
KD(L) + −→
ICD(D) = K∅(L) + −→
IC∅(D).
Since K∅(L) = 0 and −→
ICD(D) = 0,
KD(L) = −→
IC∅(D)
=PZ∈~
∇(~
I(D−∅)) |Z|
=PZ∈~
∇(~
I(D)) |Z|
=PZ∈Z(S)|Z|
=|C(S)|.
Thus when quantity(D) = 100%,KD(L) = |C(S)|.
Theorem 3. The target concept can not be learnt with
less than |Z(S)|number of data samples.
Proof. Consider a data sample set D={D1, . . . , Dd}
having quantity(D) = 100% and |D|<|Z(S)|.
Let P=Z(S)−~
∇(~
I(D)) = {P1, . . . , Pl}, l ≥1.As-
sume that Lhas learned the target concept completely
from D. Then, by theorem 2, KD(L) = |C(S)|,and by
theorem 1,
KD(L) + −→
ICD(D) = K∅(L) + −→
IC∅(D).
Since K∅(L) = 0 and −→
ICD(D) = 0,it leads to
|C(S)|=−→
IC∅(D)
=PZ∈~
∇(~
I(D−∅)) |Z|
=PZ∈~
∇(~
I(D)) |Z|
=PZ∈(Z(S)−P)|Z|
=PZ∈Z(S)|Z| − PZ∈P|Z|
=|C(S)| − PZ∈P|Z|.
This is not possible unless P=∅,in which case it
would become Z(S) = ~
∇(~
I(D)),and |D| ≥ |Z(S)|.
Hence proved.
Data-dependencies and Learning in Artificial Systems
2.2. A Learning Mechanism Based on
Data-dependencies
Consider a task instance T1with end points (~s1, ~s2) in
the configuration space. If we express T1as a point
function f, then we could write ~s2=f(~s1).An algo-
rithm instance A1that solves T1would typically im-
plement the functionality of fthereby representing a
path between ~s1and ~s2.If there exists more than one
way to implement f, then there exists more than one
path between ~s1and ~s2.Such a set of paths might be
denoted by f(~s1, α) with f(~s1,0) representing some
arbitrary path chosen to be treated as reference path.
Further, if we select some function η(~x) that vanishes
at ~x =~s1and ~x =~s2,then a possible set of varied
paths is given by
f(~x, α) = f(~x, 0) + α η(~x).
It should be noted that all these varied paths terminate
at the same end points, that is, f(~x, α) = f(~x, 0) for
all values of α.
However, when we try to consider another task in-
stance T2to be represented with these variations, we
need to make them less constrained. The tasks T1and
T2would not have the same end points in the con-
figuration space and hence there would be a variation
in the coordinates at those points. We can, however,
continue to use the same parameterization as in the
case of single task instance, and represent the family
of possible varied paths by
fi(~x, α) = fi(~x, 0) + α ηi(~x),
where αis an infinitesimal parameter that goes to zero
for some assumed reference path. Here the functions
ηido not necessarily have to vanish at the end points,
either for the reference path or for the varied paths.
Upon close inspection, one could realize that the varia-
tion in these family of paths is composed of two parts.
1. Variations within a task instance due to different
algorithmic implementations.
2. Variations across task instances due to different
initial system point configurations.
The learner can overcome the first type of variations by
observing that the end points, and their corresponding
zones, are invariant to the paths between them. In this
regard, all the system points that belong to the same
initial, final zone pair could be learned with a single
algorithm instance. However, for the second type of
variations, the learner may not be able to overcome
them without any prior knowledge of the task. All the
Figure 2. Schematic illustration of path variations across
task instances in configuration space
different instances of the task would have different cor-
responding zones for their end points and hence they
need to be remembered as they are.
Below we present a mechanism that uses these con-
cepts of variations in the configurations paths to infer
solutions to the unknown problem instances based on
the solutions to the known problem instances. This
results in a learning like behavior where the known
problem-solution configuration pairs form the training
set samples. Such samples could be collected by the
following procedure.
1. For a given problem identify the appropriate con-
figuration space and number of dimensions.
2. Express the solution as a logical relation Rsin
terms of coordinates of the configuration space.
3. Use Rsto identify an appropriate characteristic
function Fsto form a solution zone.
4. Use Fsin deciding the characteristic functions
for other zones and the number of dimensions for
zone-space.
5. Define the operator ∇to map the system points
from configuration space to the zones in zone-
space.
6. Define an appropriate variation operator δin the
configuration space such that variations in the
known problem configurations would give clue to
the variations in the solution configurations, such
as, solution(x+δx) = solution(x) + δx.
7. Construct the sample problem-solution configura-
tion pairs by using any traditional algorithm. The
Data-dependencies and Learning in Artificial Systems
samples should be such that all zones are repre-
sented.
Once we have all the required data samples with us,
the training procedure is simple and straightforward in
that all that is needed is to mark each of the sample
problem configurations as the reference configuration
for the corresponding zone and remembering the re-
spective solution configurations for those references.
We can use a memory lookup table to store these ref-
erence solutions. The procedure is as follows.
For each data sample Di= (~pi, ~si)
{
Let Z=∇(pi);
RefProbConfig[Z] = ~pi;
RefSolConfig[Z] = ~si;
}
Once the training is over, we can compute the solution
configuration ~s for any given problem configuration ~p
in the configuration space as follows.
1. For the given problem configuration ~p, apply the
operator ∇and find the zone Z=∇(~p);
2. Get the reference problem configuration ~pi=
RefProbConfig[Z],and compute the variation
δ(~p, ~pi);
3. Compute the required solution configuration from
the reference solution configuration by applying
the variation parameter as:
~s = RefSolConfig[Z] + δ(~p, ~pi);
2.3. An Example Problem
To explain how these concepts of variations in the con-
figuration paths could be used in practice, we consider
an example problem of sorting. We outline a proce-
dure that implements sorting based on the concepts
we have discussed till now.
The reason behind choosing sorting as opposed to any
other alternative is that the problem of sorting has
been well studied and well understood, and requires no
additional introduction. However, it should be noted
that our interest here is, rather to explain how zones
can be constructed and used for the sorting problem,
than to propose a new sorting algorithm; and hence
we do not consider any performance comparisons. In
fact, the procedure we outline below runs with O(n2)
time complexity requiring O(2n2) memory, thus any
performance comparisons would be futile.
To start with, we can consider the task of sort-
ing as being represented by its instances such as
{(3,5,4),(3,4,5)},where the second element (3,4,5)
represents the sorted result of first element (3,5,4).
We can consider these elements as points in a
3−dimensional space.
Thus in general given an array of nintegers to be
sorted, we can form a system with n−coordinate axes
resulting in an n−dimensional configuration space. If
we assume that each element of the array can take
a value in the range [0, N],where Nis some maxi-
mum integer value, then there would be a total of Nn
system points in the configuration space. That is, fol-
lowing our notation from section 2.1, |C(S)|=Nn.To
construct the corresponding zone-space for this con-
figuration space, consider the following mathematical
specification for sorting,
∀n
i=1∀n
j=1 [i < j ⇒a[i]< a[j] ],
where ais an array with nintegers. This specifica-
tion represents a group of conditions that need to be
satisfied by the array if it has to considered as being
in sorted order. Now, we can use this specification in
identifying the following.
1. Number of dimensions of zone-space: The spec-
ification involves two quantifiers ∀n
i=1 and ∀n
j=1,
with an additional constraint i < j. Thus the valid
values could be i= 1, . . . , n, j =i+ 1, . . . , n, re-
sulting in a group of n×(n−1)/2 conditions to
be accounted for. Each condition would form one
coordinate axis in the zone-space and hence we
have n×(n−1)/2 axes.
2. Range of each axis of zone-space: Since each axis
is formed out of the condition (a[i]< a[j]),with
various values of i, j representing various axes, the
range of each axis would be defined by the number
of possible conditions (a[i]< a[j]),(a[i] = a[j])
and (a[i]> a[j]),which is three. Hence the range
of each axis ri= 3.
3. Operator ∇: Each zone is a point in zone-space
with n×(n−1)/2 coordinates. To find these
coordinate values we need to evaluate n×(n−1)/2
conditions (one for each axis) as below.
for(int i=0,r=0; i<n; ++i)
for(int j=i+1; j<n; ++j,++r)
ZCoord[r]=((a[i]=a[j])?0:((a[i]<a[j])?1:2));
4. Variation operator δ: We implement the variation
operator using the differences between relative ar-
ray positions of numbers before and after sorting.
We can use the array indexing and de-indexing
Data-dependencies and Learning in Artificial Systems
operations for this purpose. For example, sorting
a={3,5,4}produces ~a ={3,4,5},which gives us
a variation in the indices of elements from (0,1,2)
to (0,2,1).Thus we can use our variation operator
to express ~a as, ~a ={a[0], a[2], a[1] }.
Once we have these necessary operators with us, we
can start assigning the reference (unsorted, sorted)
configuration pairs for each zone by using any tradi-
tional sorting algorithm such as heapsort or quicksort,
as shown below.
for(int i=0; i <nSamples; ++i)
{
GetZCoord(Unsorted[i], ZCoord);
quicksort(Unsorted[i], Sorted[i]);
SetRefConfig(ZCoord, Unsorted[i], Sorted[i]);
}
It should be noted that we have 3n×(n−1)/2zones in
the zone-space and hence we need so many sample
(unsorted, sorted) pairs as well. However, once we
complete the training with all those samples, we can
use the following procedure to sort any of the Nnpos-
sible arrays.
void LSort(int nArray[], int nSize, int nSorted[])
{
GetZCoord( nArray, ZCoord );
GetRefConfig( ZCoord, RefProb, RefSol);
for(int i=0; i <nSize; ++i)
nSorted[i] = nArray[RefSol[i]];
}
2.4. Limitations
Having presented the mechanism for learning based on
the concepts of variations in the system configuration
paths, here we discuss the limitations of this approach.
Disadvantages: 1. As could be easily understood,
the concepts of configuration space and zone-
space form the central theme of this ap-
proach. However, it may not be always pos-
sible to come up with appropriate configura-
tion space or zone-space for any given prob-
lem. In fact, for many tasks such as face
recognition etc. . . we readily do not have any
clues for logical relations among the data at-
tributes. This is one of the biggest limitations
of this approach.
2. To present the learner with some sample con-
figurations, we assumed the existence of an
algorithm that could solve the task at hand.
However, this assumption may not hold at
all times. Once again, face recognition is an
example.
3. The memory requirements are too high. We
have already seen that we need 3n×(n−1)/2
samples to correctly learn the sorting task.
However, given the goal of mimicking a human being
and the scope of abstract concepts the agents have
to learn from human beings, and given the virtually
unlimited number of problem instances that could be
solved by this learning mechanism, the memory re-
quirements should not become a problem at all (note
that the memory requirements do not depend on N
but on n, so there is no upper limit to the number of
problem instances that can be solved correctly). Fur-
ther advantages are as follows.
Advantages: 1. Independent of the order of train-
ing data samples. In this method, the learner
is invariant to the order in which it receives
the data samples. All that a learner does with
a data sample is, compute the corresponding
zone and mark the sample as a reference for
that zone. This process clearly is indepen-
dent of the order of the data samples and
hence gives the same results in all circum-
stances. It should be noted that the tradi-
tional learning algorithms does not guarantee
any such invariance.
2. Additional samples do not create any bias. If
there exists more than one sample per zone,
the characteristic functions of zones guaran-
tee that they all would produce the same re-
sults as that of first sample. Hence the train-
ing would not be biased by the presence of ad-
ditional samples. Further, a sample could be
repeated as many times as one wants without
affecting the training results. This is useful
for situations where a robot might be learn-
ing from real world, where some typical ob-
servations (such as the changing traffic lights,
flow of vehicles etc.. . ) would get repeated
more frequently compared with some rare ob-
servations (such as earth quakes or accidents
etc. . . ). Traditional learning algorithms fail
to provide unbiased results in such situations.
3. Non-probabilistic metrics and accurate re-
sults. To meet the demands of artificial sys-
tems, the metrics we have devised are com-
pletely deterministic and are void of any
probabilistic assumptions and thus can be
adapted to any suitable system.
Data-dependencies and Learning in Artificial Systems
4. Expandable to multi-task learning. Though
we have concentrated on learning a single
task in this discussion, there is nothing in this
method that could prevent the learner from
learning more than one task at the same time.
For example, once an agent learns to sort
in ascending order (SASC), it can further
learn to sort in descending order (SDSC )
simply by computing the new variation op-
erator δSDS C directly from (δSAS C ),instead
of from new sample problem-solution config-
uration pairs. This saves the training time
and cost for SDSC. However, to implement
this feature the agent should be informed of
the relation between the tasks a priori. Smart
agents that can automatically recognize the
relation among tasks based on their configu-
ration spaces should be an interesting option
to explore further in this direction.
5. Knowledge transfer. All the knowledge of
the learner is represented in terms of refer-
ence configurations for individual zones. Any
learner who has access to these reference con-
figurations can perform equally well as the
owner of the knowledge itself, without the
need to go through all the training again.
This could lead to the concept of tradable
knowledge resources for agents.
6. Perfect partial learning. Just as additional
samples do not create bias, lack of samples
also would not create problems for learning.
A training set with quantity less than 100%
would still give correct results as long as the
problem instance at hand is from one of the
learnt zones. That is, whatever the agent
learns, it learns perfectly. This feature comes
handy to implement low cost bootstrapping
robots with reduced features and functional-
ity which can be used as ”data sample sup-
pliers” for other full-blown implementaions.
This concept of bootstrapping robots is one
of the fundamental concepts of artificial life
study in that it might invoke the possibility of
self-replicating robots (Freitas & Gilbreath,
1980; Freitas & Merkle, 2004).
3. Conclusions
The notion of data-independence for an algorithm
speaks for constant execution paths across all its in-
stances. A variation in execution path is generally
attributable to the variations in the nature of data.
When a problem and its solution are viewed as points
in a system configuration, variations in the problem
configurations can be used to study the variations in
the solution configurations and vice versa. These vari-
ations could be used to infer solutions to unknown
instances of problems based on the solutions to the
known instances.
This paper analyzed the problem of data-dependencies
in the learning process and presented a learning mech-
anism based on the relations among data attributes.
The mechanism constructs a Cartesian hyperspace,
namely the configuration space, for any given task, and
finds a set of paths from the initial configuration to fi-
nal configuration that represents different instances of
the task. As part of the learning process the learner
gradually gains information from data samples one by
one, till all data samples were processed. Once such
a transfer of information is complete, the learner can
solve any instance of the task without any restrictions.
The mechanism presented is independent of the order
of data samples and has the flexibility to be expand-
able to multi-task learning. However, the practical-
ity of this approach may be hindered by the lack of
appropriate algorithms that could provide sample in-
stances. Further study to eliminate such bottlenecks
could make this a perfect choice to implement learning
behavior in artificial agents.
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