T o app ear in Pr o c ICML

Bla z Zupan Mark o Bohanec Iv an Bratk o Janez Dem sar

Jo zef Stefan Institute Jo zef Stefan Institute F acult y of Computer and F acult y of Computer

Ljubljana Slo v enia Ljubljana Slo v enia Information Sciences and and Information Sciences

blazupanjsi mark o ohanecjsi Jo zef Stefan Institute Univ ersit y of Ljubljana

Ljubljana Slo v enia Ljubljana Slo v enia

iv anratk oriniji janezemsarriniji

Abstract selection of examples to induce the classiation rules

usually this is a tiresome pro cess that requires activ e

a v ailabilit y of a domain exp ert o v er long p erio ds of

W e presen t a new mac hine learning metho d

time Considerable impro v emen ts in this resp ect ma y

that giv en a set of training examples induces

b e exp ected from metho ds that automate or at least

a deition of the target concept in terms of a

activ ely supp ort the user in the problem decomp osi

hierarc h y of in termediate concepts and their

tion task

deitions This ectiv ely decomp oses the

problem in to smaller less complex problems

In this pap er w e presen t a metho d for dev eloping a

The metho d is inspired b y the Bo olean func

problem decomp osition hierarc h y from examples and

tion decomp osition approac h to the design

in v estigate its applicabilit y in mac hine learning The

of digital circuits T o cop e with high time

metho d is based on function decomp osition an ap

complexit y of ding an optimal decomp osi

proac h originally dev elop ed for the design of digital

tion w e prop ose a sub optimal heuristic al

circuits shenh urst Curtis The goal is to

gorithm The metho d implemen ted in pro

decomp ose a function y F X in to y G A H B

gram Ierarc h y Induction T o ol is ex

where X is a set of input attributes x x and y is

p erimen tally ev aluated using a set of arti

the class v ariable F G and H are functions partially

cial and real orld learning problems It is

sp ecid b y examples i b y sets of attribute alue

sho wn that the metho d p erforms w ell b oth in

v ectors with assigned classes A and B are subsets of

terms of classiation accuracy and disco v ery

input attributes suc h that A B X The functions

of meaningful concept hierarc hies

G and H are determined in the decomp osition pro

cess and are not predeed in an y w a y Their join t

complexit y etermined b y some complexit y measure

should b e lo w er than the complexit y of F Suc h a de

INTR ODUCTION

comp osition also disco v ers a new in termediate concept

c H B Since the decomp osition can b e applied

T o solv e a complex problem one of the most general

recursiv ely on H and G the result in general is a hier

approac hes is to decomp ose it in to smaller less com

arc h y of concepts F or eac h concept in the hierarc h y

plex and more manageable subproblems In mac hine

there is a corresp onding function uc h as H B that

learning this principle is a foundation for structured

determines the dep endency of that concept on its im

induction hapiro instead of learning a sin

mediate descendan ts in the hierarc h y

gle complex classiation rule from examples dee a

goalubgoal hierarc h y and learn the rules for eac h of

The prop osed decomp osition metho d is limited to

the subgoals Originally Shapiro used structured in

nominal alued attributes and classes It w as imple

duction for the classiation of a fairly complex c hess

men ted in program Ierarc h y Induction T o ol

endgame and demonstrated that the complexit y and

In this pap er w e do not describ e the sp eci noise han

comprehensibilit y brainompatibilit y of the ob

dling mec hanism in

tained solution w as sup erior to the unstructured one

The reminder of the pap er is organized as follo ws

T ypically applications of structured induction in v olv e

Section o v erviews the related w ork The learning

a man ual dev elopmen t of the hierarc h y and a man ualmetho d is describ ed in detail in section and exp er applied b y Shapiro Their approac h is based on

imen tally ev aluated in section on sev eral domains a man ual decomp osition of the problem and an exp ert

of diren t complexit y The pap er is concluded b y a assisted selection of examples to construct rules for the

summary and p ossible directions of further w ork concepts in the hierarc h y In comparison with stan

dard decision tree induction tec hniques structured in

duction exhibits ab out the same classiation accuracy

RELA TED W ORK

with the increased transparency and lo w er complexit y

of the dev elop ed mo dels Mic hie emphasized

The decomp osition approac h to mac hine learning w as

the imp ortan t role of structured induction and listed

used b y a pioneer of artiial in telligence A Sam uel

sev eral real problems that w ere solv ed in this w a y

He prop osed a metho d based on a signature table sys

The concept hierarc h y has also b een used b y a

tem am uel and successfully used it as an ev al

m ultittribute decision supp ort exp ert system shell

uation mec hanism for his c hec k ers pla ying programs

DEX ohanec and Ra jk o vi c There a treeik e

This approac h w as later impro v ed b y Biermann et al

structure of v ariables is deed b y a domain exp ert

Their metho d ho w ev er did not address the

DEX has b een successfully applied in more than

problem of deriving the structure of concepts

realistic decision making problems

A similar approac h had b een deed ev en earlier

The metho d presen ted in this pap er therefore b or

within the area of switc hing circuit design Ashenh urst

ro ws from three diren t researc h areas it shares

rep orted on a unid theory of decomp osition

the motiv ation with structured induction and struc

of switc hing functions His decomp osition metho d w as

tured approac h to decision supp ort while the core

essen tially the same as that of Biermann et al except

of the metho d is based on Ashenh ursturtis func

that it w as used to decomp ose a truth table of a sp eci

tion decomp osition In comparison with related w ork

Bo olean function to b e then realized with standard bi

the presen t pap er is original in the follo wing asp ects

nary gates Most of other related w ork of those times

new metho d for handling m ulti alued attributes and

is rep orted and reprin ted b y Curtis

classes impro v ed decomp osition heuristics empha

Recen tly the Ashenh ursturtis approac h w as sub

sis on generalization ects of decomp osition pa ying

stan tially impro v ed b y researc h groups of M A

strong atten tion to the disco v ery of meaningful con

P erk o wski T Luba and T D Ross P erk o wski et al

cept hierarc hies and exp erimen tal ev aluation on ma

rep ort on the decomp osition approac h for in

c hine learning problems

completely sp ecid switc hing functions Luba

prop oses a metho d for the decomp osition of m ulti

DECOMPOSITION METHOD

v alued switc hing functions in whic h eac h m ulti alued

v ariable is enco ded b y a set of Bo olean v ariables The

authors iden tify the p oten tial usefulness of function

This section presen ts the decomp osition metho d

decomp osition for mac hine learning Goldman et al

First w e in tro duce the metho d b y an example Next

ev aluate FLASH a Bo olean function decom

w e formally presen t the decomp osition algorithm and

p oser on a set of eigh tttribute binary functions and

conclude with a note on the implemen tation

sho w its robustness in comparison with C decision

tree inducer

INTR ODUCTOR Y EXAMPLE

F eature disco v ery has b een at large in v estigated b y

constructiv e induction ic halski P erhaps clos Supp ose a function y F x x x is giv en where x

est to the function decomp osition metho d are the con x and x are attributes and y is the target concept

structiv e induction systems that use a set of existing y x and x can tak e the v alues lo med hi x can

attributes and a set of predeed constructiv e op era tak e the v alues lo hi The function F is partially

tors to deriv e new attributes fahringer Raga sp ecid with a set of examples in T able

v an and Rendell

There are three nonrivial partitions of the at

Within mac hine learning there are other approac hes tributes h x ijh x x i h x ijh x x i and h x ijh x x i

that are based on problem decomp osition but where and three corresp onding decomp ositions y

the problem is decomp osed b y the exp ert and not dis G x H x x y G x H x x and y

co v ered b y a mac hine A w ellno wn example is struc G x H x x These decomp ositions are giv en

tured induction term in tro duced b y Donald Mic hie in Figure The comparison sho ws that

y y

y

lo lo

lo lo

lo lo

lo lo lo lo

lo med

lo hi lo lo

lo hi

lo med

med lo

med med

lo hi

med med

med hi

med med hi lo

hi hi

hi lo hi med

hi hi

hi hi

I

hi med hi hi

hi hi

x c

I

I

c x

c x

lo lo

lo hi

med lo

lo lo

med hi

lo med

hi lo

lo lo

lo hi

hi hi

lo hi

med med

med lo

med hi

I

med hi

hi lo

hi lo

x x

hi hi

I I

x x x x

Figure Three diren t decomp ositions of the example set from T able

x x x y

SINGLETEP DECOMPOSITION

lo lo lo lo

lo lo hi lo

The core of the decomp osition algorithm is a single

lo med lo lo

step de c omp osition whic h giv en a set of examples E

lo med hi med

that partially sp ecify the function c F X and a

lo hi lo lo

lo hi hi hi partition of attributes X to sets A and B decomp oses

med med lo med

F in to c G A c and c H B This is done b y

med hi lo med

constructing the example sets E and E that par

med hi hi hi

tially sp ecify G and H resp ectiv ely X is a set of

hi lo lo hi

attributes x x and c is a new in termediate

hi hi lo hi

concept A is called a fr e e set and B a b ound set suc h

T able Set of examples that partially describ e the that A B X and A B E and E are

function y F x x x disco v ered in the decomp osition pro cess and are not

predeed in an y w a y

The singletep decomp osition starts with the deriv a

Example sets in the decomp osition y

tion of partition matrix

G x H x x are o v erall smaller than those

for the other t w o decomp ositions

Deition Giv en a disjoin t partition of X to A j B

a p artition matrix P is a tabular represen tation of

j

The new concept c H x x uses only three

example set E with all com binations of v alues of at

v alues whereas that for H x x uses four and

tributes in A as ro w lab els and of B as column lab els

that for H x x uses e

Eac h example e E has its corresp onding en try

By insp ecting the example sets for H and G it in P with a ro w index A e and a column index

j

is easy to see that c corresp onds to MIN x x B e P en tries with no corresp onding examples in

j

and y to MAX x c It is harder to in terpret E are denoted with A column a of P is called

j

the sets of examples for G H G and H nonmpt y if there exists e E suc h that B e a

Among the three attribute partitions it is therefore Eac h column in the partition matrix denotes the b e

b eneial to decide for h x ijh x x i and decomp ose ha vior of F when the attributes in the b ound set are

y F x x x to y G x c and c constan t Columns that exhibit the same b eha vior

H x x are called compatible and can b e represen ted with the

same v alue of c An example partition matrix is giv en

x lo lo med med hi hi

in Figure a

x x lo hi lo hi lo hi

lo lo lo lo med lo hi

Deition Columns a and b of partition matrix med med med hi

hi hi hi

P are c omp atible if F e F e for ev ery pair

j

c

of examples e e E with A e A e and

B e a B e b The n um b er of suc h pairs is

denoted d a b

Note that according to this deition the unsp eci

loo

d P en tries are compatible with an y v alue The

j

n um b er of v alues for c corresp onds to the n um b er of

groups of m utually compatible columns The lo w est loi

hii

n um b er of suc h groups is called c olumn multiplicity

and denoted b y A j B It is deriv ed b y the coloring hio medo

of column incompatibilit y graph

medi

Deition Column inc omp atibility gr aph I is a

j

pair V E where eac h nonmpt y column i of P

j

is represen ted with a v ertex v V and an edge

v v E connects t w o v ertices if the corresp ond

Figure P artition matrix with column lab els c for

ing columns of v and v are incompatible

the attribute partition h x ijh x x i and set of exam

ples from T able and corresp onding column in

Then A j B is the n um b er of colors needed to color

compatibilit y graph Colors ab els of the v ertices

I Namely the prop er coloring guaran tees that t w o

j

are circled

v ertices represen ting incompatible columns are not as

signed the same color The same colors are only as

signed to the columns that are compatible Therefore

the v alue of c It is therefore straigh tforw ard to deriv e

the optimal coloring disco v ers the lo w est n um b er of

an example set E from the colored I A ttribute

j

groups of compatible P columns An example of

j

set for these examples is B Eac h v ertex in I is an

j

colored incompatibilit y graph is giv en in Figure b

example in set E Color c of the v ertex is the class

Graph coloring is an NPard problem and the com of the example

putation time of an exhaustiv e searc h algorithm is pro

E is deriv ed as follo ws F or an y v alue of c and com

hibitiv e ev en for small graphs with ab out v ertices

bination of v alues of attributes in A c G A c is

Instead P erk o wski et al suggested a Color In

determined b y lo oking for an example e in ro w A e

ence Metho d of p olynomial complexit y and sho w ed

and in an y column lab eled with the v alue of c If suc h

that the metho d p erformed w ell compared to the opti

example exists an example with attribute set A f c g

mal algorithm The Color Inence Metho d sorts the

and class c F e is added to E

v ertices to color b y their decreasing connectivit y and

then assigns to eac h v ertex a color that is diren t from Decomp osition generalizes ev ery undeed en try

the colors of its neigh b ors so that a minimal n um b er of P in ro w a and column b if a corresp onding

j

of colors is used W e use the same coloring metho d example e with a A e and column B e with

with the follo wing impro v emen t when a color is to the same lab el as b is found F or example an en try

b e assigned to v ertex v and sev eral compatible v er P ioi of partition matrix in Figure a

j

tices ha v e already b een colored with diren t colors w as generalized to hi b ecause the column oi has

the color is c hosen that is used for a group of colored the same lab el as columns oo and io

v ertices v v that are most c omp atible to v The

P

In our implemen tation the incompatibilit y graph is

degree of compatibilit y is estimated as d v v ee

constructed directly from the set of examples a v oiding

Deition for d

the construction of partition matrix for eiency rea

Eac h v ertex in I denotes a distinct com bination of sons The algorithm st sorts the examples E based

j

v alues of attributes in B and its lab el olor denotes on the v alues of attributes in A and v alues of c The

Input Initial set of examples describing

The decomp osition algorithm will decomp ose E and

a single output concept

the function F it partially represen ts only if its decom

Output Its hierarc hical decomp osition

p osed functions G and H are o v erall less complex than

get an initial example set E and mark it decomp osable

F Therefore the partition A j B can b e used to decom

j

p ose E to E and E if and only if A j B F

while decomp osable example set E that partially

W e sa y that example set E is decomp osable if there

sp ecis c F x x with m do

exists a partition A j B with this prop ert y

ev aluate all p ossible partitions A j B of X h x x i

suc h that A B X A B and jj B jj b

select the b est partition A j B

COMPLEXITY OF DECOMPOSITION

if E is decomp osable using A j B then

ALGORITHM

decomp ose E to E and E suc h that

c G A c and c H B where G and H

The time complexit y of single step decomp osition of

are partially sp ecid b y E and E

E to E and E whic h consists of sorting of E

mark E and E decomp osable

j j

deriving the incompatibilit y graph and coloring it is

else mark E nonecomp osable

O N log N O N k O k where N is the n um b er

of examples in E and k is the n um b er of v ertices in

Algorithm The decomp osition algorithm

I F or an y b ound set B the upp er b ound of k is

j

k ax jj x jj where b jj B jj The n um

b er of disjoin t partitions considered b y decomp osition

examples with the same A e constitute groups that

when decomp osing E with m attributes is

corresp ond to ro ws in partition matrix P Within

j

eac h group examples with the same v alue of c con X X

m e m

O m

stitute subgroups Tw o examples that are in the same

j j

group but in diren t subgroups ha v e a corresp onding

edge in I

The highest n um b er of n decomp ositions is required

j

when the hierarc h y is a binary tree where n is the

n um b er of attributes in the initial example set The

DECOMPOSITION ALGORITHM

running time of the decomp osition algorithm is th us

The decomp osition aims to disco v er a hierarc h y of

X

O N log N N k k m

concepts describ ed with example sets that are o v er

all less complex than the initial one Since an exhaus

tiv e searc h is prohibitiv ely complex the decomp osition

O n N log N N k k

uses a sub optimal iterativ e algorithm lgorithm

Therefore the algorithm complexit y is p olynomial

In eac h step the algorithm tries to decomp ose a single

in N n and k Note that the b ound b is a user

example set of the ev olving structure It ev aluates all

deed constan t This analysis clearly illustrates the

p ossible disjoin t partitions of the attributes and selects

b enes of setting b to a suien tly lo w v alue In our

the b est one This step requires a soalled p artition

exp erimen ts b w as set to

sele ction me asur e A p ossible measure is the n um b er of

v alues of the new concept A j B The b est partition

IMPLEMENT A TION

A j B is the one with the lo w est A j B

An alternativ e measure for the selection of partitions The mac hine learning metho d based on function de

is based on the complexit y of function F Let F comp osition w as implemen ted in the C language as a

b e deed on attributes x X with class v ari system called ierarc h y INduction T o ol The

able y In this attributelass space there are a to system runs on sev eral UNIX platforms including HP

Q

jj jj

UX SGI Iris and SunOS The deition of domain

tal of N X y jj y jj p ossible func

names and examples and the guidance of the decom

tions where jj y jj and jj x jj represen t the cardinal

p osition is managed through a script language

ities of v alue sets of y and x resp ectiv ely The

n um b er of bits to enco de F is therefore F

Q

log N X y og jj y jj jj x jj Decom EXPERIMENT AL EV ALUA TION

p osition prefers to disco v er functions of lo w complex

it y so the measure is therefore deed as A j B W e exp erimen tally ev aluated the decomp osition

G H metho d using the follo wing datasetsDomain n N Class names and their

MM A function y MIN x A V G x MAX x

relativ e frequencies

x x with v alued attributes and class

MM

While the deition of MIN and MAX is stan

LENSES hard soft no

dard the function A V G computes the a v erage of

MONK

its argumen ts and rounds it to the closest in teger MONK

CAR unacc acc

good vood

LENSES A small domain tak en from UCI mac hine

NURSER Y unacc acc

learning rep ository urph y and Aha Us

vcc prior

ing patien t age sp ectacle prescription astigma

hrior

tism and tear pro duction rate eac h example de

T able Some c haracteristics of domains used in the

scrib es whether the patien t should w ear soft or

exp erimen ts n is the n um b er of attributes and N the

hard con tact lenses or no lenses at all

dataset size

MONK and MONK W ellno wn sixttribute

binary classiation problems tak en from the

same rep ository urph y and Aha Thrun

w ere split to training and test sets of sizes p and p

et al A ttributes are to v alued

resp ectiv ely for p from to deriv ed

MONK has an underlying concept x x

a concept hierarc h y and corresp onding classir using

OR x and MONK the concept x for

the examples in the training set and w as tested for

exactly t w o c hoices of i f g

classiation accuracy on the test set F or eac h p the

results are the a v erage of randomly c hosen splits

CAR and NURSER Y F or these t w o domains hi

The learning curv e is compared to the one obtained

erarc hical classirs in DEX ohanec and Ra

b y C inductiv e decision tree learner uinlan

jk o vi c formalism already existed These

run on the same data C used the default options

w ere used to obtain a set of examples from whic h

except for whic h w as observ ed to obtain a b etter

decomp osition tried to reconstruct the original hi

classiation accuracy than the default Accuracy

erarc hies CAR ev aluates cars based on their price

is measured on unpruned decision trees for the same

and tec hnical c haracteristics This simple mo del

reason F or eac h p the signiance of the dirence

w as dev elop ed for educational purp oses and is de

b et w een C and is determined using a paired

scrib ed in ohanec and Ra jk o vi c NURS

t est with conence lev el

ER Y is a real orld mo del dev elop ed to rank ap

plications for n ursery sc ho ols la v e et al

The learning curv es are giv en in Figure F or all the

domains other than LENSES outp erforms C

The original datasets are noiseless They completely With more than of examples in the training set

co v er the attribute space for all domains other than this dirence is alw a ys signian t Moreo v er

MONK and MONK where the co v erage is learning curv es con v erge faster to the desired

and resp ectiv ely Some other domain c harac whic h is in turn nev er reac hed b y C F or LENSES

teristics are giv en in T able there are no signian t dirences in the classiation

accuracy of the t w o learners It is also in teresting

The decomp osition used column m ultiplicit y as a par

to note that in MM Cs accuracy decreases with

tition selection measure When the complexit y mea

higher co v erage of example space whic h ma y b e ex

sure w as used instead the results w ere similar and are

plained with decreased generalization

not sho wn here

w as further tested on the data sets for MONK

The b ound set size b w as limited to the maxim um of

and MONK used in the detailed study of ma

three elemen ts The decomp osition times on HP J

c hine learning algorithms hrun et al F or

w orkstation w ere all b elo w seconds for all the do

b oth MONK and MONK the training set w as the

mains other than NURSER Y for whic h required

same as our original data set describ ed ab o v e The

ab out seconds for the largest training sets

t w o test sets used in the study consisted of ex

The exp erimen tal ev aluation addressed the classia amples that completely co v ered the attribute space

tion accuracy of and its abilit y to deriv e a com F or MONK the accuracy of is In the

prehensible and meaningful structure p ossibly simi study hrun et al this score w as ac hiev ed b y

lar to the an ticipated one The classiation accuracy learners three v arian ts of A Q Assistan t Profes

learning curv es w ere computed where the datasets sional mF OIL CN t w o v arian ts of Bac kpropagationclcc clcc clcc

p

p p

MM MONK CAR

clcc

clcc

clcc

p

p

p

LENSES MONK NURSER Y

Figure Learning curv es for olid line with and for C ashed line with When for a sp eci

relativ e training set size p the classiation accuracy of is signian tly b etter than that of C

data p oin ts are mark ed with

and Cascade Correlation F or MONK the accuracy tiv e condition on a b ound and free set it w as imp ossible

of is In the same study four learners to deriv e concepts comparable to the original concept

p erformed b etter A QCI t w o v arian ts of Bac k deition Ho w ev er the disco v ered concept hierarc h y

propagation and Cascade Correlation It should b e is a reform ulation of the target concept using func

noted that these results w ere obtained b y with tions that coun t s F or LENSES disco v ered

out tuning in less than seconds of CPU time on the structure in Figure whic h w e did not try to in ter

HP J w orkstation pret without the domain exp ert F or CAR and NURS

ER Y igures and the structures disco v ered w ere

F or eac h of the domains and with increasing p

v ery similar to the original DEX mo dels In fact they

con v erged to a single concept structure These are

w ere the same except that some original DEX in terme

sho wn in Figures to with the names of attributes

diate concepts w ere further decomp osed It should b e

and concepts and cardinalit y of their v alue sets F or

emphasized that w e consider this similarit y of concept

MM this is the an ticipated structure except for the

structures as a most signian t indicator of success of

concept A V G x MAX x x x whic h ad

our decomp ositionased learning metho d

ditionally decomp osed b y in tro ducing an in termediate

concept c F or MONK disco v ered the an tici

pated hierarc h y MONK F c x c F x x

with F and F matc hing the exp ected disjunctiv e and

equalit y functions F or MONK b ecause of disjuncMM4/4 LENSES/3 MONK2/2

x1/4 c2/4 c1/3 c2/3 c4/4 c3/3

c1/3 c2/3 e/4 f/2

x2/4 c3/7

age/3 prescr/2 astigm/2 tears/2

a/3 b/3 c/2 d/3

x5/4 c1/4

x3/4 x4/4

Figure Structures disco v ered for MM LENSES and MONK domains

car/4 CAR/4

c2/4 c3/4

price/4 tech/4

buying/4 maint/4 safety/3 c1/3

buying/4 maint/4 safety/3 comfort/4

lug_boot/3 c4/4

doors/4 persons/3 lug_boot/3

doors/4 persons/3

Figure Original eft and disco v ered structure igh t for CAR

CONCLUSION The classiation accuracy results ma y b e biased b e

cause w e ha v e mostly used the domains where w e an

W e in tro duced a new mac hine learning approac h based ticipated the hierarc hies disco v erable b y decomp osi

on function decomp osition A distinguishing feature tion Ho w ev er MONK is a coun ter example where

of this approac h is its capabilit y to disco v er new in

decomp osition w as not able to disco v er the original

termediate concepts organize them in to a hierarc hi

deition of the target concept but rather unexp ect

cal structure and dee the relationships b et w een the

edly its reform ulation

attributes newly disco v ered concepts and target con

The decomp osition approac h as presen ted in this pa

cept In their basic form these relationships are sp ec

p er is limited b y that there is no sp ecial mec hanism

id b y newly constructed example sets In a w a y

for handling noise and con tin uous attributes Ho w

the learning pro cess can th us b e view ed as a pro cess

ev er preliminary results on using an extended v ersion

of generating new equiv alen t example sets whic h are

of decomp osition for con tin uously alued data sets in

consisten t with the original example set The new sets

em sar et al and preliminary results on noise

are smaller ha v e smaller n um b er of attributes and

handling extension strongly encourage further dev el

in tro duce in termediate concepts Generalization also

opmen ts in this direction

o ccurs in this pro cess

W e ha v e ev aluated the decomp ositionased learning

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