approximate reasoning using anytime algorithms

19 Pages·1994·0.12 MB·English

1
APPROXIMATE REASONING
USING ANYTIME ALGORITHMS
ShlomoZilbersteinandStuartRussell*
DepartmentofComputerScience
UniversityofMassachusetts,Amherst,MA01003
*DepartmentofEECS,ComputerScienceDivision
UniversityofCalifornia,Berkeley,CA94720
ABSTRACT
The complexityof reasoning in intelligent systemsmakes it undesirable, and some-
times infeasible, to find the optimal action in every situation since the deliberation
process itself degrades the performanceof the system. The problem is then to con-
structintelligentsystemsthatreacttoasituationafterperformingthe“right”amount
of thinking. It is by now widely accepted that a successful system must trade off
decisionqualityagainstthecomputationalrequirementsofdecision-making. Anytime
algorithms,introducedbyDean,Horvitzandothersinthelate1980’s,weredesigned
tooffersuchatrade-off. Wehaveextendedtheirworktotheconstructionofcomplex
systemsthatarecomposedofanytimealgorithms. Thispaperdescribesthecompila-
tionandmonitoringmechanismsthatarerequiredtobuildintelligentsystemsthatcan
efficientlycontroltheirdeliberationtime. Wepresenttheoreticalresultsshowingthat
thecompilationandmonitoringproblemsare tractableinawiderangeofcases, and
providetwoapplicationstoillustratetheideas.
1 INTRODUCTION
A fundamental problem in computer science and artificial intelligence is the
construction of systems that can operate robustly in a variety of real-time
environments. A real-time environment can be characterized by a time-
dependent utility function. In almost all cases, the deliberation required to
select optimal actions will degrade the system’s overall utility. It is by now
well-understood that a successful system must trade off decision quality for
deliberationcost[1,12,14,19,21,22].
1
2 (cid:0)(cid:2)(cid:1)(cid:4)(cid:3)(cid:6)(cid:5)(cid:8)(cid:7)(cid:10)(cid:9)(cid:12)(cid:11)(cid:14)(cid:13)
The problem of deliberation cost has been widely discussed in artificial in-
telligence, economics, engineering and philosophy. In artificial intelligence
in particular, researchershave proposeda numberof meta-levelarchitectures
to control the cost of base-level reasoning [4, 7, 9, 12, 18]. One promising
approach is to use anytime [3] or flexible [10] algorithms, which allow the
execution time to be specified, either as a parameter or by an interrupt, and
exhibit a time/quality tradeoff defined by a performance profile. They pro-
vide a simple means by which a system can control its deliberation without
significantoverhead.
Soonaftertheintroductionofanytimealgorithms,itbecameapparentthattheir
compositionpresents a vital, non-trivial problem [3]. We show that modular
composition of anytime algorithms can be achieved, hence the advantages
of anytime algorithms can be extended to the design of complex real-time
systems with many components. In standard algorithms, the fixed quality of
the output allows for composition to be implemented by a simple call-return
mechanism. Whenalgorithmshaveresourceallocationasadegreeoffreedom,
and can be interrupted at any time, the situation becomes more complex.
Consider the following simple example: a real-time medical expert system
containing a diagnosis component which passes its results to a treatment-
planningcomponent. Thefollowingissuesarise:
1. Howcantheindividualcomponentsbedesignedasanytimealgorithms?
2. How can their performance be described as a function of time and the
natureoftheinputs?
3. How does the output quality of the treatment component depend on the
accuracyofthediagnosisitreceives?
4. Whatsortofprogramminglanguageconstructsareneededtospecifyhow
thesystemisbuiltfromitscomponents?
5. Foranygivenamountoftime,howshouldthattimebeallocatedtoeach
ofthecomponents?
6. What if theconditionof thepatient suddenlyrequiresinterventionwhile
the diagnosis component is still running and no treatment has been con-
sidered?
7. How should the execution of the compositesystem be managed so as to
optimize overall utility, particularlywhen the total execution time is not
knowninadvance?
ApproximateReasoningUsingAnytimeAlgorithms 3
In several publications, particularly [23], we address these issues in some
depth. Here, we sketch our general approach, focusing in particular on the
compilationproblem. Givenasystemcomposedofanytimealgorithms,com-
pilation determines off-line the optimal allocationof time to the components
for any given total allocation. The crucial meta-level knowledge for solving
this problem is kept in the anytime library in the form of conditionalperfor-
manceprofiles. Theseprofilescharacterizetheperformanceofeachelementary
anytimealgorithmasafunctionofrun-timeandinputquality.
In Section2, we define thebasic propertiesof anytimealgorithms. We show
howtoconstructanytimealgorithmsandhowtocharacterizethetrade-offthat
theyofferbetweenqualityofresultsandcomputationtime. Section3explains
thebenefitsanddifficultiesinvolvedinthecompositionofanytimealgorithms.
Sections 4 and 5 describe the two main components of our solution to the
composition problem, namely off-line compilation and run-time monitoring.
Section 6 describes briefly two applications of this approach. Finally, in
Section 7, we summarize the benefits of our approach and discusses some
directionsforfurtherworkinthisfield.
2 ANYTIMEALGORITHMS
The term “anytime algorithm” was coined by Dean in the late 1980’s in the
context of his work on time-dependent planning. Anytime algorithms are
algorithms whose quality of results improves gradually as computation time
increases,hencetheyofferatradeoffbetweenresourceconsumptionandoutput
quality.
Variousmetricscanbeusedtomeasurethequalityofaresultproducedbyan
anytime algorithm. From a pragmatic point of view, it may seem useful to
defineasingletypeofqualitymeasuretobeappliedtoallanytimealgorithms.
Such a unifying approach may simplify the meta-level control. However, in
practice,differenttypesofanytimealgorithmstendtoapproachtheexactresult
incompletelydifferentways. Thefollowingmetricshavebeenproveduseful
inanytimealgorithmconstruction:
1. Certainty – this metric reflects the degree of certainty that the result is
correct. The degree of certainty can be expressed using probabilities,
certaintyfactors,oranyotherapproach.
4 (cid:0)(cid:2)(cid:1)(cid:4)(cid:3)(cid:6)(cid:5)(cid:8)(cid:7)(cid:10)(cid:9)(cid:12)(cid:11)(cid:14)(cid:13)
2. Accuracy – this metric reflects a measure of the difference between the
approximate result and the exact answer. Many anytime algorithms can
provideaguaranteeaboundontheerror,wheretheboundisreducedover
time.
3. Specificity – this metric reflects the level of detail of the result. In this
case,theanytimealgorithmalwaysproducescorrectresults,butthelevel
ofdetailisincreasedovertime.
Many existing programming techniques produce useful anytime algorithms.
Examplesincludeiterativedeepeningsearch,iterativeimprovementalgorithms
innumericalcomputation,variableprecisionlogic,andrandomizedtechniques
suchasMonteCarloalgorithmsorfingerprintingalgorithms. Forasurveyof
suchprogrammingtechniquesandexamplesofalgorithmssee[23].
The notion of interrupted computationis almost as old as computationitself.
Traditionally, interruptionwas used primarily for two purposes: aborting the
executionof an algorithmwhose resultsare no longernecessary,or suspend-
ing the execution of an algorithm for a short time because a computation of
higher priority must be performed. Anytime algorithms offer a third type of
interruption: interruption of the execution of an algorithm whose results are
considered “good enough” by their consumer. Similar ideas motivated the
developmentoftheCONCORDsystemforimprecisecomputation[15].
2.1 Conditionalperformanceprofiles
Toallowforefficientmeta-levelcontrolofanytimealgorithms,wecharacterize
theirbehaviorbyconditionalperformanceprofiles(CPP)[24]. Aconditional
performanceprofilecapturesthedependencyofoutputqualityontimealloca-
tionaswellasoninputquality. In[23],thereadercanfindadetaileddiscussion
of varioustypes of conditionalperformanceprofiles and their representation.
To simplify the discussionof compilation, we will refer only to the expected
CPPthatmapscomputationtimeandinputqualitytotheexpectoutputquality.
Definition1 The conditional performance profile (CPP), of an algorithm
(cid:0) is a function that maps input quality and
(cid:1)(cid:3)(cid:2)(cid:4)(cid:2)(cid:6)(cid:5) (cid:7)(cid:9)(cid:8)(cid:11)(cid:10)(cid:13)(cid:12)(cid:15)(cid:14)(cid:17)(cid:16)(cid:19)(cid:18)(cid:21)(cid:20) (cid:8)(cid:23)(cid:22)(cid:25)(cid:24)(cid:27)(cid:26)
computationtimetotheexpectedqualityoftheresults.
ApproximateReasoningUsingAnytimeAlgorithms 5
q
1.00
0.90
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.00 1.00 2.00 3.00 4.00 t
Figure1 GraphicalrepresentationofaCPP
Figure1showsatypicalCPP.Eachcurverepresentstheexpectedoutputquality
asafunctionoftimeforagiveninputquality.
2.2 Interruptibleandcontractalgorithms
In [20] we make an important distinction between two types of anytime al-
gorithms, namely interruptible and contract algorithms. An interruptible al-
gorithm can be interrupted at any time to produce results whose quality is
described by its performance profile. A contract algorithm offers a similar
trade-off between computation time and quality of results, but it must know
the total allocation of time in advance. If interrupted at any point before the
termination of the contract time, it may yield no useful results. Interruptible
algorithms are in many cases more appropriate for the application, but they
are also more complicated to construct. In [20] we show that a simple, gen-
eral construction can produce an interruptible version for any given contract
algorithm. Furthermore, the interruptible algorithm requires at most sec-
(cid:0)(cid:2)(cid:1)
onds to produceresults of the qualityachievedby the contractalgorithmin
(cid:1)
seconds,forall . Thistheoremallowsustoconcentrateontheconstructionof
(cid:1)
contractalgorithmsforcomplexdecision-makingtasksandthenconvertthem
intointerruptiblealgorithmsusingastandardtransformation.
6 (cid:0)(cid:2)(cid:1)(cid:4)(cid:3)(cid:6)(cid:5)(cid:8)(cid:7)(cid:10)(cid:9)(cid:12)(cid:11)(cid:14)(cid:13)
speech recognizer
classify speaker recognize utterance linguistic validity
keyword meaning
gender accent
syntax
Figure2 Acompositemoduleforspeechrecognition
3 COMPOSINGANYTIMEALGORITHMS
Modularity is widely recognized as an important issue in system design and
implementation. However,theuseofanytimealgorithmsasthecomponentsof
amodularsystempresentsaspecialtypeofschedulingproblem. Thequestion
is how much time to allocate to each component in order to maximize the
outputqualityofthecompletesystem. Werefertothisproblemastheanytime
algorithmcompositionproblem.
Consider for example a speech recognition system whose structure is shown
in Figure 2. Each box represents an elementary anytime algorithm whose
conditional performance profile is given. The system is composed of three
maincomponents. First,thespeakerisclassifiedintermsofgenderandaccent.
Then a recognition algorithm suggests several possible matching utterances.
Andfinally,thelinguisticvalidityofeachpossibleutteranceisdeterminedand
thebestinterpretationisselected. Thecompositionproblemistheproblemof
calculating how much time to allocate to each elementary component of the
compositesystem,soastomaximizethequalityoftheutterancerecognition.
Solving the composition problem is important for several reasons. First, it
introduces a new kind of modularity into real-time system development by
allowing for separation between the development of the performance com-
ponents and the optimization of their performance. In traditional design of
real-time systems, the performance components must meet certain time con-
straintsthatarenotalwaysknownatdesigntime. Theresultisahand-tuning
process that may, or may not, culminate with a working system. Anytime
computation offers an alternative to this approach. By developing perfor-
mancecomponentsthatareresponsivetoawiderangeoftimeallocations,one
ApproximateReasoningUsingAnytimeAlgorithms 7
avoids the commitment to a particular performance level that might fail the
system.
The second reason why the composition problem is important relates to the
difficulty of programming with anytime algorithms. To make a composite
system optimal (or even executable), one must control the activation and
interruptionofthecomponents. Insolvingthecompositionproblem,ourgoal
istominimizetheresponsibilityoftheprogrammerregardingthisoptimization
problem. Oursolutionisdescribedinthefollowingtwosections.
4 COMPILATION
Given a system composed of anytime algorithms, the compilation process is
designed to: (a) determine the optimal performance profile of the complete
system;and(b)insertintothecompositemodulethenecessarycodetoachieve
that performance. The precise definitionand solutionof the problem depend
onthefollowingfactors:
1. Composite program structure – what type of programming operators
areusedtocomposeanytimealgorithms?
2. Type of performance profiles – what kind of performance profiles are
usedtocharacterizeelementaryanytimealgorithms?
3. Type of anytime algorithms – what type of elementary anytime algo-
rithms are used as input? what type of anytime algorithm should the
resultingsystembe?
4. Typeofmonitoring–whattypeofrun-timemonitoringisusedtoactivate
andinterrupttheexecutionoftheelementarycomponents?
5. Qualityofintermediateresults–whataccessdoesthemonitoringcom-
ponenthavetointermediateresults? istheactualqualityofanintermediate
resultknowntothemonitor?
Depending on these factors, different types of compilation and monitoring
strategies are needed. To simplify the discussion in this paper, we will con-
sideronlytheproblemofproducingcontractalgorithmswhentheconditional
performance profiles of the components are given. We will assume that no
8 (cid:0)(cid:2)(cid:1)(cid:4)(cid:3)(cid:6)(cid:5)(cid:8)(cid:7)(cid:10)(cid:9)(cid:12)(cid:11)(cid:14)(cid:13)
activemonitoringisallowedoncethesystemisactivated. Abroader,in-depth
analysisofcompilationandmonitoringcanbefoundin[23].
Let be a setof anytimefunctions. Assumethat all functionparametersare
(cid:0)
passedby valueandthat functionshavenoside-effects(asinpurefunctional
programming). Let be a set of input variables. Then, the notion of a
(cid:1)
compositeexpressionisdefinedasfollows:
Definition2 Acompositeexpressionover withinput is:
(cid:0) (cid:1)
1. An expression where is a functionof argumentsand
(cid:2)(cid:4)(cid:3)(cid:6)(cid:5)(cid:8)(cid:7)(cid:10)(cid:9)(cid:12)(cid:11)(cid:13)(cid:11)(cid:14)(cid:11)(cid:13)(cid:9)(cid:8)(cid:5) (cid:16)(cid:15) (cid:2)(cid:18)(cid:17)(cid:19)(cid:0) (cid:20)
(cid:12)
.
(cid:5) (cid:7) (cid:9)(cid:12)(cid:11)(cid:13)(cid:11)(cid:13)(cid:11)(cid:14)(cid:9)(cid:21)(cid:5) (cid:17)(cid:22)(cid:1)
(cid:12)
2. Anexpression where isa functionof argumentsand
(cid:2)(cid:4)(cid:3)(cid:24)(cid:23) (cid:7) (cid:9)(cid:25)(cid:11)(cid:14)(cid:11)(cid:13)(cid:11)(cid:13)(cid:9)(cid:26)(cid:23) (cid:15) (cid:2)(cid:27)(cid:17)(cid:28)(cid:0) (cid:20)
(cid:12)
each isacompositeexpressionoraninputvariable.
(cid:23)
(cid:10)
Forexample,theexpression isacompositeexpressionover
(cid:29)(cid:30)(cid:3) (cid:31)!(cid:3)#"$(cid:15)%(cid:9) (cid:22)(cid:3)(cid:6)&(cid:28)(cid:3)(cid:6)’((cid:15)(cid:21)(cid:15)(cid:8)(cid:15)
) with input ) . Supp(cid:1) ose that each function in has a con-
(cid:29)*(cid:9)+(cid:31),(cid:9) -(cid:9)(cid:21)&/. "0(cid:9)(cid:21)’1. (cid:0)
(cid:1)
ditional performanceprofile associatedwith it that specifiesthe quality of its
outputasafunctionoftimeallocationtothatfunctionandthequalitiesofitsin-
puts. Givenacompositeexpressionofsize ,themainpartofthecompilation
(cid:20)
processistodetermineamapping:
2 (1.1)
3(cid:3) (cid:8)(cid:7)4(cid:9)(cid:12)(cid:11)(cid:13)(cid:11)(cid:13)(cid:11)(cid:14)(cid:9) (cid:16)(cid:15)
(cid:7) (cid:1) (cid:20) (cid:1) (cid:1) (cid:12)
This mapping determines for each total allocation, , the allocation to the
(cid:1)
componentsthatmaximizestheoutputquality.
4.1 Acompilationexample
Letuslookfirstatasimpleexampleofcompilationinvolvingonlytwoanytime
algorithms. Supposethatonealgorithmtakestheinputandproducesaninter-
mediateresult. Thisresultisthenusedasinputtoanotheranytimealgorithm
which, in turn, produces the final result. Many systems can be implemented
by a compositionof a sequenceof two or more algorithms. For example, an
automated repair system can be composed of two algorithms: diagnosis and
treatment. Thiscanberepresentedingeneralbythefollowingexpression:
5-6 6 *; 6
(cid:0) (cid:0)
87 :9 (cid:3) (cid:7)<(cid:3)(cid:6)=>(cid:20)?7 (cid:26)(cid:15)(cid:8)(cid:15)
(cid:1) (cid:1) (cid:1)
ApproximateReasoningUsingAnytimeAlgorithms 9
Q Q
1 1
x Time t−x Time
Figure3 Theperformanceprofilesof and
(cid:0)(cid:2)(cid:1) (cid:0)(cid:4)(cid:3)
Figure 3 shows the performance profiles of (cid:0) and (cid:0)*; . These performance
(cid:7)
profilesaredefinedby:
(cid:5) (cid:5) ;
(cid:7)(cid:10)(cid:3) (cid:1)(cid:8)(cid:15)(cid:7)(cid:6)(cid:9)(cid:8)(cid:11)(cid:10)(cid:13)(cid:12)(cid:15)(cid:14)(cid:17)(cid:16)(cid:19)(cid:18) (cid:26) (cid:3) (cid:1)(cid:8)(cid:15)(cid:20)(cid:6)(cid:21)(cid:8)(cid:11)(cid:10)(cid:13)(cid:12)(cid:15)(cid:14)(cid:17)(cid:16)(cid:23)(cid:22) (cid:26)
Assumethattheoutputqualityisthesumofthequalitiesof (cid:0) and (cid:0)*; andis
(cid:7)
expressedby:
(cid:5) (cid:3)(cid:6)" (cid:15)(cid:20)(cid:6)(cid:24)(cid:8)(cid:11)(cid:10)(cid:13)(cid:12)(cid:25)(cid:14)(cid:26)(cid:16)(cid:19)(cid:18)(cid:28)(cid:27)(cid:2)(cid:29)(cid:30)(cid:8)(cid:11)(cid:10)(cid:31)(cid:12)(cid:15)(cid:14)(cid:17)(cid:16)(cid:23)(cid:22)! (cid:26) (cid:14)"(cid:27)$# (1.2)
Themaximalqualityisachievedwhen . Inotherwords:
%$& (cid:6)(cid:30)’
% (cid:27)
( ( ; (1.3)
(cid:7) (cid:12)(cid:15)(cid:14)(cid:26)(cid:16)$(cid:18))(cid:27)*(cid:10) (cid:12)(cid:15)(cid:14)(cid:17)(cid:16)(cid:23)(cid:22)+ (cid:26) (cid:14),(cid:27)(cid:23)#(cid:7)(cid:6)-’
The solution of this equation yields the following optimal time allocation
mapping:
.0/(cid:2)( .0/(cid:2)( ; ( ; .2/(cid:2)( ; .2/(cid:2)( (
2 (cid:7)(cid:2)(cid:1)(cid:6)(cid:20)3(cid:3) (cid:7)1(cid:10)( (cid:7) (cid:29) ( ; (cid:29) (cid:1) (cid:9) (cid:10)( (cid:7) (cid:29) ((cid:7)(cid:20); (cid:29) (cid:7) (cid:1) (cid:15) (1.4)
Tocompletethecompilationprocess,thecompilerneedstoinsertcodeinthe
originalexpressionfor properactivationof (cid:0) (cid:7) and (cid:0) ; as contractalgorithms
with the appropriate time allocation. This is done by replacing the simple
function call by an anytime function call [23]. The implementation of an
anytimefunctioncalldependsontheparticularprogrammingenvironmentand
willnotbediscussedinthispaper.
4.2 Thecomplexityofcompilation
The compilation problem is defined as an optimization problem, that is, a
problem of finding a schedule of a set of components that yields maximal
outputquality. Inordertoanalyzeitscomplexity,itismoreconvenienttorefer
tothedecisionproblemvariantofthecompilationproblem. Givenacomposite
10 (cid:0)(cid:2)(cid:1)(cid:10)(cid:3) (cid:5)(cid:8)(cid:7)(cid:10)(cid:9)(cid:12)(cid:11)(cid:14)(cid:13)
expression ,theconditionalperformanceprofilesofitscomponents,andatotal
(cid:12)
allocation , the decision problem is whether there exists a schedule of the
(cid:31)
components that yields output quality greater than or equal to . To begin,
(cid:0)
considerthegeneralproblemofglobalcompilationofcompositeexpressions,
orGCCE.In[23],weprovethefollowingresult:
Theorem3 TheGCCEproblemisNP-completeinthestrongsense.
TheproofisbasedonareductionfromthePARTIALLYORDEREDKNAP-
SACK problem which is known to be NP-completein the strong sense. The
meaningofthisresultisthattheapplicationofthecompilationtechniquemay
be limited to small programs. To address the complexity problem of global
compilation,wehavedevelopedanefficientlocalcompilationtechnique.
4.3 Localcompilation
Local compilation is the process of finding the best performance profile of a
module based on the performance profiles of its immediate components. If
those components are not elementary anytime algorithms, then their perfor-
mance profiles are determined using local compilation. Local compilation
replacestheglobaloptimizationproblemwithasetofsimpler,localoptimiza-
tion problems and reduce the complexity of the whole problem. Unfortu-
nately,localcompilationcannotbeappliedtoeverycompositeexpression. If
theexpressionhasrepeatedsubexpressions,thencomputationtimeshouldbe
allocated only once to evaluate all identical copies. Local compilation can-
nothandlesuchcases. However,thefollowingthreeassumptionsmakelocal
compilationbothefficientandoptimal[23]:
1. The tree-structured assumption – the input composite expression has
norepeatedsubexpressions,thusitsDAG(directedacyclicgraph)repre-
sentationisatree.
2. Theinput-monotonicityassumption–theoutputqualityofeachmodule
increaseswhenthequalityoftheinputimproves.
3. Thebounded-degreeassumption–thenumberofinputstoeachmodule
isboundedbyaconstant, .
(cid:1)