Introduction to Artificial Intelligence

232 Pages·2017·7.83 MB·English

by  Marc Toussaint

Introduction to Artificial Intelligence
MarcToussaint
February6,2017
Themajorityofslidesonsearch,CSPandlogicareadaptedfromStuartRussell.
This is a direct concatenation and reformatting of all lecture slides and exercises from
theArtificialIntelligencecourse(winterterm2016/17,UStuttgart),includingindexing
tohelpprepareforexams.
Contents
1 Introduction 6
2 Search 14
Motivation&Outline
2.1 ProblemFormulation&Examples . . . . . . . . . . . . . . . . . . . . 14
Example:Romania(2:3)ProblemDefinition:Deterministic,fullyobservable(2:5)
2.2 BasicTreeSearchAlgorithms . . . . . . . . . . . . . . . . . . . . . . . 16
Treesearchimplementation: statesvsnodes(2:12)TreeSearch: GeneralAlgorithm
(2:13)Breadth-firstsearch(BFS)(2:16)ComplexityofBFS(2:17)Uniform-costsearch
(2:18)Depth-firstsearch(DFS)(2:19)ComplexityofDFS(2:20)Iterativedeepening
search(2:22)ComplexityofIterativeDeepeningSearch(2:24)Graphsearchandre-
peatedstates(2:26)
2.3 GreedyandA∗Search . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Best-firstSearch(2:30)GreedySearch(2:32)ComplexityofGreedySearch(2:35)A∗
search(2:36)A∗: Proof1ofOptimality(2:38)ComplexityofA∗(2:39)A∗: Proof2of
Optimality(2:40)Admissibleheuristics(2:42)Memory-boundedA∗(2:45)
1
2 IntroductiontoArtificialIntelligence,MarcToussaint
3 Probabilities 34
Motivation & Outline Probabilities as (subjective) information calculus (3:3) Infer-
ence:generalmeaning(3:5)FrequentistvsBayesian(3:6)
3.1 Basicdefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Definitionsbasedonsets(3:8)Randomvariables(3:9)Probabilitydistribution(3:10)
Jointdistribution(3:11)Marginal(3:11)Conditionaldistribution(3:11)Bayes’Theo-
rem(3:13)MultipleRVs,conditionalindependence(3:14)
3.2 Probabilitydistributions . . . . . . . . . . . . . . . . . . . . . . . . . . 39
BernoulliandBinomialdistributions(3:16)Beta(3:17)Multinomial(3:20)Dirichlet
(3:21)Conjugatepriors(3:25)
3.3 Distributionsovercontinuousdomain . . . . . . . . . . . . . . . . . . 44
Dirac distribution (3:28) Gaussian (3:29) Particle approximation of a distribution
(3:33) Utilities and Decision Theory (3:36) Entropy (3:37) Kullback-Leibler diver-
gence(3:38)
3.4 MonteCarlomethods. . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Monte Carlo methods (3:40) Rejection sampling (3:41) Importance sampling (3:42)
Student’st,Exponential,Laplace,Chi-squared,Gammadistributions(3:44)
4 Bandits,MCTS,&Games 51
Motivation&Outline
4.1 Bandits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Multi-armedBandits(4:2)
4.2 UpperConfidenceBounds(UCB) . . . . . . . . . . . . . . . . . . . . . 53
Exploration,Exploitation(4:7)UpperConfidenceBound(UCB1)(4:8)
4.3 MonteCarloTreeSearch . . . . . . . . . . . . . . . . . . . . . . . . . . 55
MonteCarloTreeSearch(MCTS)(4:14)UpperConfidenceTree(UCT)(4:19)MCTS
forPOMDPs(4:20)
4.4 MCTSappliedtoPOMDPs* . . . . . . . . . . . . . . . . . . . . . . . . 58
4.5 GamePlaying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Minimax(4:29)Alpha-BetaPruning(4:32)Evaluationfunctions(4:37)UCTforgames
(4:38)
4.6 Beyondbandits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
GlobalOptimization(4:43)GP-UCB(4:46)ActiveLearning(4:50)
4.7 ActiveLearning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
IntroductiontoArtificialIntelligence,MarcToussaint 3
5 Formalmodelsofinteractivedomains 72
5.1 BasicTaxonomyofdomainmodels . . . . . . . . . . . . . . . . . . . . 72
PDDL (5:10) Noisy Deictic Rules (5:13) Markov Decision Process (5:15) POMDP
(5:18)Dec-POMDP(5:22)Control(5:23)
6 DynamicProgramming 82
Motivation&Outline
6.1 DynamicProgramming . . . . . . . . . . . . . . . . . . . . . . . . . . 82
Value Function (6:3) Bellman optimality equation (6:7) Value Iteration (6:9) Q-
Function(6:10)Q-Iteration(6:11)ProofofconvergenceofQ-Iteration(6:12)
6.2 DynamicProgramminginBeliefSpace . . . . . . . . . . . . . . . . . . 88
7 ReinforcementLearning 94
Motivation&Outline
7.1 LearninginMDPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Temporaldifference(TD)(7:10)Q-learning(7:11)ProofofconvergenceofQ-learning
(7:14)Eligibilitytraces(7:16)Model-basedRL(7:28)
7.2 Exploration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Epsilon-greedyexplorationinQ-learning(7:31)R-Max(7:32)BayesianRL(7:34)Op-
timisticheuristics(7:35)
7.3 PolicySearch,Imitation,&InverseRL*. . . . . . . . . . . . . . . . . . 109
Policygradients(7:39)ImitationLearning(7:43)InverseRL(7:46)
8 ConstraintSatisfactionProblems 117
Motivation&Outline
8.1 ProblemFormulation&Examples . . . . . . . . . . . . . . . . . . . . 117
Inference (8:2) Constraint satisfaction problems (CSPs): Definition (8:4) Map-
ColoringProblem(8:5)
8.2 MethodsforsolvingCSPs . . . . . . . . . . . . . . . . . . . . . . . . . 121
Backtracking (8:11) Variable order: Minimum remaining values (8:16) Variable or-
der: Degreeheuristic(8:17)Valueorder: Leastconstrainingvalue(8:18)Constraint
propagation(8:19)Tree-structuredCSPs(8:25)
4 IntroductiontoArtificialIntelligence,MarcToussaint
9 GraphicalModels 130
Motivation&Outline
9.1 BayesNetsandConditionalIndependence . . . . . . . . . . . . . . . 130
Bayesian Network (9:4) Conditional independence in a Bayes Net (9:8) Inference:
generalmeaning(9:13)
9.2 InferenceMethodsinGraphicalModels . . . . . . . . . . . . . . . . . 136
Inferenceingraphicalmodels: overview(9:18)MonteCarlo(9:20)Importancesam-
pling(9:23)Gibbssampling(9:25)Variableelimination(9:28)Factorgraph(9:31)Be-
liefpropagation(9:37)Messagepassing(9:37)Loopybeliefpropagation(9:41)Junc-
tiontreealgorithm(9:43)Maximuma-posteriori(MAP)inference(9:47)Conditional
randomfield(9:48)
10 DynamicModels 150
Motivation&OutlineMarkovProcess(10:1)HiddenMarkovModel(10:2)Filtering,
Smoothing,Prediction(10:3)HMM:Inference(10:4)HMMinference(10:5)Kalman
filter(10:8)
11 PropositionalLogic 157
Motivation&Outline
11.1 Syntax&Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Knowledgebase:Definition(11:3)WumpusWorldexample(11:4)Logic:Definition,
Syntax, Semantics (11:7) Propositional logic: Syntax (11:9) Propositional logic: Se-
mantics(11:10)Logicalequivalence(11:12)
11.2 InferenceMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Inference(11:19)HornForm(11:23)ModusPonens(11:23)Forwardchaining(11:24)
CompletenessofForwardChaining(11:27)BackwardChaining(11:28)Conjunctive
NormalForm(11:31)Resolution(11:31)ConversiontoCNF(11:32)
12 First-OrderLogic 183
Motivation&Outline
12.1 TheFOLlanguage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
FOL:Syntax(12:4)Universalquantification(12:6)Existentialquantification(12:6)
12.2 FOLInference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Reductiontopropositionalinference(12:16)Unification(12:19)GeneralizedModus
Ponens(12:20)ForwardChaining(12:21)BackwardChaining(12:27)Conversionto
CNF(12:33)Resolution(12:35)
IntroductiontoArtificialIntelligence,MarcToussaint 5
13 RelationalProbabilisticModellingandLearning 200
Motivation&Outline
13.1 STRIPS-likerulestomodelMDPtransitions . . . . . . . . . . . . . . . 200
MarkovDecisionProcess(MDP)(13:2)STRIPSrules(13:3)PlanningDomainDefini-
tionLanguage(PDDL)(13:3)Learningprobabilisticrules(13:9)Planningwithprob-
abilisticrules(13:11)
13.2 RelationalGraphicalModels . . . . . . . . . . . . . . . . . . . . . . . . 204
Probabilistic Relational Models (PRMs) (13:20) Markov Logic Networks (MLNs)
(13:24)TheroleofuncertaintyinAI(13:31)
14 Exercises 212
14.1 Exercise1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
14.2 Exercise2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
14.3 Exercise3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
14.4 Exercise4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
14.5 Exercise5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
14.6 Exercise6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
14.7 Exercise6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
14.8 Exercise9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
14.9 Exercise7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
Index 230
sdeeqcuiseinotniasl propositional relational
spedrqeoucbielseniomtniasl sequential assignment CSP proploosgiitcional FOL
constraint
deterministic seBaFrSch gamesbacktarmalppicnrhkuiaimnn/ibganexgta ptrrooenpeasgation cfwhdai/nbiwndg
MCTS UCB bandits
utilities graphical FOL relational
probabilistic Decision Theory muMltiD-aPgsent pVrod(gsy)rn,a aQmm(msi,cMain)DgPsreMlaDtioPnsal models mprsHogMpb. aeMpglaismeasftssfiigownn.dg p/bawssding gmraopdheiclsal
ML
Reinforcement
Learning
learning LeAacrtnivineg
6 IntroductiontoArtificialIntelligence,MarcToussaint
1 Introduction
(someslidesbasedonStuartRussell’sAIcourse)
• ThecurrenthypeaboutAI
– Thesingularity
– Ethics
– Thevalueproblem
– Theoutrageousinabilityofhumanstodefinewhatis“good”
– Paperclips
• What’stheroutetoAI?
– Neuroscience?(EUBigBrainproject)
– DeepLearning?(PureMachineLearning?,DeepMind(London))
– Social/Emotional/conciousnes/Feelingsstuff?
– HardcoreclassicalAI?Modernprobabilistic/learningAI?
– Robotics?
• WhatarethingsAIneverbeabletodo?
• WhyistherenouniversitydepartmentforIntelligenceResearch?!
1:1
UpsanddownsofAI–history
1943 McCulloch&Pitts:Booleancircuitmodelofbrain
1950 Turing’s“ComputingMachineryandIntelligence”
1952–69 Look,Ma,nohands!
1950s EarlyAIprograms,includingSamuel’scheckersprogram,
Newell&Simon’sLogicTheorist,Gelernter’sGeometryEngine
1956 Dartmouthmeeting:“ArtificialIntelligence”adopted
1965 Robinson’scompletealgorithmforlogicalreasoning
1966–74 AIdiscoverscomputationalcomplexity
Neuralnetworkresearchalmostdisappears
1969–79 Earlydevelopmentofknowledge-basedsystems
1980–88 Expertsystemsindustrybooms
1988–93 Expertsystemsindustrybusts:“AIWinter”
1985–95 Neuralnetworksreturntopopularity
1988– Resurgenceofprobability;generalincreaseintechnicaldepth
“NouvelleAI”:ALife,GAs,softcomputing
1995– Agents,agents,everywhere...
2003– Human-levelAIbackontheagenda
1:2
Whatisintelligence?
• Maybeitiseasiertofirstaskwhatsystemsweactuallytalkabout:
IntroductiontoArtificialIntelligence,MarcToussaint 7
– Decisionmaking
– Interactingwithanenvironment
• Thendefineobjectives!
– Quantifywhatyouconsidergoodorsuccessful
– Intelligencemeanstooptimize...
1:3
IntelligenceasOptimization?
• A cognitive scientist or psychologist: “Why are you AI people always so ob-
sessedwithoptimization? Humansarenotoptimal!”
• That’satotalmisunderstandingofwhat“beingoptimal”means.
• Optimizationprinciplesareameanstodescribesystems:
– Feynman’s“unworldlinessmeasure”objectivefunction
– Everythingcanbecastoptimal–undersomeobjective
– Optimalityprinciplesarejustascientificmeansofformallydescribingsystemsand
theirbehaviors(esp.inphysics,economy,...andAI)
– Toussaint, Ritter&Brock: TheOptimizationRoutetoRobotics–andAlternatives. Ku¨nstliche
Intelligenz,2015
• Generally,Iwouldroughlydistinguishthreebasictypesofproblems:
– Optimization
– Logical/categorialInference (CSP,findfeasiblesolutions)
– ProbabilisticInference
1:4
Whatareinterestinggenericobjectives
• Learntocontrolalldegreesoffreedomoftheenvironmentthatarecontrollable
– DOFs are mechanical/kinematics DOFs, objects, light/temperature, mood of hu-
mans
– Thisobjectiveisgeneric:nopreferences,notlimits
– ImpliestoactivelygoexploringandfindingcontrollableDOFs
– ActingtoLearning(insteadof’LearningtoAct’forafixedtask)
– Relatednotionsinotherfields: (Bayesian)ExperimentalDesign,ActiveLearning,cu-
riosity,intrinsicmotivation
• AttimeT,thesystemwillbegivenarandomtask(e.g.,randomgoalconfigu-
rationofDOFs);theobjectivethenistoreachitasquicklyaspossible
1:5
8 IntroductiontoArtificialIntelligence,MarcToussaint
Interactivedomains
• Weassumetheagentisininteractionwithadomain.
– Theworldisinastatest ∈S (seebelowonwhatthatmeans)
– Theagentsensesobservationsyt ∈O
– Theagentdecidesonanactionat ∈A
– Theworldtransitionstoanewstatest+1
• Theobservationy describesallinformationreceivedbytheagent(sensors,also
t
rewards,feedback,etc)ifnotexplicitlystatedotherwise
(ThetechnicaltermforthisisaPOMDP)
1:6
State
• Thenotionofstateisoftenusedimprecisely
• Atanytimet,weassumetheworldisinastates ∈S
t
• s isastatedescriptionofadomainifffutureobservationsy ,t+ >tarecondi-
t t+
tionallyindependentofallhistoryobservationsy ,t− <tgivens andfuture
t− t
actionsa :
t:t+
agent
y0 a0 y1 a1 y2 a2 y3 a3
s0 s1 s2 s3
• Notes:
– Intuitively,stdescribeseverythingabouttheworldthatis“relevant”
– Worldsdonothaveadditionallatent(hidden)variablestothestatest
1:7
Examples
IntroductiontoArtificialIntelligence,MarcToussaint 9
• Whatisasufficientdefinitionofstateofacomputerthatyouinteractwith?
• Whatisasufficientdefinitionofstateforathermostatscenario?
(First,assumethe’room’isanisolatedchamber.)
• Whatisasufficientdefinitionofstateinanautonomouscarcase?
→inrealworlds,theexactstateispracticallynotrepresentable
→allmodelsofdomainswillhavetomakeapproximatingassumptions(e.g.,
aboutindependencies)
1:8
Howcanagentsbeformallydescribed?
...or,whatformalclassesofagentsdoexist?
• Basicalternativeagentmodels:
– Theagentmapsyt (cid:55)→at
(stimulus-responsemapping..non-optimal)
– Theagentstoresallpreviousobservationsandmaps
f : y ,a (cid:55)→a
0:t 0:t-1 t
f iscalledagentfunction. Thisisthemostgeneralmodel,includingtheothersas
specialcases.
– Theagentstoresonlytherecenthistoryandmaps
yt−k:t,at−k:t-1 (cid:55)→at(crude,butmaybeagoodheuristic)
– Theagentissomemachinewithitsowninternalstatent,e.g.,acomputer,afinite
statemachine,abrain... Theagentmaps(nt-1,yt)(cid:55)→nt(internalstateupdate)and
n (cid:55)→a
t t
– The agent maintains a full probability distribution (belief) bt(st) over the state,
maps(bt-1,yt)(cid:55)→bt(Bayesianbeliefupdate),andbt (cid:55)→at
1:9
POMDPcoupledtoastatemachineagent
10 IntroductiontoArtificialIntelligence,MarcToussaint
agent
n0 n1 n2
y0 a0 y1 a1 y2 a2
s0 s1 s2
r0 r1 r2
1:10
Multi-agentdomainmodels
(ThetechnicaltermforthisisaDecentralizedPOMDPs)
(fromKumaretal.,IJCAI2011)
• Thisisaspecialtype(simplification)ofageneralDEC-POMDP
• Generally,thislevelofdescriptionisverygeneral,butNEXP-hard
Approximatemethodscanyieldverygoodresults,though
1:11
• Wegaveaverygeneralmodel(formalization)ofwhatitmeansthatanagent
takesdecisionsinaninteractiveenvironment
• Therearemanyflavorsofthis:
– Fullyobservablevs.partiallyobservable
– Singleagentvs.multiagent
– Deterministicvs.stochastic
– Structureofthestatespace:Discrete,continuous,hybrid;factored;relational
– Discretevs.continuoustime
1:12
Organisation