Exact Lifted Inference on Relational Temporal Modelsbraun/Tutorial/iccs-19/... · 2019. 7. 1. ·...
Transcript of Exact Lifted Inference on Relational Temporal Modelsbraun/Tutorial/iccs-19/... · 2019. 7. 1. ·...
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Exact Lifted Inference on Relational Temporal Models
Marcel Gehrke, University of Lübeck
Statistical Relational AITutorial at ICCS 2019
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Propositional: Dynamic Model
2
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• Temporal pattern• Instantiate and unroll pattern• Infer on unrolled model
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Dynamic Model: Inference Problems
3
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• Marginal distribution query: : ;<= 2>:") w.r.t. the model:• Hindsight: A < . (was there an epidemic A − . days ago?) • Filtering: A = . (is there an currently an epidemic?)• Prediction: A > . (is there an epidemic in A − . days?)
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Propositional: Dynamic Model
4
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• Problems with unrolling:• Huge model (unrolled for T timesteps)• Redundant temporal information and calculations• Redundant calculations to answer multiple queries
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Propositional: Interface Algorithm
5
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• Main idea: Use temporal conditional independences to perform inference on smaller model
Murphy (2002)
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Propositional: Interface Algorithm
6
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• Main idea: Use temporal conditional independences to perform inference on smaller model• Normally only a subset of random variables influence next time step
Murphy (2002)
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Propositional: Interface Algorithm
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• Main idea: Use temporal conditional independences to perform inference on smaller model• Normally only a subset of random variables influence next time step• State description of interface variables (!"#$%&' and (#)*. ,-,%&')
suffice to perform inference on time slice t• Proceed forward one time step at a time, using the same structure
Murphy (2002)
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Propositional: Interface Algorithm
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• Build Junction Tree from reoccurring structure• Ensure that interface variable for time slice ! − 1
occur in one cluster and that interface variable for time slice ! occur in one cluster
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Murphy (2002)
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Propositional: Interface Algorithm
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• Perform inference on time slice 3• How to perform inference on time slice 4?• Store state descriptions of interface variables in !• Distribute ! with message pass
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Murphy (2002)
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Propositional: Dynamic Model
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Propositional: Interface Algorithm
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Murphy (2002)
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Propositional: Interface Algorithm
12
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Propositional: Interface Algorithm
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Murphy (2002)
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Propositional: Interface Algorithm
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• For the static example increasing the domain size only increased the number of clusters
• Increasing the domain size of interface variables, increases the number of clusters and cluster size
• Inference is exponential in the largest cluster• Can we lift the interface algorithm?
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Lifted: Dynamic Model
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Gehrke et al. (2018)• Marginal distribution query: < =>? 3@:") w.r.t. the
model:• Hindsight: B < 0 (was there an epidemic B − 0 days ago?) • Filtering: B = 0 (is there an currently an epidemic?)• Prediction: B > 0 (is there an epidemic in B − 0 days?)
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Lifted Dynamic Junction Tree Algorithm: LDJT
• Input• Temporal model !• Evidence "• Queries #
• Algorithm1. Identify interface variables2. Build FO jtree structures $ for !3. Instantiate $%4. Restore state description of interface variables from &%'(5. Enter evidence "% into $%6. Pass messages in $%7. Answer queries #)8. Store state description of interface variables in &%9. Proceed to next time step (step 3)
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LDJT: Identify Interface Variables
• !"#$ = {'"#$( | ∃ + , |- ∈ / ∶ '"#$( ∈ , ∧ '"#$2 ∈ ,}
• Set of interface variable !"#$ consists of all PRVs from time slice 4 − 1 that occur in a parfactor with PRVs from time slice 4
17
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Gehrke et al. (2018)
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LDJT: Construct FO jtree Structure• Turn model in 1.5 time slice model• Suffices to perform inference over time slice !• From 1.5 time slice model construct FO jtree structure• Ensure "#$%is contained in a parcluster and "#is contained in
a parcluster• Label parcluster with "#$% as in-cluster and parcluster with "# as out-cluster
18
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Gehrke et al. (2018)
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LDJT: Query answering• Instantiate FO jtree structure
• Restore state description of interface variables
• Enter evidence
• Pass messages
• Query answering:• Find parcluster contain query term
• Extract submodel
• Answer query with LVE
19
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LDJT: Proceed in time• Calculate !" using out-cluster (#"$)• Eliminate %&'()* + " from #"$’s local model• Instantiate next FO jtree and enter !"• Enter evidence and pass messages
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Gehrke et al. (2018)
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LDJT: Intermediate Overview• So far only a temporal forward pass• Reason over one time step• Keep only one time step in memory• Filtering queries• Prediction queries (filtering without new evidence)• Hindsight queries
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LDJT: Forward and Backward Pass• Use same FO jtree structures for backward pass• Calculate a message ! using an in-cluster over
interface variables and pass ! to previous time step• LDJT needs to keep FO jtrees of previous time steps• Different instantiation approaches during a
backward pass • Keep all computations for all time steps in memory (not
always feasible) • Instantiate time steps on demand (same as for the
forward pass, possible due to the separation between time steps)
22
Gehrke et al. (2019)
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LDJT: Backward Pass• Calculate !" using in-cluster (#"$)• Eliminate %&'(" from #"$’s local model, without )"*$• Add !" to local model of out-cluster #"*$+
• Pass messages for , − 1 to account for !"
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Gehrke et al. (2019)
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LDJT: Instantiations during a Backward Pass
24
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Keep Instantiations Instantiate on demandMessages to prepare for queriesMessages to solely calculate ("#$Additional memory for each time step
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LDJT: Instantiations during a Backward Pass
25
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Gehrke et al. (2019)
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LDJT: Instantiations during a Backward Pass
26
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Gehrke et al. (2019)
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LDJT: Instantiations during a Backward Pass
27
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Gehrke et al. (2019)
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LDJT: Instantiations during a Backward Pass
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LDJT: Instantiations during a Backward Pass
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LDJT: Instantiations during a Backward Pass
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LDJT: Relational Forward Backward Algorithm • LDJT can answer hindsight queries, even to the first
time step • By combining the instantiation approaches, LDJT
can trade off memory consumption and reusing computations • LDJT is in the worst case quadratic to T, but
normally remains linear w.r.t. T (T max # time steps)• But does it really suffice to lift the interface
algorithm?
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Gehrke et al. (2019)
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LDJT: Preventing Unnecessary Groundings• Groundings in inter time slice messages (especially
forward messages) can lead to grounding the
model for all time steps
• Elimination order predetermined in FO jtree
• Non-ideal elimination order leads to groundings
• Minimal set of interface variables not always ideal
• Delay eliminations for inter time slice messages to
prevent unnecessary groundings
• Simply lifting the interface algorithm does not suffice,
one also needs to ensure preconditions of lifting
• Trade off between lifting and handling temporal
aspects due to restrictions on elimination orders
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Gehrke et al. (2018b,c)
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0 200 400 600 800 1000
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LDJTLDJT GroundingsLJT Model
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LDJT: Preven,ng Unnecessary Groundings• Depending on the settings, either lifting or handling
of temporal aspects is more efficient• Preventing groundings to calculate a lifted solution
pays off
Gehrke et al. (2018b,c)
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LDJT: Additional Queries• Conjunctive queries over different time steps
• Can be used for event detection
• What is the probability that someone travelled from X to
Y and that afterwards there is a epidemic in Y given
there is an epidemic in X?
• Maximum expected utility
• Decision support
• Well studied within one time step (Apsel and Brafman
(2011), Nath and Domingos (2009))
• Assignment queries
• Most likely state sequence
• Well studied for static models (Dawid (1992), Dechter
(1999), de Salvo Braz et al. (2006), Apsel and Brafman
(2012), Braun and Möller (2018))
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LDJT: Maximum expected utility• Extend representation with actions and utilities• Problem: Find the action sequence that maximises
the expected utility value w.r.t. a utility function.• Only possible for a finite horizon• Combinatorial in horizon • Combinatorial in splits of logvar(s) of action PRV(s)
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Outlook• Continue optimising
• Parallelisation• Caching
• From discrete time interval to time continuous• Preserving symmetries• Learning?
• Structure• Potentials (Idea of Baum Welch now possible)• Symmetries• Transfer learning
• Open world?• Unknown domains• Unknown behaviour
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Wrap-up Exact Lifted Dynamic Inference• Parfactor models for sparse encoding• Factorisation of full joint distribution• Logical variables to model objects
• Algorithms for exact query answering• LDJT for repeated inference• Extensions possible
• Parameterised, conjunctive queries• Maximum expected utility• Assignment queries (Tomorrow)
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Mission and Schedule of the Tutorial*
• Introduction 20 min• StaR AI
• Overview: Probabilistic relational modeling 30 min• Semantics (grounded-distributional, maximum entropy)• Inference problems and their applications• Algorithms and systems
• Scalable static inference 40 + 30 min• Exact propositional inference• Exact lifted inference
• Scalable dynamic inference 50 min• Exact propositional inference• Exact lifted inference
• Summary 10 min
Providing an introduction into inference in StaRAI
*Thank you to the SRL/StaRAI crowd for all their exciting contributions! The tutorial is necessarily incomplete. Apologies to anyone whose work is not cited
✓
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✓
✓✓
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References
• Apsel and Brafman (2012) Udi Apsel and Ronen I. Brafman. Exploiting Uniform Assignments in First-Order MPE. Proceedings of the 28th Conference on Uncertainty in Artificial Intelligence, 2012.
• Apsel and Brafman (2011)Udi Apsel and Ronen I. Brafman. Extended Lifted Inference with Joint Formulas. Proceedings of the 27th Conference on Uncertainty in Artificial Intelligence. pp. 11–18, 2011.
• Dawid (1992)Alexander Philip Dawid. Applications of a General Propagation Algorithm for Probabilistic Expert Systems. Statistics and Computing, 2(1):25–36, 1992.
• Dechter (1999)Rina Dechter. Bucket Elimination: A Unifying Framework for Probabilistic Inference. In Learning and Inference in Graphical Models, pages 75–104. MIT Press, 1999.
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References
• De Salvo Braz et al. (2006)Rodrigo de Salvo Braz, Eyal Amir, and Dan Roth. MPE and Partial Inversion in Lifted Probabilistic Variable Elimination. AAAI-06 Proceedings of the 21st Conference on Artificial Intelligence, 2006.
• Murphy (2002)Kevin P. Murphy. Dynamic Bayesian Networks: Representation, Inference and Learning. PhD Thesis University of California, Berkeley, 2002.
• Nath and Domingos (2009)Aniruddh Nath and Pedro Domingos, A language for relational decision theory, Proceedings of the International Workshop on Statistical Relational Learning, 2009.
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Work @ IFIS• Braun and Möller (2018b)
Tanya Braun and Ralf Möller. Lifted Most Probable Explanation. In Proceedings of the International Conference on Conceptual Structures, 2018.
• Gehrke et al. (2018)Marcel Gehrke, Tanya Braun, and Ralf Möller. Lifted Dynamic Junction Tree Algorithm. In Proceedings of the International Conference on Conceptual Structures, 2018.
• Gehrke et al. (2018b)Marcel Gehrke, Tanya Braun, and Ralf Möller. Towards Preventing UnnecessaryGroundings in the Lifted Dynamic Junction Tree Algorithm. In Proceedings of KI 2018: Advances in Artificial Intelligence, 2018.
• Gehrke et al. (2018c)Marcel Gehrke, Tanya Braun, and Ralf Möller. Preventing UnnecessaryGroundings in the Lifted Dynamic Junction Tree Algorithm. In Proceedings of theAI 2018: Advances in Artificial Intelligence, 2018.
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Work @ IFIS• Gehrke et al. (2019)
Marcel Gehrke, Tanya Braun, and Ralf Möller. Rela=onal Forward Backward Algorithm for Mul=ple Queries. In FLAIRS-32 Proceedings of the 32nd
Interna9onal Florida Ar9ficial Intelligence Research Society Conference, 2019.
• Gehrke et al. (2019b)Marcel Gehrke, Tanya Braun, Ralf Möller, Alexander Waschkau, Christoph Strumann, and Jost Steinhäuser. LiQed Maximum Expected U=lity. In Ar=ficialIntelligence in Health, 2019.
• Gehrke et al. (2019c)Marcel Gehrke, Tanya Braun, and Ralf Möller. LiQed Temporal Maximum Expected U=lity. In Proceedings of the 32nd Canadian Conference on Ar=ficialIntelligence, Canadian AI 2019, 2019.
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