generating and sampling orbits for lifted probabilistic...
TRANSCRIPT
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UAI 2019 1
Generating and Sampling Orbits for Lifted Probabilistic Inference
Steven Holtzen, Todd Millstein, Guy Van den Broeck
Computer Science Department, University of California, Los Angeles
{sholtzen, todd, guyvdb}@cs.ucla.edu
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UAI 2019 2
Motivation: The Pigeonhole Distribution
• Suppose there are 3 pigeons…
• … that want to hide in 2 holes
Each dislikes being placed into the same hole…
…no quantum pigeons, pigeons hiding in multiple
holes simultaneously
What is the probability that 𝑘 pigeons are placed into the same hole? Requires computing partition (i.e., counting); does this seem hard?
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UAI 2019 3
• One way to answer queries: convert to factor graph
• Problem: Factor graph is dense; little conditional independence • Join-tree, variable elimination, etc. fail
• Is hope lost? What kind of structure is there to exploit?
Motivation: Encoding to Factor Graphs
1 2 3
A B
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UAI 2019 4
• Pigeons and holes are exchangeable: relabeling them does not change the probability
• These two states are in the same orbit
• Dramatically reduces state space of the problem
Symmetry is Structure Too
1 2 3
A B
1 2 3
B A
≅
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UAI 2019 5
• Lifted inference scales in degree of symmetry • Scales to large dense problems
• Orthogonal to independence
• Problem: Exact lifted inf. requires relational representation • Cannot handle factor graphs
Related Work: Lifted Inference
[Richardson, Matthew, and Pedro Domingos. "Markov logic networks." Machine learning 62.1-2 (2006): 107-136.]
✓
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UAI 2019 6
• Lifted inference scales in degree of symmetry • Scales to large dense problems
• Orthogonal to independence
• Problem: Exact lifted inf. requires relational representation • Cannot handle factor graphs
Related Work: Lifted Inference
[Richardson, Matthew, and Pedro Domingos. "Markov logic networks." Machine learning 62.1-2 (2006): 107-136.]
✓ How can we exploit symmetry in exact factor graph inference?
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UAI 2019 7
• Assignments have a natural colored encoding
• Black factors: Each pigeon dislikes being placed into the same hole
• Red factors: no quantum pigeons
• Green = true variable, red = false variable
Our Key Insight: Colored Assignment Encodings
1 2 3
A B
Assignment Encoding
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UAI 2019 8
• Assignments have a natural colored encoding
• Black factors: Each pigeon dislikes being placed into the same hole
• Red factors: no quantum pigeons
• Green = true variable, red = false variable
Our Key Insight: Colored Assignment Encodings
1 2 3
A B
Assignment Encoding
Represent symmetries of distribution through isomorphisms of graph [Kersting et al., 2009, Niepert, 2012, 2013, Bui et al., 2013]
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UAI 2019 9
• Two new algorithms:
Contribution
Orbit Generation First example of exact lifted inference for
arbitrary discrete factor graphs
Orbit-Jump MCMC Approximate lifted inference that mixes
rapidly* in number of orbits
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UAI 2019 10
Orbit Generation Exact lifted inference for factor graphs
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UAI 2019 11
A Simpler Example
• Consider a complete factor graph
• If all factors identical and symmetric, then Pr = Pr = Pr = Pr
• Probability is determined by number of true states
𝐴
𝐶 𝐷
𝐵
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UAI 2019 12
Orbit # Elements of the Orbit
0
1
2
3
4
Orbits of Factor Graphs
• Pr = Pr = Pr = Pr
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UAI 2019 13
Exact lifted inference algorithm
• If we can:
1. Efficiently generate one element of each orbit,
2. Efficiently compute the size of each orbit
• Then, the partition function can be computed efficiently in the number of orbits (Theorem 4.1)
Let’s see an example…
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UAI 2019 14
1. Efficiently find one representative of each orbit
2. Compute the size of the orbit
• 𝑍 = ∑unnormalized = 5 + 52 + 126 + 8 + 3 = 193
Orbit # Orbit Repr.
0
1
2
3
4
Exact Lifted Inference
Unnormalized State Probability
5
13
21
2
3
Total Orbit Unnormalized
5 × 1 = 5
13 × 4 = 52
21 × 6= 126
2 × 4 = 8
3 × 1 = 3
Orbit Size
1
4
6
4
1
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UAI 2019 15
1. Efficiently generate one element of each orbit,
2. Efficiently compute the size of each orbit
Exact lifted inference algorithm
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UAI 2019 16
Orbit Generation: Breadth-First Search
Start with all-false assignment
These are isomorphic to the first, prune them
Requires linear (in #orbits × #variables ) calls to graph
isomorphism tool …
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UAI 2019 17
1. Efficiently generate one element of each orbit,
2. Efficiently compute the size of each orbit
Exact lifted inference algorithm
✓
• Seems #P-hard at first, but in fact is not • Use graph isomorphism tools to count things
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UAI 2019 18
Orbit Size Pipeline
Assignment- Encoded Colored
Factor Graph
Graph Iso. Tool
Stabilizer Group
Group Order
Orbit Size
Efficient to compute!
Orbit-Stabilizer Theorem (Group Theory)
• Avoid enumerating the whole orbit
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UAI 2019 19
Stabilizer Group
• Question: Which isomorphisms preserve (stabilize) this coloring?
• Graph isomorphism tools can compute this set of permutations • Represented in a compact way (generators)
𝐴
𝐶 𝐷
𝐵
Answer: Any permutation of {𝐵, 𝐶, 𝐷}
Assignment- Encoded Colored
Factor Graph
Graph Iso. Tool
Stabilizer Group: Small set of generators
Group Order
Orbit Size ✓
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UAI 2019 20
Orbit-Stabilizer Theorem
• Relates size of orbit to order (size) of stabilizer
• Computing the order of a group is a standard problem in computational group theory • Efficient to compute (in size of graph)
=#ways of permuting 𝐴, 𝐵, 𝐶, 𝐷
#ways of permuting 𝐵, 𝐶, 𝐷=
4!
3!= 4
𝐴
𝐶 𝐷
𝐵
Orbit size
Group Theory
GAP
Stabilizer Group: Small set of generators
Order of group 4 × 1084 states
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UAI 2019 21
Exact Inference Experiments
• Proof of concept: Compared against existing exact inference tool, ACE
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UAI 2019 22
Orbit-Jump MCMC Approximate lifted inference with mixing time guarantees
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UAI 2019 23
Motivation
• Local-search (e.g. Gibbs sampling) can get stuck
Propose local move
😐 😐
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UAI 2019 24
Related Work: Within-Orbit Jumps
• Lifted MCMC [Niepert, 2012, 2013] jumps within orbits, unfortunately doesn’t help here
😐
Lifted MCMC step
😐
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UAI 2019 25
Jumping Between Orbits
• Orbit-Jump MCMC proposes jumps between orbits
• Non-local moves: can skip over low-probability regions
• Exploits orbit structure: better than random restarts
Orbit-Jump Proposal
• We show how to jump between orbits using the Burnside process • Requires multiple graph isomorphism + group order computations • More expensive than lifted MCMC, with mixing rate guarantees
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UAI 2019 26
Orbit-Jump MCMC Mixing Time
• Empirical mixing time, 5 pigeons 2 holes • Total variation distance from stationary dist.
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UAI 2019 27
Conclusion
• Some distributions have little independence, but inference remains tractable • Symmetry complements independence
• This work develops symmetry as a source of tractability for factor graph inference • First exact lifted inference for factor graphs
• Orbit-Jump MCMC algorithm, mixes rapidly in #orbits (with some caveats)
Grand challenge: Integrating independence and
symmetry into a single algorithm for factor graphs