Generating and Sampling Orbits for Lifted Probabilistic Inference - PowerPoint PPT Presentation

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 UAI 2019 1

Motivation: The Pigeonhole Distribution • Suppose there are 3 pigeons… Each dislikes being placed into the same hole… …no quantum pigeons , pigeons hiding in multiple holes simultaneously What is the probability that 𝑙 pigeons are placed into the same hole? • … that want to hide in 2 holes Requires computing partition (i.e., counting); does this seem hard ? UAI 2019 2

Motivation: Encoding to Factor Graphs • One way to answer queries: convert to factor graph 2 3 1 A B • 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? UAI 2019 3

Symmetry is Structure Too • Pigeons and holes are exchangeable : relabeling them does not change the probability 2 1 3 2 1 3 ≅ B A A B • These two states are in the same orbit • Dramatically reduces state space of the problem UAI 2019 4

Related Work: Lifted Inference • 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 Machine learning 62.1-2 (2006): 107-136.] ✓ [Richardson, Matthew, and Pedro Domingos. "Markov logic networks." UAI 2019 5

Related Work: Lifted Inference • 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 Machine learning 62.1-2 (2006): 107-136.] ✓ How can we exploit symmetry in exact factor graph inference? [Richardson, Matthew, and Pedro Domingos. "Markov logic networks." UAI 2019 6

Our Key Insight: Colored Assignment Encodings • Assignments have a natural colored encoding 2 1 3 Assignment Encoding A B • Black factors: Each pigeon dislikes being placed into the same hole • Red factors: no quantum pigeons • Green = true variable, red = false variable UAI 2019 7

Our Key Insight: Colored Assignment Encodings • Assignments have a natural colored encoding 2 1 3 Assignment Encoding A B Represent symmetries of • Black factors: Each pigeon dislikes being placed into distribution through the same hole isomorphisms of graph • Red factors: no quantum pigeons [Kersting et al., 2009, Niepert, • Green = true variable, red = false variable 2012, 2013, Bui et al., 2013] UAI 2019 8

Contribution • Two new algorithms: 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 UAI 2019 9

Orbit Generation Exact lifted inference for factor graphs UAI 2019 10

A Simpler Example • Consider a complete factor graph 𝐵 𝐶 𝐷 𝐸 • If all factors identical and symmetric , then Pr = Pr = Pr = Pr • Probability is determined by number of true states UAI 2019 11

Orbits of Factor Graphs • Pr = Pr = Pr = Pr Orbit # Elements of the Orbit 0 1 2 3 4 UAI 2019 12

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… UAI 2019 13

Exact Lifted Inference 1. Efficiently find one representative of each orbit 2. Compute the size of the orbit Orbit Unnormalized Total Orbit Orbit # Orbit Size Repr. State Probability Unnormalized 5 1 5 × 1 = 5 0 13 4 13 × 4 = 52 1 21 6 21 × 6 2 = 126 2 4 3 2 × 4 = 8 3 1 4 3 × 1 = 3 • 𝑎 = ∑unnormalized = 5 + 52 + 126 + 8 + 3 = 193 UAI 2019 14

Exact lifted inference algorithm 1. Efficiently generate one element of each orbit, 2. Efficiently compute the size of each orbit UAI 2019 15

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 UAI 2019 16

Exact lifted inference algorithm ✓ 1. Efficiently generate one element of each orbit, 2. Efficiently compute the size of each orbit • Seems #P-hard at first, but in fact is not • Use graph isomorphism tools to count things UAI 2019 17

Orbit Size Pipeline • Avoid enumerating the whole orbit Assignment- Stabilizer Group Graph Orbit Size Encoded Colored Order Iso. Tool Group Factor Graph Efficient to compute! Orbit-Stabilizer Theorem (Group Theory) UAI 2019 18

Stabilizer Group • Question: Which isomorphisms preserve ( stabilize ) this coloring? 𝐵 𝐶 Answer: Any permutation of {𝐶, 𝐷, 𝐸} 𝐷 𝐸 ✓ • Graph isomorphism tools can compute this set of Assignment- Stabilizer Group: Group Graph permutations Orbit Size Encoded Colored Small set of Order Iso. Tool • Represented in a compact way (generators) Factor Graph generators UAI 2019 19

Group Orbit-Stabilizer Theorem Theory • Relates size of orbit to order (size) of stabilizer 𝐵 𝐶 Orbit size = #ways of permuting 𝐵, 𝐶, 𝐷, 𝐸 = 4! 3! = 4 #ways of permuting 𝐶, 𝐷, 𝐸 𝐸 𝐷 • Computing the order of a group is a standard problem in computational group theory • Efficient to compute (in size of graph) Stabilizer Group: Order of group Small set of GAP 4 × 10 84 states generators UAI 2019 20

Exact Inference Experiments • Proof of concept: Compared against existing exact inference tool, ACE UAI 2019 21

Orbit-Jump MCMC Approximate lifted inference with mixing time guarantees UAI 2019 22

Motivation • Local-search (e.g. Gibbs sampling) can get stuck Propose local move ฀ ฀ UAI 2019 23

Related Work: Within-Orbit Jumps • Lifted MCMC [ Niepert, 2012, 2013 ] jumps within orbits, unfortunately doesn’t help here Lifted MCMC step ฀ ฀ UAI 2019 24

Jumping Between Orbits • Orbit-Jump MCMC proposes jumps between orbits Orbit-Jump Proposal • Non-local moves: can skip over low-probability regions • We show how to jump between orbits using the Burnside process • Exploits orbit structure: better than random restarts • Requires multiple graph isomorphism + group order computations • More expensive than lifted MCMC, with mixing rate guarantees UAI 2019 25

Orbit-Jump MCMC Mixing Time • Empirical mixing time, 5 pigeons 2 holes • Total variation distance from stationary dist. UAI 2019 26

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 UAI 2019 27

Thank you! Questions? Comments? sholtzen@cs.ucla.edu UAI 2019 28