Automorphism Groups of Graphical Models and Lifted Variational - - PowerPoint PPT Presentation

Automorphism Groups of Graphical Models and Lifted Variational Inference

Hung Hai Bui1 Tuyen N. Huynh2 Sebastian Riedel3

1Laboratory for Natural Language Understanding, Nuance Communications 2AI Center, SRI International 3Department of Computer Science, University College London

July 22, 2013

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Motivations

• Probabilistic inference – exploit low tree-width, local graph

sparsity

• What about the above graphical models?
• Inference can be done efficiently using lifted inference.
• Why? “Symmetry” is essential.
• What is symmetry?
• How do we exploit symmetry in variational inference?

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Motivations (cont.)

• Lifted inference methods mostly are derived procedurally:
• identify same duplicated computational steps that can be

performed once.

• Not clear what form of symmetry is being exploited.
• Hard to generalize.
• We propose a declarative approach
• Formalize symmetry in graphical models
• Lift variational optimization formulations rather than lift

inference algorithms

• Lifted problems can be solved with the usual optimization

tool-box (LP solvers, cutting plane, dual decomposition, etc)

• Connect lifted inference and mainstream variational

approximations

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Outline

1 Motivation 2 Symmetry of Graphical Models 3 A General Framework for Lifted Convex Variational Inference 4 Lifted MAP with Local and Cycle constraints 5 Experiments

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Symmetry in Graphs

• Symmetry is formalized as the set of transformations that

preserve the object.

• The set of all such transformations (permutations) π form the

automorphism group.

S5 S4 x S3 D5

(a) (b) (c) (d) 1

• The automorphism group partitions the set of nodes into
• rbits.
• Node orbit is a set of nodes equivalent up to symmetry.
• Similarly for edge orbits and configuration (node-value

assignment) orbits.

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Symmetry of Graphical Models

• Exponential family

F(x | θ) = exp (Φ(x), θ − A(θ)) x ∈ Rn, θ ∈ Rm

• Automorphisms are permutations of variables and features

preserving Φ.

• Formally, a pair (π, γ) ∈ Sn × Sm such that

Φγ−1(xπ) ≡ Φ(x) i.e., Φi(xπ) ≡ Φγ(i)(x)

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Symmetry of Graphical Models (cont.)

G

1 3 4 2 f1 f2 f3 f4 f5 G Colored G Orbits of G

• Automorphisms of Colored G are automorphisms of F
• So far we’ve ingored parameters. If parameters are tied
• Refine colors of features to be consistent with parameter tying
• Similarly compute the automophorism group consistent with

parameter tying

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Properties of Exponential Family Automorphisms Theorem

If (π, γ) ∈ Aut[F] then

• Pr(xπ|θγ) = Pr(x|θ)
• π is an automorphism of the structure graph G[F].
• Θγ = Θ and A(θγ) = A(θ)
• Mγ = M and A∗(µγ) = A∗(µ)
• mγ(θ) = m(θγ)

Proof tools: theory of group actions, orbit-stabilizer theorem

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Partitions and Symmetrized Subspaces

A partition ∆ of {1 . . . m} induces a symmetrized subspace Rm

∆ of

points r in Rm such that if j and j′ are in the same cell of ∆, then rj = rj′.

1 2 3

RΔ

3

Δ = {1,{2,3}}

S∆ denotes the intersection of S with the subspace: S ∩ Rm

∆

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Variational Inference with Parameter Tying

Given parameter-tying partition ∆ and θ ∈ Θ∆. Solve sup

µ∈M

θ, µ − A∗(µ) How does symmetry help? Intuitions:

• some features have the same expectations, so some µi’s are

the same.

• optimal µ is trapped in a symmetrized subspace Rm

ϕ for some

lifting partition ϕ

• this generalizes to general convex optimization problem
• How to find ϕ?

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Lifted Variational Inference Framework

Mean parameter space Symmetry of the family Parameter-tying

Δ Aut[F] AutΔ[F]

Lifting group

ϕ

Lifting partition (feature orbit partition)

M Rϕ

m

Symmetrized subspace

Mϕ

Lifted mean parameter space

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Lifted Variational Inference Framework (cont.) Theorem

Marginal inference sup

µ∈M

θ, µ − A∗(µ) = sup

µ∈Mϕ

θ, µ − A∗(µ) MAP inference sup

µ∈M

θ, µ = sup

µ∈Mϕ

θ, µ

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Lifted Marginal Polytope Mϕ

Configuration orbit Centroid

Mϕ

• M projected to Rm

ϕ = M intersects with Rm ϕ

• Projection view:
• Ground configurations in the same orbit project to the same

point, which is the centroid of the orbit

• num. extreme points ≤ num. of configuration orbits (typically

still exponential)

• For complete symmetric graphical models with N binary

variables:

• num. configuration orbits = N + 1

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Lifted Marginal Polytope Mϕ

• Intersection view
• Let ρ be the orbit mapping function ρ : i → orb(i)
• M described by a system of constraints T1(µ) . . . TK(µ)
• Intersect Tk with Rm

ϕ means substitute µi by ¯

µρ(i)

• num. variables reduced to num. feature orbits
• num. constraints unchanged
• However, many constraints become redundant and can be

removed

• This view is useful in practice for lifting outer bounds of M

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Lifted Approximate MAP Inference Theorem

Lifted Relaxed MAP: If OUTER = OUTER(G) depending only on the graphical model structure G sup

µ∈OUTER

θ, µ = sup

µ∈OUTERϕ

θ, µ

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Lifted MAP on LOCAL

LOCAL=

           τ ≥ 0

• τv:0 + τv:1 = 1

∀v ∈ V(G) τ{u:0,v:0} + τ{u:0,v:1} = τu:0 τ{u:0,v:0} + τ{v:0,u:1} = τv:0 ∀ {u, v} ∈ E(G) τ{u:1,v:1} + τ{u:0,v:1} = τv:1 τ{u:1,v:1} + τ{v:0,u:1} = τu:1           

LOCALϕ =

           ¯ τ ≥ 0

• ¯

τv:0 + ¯ τv:1 = 1 ∀ node orbit v ¯ τe:00 + ¯ τ(u,v):01 = ¯ τ¯

u:0

¯ τe:00 + ¯ τ(v,u):01 = ¯ τ¯

v:0

∀ edge orbit e with ¯ τe:11 + ¯ τ(u,v):01 = ¯ τ¯

v:1

{u, v} a representative of e ¯ τe:11 + ¯ τ(v,u):01 = ¯ τ¯

u:1

          

• num. constraints is O(#node orbits + #edge orbits).
• Lifted LOCAL LP can be solved by a generic solver, or a

message-passing variant, e.g., MPLP.

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Tightening Bound: Cycle Polytope

Cycle constraints (Sontag & Jaakkola 07):

1 1 1 X X X X =

• {u,v}∈F nocut({u, v}, τ) +

{u,v}∈C\F cut({u, v}, τ) ≥ 1

C a cycle in G, F ⊂ C, |F| is odd cut({u, v}, τ) = τu:0,v:1 + τu:1,v:0 nocut({u, v}, τ) = τu:0,v:0 + τu:1,v:1

• num. constraints is large
• Iteratively add violated cycle contraint via cutting plane
• Run shortest path on mirror graph of G to find violated

constraint

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Lifted Cycle Polytope

• Substituting ground variables τ by lifted variables ¯

τ yields lifted cycle constraints

• {u,v}∈F

nocut({u, v}, ¯ τ) +

• {u,v}∈C\F

cut({u, v}, ¯ τ) ≥ 1

• However
• these constraints are still defined on cycles of the ground

graphical model G.

• many constraints are duplicates
• Can we characterize cycle constraints using the lifted graph ¯

G?

• not straightforward, since cycles in ¯

G might not correspond to cycles in G

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Lifted Cycle Polytope (cont.)

• Fix a node i (mark i distinct), let ¯

G[i] be the new lifted graph

G[i]

Node Orbits of G i Node Orbits of G fixing i {i}

• A cycle through {i} on ¯

G[i] always corresponds to a set of cycles throuh i on G.

• Violated constraints on ground cycles through i can be found

by

• forming mirror graph of ¯

G[i]

• running shortest path from {i} to its mirror node

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Lifted MAP with Cycle Constraints

Algorithm

1 constraints ← LOCALϕ 2 ¯

τ ←Solve max¯

t

¯ t, ¯ θ

• given current constraints

3 ∀ node orbit v, pick a representative i ∈ v. Find the shortest path

from {i} to {i}-mirror on the mirror graph of ¯ G[i].

4 If shortest-path < 1, add violated constraint to the current set of

constraints.

5 If no more violated constraint, exit; else goto 2.

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Markov Logic Networks (MLNs)

• “Lovers & Smokers” Markov Logic Network

100 Male(x) ⇔ ¬Female(x) 2 Male(x) ∧ Smokes(x) 2 Female(x) ∧ ¬Smokes(x) 0.5 x = y ∧ Male(x) ∧ Female(y) ∧ Loves(x, y) 0.5 x = y ∧ Loves(x, y) ⇒ (Smokes(x) ⇔ Smokes(y)) −100 x = y ∧ y = z ∧ z = x ∧ Loves(x, y) ∧ Loves(y, z) ∧ Loves(x, z)

• How to find lifting partition?
• Ground the model, construct colored factor graph and run

graph automorphism tool (e.g. nauty)

• Can we do this without grounding?

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Renaming Symmetry of MLNs

• Let D∗ be the set of individuals that do not appear as

constants in MLN.

• Intuition: every permutation (renaming) of individuals in D∗

preserves the probabilistic model.

Theorem

The renaming group S(D∗) is (isomorphic to) a subgroup of the MLN’s lifting group

• The number of orbits induced by this group does not depend
• n domain size |D|.
• Example: observed one constant a
• Renaming group induces 5 orbits for Love predicate

{L(a, a)}, {L(x, x)}, {L(a, x)}, {L(x, a)}, {L(x, y)} x, y = a, x = y

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Experiment

Finding symmetry: renaming group versus graph automorphism (nauty)

domain size 10 20 50 100 200 1000 Nauty #Orbits 12 23 25 27 * * Time(s) .49 1.77 172.79 9680.48 * * Renaming #Orbits 12 23 80 255 905 20505 Time(s) .08 .09 .221 .4 .84 2.19

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Experiment (cont.)

Lifted MAP (LOCAL, CYCLE) inference versus ground counterparts

515 ¡ 516 ¡ 517 ¡ 518 ¡ 519 ¡ 520 ¡ 521 ¡ 522 ¡ 0 ¡ 10 ¡ 20 ¡ 30 ¡ 40 ¡ Objec&ve ¡ ¡ Time(s) ¡

(b) ¡Objec&ve ¡over ¡&me ¡for ¡domain ¡size ¡5 ¡

True ¡Op2mal ¡ Local ¡ Cycle ¡ Renaming-­‑Local ¡ Renaming-­‑Cycle ¡ 0.001 ¡ 0.01 ¡ 0.1 ¡ 1 ¡ 10 ¡ 100 ¡ 1000 ¡ 10000 ¡ 100000 ¡ 5 ¡ 10 ¡ 15 ¡ 20 ¡ 100 ¡ 1000 ¡ Time ¡(s) ¡ Domain ¡size ¡

(a) ¡Run&me ¡vs. ¡domain ¡size ¡

“Lovers & Smokers” MLN without evidence, varying domain size

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Experiment (cont.)

Lifted CYCLE versus lifted LOCAL

0.001 ¡ 0.01 ¡ 0.1 ¡ 1 ¡ 10 ¡ 100 ¡ 1000 ¡ 10000 ¡ 100000 ¡ 2 ¡ 4 ¡ 6 ¡ 8 ¡ 10 ¡ Time(s) ¡ Number ¡of ¡observed ¡constants ¡ Renaming-­‑Local ¡ Renaming-­‑Cycle ¡ 14500 ¡ 15000 ¡ 15500 ¡ 16000 ¡ 16500 ¡ 2 ¡ 4 ¡ 6 ¡ 8 ¡ 10 ¡ Objec7ve ¡ Number ¡of ¡observed ¡constants ¡

“Lovers & Smokers” MLN with varying amount of soft evidence, fixing domain size to 100

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Conclusion and Future Direction

• Summary:
• First rigorous formalization of symmetry in graphical models
• A general framework for lifting convex outer bound variational

approximations

• First lifted algorithm that works with a bound tighter than the

local polytope

• Related and Future work:
• Exploit symmetry of the log-partition function A (Bui et al.,

AAAI’12)

• Exploit symmetry in sampling methods (Niepert UAI’12,

AAAI’13)

• Lift convex variational marginal inference

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