Automorphism Groups of Graphical Models and Lifted Variational Inference Hung Hai Bui 1 Tuyen N. Huynh 2 Sebastian Riedel 3 1 Laboratory for Natural Language Understanding, Nuance Communications 2 AI Center, SRI International 3 Department of Computer Science, University College London July 22, 2013 UAI 2013 1/26
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? UAI 2013 2/26
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 UAI 2013 3/26
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 UAI 2013 4/26
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 . S 5 S 4 x S 3 D 5 1 (d) (a) (b) (c) • The automorphism group partitions the set of nodes into orbits. • Node orbit is a set of nodes equivalent up to symmetry. • Similarly for edge orbits and configuration (node-value assignment) orbits. UAI 2013 5/26
Symmetry of Graphical Models • Exponential family x ∈ R n , θ ∈ R m F ( x | θ ) = exp ( � Φ( x ) , θ � − A ( θ )) • Automorphisms are permutations of variables and features preserving Φ . • Formally, a pair ( π, γ ) ∈ S n × S m such that Φ γ − 1 ( x π ) ≡ Φ( x ) i.e., Φ i ( x π ) ≡ Φ γ ( i ) ( x ) UAI 2013 6/26
Symmetry of Graphical Models (cont.) 1 f 1 f 2 f 3 2 3 f 4 f 5 4 G Colored G Orbits of G 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 UAI 2013 7/26
Properties of Exponential Family Automorphisms Theorem If ( π, γ ) ∈ Aut [ F ] then • Pr ( x π | θ γ ) = Pr ( x | θ ) • π is an automorphism of the structure graph G [ F ] . • Θ γ = Θ and A ( θ γ ) = A ( θ ) • M γ = M and A ∗ ( µ γ ) = A ∗ ( µ ) • m γ ( θ ) = m ( θ γ ) Proof tools: theory of group actions, orbit-stabilizer theorem UAI 2013 8/26
Partitions and Symmetrized Subspaces A partition ∆ of { 1 . . . m } induces a symmetrized subspace R m ∆ of points r in R m such that if j and j ′ are in the same cell of ∆ , then r j = r j ′ . 1 Δ = {1,{2,3}} 3 R Δ 2 3 S ∆ denotes the intersection of S with the subspace: S ∩ R m ∆ UAI 2013 9/26
Variational Inference with Parameter Tying Given parameter-tying partition ∆ and θ ∈ Θ ∆ . Solve � θ, µ � − A ∗ ( µ ) sup µ ∈M How does symmetry help? Intuitions: • some features have the same expectations, so some µ i ’s are the same. • optimal µ is trapped in a symmetrized subspace R m ϕ for some lifting partition ϕ • this generalizes to general convex optimization problem • How to find ϕ ? UAI 2013 10/26
Lifted Variational Inference Framework Parameter-tying Symmetry of the family Aut [ F ] Δ Lifting group Aut Δ [ F ] Lifting partition Mean parameter space (feature orbit partition) M ϕ Symmetrized subspace Lifted mean parameter space m R ϕ M ϕ UAI 2013 11/26
Lifted Variational Inference Framework (cont.) Theorem � θ, µ � − A ∗ ( µ ) = sup � θ, µ � − A ∗ ( µ ) Marginal inference sup µ ∈M µ ∈M ϕ MAP inference sup � θ, µ � = sup � θ, µ � µ ∈M µ ∈M ϕ UAI 2013 12/26
Lifted Marginal Polytope M ϕ Configuration orbit M ϕ Centroid • M projected to R m ϕ = M intersects with R m 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 UAI 2013 13/26
Lifted Marginal Polytope M ϕ • Intersection view • Let ρ be the orbit mapping function ρ : i �→ orb ( i ) • M described by a system of constraints T 1 ( µ ) . . . T K ( µ ) • Intersect T k with R m ϕ means substitute µ i by ¯ µ ρ ( 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 UAI 2013 14/26
Lifted Approximate MAP Inference Theorem Lifted Relaxed MAP: If OUTER = OUTER ( G ) depending only on the graphical model structure G sup sup � θ, µ � = � θ, µ � µ ∈ OUTER µ ∈ OUTER ϕ UAI 2013 15/26
Lifted MAP on LOCAL � τ v : 0 + τ v : 1 = 1 ∀ v ∈ V ( G ) � � τ { u : 0 , v : 0 } + τ { u : 0 , v : 1 } = τ u : 0 � LOCAL= � τ ≥ 0 τ { u : 0 , v : 0 } + τ { v : 0 , u : 1 } = τ v : 0 ∀ { u , v } ∈ E ( G ) � � τ { u : 1 , v : 1 } + τ { u : 0 , v : 1 } = τ v : 1 � � τ { u : 1 , v : 1 } + τ { v : 0 , u : 1 } = τ u : 1 � � τ v : 0 + ¯ ¯ τ v : 1 = 1 ∀ node orbit v � � τ e : 00 + ¯ ¯ τ ( u , v ): 01 = ¯ τ ¯ u : 0 � LOCAL ϕ = � τ ≥ 0 ¯ τ e : 00 + ¯ ¯ τ ( v , u ): 01 = ¯ ∀ edge orbit e with τ ¯ v : 0 � � τ e : 11 + ¯ ¯ τ ( u , v ): 01 = ¯ { u , v } a representative of e τ ¯ v : 1 � � τ e : 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. UAI 2013 16/26
Tightening Bound: Cycle Polytope Cycle constraints (Sontag & Jaakkola 07): 0 X X 1 1 � { u , v }∈ F nocut ( { u , v } , τ ) + � { u , v }∈ C \ F cut ( { u , v } , τ ) ≥ 1 = X C a cycle in G , F ⊂ C , | F | is odd X 0 1 cut ( { u , v } , τ ) = τ u : 0 , v : 1 + τ u : 1 , v : 0 nocut ( { u , v } , τ ) = τ u : 0 , v : 0 + τ u : 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 UAI 2013 17/26
Lifted Cycle Polytope • Substituting ground variables τ by lifted variables ¯ τ yields lifted cycle constraints � � nocut ( { u , v } , ¯ τ ) + cut ( { u , v } , ¯ τ ) ≥ 1 { u , v }∈ F { u , v }∈ C \ F • 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 UAI 2013 18/26
Lifted Cycle Polytope (cont.) • Fix a node i (mark i distinct), let ¯ G [ i ] be the new lifted graph i {i} Node Orbits of G fixing i Node Orbits of G G [ 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 UAI 2013 19/26
Lifted MAP with Cycle Constraints Algorithm 1 constraints ← LOCAL ϕ t , ¯ � ¯ � 2 ¯ τ ← Solve max ¯ θ given current constraints t 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. UAI 2013 20/26
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? UAI 2013 21/26
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