Introduction to the Correlation Decay Method Yitong Yin Nanjing - PowerPoint PPT Presentation

Introduction to the Correlation Decay Method Yitong Yin Nanjing University orkshop, July 20, 2018 Introduction to Partition Functions W Computational Aspects of Partition Functions, CIB@EPFL

• July 20: Introduction to the correlation decay method ( Yitong ) • July 23: Correlation decay for distributed counting ( Yitong ) • July 25: Beyond bounded degree graphs ( Piyush ) • July 26: Correlation decay, zeros of polynomials, and the Lovász local lemma ( Piyush )

Counting Independent Set undirected graph G ( V , E ) fugacity λ > 0 hardcore model: ( λ | I | I is an independent set in G ∀ I ⊆ V : w ( I ) = 0 otherwise partition function: X Z = Z G ( λ ) = w ( I ) I ⊆ V λ c ( ∆ ) = ( ∆ − 1) ∆ − 1 e uniqueness threshold: ≈ ( ∆ − 2) ∆ ∆ − 2 Computing Z G ( λ ) in graphs with constant max-degree ≤ ∆ • λ < λ c ( Δ ) ⇒ FPTAS ⟸ strong spatial mixing [Weitz 06] • λ > λ c ( Δ ) ⇒ no FPRAS unless NP=RP [Galanis Š tefankovi č Vigoda 12] [Sly Sun 12]

Spin System finite integer q ≥ 2 undirected graph G = ( V , E ) configuration σ ∈ [ q ] V Y Y weight: w ( σ ) = A ( σ u , σ v ) b ( σ v ) { u,v } ∈ E v ∈ V symmetric q × q matrix A : [ q ] × [ q ] → R ≥ 0 (symmetric binary constraint) (unary constraint) q -vector b : [ q ] → R ≥ 0 partition function: X Z G = w ( σ ) σ ∈ [ q ] V µ G ( σ ) = w ( σ ) Gibbs distribution: Z G

undirected graph G = ( V , E ) finite integer q ≥ 2 configuration σ ∈ [ q ] V Y Y weight: w ( σ ) = A ( σ u , σ v ) b ( σ v ) { u,v } ∈ E v ∈ V • 2-spin system: q =2, σ ∈ { 0 , 1 } V  �  � a 00 a 01 b 0 symmetric A = b = a 10 a 11 b 1  λ � • hardcore model:  0 � 1 A = b = 1 1 1 • Ising model:  � λ  β � 1 b = A = β 1 1 • multi-spin system: general q ≥ 2 • Potts model:     β 1 1 β .   A = b = .   • q -coloring: β =0   . 1   ...     1 β

Marginal Probability finite integer q ≥ 2 undirected graph G = ( V , E ) Gibbs distribution µ G over all configurations in [ q ] V ∀ possible boundary condition σ ∈ [ q ] Λ specified on an arbitrary subset Λ ⊂ V marginal distribution at vertex v ∈ V : µ σ v ∀ x ∈ [ q ] : v ( x ) = X ∼ µ G [ X v = x | X Λ = σ ] Pr µ σ n w ( σ ) Y µ σ 1 ,..., σ i − 1 = µ ( σ ) = ( σ i ) v i Z G i =1

Marginal Probability finite integer q ≥ 2 undirected graph G = ( V , E ) Gibbs distribution µ G over all configurations in [ q ] V ∀ possible boundary condition σ ∈ [ q ] Λ specified on an arbitrary subset Λ ⊂ V marginal distribution at vertex v ∈ V : µ σ v ∀ x ∈ [ q ] : v ( x ) = X ∼ µ G [ X v = x | X Λ = σ ] Pr µ σ approximately approximately computing µ σ computing Z G v approx . inference approx . counting

Spatial Mixing (Decay of Correlation) v : marginal distribution at vertex v conditioning on σ µ σ weak spatial mixing (WSM) at rate δ ( ) : ∀ σ , τ ∈ [ q ] Λ : k µ σ v � µ τ v k T V  δ (dist G ( v, Λ )) G µ σ boundary v conditions σ dist( v , Λ ) v Λ on infinite graphs: uniqueness of infinite- WSM volume Gibbs measure WSM of hardcore model λ ≤ λ c ( ∆ ) on infinite Δ -regular tree

Spatial Mixing (Decay of Correlation) v : marginal distribution at vertex v conditioning on σ µ σ weak spatial mixing (WSM) at rate δ ( ) : weak spatial mixing (WSM) at rate δ ( ) : ∀ σ , τ ∈ [ q ] Λ : k µ σ v � µ τ v k T V  δ (dist G ( v, Λ )) strong spatial mixing (SSM) at rate δ ( ) : ∀ σ, τ ∈ [ q ] Λ that differ on Δ : k µ σ v � µ τ v k T V  δ (dist G ( v, ∆ )) SSM G marginal probabilities dist( v , Δ ) v Δ are well approximated Λ \ Δ by the local information

Tree Recurrence hardcore model: µ T ( I ) ∝ λ | I | independent set I in T p T , I ∼ µ T [ v is unoccupied by I ] Pr T v u 1 u i u d T 1 T i T d p T i , Pr [ u i is unoccupied by I ] I ∼ µ Ti

p T , I ∼ µ T [ v is unoccupied by I ] Pr Z T ( v is unoccupied) = Z T ( v is unoccupied) + Z T ( v is occupied) T v where u i X Z T (event A ) , w ( I ) I : A holds T i

p T , I ∼ µ T [ v is unoccupied by I ] Pr Z T ( v is unoccupied) = Z T ( v is unoccupied) + Z T ( v is occupied) T v Product rule: d u i Y Z T ( v is unoccupied) = Z T i i =1 d T i Y Z T ( v is occupied) = λ Z T i ( u i is unoccupied) i =1 Q d i =1 Z T i = Q d I =1 Z T i + λ Q d i =1 Z T i ( u i is unoccupied) 1 = 1 + λ Q d i =1 p T i

p σ , I ∼ µ T [ v is unoccupied by I | σ ] Pr T Z T ( v is unoccupied ∧ σ ) = Z T ( v is unoccupied ∧ σ ) + Z T ( v is occupied ∧ σ ) T v Product rule: d u i Y Z T ( v is unoccupied ∧ σ ) = Z T i ( σ i ) i =1 T i d Y Z T ( v is occupied ∧ σ ) = λ Z T i ( u i is unoccupied ∧ σ i ) σ i i =1 Q d i =1 Z T i ( σ i ) = Q d I =1 Z T i + λ Q d i =1 Z T i ( σ i )( u i is unoccupied ∧ σ i ) 1 = 1 + λ Q d i =1 p σ i T i

Tree Recurrence µ T ( I ) ∝ λ | I | hardcore model: independent set I in T p σ , I ∼ µ T [ v is unoccupied by I | σ ] Pr T T Occupancy ratio: v Pr T [ v is occupied | σ ] R σ T , u i Pr T [ v is unoccupied | σ ] = (1 − p σ T ) /p σ T i T σ i d 1 1 Y R σ p σ T = λ = T R σ i 1 + λ Q d T i + 1 i =1 p σ i T i i =1

Tree Recurrence  a 00 �  � 2-spin system: a 01 Y Y b 0 µ T ( σ ) ∝ a σ ( u ) , σ ( v ) b σ ( v ) A = b = a 10 a 11 b 1 uv ∈ E v ∈ V T v Occupancy ratio: v i T , Pr X ∼ µ T [ X v = 0 | σ ] R σ Pr X ∼ µ T [ X v = 1 | σ ] T i σ i d a 00 R σ i T i + a 01 T = b 0 Y R σ a 10 R σ i b 1 T i + a 11 i =1 a Möbius transformation

Tree Recurrence  λ �  0 � hardcore model: 1 A = b = 1 1 1 T v Occupancy ratio: v i T , Pr X ∼ µ T [ X v = 0 | σ ] R σ Pr X ∼ µ T [ X v = 1 | σ ] T i σ i d 1 Y R σ T = λ R σ i T i + 1 i =1

Z G ( v is occupied) Pr I ∼ µ G [ v is occupied by I ] Pr I ∼ µ G [ v is unoccupied by I ] = R v = Z G ( v is unoccupied) G v u i

Z G ( v is occupied) Pr I ∼ µ G [ v is occupied by I ] Pr I ∼ µ G [ v is unoccupied by I ] = R v = Z G ( v is unoccupied) = Z G 0 ( v 1 , . . . , v d = • · · · • ) G’ Z G 0 ( v 1 , . . . , v d = � · · · � ) λ 1 /d λ 1 /d λ 1 /d v i v d v 1 u i

Z G ( v is occupied) Pr I ∼ µ G [ v is occupied by I ] Pr I ∼ µ G [ v is unoccupied by I ] = R v = Z G ( v is unoccupied) = Z G 0 ( v 1 , . . . , v d = • · · · • ) G’ Z G 0 ( v 1 , . . . , v d = � · · · � ) λ 1 /d v i i − 1 d − i z }| { z }| { d Z G 0 ( v 1 , . . . , v d = • · · · • ) Y � · · · � • u i = Z G 0 ( v 1 , . . . , v d = � · · · � ) � • · · · • i =1 | {z } | {z } i − 1 d − i d τ i : v 1 , . . . , v i − 1 = � · · · � Y : unoccupied R τ i = G 0 ,v i v i +1 , . . . , v d = • · · · • : occupied i =1

Z G ( v is occupied) Pr I ∼ µ G [ v is occupied by I ] Pr I ∼ µ G [ v is unoccupied by I ] = R v = Z G ( v is unoccupied) = Z G 0 ( v 1 , . . . , v d = • · · · • ) G i Z G 0 ( v 1 , . . . , v d = � · · · � ) i − 1 d − i z }| { z }| { d Z G 0 ( v 1 , . . . , v d = • · · · • ) Y � · · · � • u i = Z G 0 ( v 1 , . . . , v d = � · · · � ) � • · · · • i =1 | {z } | {z } i − 1 d − i d τ i : v 1 , . . . , v i − 1 = � · · · � Y : unoccupied R τ i = G 0 ,v i v i +1 , . . . , v d = • · · · • : occupied i =1 d d λ 1 /d 1 Y Y G i ,u i + 1 = λ = R τ i R τ i G i ,u i + 1 i =1 i =1

Tree Recurrence Pr[ v is occupied | σ ] hardcore model: v = R σ Pr[ v is unoccupied | σ ] v independent set I in G d 1 µ ( I ) ∝ λ | I | Y R σ v = λ 1 + R σ i i i =1 v v v v 3 v 1 v 2 Pr[ v 2 is occupied | σ 2 ] Pr[ v 3 is occupied | σ 3 ] Pr[ v 1 is occupied | σ 1 ] 2 = 3 = R σ 1 1 = R σ 2 R σ 3 Pr[ v 1 is unoccupied | σ 1 ] Pr[ v 2 is unoccupied | σ 2 ] Pr[ v 3 is unoccupied | σ 3 ] : occupied : unoccupied

Self-Avoiding Walk Tree (Godsil 1981; Weitz 2006) = T = T ��� ( G, v ) µ σ µ σ in T in G v root G v d 1 1 1 Y R σ T = λ 1 + R σ i 4 4 T i 2 3 2 i =1 3 σ 6 5 6 6 6 5 3 4 5 6 6 5 6 6 SSM in trees of max-deg ≤Δ : 6 6 5 SSM and approx. inference in graphs with max-deg ≤Δ if cycle closing edge > cycle starting edge if cycle closing edge < cycle starting edge

Self-Avoiding Walk Tree (Godsil 1981; Weitz 2006) T = T ��� ( G, v ) = µ σ µ σ in T in G d a 00 R σ i T i + a 01 v T = b 0 root G v Y R σ 1 1 a 10 R σ i b 1 T i + a 11 i =1 4 4 2 3 2 3 σ 6 5 6 6 6 5 3 4 5 6 6 5 6 6 SSM in trees of max-deg ≤Δ : 6 6 5 SSM and approx. inference in graphs with max-deg ≤Δ hold for 2-spin systems if cycle closing edge > cycle starting edge if cycle closing edge < cycle starting edge  a 00 �  � a 01 b 0 A = b = a 10 a 11 b 1

Correlation Decay hardcore model: v ` → ∞ regular tree Δ - σ : all leaves are occupied τ : all leaves are unoccupied λ c ( ∆ ) = ( ∆ − 1) ∆ − 1 uniqueness threshold: ( ∆ − 2) ∆ ∆ − 1 hardcore recurrence: 1 Y R σ T = λ 1 + R σ i T i i =1 λ single-variate dynamical system: f ( x ) = (1 + x ) ∆ − 1