T wo-State Spin System graph G =( V , E ) 2 states {0,1} - PowerPoint PPT Presentation

Approximate Counting v ia Correlation Decay i n Spin Systems Yitong Yin Nanjing University Joint work with Liang Li ( Peking U ) and Pinyan Lu ( MSRA )

T wo-State Spin System graph G =( V , E ) 2 states {0,1} configuration σ : V → { 0 , 1 } contributions of local interactions: 1 β γ weight: � ������������ ( e ) w ( σ ) = e ∈ E

T wo-State Spin System graph G =( V , E ) 2 states {0,1} configuration σ : V → { 0 , 1 } � A 0 , 0 � � β � 1 A 0 , 1 A = = 1 A 1 , 0 A 1 , 1 γ contributions of local interactions: 1 β γ weight: � w ( σ ) = A σ ( u ) , σ ( v ) ( u,v ) ∈ E

T wo-State Spin System graph G =( V , E ) 2 states {0,1} configuration σ : V → { 0 , 1 } � A 0 , 0 � � β � 1 A 0 , 1 A = = 1 A 1 , 0 A 1 , 1 γ weight: � w ( σ ) = A σ ( u ) , σ ( v ) ( u,v ) ∈ E µ ( σ ) = w ( σ ) Gibbs measure: Z A ( G ) partition function: � Z A ( G ) = w ( σ ) σ ∈ { 0 , 1 } V

Partition Function graph G =( V , E ) 2 states {0,1} configuration σ : V → { 0 , 1 } � A 0 , 0 � � β � 1 A 0 , 1 A = = 1 A 1 , 0 A 1 , 1 γ partition function: � � Z A ( G ) = A σ ( u ) ,σ ( v ) σ ∈ { 0 , 1 } V ( u,v ) ∈ E # independent set # vertex cover β = 0 , γ = 1 weighted Boolean CSP with one symmetric relation

Approximate Counting � A 0 , 0 � � β � 1 A 0 , 1 fix A = = 1 A 1 , 0 A 1 , 1 γ partition function: � � Z A ( G ) = A σ ( u ) ,σ ( v ) σ ∈ { 0 , 1 } V ( u,v ) ∈ E is a well-define computational problem or poly-time computable if βγ = 1 ( β , γ ) = (0 , 0) #P-hard if otherwise Approximation!

[JS93] Jerrum-Sinclair’93 [GJP03] Goldberg-Jerrum-Paterson’03 3 γ ferromagnetic 2-state spin FPRAS [GJP03] 2.5 �� = 1 uniqueness threshold threshold achieved by heatbath random walk 2 ferromagnetic Ising Model 1.5 anti- FPRAS [JS93] � ferromagnetic 1.11017 1 0 ≤ β , γ ≤ 1 0 < � , � < 1 no FPRAS 0.5 FPRAS [GJP03] unless NP ⊆ RP heat-bath [GJP03] 0 β 0 0.5 1 1.5 2 2.5 3 �

Uniqueness Threshold � �� � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � �� � d � � � � � � � � � � � � � � �� � � � � β x + 1 � f ( x ) = x + γ �� � � 1.11017 x = f (ˆ ˆ x ) � � � � � � � � � � � �� � | f � (ˆ x ) | < 1 for all d � � �� � � �� � � �� � � � 0 ≤ β < 1 < γ

Our Result 3 γ 2.5 �� = 1 uniqueness threshold threshold achieved by heatbath random walk 2 1.5 � 1.11017 1 0 ≤ β , γ ≤ 1 0 < � , � < 1 0.5 FPTAS 0 β 0 0.5 1 1.5 2 2.5 3 �

Marginal Distribution weight: � w ( σ ) = A σ ( u ) , σ ( v ) ( u,v ) ∈ E µ ( σ ) = w ( σ ) Gibbs measure: Z A ( G ) � � Z A ( G ) = A σ ( u ) ,σ ( v ) σ ∈{ 0 , 1 } V ( u,v ) ∈ E marginal distributions at vertex v : p v = �� σ ∼ µ [ σ ( v ) = 0] σ Λ ∈ { 0 , 1 } Λ fixed free v �� Λ v ∈ Λ Λ ⊂ V p σ Λ = �� σ ∼ µ [ σ ( v ) = 0 | σ ( Λ ) = σ Λ ] v

Self-reduction (Jerrum-Valiant-Vazirani) p σ Λ σ Λ ∈ { 0 , 1 } Λ = �� σ [ σ ( v ) = 0 | σ (Λ) = σ Λ ] Λ ⊂ V v σ i : v 1 , v 2 , . . . , v i �� 1 V = { v 1 , v 2 , . . . , v n } �� ( σ n ) = �� σ [ σ : v 1 , . . . , v n �� 1] n � �� = σ [ σ ( v i ) = 1 | σ : v 1 , . . . , v i − 1 �� 1] i =1 n � = (1 − p σ i − 1 ) v i i =1 γ | E | = γ | E | �� ( σ n ) = w ( σ n ) Z ( G ) = � n i =1 (1 − p σ i − 1 ) Z ( G ) Z ( G ) v i

Correlation Decay ∀ σ ∂ B , τ ∂ B ∈ { 0 , 1 } ∂ B �� σ [ σ ( v ) = 0 | σ ∂ B ] ≈ �� σ [ σ ( v ) = 0 | τ ∂ B ] p σ Λ �� σ [ σ ( v ) = 0 | σ ∂ B , σ Λ ] ≈ �� σ [ σ ( v ) = 0 | τ ∂ B , σ Λ ] ≈ v G error < exp (- t ) ∂ B B exponential t correlation decay v Λ “ strong spatial mixing ” in [Weitz’06]

Recursion for Tree σ Λ ∈ { 0 , 1 } Λ Λ ⊂ V T v p σ Λ v R σ Λ = T 1 − p σ Λ v v 1 v 2 v d T d T 1 = �� σ ∼ µ | σ Λ [ σ ( v ) = 0] �� σ ∼ µ | σ Λ [ σ ( v ) = 1] d β R σ Λ T i + 1 R σ Λ � T = R σ Λ T i + γ i =1 � d i =1 ( β w ( σ T i : v i �� 0) + w ( σ T i : v i �� 1)) w ( σ T : v �� 0) w ( σ T : v �� 1) = � d i =1 ( w ( σ T i : v i �� 0) + γ w ( σ T i : v i �� 1))

Self-Avoiding Walk Tree due to Weitz (2006) G =( V , E ) T = T ��� ( G, v ) 1 v 1 4 2 3 4 3 2 6 6 4 3 5 σ Λ 5 6 6 5 1 6 6 1 5 6 6 p σ Λ v R σ Λ G,v = 4 5 4 1 − p σ Λ 6 6 v 4 4 Weitz (2006) if cycle closing > cycle starting R σ Λ G,v = R σ Λ T if cycle closing < cycle starting

Approximation Algorithm exponential T = T ��� ( G, v ) 1 correlation decay: preserve degrees 4 2 3 error= ����� − R σ Λ ∩ B R σ Λ ∩ B ����� 6 4 3 5 = ��� ( − �������� ∂ B ) 5 error decreases 1 6 1 5 6 exponentially in depth 4 4 ∂ B 5 5 4 4 6 6 poly-time on 4 4 O(1) -degree graphs 4 4 Correlation Decay!

T echnique • amortized analysis of decay: • the potential method; • Computationally Efficient Correlation Decay : dealing with unbounded-degree graphs;

Uniqueness Threshold T d infinite ( d +1) -regular tree � Uniqueness of Gibbs measure (Bethe lattice, Cayley tree) � d � β x + 1 f ( x ) = x + γ x = f (ˆ ˆ x ) | f � (ˆ x ) | < 1

Uniqueness Threshold � �� � �� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �� � � � � � � � �� � d � � � � � � � � � � � � � � �� � � � � β x + 1 � f ( x ) = x + γ �� � � 1.11017 x = f (ˆ ˆ x ) � � � � � � � � � � � �� � | f � (ˆ x ) | < 1 for all d � � �� � � �� � � �� � � � 0 ≤ β < 1 < γ

Correlation Decay T = T ��� ( G, v ) anti-ferromagnetic βγ < 1 1 δ = R ����� − R ����� d β R σ Λ T i + 1 = f ( R σ Λ T 1 , . . . , R σ Λ � T d ) R σ Λ = R σ Λ T T i + γ i =1 4 monotonically decreasing 2 3 6 6 4 3 5 upper bound = f ( lower bounds ) lower bound = f ( upper bounds ) 5 1 6 1 5 6 v ∈ Λ fixed to be 0 4 4 ∆ 5 5 4 4 6 6 R ∈ [0 , ∞ ) lower=upper= ∞ 4 4 4 4 v ∈ Λ fixed to be 1 lower=upper= 0 Goal: δ = ��� ( − Ω ( �������� ∆ ))

Cheating: R v ≤ R σ Λ ≤ R v + δ v T � d � β x + 1 f ( x ) = x ∈ [ R v , R v + δ v ] x + γ v T v 1 v 2 v d T d T 1 x d x 1 [ R v d , R v d + δ v d ] [ R v 1 , R v 1 + δ v 1 ] d β x i + 1 we do not always have � f ( x 1 , . . . , x d ) = | f � ( x ) | < 1 x i + γ i =1 δ ������ δ ����� | f � ( x ) | Φ ( x ) Φ ( x ������ ) ≤ α · α = < 1 Φ ( x ����� ) Φ ( f ( x )) D +1 +1 2 D ( β x + 1) Φ ( x ) = x

D +1 +1 2 D ( β x + 1) Φ ( x ) = x x ∈ [ R v , R v + δ v ] v T v 1 v 2 v d T d T 1 x d x 1 [ R v d , R v d + δ v d ] [ R v 1 , R v 1 + δ v 1 ] Mean Value Thms d β x i + 1 � f ( x 1 , . . . , x d ) = x i + γ i =1

Jensen’s Inequality d ( D − 1) D +1 2 D ( β x + 1) d (1 − βγ ) x 2 D α ( d, x ) = � � � d � ( x + γ ) 1+ d ( D − 1) β x +1 + 1 β 2 D x + γ Φ ( x ) Φ ( f ( x )) | f � ( x ) | =

x ∈ [ R v , R v + δ v ] D +1 +1 2 D ( β x + 1) Φ ( x ) = x v T v 1 v 2 v d T d T 1 x d x 1 [ R v d , R v d + δ v d ] [ R v 1 , R v 1 + δ v 1 ] d ( D − 1) D +1 2 D ( β x + 1) d (1 − βγ ) x 2 D α ( d, x ) = � � � d � ( x + γ ) 1+ d ( D − 1) β x +1 + 1 β 2 D x + γ if

d ( D − 1) D +1 2 D ( β x + 1) d (1 − βγ ) x 2 D α ( d, x ) = � � � d � ( x + γ ) 1+ d ( D − 1) β x +1 + 1 β 2 D x + γ

R v ≤ R σ Λ ≤ R v + δ v T T = T ��� ( G, v ) x ∈ [ R v , R v + δ v ] 1 δ = R ����� − R ����� v T 4 2 3 v 1 v 2 v d T d T 1 x d x 1 6 6 4 3 5 [ R v d , R v d + δ v d ] [ R v 1 , R v 1 + δ v 1 ] 5 1 6 1 5 6 4 4 5 5 4 4 6 6 R ∈ [0 , ∞ ) 4 4 4 4 δ = ��� ( − Ω ( �������� ∆ ))

Computationally Efficient Correlation Decay v T x ∈ [ R v , R v + δ v ] v 1 v 2 v d T d T 1 x d x 1 [ R v d , R v d + δ v d ] [ R v 1 , R v 1 + δ v 1 ] d ( D − 1) D +1 2 D ( β x + 1) d (1 − βγ ) x 2 D α ( d, x ) = � � � d � ( x + γ ) 1+ d ( D − 1) β x +1 + 1 β 2 D x + γ for some

Computationally Efficient Correlation Decay v T x ∈ [ R v , R v + δ v ] v 1 v 2 v d T d T 1 x d x 1 [ R v d , R v d + δ v d ] [ R v 1 , R v 1 + δ v 1 ] for some α ( d, x ) for small one-step recursion decays for large one-step recursion decays steps! behaves like