Dynamic Sampling fs om Graphical Models Yitong Yin Nanjing - PowerPoint PPT Presentation

Dynamic Sampling fs om Graphical Models Yitong Yin Nanjing University Joint work with W eiming Feng ( Nanjing ) Nisheeth Vishnoi ( EPFL )

Graphical Model instance of graphical model: I = ( V, E, [ q ] , Φ ) • V : variables ϕ v • E ⊂ 2 V : constraints • [ q ] = {0,1, …, q -1} : domain constraint ϕ e • Φ = ( 𝜚 v ) v ∈ V ∪ ( 𝜚 e ) e ∈ E : factors e • Gibbs distribution µ over all σ ∈ [ q ] V : μ ( σ ) ∝ ∏ V ϕ v ( σ v ) ∏ ϕ e ( σ e ) v ∈ V e ∈ E

Graphical Model instance of graphical model: I = ( V, E, [ q ] , Φ ) • Each v ∈ V is a variable with domain [ q ] ϕ v over [ q ] and has a distribution ϕ v ϕ v : [ q ] → [0,1] • Each e ∈ E is a set of variables and constraint ϕ e corresponds to a constraint (factor) e ϕ e : [ q ] e → [0,1] • Gibbs distribution µ over all σ ∈ [ q ] V : μ ( σ ) ∝ ∏ V ϕ v ( σ v ) ∏ ϕ e ( σ e ) v ∈ V e ∈ E

Graphical Model • Gibbs distribution µ over all σ ∈ [ q ] V : μ ( σ ) ∝ ∏ ϕ v ( σ v ) ∏ ϕ e ( σ e ) v ∈ V e ∈ E ϕ v is a distribution over [ q ] ϕ e : [ q ] e → [0,1] • each v ∈ V independently samples X v ∈ [ q ] according to 𝜚 v ; • each e ∈ E is passed independently with probability 𝜚 e ( X e ) ; • X is accepted if all constraints e ∈ E are passed.

Graphical Model • Gibbs distribution µ over all σ ∈ [ q ] V : μ ( σ ) ∝ ∏ G ( V , E ) ∏ ϕ v ( σ v ) ϕ e ( σ u , σ v ) v ∈ V e =( u , v ) ∈ E ϕ v ϕ e • hardcore morel: v u [ q ] = {0,1} ϕ e ( σ u , σ v ) = { if σ u = σ v = 1 0 otherwise 1 1 if σ v = 0 1 + λ v λ v > 0 is (local) fugacity ϕ v ( σ v ) = λ v if σ v = 1 1 + λ v

Graphical Model • Gibbs distribution µ over all σ ∈ [ q ] V : μ ( σ ) ∝ ∏ G ( V , E ) ∏ ϕ v ( σ v ) ϕ e ( σ u , σ v ) v ∈ V e =( u , v ) ∈ E ϕ v ϕ e • Ising/Potts model: v u or { (ferromagnetic) ϕ e ( σ u , σ v ) = { if σ u = σ v 1 β e ∈ [0,1] otherwise (anti-ferromagnetic) ϕ e ( σ u , σ v ) = { β e ∈ [0,1] if σ u = σ v otherwise 1 ϕ v is a distribution over [ q ] (arbitrary local fields)

Dynamic Sampling • Gibbs distribution µ over all σ ∈ [ q ] V : ϕ ′ � v μ ( σ ) ∝ ∏ ϕ v ( σ v ) ∏ ϕ v ϕ e ( σ e ) ϕ e v u v ∈ V e ∈ E ϕ ′ � e current sample: X ~ µ dynamic update: • adding/deleting a constraint e } new distribution • changing a factor 𝜚 v or 𝜚 e µ’ • adding/deleting an independent variable v Question: Obtain X’ ~ µ’ from X ~ µ with small incremental cost.

Dynamic Sampling instance of graphical model: I = ( V, E, [ q ] , Φ ) • Gibbs distribution µ over all σ ∈ [ q ] V : ϕ v μ ( σ ) ∝ ∏ ϕ v ( σ v ) ∏ ϕ e ( σ e ) v ∈ V e ∈ E constraint ϕ e e current sample: X ~ µ • V : variables • E ⊂ 2 V : constraints V • [ q ] = {0,1, …, q -1} : domain • Φ = ( 𝜚 v ) v ∈ V ∪ ( 𝜚 e ) e ∈ E : factors

Dynamic Sampling instance of graphical model: I = ( V, E, [ q ] , Φ ) update: ( D , 𝜚 D ) is the set of changed variables and constraints D ⊂ V ∪ 2 V Φ D = ( ϕ v ) v ∈ V ∩ D ∪ ( ϕ e ) e ∈ 2 V ∩ D specifies the new factors ( D , Φ D ) ( V , E ′ � , [ q ], Φ′ � ) ( V , E , [ q ], Φ ) E ′ � = E ∪ (2 V ∩ D ) Φ′ � = ( ϕ ′ � a ) a ∈ V ∪ E ′ � Φ D if a ∈ D a is as specified in { where each ϕ ′ � otherwise Φ

Dynamic Sampling instance of graphical model: I = ( V, E, [ q ] , Φ ) update: ( D , 𝜚 D ) is the set of changed variables and constraints D ⊂ V ∪ 2 V Φ D = ( ϕ v ) v ∈ V ∩ D ∪ ( ϕ e ) e ∈ 2 V ∩ D specifies the new factors ( D , Φ D ) ( V , E ′ � , [ q ], Φ′ � ) ( V , E , [ q ], Φ ) Input : a graphical model with Gibbs distribution µ a sample X ~ µ, and an update ( D , 𝜚 D ) Output : X’ ~ µ’ where µ’ is the new Gibbs distribution ( D , 𝜚 D ) is fixed by an offline adversary independently of X ~ µ

Dynamic Sampling Input : a graphical model with Gibbs distribution µ a sample X ~ µ, and an update ( D , 𝜚 D ) Output : X’ ~ µ’ where µ’ is the new Gibbs distribution • inference/learning tasks where the graphical model is changing dynamically • video de-noising • online learning with dynamic or streaming data • sampling/inference/learning algorithms which adaptively and locally change the joint distribution • stochastic gradient descent • JSV algorithm for perfect matching

Dynamic Sampling Input : a graphical model with Gibbs distribution µ a sample X ~ µ, and an update ( D , 𝜚 D ) Output : X’ ~ µ’ where µ’ is the new Gibbs distribution Goal: transform a X ~ µ to a X’ ~ µ’ by local changes Current sampling techniques are not powerful enough: • µ may be changed significantly by dynamic updates; • Monte Carlo sampling does not know when to stop; • notions such as mixing time give worst-case estimation.

Graphical Model instance of graphical model: I = ( V, E, [ q ] , Φ ) • V : variables ϕ v • E ⊂ 2 V : constraints • [ q ] = {0,1, …, q -1} : domain constraint ϕ e • Φ = ( 𝜚 v ) v ∈ V ∪ ( 𝜚 e ) e ∈ E : factors e • Gibbs distribution µ over all σ ∈ [ q ] V : μ ( σ ) ∝ ∏ V ϕ v ( σ v ) ∏ ϕ e ( σ e ) v ∈ V e ∈ E

Notations instance of graphical model: I = ( V, E, [ q ] , Φ ) for D ⊆ V ∪ 2 V 𝗐𝖼𝗆 ( D ) ≜ ( V ∩ D ) ∪ ( ⋃ e ∈ D ∩ E e ) (involved variables) R ⊆ V for E ( R ) ≜ { e ∈ E ∣ e ⊆ R } (internal constraints) δ ( R ) ≜ { e ∈ E ∖ E ( R ) ∣ e ∩ R ≠ ∅ } (boundary constraints) E + ( R ) ≜ { e ∈ E ∣ e ∩ R ≠ ∅ } (incident constraints) = E ( R ) ∪ δ ( R )

Dynamic Sampler Input : a graphical model with Gibbs distribution µ a sample X ~ µ, and an update ( D , 𝜚 D ) Output : X’ ~ µ’ where µ’ is the new Gibbs distribution Upon receiving update ( D , 𝜚 D ): • apply changes ( D , 𝜚 D ) to the current graphical model; R ← 𝗐𝖼𝗆 ( D ) ≜ ( V ∩ D ) ∪ ( ⋃ e ∈ D ∩ E e ) ; • • while R ≠ ∅ : • ( X , R ) ← 𝚂𝚏𝚝𝚋𝚗𝚚𝚖𝚏 ( X , R );

Dynamic Sampler Upon receiving update ( D , 𝜚 D ): • apply changes ( D , 𝜚 D ) to the current graphical model; R ← 𝗐𝖼𝗆 ( D ) ≜ ( V ∩ D ) ∪ ( ⋃ e ∈ D ∩ E e ) ; • • while R ≠ ∅ : • ( X , R ) ← 𝚂𝚏𝚝𝚋𝚗𝚚𝚖𝚏 ( X , R ); 𝚂𝚏𝚝𝚋𝚗𝚚𝚖𝚏 ( X , R ) : • each e ∈ E + ( R ) computes κ e = min ϕ e ( x e )/ ϕ e ( X e ) x e : x e ∩ R = X e ∩ R • each v ∈ R resamples X v ∈ [ q ] independently according to 𝜚 v ; • each e ∈ E + ( R ) is passed independently with prob. κ e · 𝜚 e ( X e ) ; (otherwise e is violated) R ← ⋃ e ∈ E : violated e e ; •

Resampling 𝚂𝚏𝚝𝚋𝚗𝚚𝚖𝚏 ( X , R ) : • each e ∈ E + ( R ) computes κ e = min ϕ e ( x e )/ ϕ e ( X e ) x e : x e ∩ R = X e ∩ R • each v ∈ R resamples X v ∈ [ q ] independently according to 𝜚 v ; • each e ∈ E + ( R ) is passed independently with prob. κ e · 𝜚 e ( X e ) ; (otherwise e is violated) R ← ⋃ e ∈ E : violated e e ; • • each boundary constraint e ∈ δ ( R ) is violated ind. with prob. ; 1 − min ϕ e ( x e )/ ϕ e ( X e ) R x e : x e ∩ R = X e ∩ R e • X e ∩ R each v ∈ R resamples X v ind. from 𝜚 v ; ? • each non-violated incident constraint e ∈ E + ( R ) is violated ind. with prob. 1- 𝜚 e ( X e ) ; • all violating variables form the new R ; V

Resampling 𝚂𝚏𝚝𝚋𝚗𝚚𝚖𝚏 ( X , R ) : • each e ∈ E + ( R ) computes κ e = min ϕ e ( x e )/ ϕ e ( X e ) x e : x e ∩ R = X e ∩ R • each v ∈ R resamples X v ∈ [ q ] independently according to 𝜚 v ; • each e ∈ E + ( R ) is passed independently with prob. κ e · 𝜚 e ( X e ) ; (otherwise e is violated) R ← ⋃ e ∈ E : violated e e ; • • each boundary constraint e ∈ δ ( R ) is violated ind. with prob. ; 1 − min ϕ e ( x e )/ ϕ e ( X e ) R x e : x e ∩ R = X e ∩ R e • X e ∩ R each v ∈ R resamples X v ind. from 𝜚 v ; wrong ? distribution • each non-violated incident constraint e ∈ E + ( R ) is violated ind. with prob. 1- 𝜚 e ( X e ) ; • all violating variables form the new R ; V A more “natural” algorithm?

Dynamic Sampler Upon receiving update ( D , 𝜚 D ): • apply changes ( D , 𝜚 D ) to the current graphical model; R ← 𝗐𝖼𝗆 ( D ) ≜ ( V ∩ D ) ∪ ( ⋃ e ∈ D ∩ E e ) ; • • while R ≠ ∅ : • ( X , R ) ← 𝚂𝚏𝚝𝚋𝚗𝚚𝚖𝚏 ( X , R ); 𝚂𝚏𝚝𝚋𝚗𝚚𝚖𝚏 ( X , R ) : • each e ∈ E + ( R ) computes κ e = min ϕ e ( x e )/ ϕ e ( X e ) x e : x e ∩ R = X e ∩ R • each v ∈ R resamples X v ∈ [ q ] independently according to 𝜚 v ; • each e ∈ E + ( R ) is passed independently with prob. κ e · 𝜚 e ( X e ) ; (otherwise e is violated) R ← ⋃ e ∈ E : violated e e ; •

Correctness of Sampling Correctness : Assuming input sample X ~ µ , upon termination, the dynamic sampler returns a sample from the updated distribution µ’ .