Dynamic Sampling from Graphical Models Yitong Yin Nanjing University Joint work with Weiming Feng (Nanjing) Nisheeth Vishnoi EPFL)
Dynamic Sampling fom Graphical Models Yitong Yin Nanjing University Joint work with Weiming Feng (Nanjing) Nisheeth Vishnoi (EPFL)
Graphical Model instance of graphical model:=(V,E,g,) oD:variables ●EC2:constraints [g]=(0,1,...g-1:domain ●pe constraint（ ●Φ=（中v)eyU(中e)eeE:factors Gibbs distribution u over all oE[g]v: 4(o)x中，(o,)中(o。) v∈V e∈E
Graphical Model • Gibbs distribution µ over all σ∈[q]V : • V : variables • E ⊂ 2V: constraints • [q] = {0,1, …, q-1}: domain • Φ = (�v)v∈V ∪ (�e)e∈E: factors μ(σ) ∝ ∏ v∈V ϕv(σv) ∏ e∈E ϕe(σe) constraint e V ϕv ϕe instance of graphical model: I = (V,E, [q], )
Graphical Model instance of graphical model:=(V,E,g,) ● Each vEV is a variable with domain [g] and has a distribution中，over[q] 中：[q]→[0,1] ●Each e∈E is a set of variables and corresponds to a constraint (factor) e constraint φe:[q]e→[0,1] ● Gibbs distribution u over all olg]: 4(o)x中，(o,)中(oe) v∈V e∈E
Graphical Model • Each v∈V is a variable with domain [q] and has a distribution • Each e∈E is a set of variables and corresponds to a constraint (factor) ϕe : [q] e → [0,1] μ(σ) ∝ ∏ v∈V ϕv(σv) ∏ e∈E ϕe(σe) constraint e ϕv over [q] V ϕv ϕe ϕv : [q] → [0,1] instance of graphical model: I = (V,E, [q], ) • Gibbs distribution µ over all σ∈[q]V :
Graphical Model Gibbs distribution u over all oe[g]v: 4(o)cφ，(o,)Πp(a) v∈V e∈E is a distribution over [g] φe:[q]e→[0,1] ● each vE V independently samples XvE[g]according to ● each eEE is passed independently with probability e(Xe); X is accepted if all constraints e eE are passed
Graphical Model • Gibbs distribution µ over all σ∈[q]V : μ(σ) ∝ ∏ v∈V ϕv(σv) ∏ e∈E ϕe(σe) ϕv is a distribution over [q] ϕe : [q] e → [0,1] • each v ∈ V independently samples Xv∈[q] according to �v; • each e ∈ E is passed independently with probability �e(Xe); • X is accepted if all constraints e ∈ E are passed
Graphical Model ·Gibbs distribution u over all o∈[q]': a(o)cΠ中，(o,)Π.(o,) G(V,E) v∈V e=(u,v)∈E ●hardcore morel: V [q]={0,1} grve otherwise if oy=0 A>0 is (local)fugacity if o=1
Graphical Model • Gibbs distribution µ over all σ∈[q]V : μ(σ) ∝ ∏ v∈V ϕv(σv) ∏ e=(u,v)∈E ϕe(σu, σv) ϕ ϕv e u v G(V,E) • hardcore morel: [q] = {0,1} ϕe(σu, σv) = { 0 if σu = σv = 1 1 otherwise ϕv(σv) = 1 1 + λv if σv = 0 λv 1 + λv if σv = 1 λv > 0 is (local) fugacity
Graphical Model ●Gibbs distribution u over all o∈[q]': a(o)cΠ,(a,)Πg e(ou,O) G(V,E) v∈V e=(u,v)∈E Ising/Potts model: (ferromagnetic) if ou=oy or了 10.1 otherwise (anti-ferromagnetic) a,ow={eolee otherwise is a distribution over [g]( (arbitrary local fields)
Graphical Model • Gibbs distribution µ over all σ∈[q]V : μ(σ) ∝ ∏ v∈V ϕv(σv) ∏ e=(u,v)∈E ϕe(σu, σv) ϕ ϕv e u v G(V,E) • Ising/Potts model: ϕe(σu, σv) = { βe ∈ [0,1] if σu = σv 1 otherwise ϕe(σu, σv) = { 1 if σu = σv βe ∈ [0,1] otherwise ϕv is a distribution over [q] (ferromagnetic) (anti-ferromagnetic) or { (arbitrary local fields)
Dynamic Sampling ·Gibbs distribution u over all oe∈[q]': 4(o)x,(o,)中(oe) v∈V e∈E current sample:X~u dynamic update: adding/deleting a constraint e new distribution changing a factor y or e 儿 adding/deleting an independent variable v Question: Obtain X'~u'from X~u with small incremental cost
Dynamic Sampling • Gibbs distribution µ over all σ∈[q]V : ϕ ϕv e u v • adding/deleting a constraint e • changing a factor �v or �e • adding/deleting an independent variable v current sample: X ~ µ μ(σ) ∝ ∏ v∈V ϕv(σv) ∏ e∈E ϕe(σe) dynamic update: Obtain X’ ~ µ’ from X ~ µ with small incremental cost. Question: new distribution } µ’ ϕ′ e ϕ′ v
Dynamic Sampling instance of graphical model:=(V,E,g,) Gibbs distribution u over all oeg]: D u(o)xb(G)B.(G.) v∈V e∈E Ope constraint current sample:X~u ●V:variables 。EC2:constraints ·[q]={0,1,.,q-1}:domain ●Φ=（中v)eyU(中e)eeE:factors
Dynamic Sampling instance of graphical model: I = (V,E, [q], ) • V : variables • E ⊂ 2V: constraints • [q] = {0,1, …, q-1}: domain • Φ = (�v)v∈V ∪ (�e)e∈E: factors constraint e V ϕv ϕe • Gibbs distribution µ over all σ∈[q]V : current sample: X ~ µ μ(σ) ∝ ∏ v∈V ϕv(σv) ∏ e∈E ϕe(σe)
Dynamic Sampling instance of graphical model::Z=(V,E,[gl,Φ)） update:(D,D) D C VU2V is the set of changed variables and constraints ΦD=(φ，)vEVnD U(中e)e∈2 vD specifies the new factors (Y,E,[gl,Φ)Dg(Y,E,[ql,Φ) E'=EU(2V∩D) Φ'=(pa)a∈uE' where each is as specified in {8 ifa∈D otherwise
Dynamic Sampling instance of graphical model: I = (V,E, [q], ) update: (D, �D) (V, E, [q], Φ) (D,ΦD) (V, E′ , [q], Φ′) is the set of changed variables and constraints ΦD = (ϕv)v∈V∩D ∪ (ϕe)e∈2V∩D specifies the new factors D ⊂ V ∪ 2V E′ = E ∪ (2V ∩ D) Φ′ = (ϕ′ a)a∈V∪E′ where each ϕ′ a is as specified in { ΦD if a ∈ D Φ otherwise
Dynamic Sampling instance of graphical model:Z =(V,E,g,) update:(D,D) DC VU2V is the set of changed variables and constraints D=()vEVD U(e)eE2vD specifies the new factors (V,E,Iql,)VEIql) Input:a graphical model with Gibbs distribution a sample ~u,and an update (D,D) Output:X'~u'where u'is the new Gibbs distribution (D,p)is fixed by an offline adversary independently of X~u
Dynamic Sampling Input: Output: a graphical model with Gibbs distribution µ a sample X ~ µ, and an update (D, �D) X’ ~ µ’ where µ’ is the new Gibbs distribution instance of graphical model: I = (V,E, [q], ) update: (D, �D) (V, E, [q], Φ) (D,ΦD) (V, E′ , [q], Φ′) is the set of changed variables and constraints ΦD = (ϕv)v∈V∩D ∪ (ϕe)e∈2V∩D specifies the new factors D ⊂ V ∪ 2V (D, �D) is fixed by an offline adversary independently of X ~ µ
