Discrete Sampling using Semigradient-based Product Mixtures Alkis - PowerPoint PPT Presentation

Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos Hamed Hassani Andreas Krause Stefanie Jegelka ETH Zurich UPenn ETH Zurich MIT UAI 2018

Modeling gene alterations [ cancergenome.nih.gov ] Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 1

Modeling gene alterations Patients Genes Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 2

Modeling gene alterations Patients Genes Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 3

Modeling gene alterations Patients Genes Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 4

Modeling teams in online games [ www.theverge.com ] Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 5

Modeling teams in online games [ euw.leagueoflegends.com ] Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 6

Modeling teams in online games vs Team 1 Team 2 Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 7

Modeling teams in online games Teams Characters Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 8

Discrete probabilistic models Ground set Data , Teams Characters Model higher-order interactions exp graph-cut Ising model log DPP Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 9 V = { 1 , . . . , n }

Discrete probabilistic models exp Alkis Gotovos Discrete Sampling using Semigradient-based Product Mixtures DPP log Ising model graph-cut Model higher-order interactions Ground set Characters Teams Data 9 D = { S i } m V = { 1 , . . . , n } i =0 , S i ⊆ V

Discrete probabilistic models Ground set Alkis Gotovos Discrete Sampling using Semigradient-based Product Mixtures DPP log Ising model graph-cut 9 Characters Model higher-order interactions Teams Data D = { S i } m V = { 1 , . . . , n } i =0 , S i ⊆ V 1 ( ) p ( S ; θ ) = F ( S ; θ ) Z ( θ ) exp

Discrete probabilistic models Model higher-order interactions Alkis Gotovos Discrete Sampling using Semigradient-based Product Mixtures DPP log Ground set 9 Characters Teams Data D = { S i } m V = { 1 , . . . , n } i =0 , S i ⊆ V 1 ( ) p ( S ; θ ) = F ( S ; θ ) Z ( θ ) exp ◦ F ( S ) = graph-cut ( S ) → Ising model

Discrete probabilistic models Characters Alkis Gotovos Discrete Sampling using Semigradient-based Product Mixtures Ground set Model higher-order interactions 9 Data Teams D = { S i } m V = { 1 , . . . , n } i =0 , S i ⊆ V 1 ( ) p ( S ; θ ) = F ( S ; θ ) Z ( θ ) exp ◦ F ( S ) = graph-cut ( S ) → Ising model ◦ F ( S ) = log | K S | → DPP

Discrete probabilistic models Max. likelihood #P-hard in general Approximate Sample from Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 10 Learn θ

Discrete probabilistic models Max. likelihood #P-hard in general Approximate Sample from Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 10 Learn θ

Discrete probabilistic models Max. likelihood #P-hard in general Approximate Sample from Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 10 Learn θ Compute ∇ θ Z ( θ )

Discrete probabilistic models Max. likelihood #P-hard in general Approximate Sample from Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 10 Learn θ Compute ∇ θ Z ( θ )

Discrete probabilistic models Max. likelihood #P-hard in general Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 10 Learn θ Approximate ∇ θ Z ( θ ) Sample from p ( · ; θ )

Discrete probabilistic models Max. likelihood #P-hard in general Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 10 Learn θ Approximate ∇ θ Z ( θ ) Sample from p ( · ; θ )

The Gibbs sampler Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 11 V { 1 , 2 } { 1 , 3 } { 2 , 3 } { 1 } { 2 } { 3 } {}

The Gibbs sampler Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 11 V { 1 , 2 } { 1 , 3 } { 2 , 3 } { 1 } { 2 } { 3 } {}

The Gibbs sampler Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 11 V { 1 , 2 } { 1 , 3 } { 2 , 3 } { 1 } { 2 } { 3 } {}

When Gibbs fails Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 12 Ω

When Gibbs fails Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 12 Ω 1 Ω 2

When Gibbs fails Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 12 Ω 1 Ω 2 ?

1 Mixture exp 2 Log-Modulars 3 Metropolis Target exp Accept with probability Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 13 The M 3 chain → M 3 = Mixture of Log-Modulars Metropolis

1 Mixture exp 2 Log-Modulars 3 Metropolis Target exp Accept with probability Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 13 The M 3 chain → M 3 = Mixture of Log-Modulars Metropolis

1 Mixture exp 2 Log-Modulars 3 Metropolis Target exp Accept with probability Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 13 The M 3 chain → M 3 = Mixture of Log-Modulars Metropolis

1 Mixture exp 2 Log-Modulars 3 Metropolis Accept with probability Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 13 The M 3 chain → M 3 = Mixture of Log-Modulars Metropolis ◦ Target p ( S ) ∝ exp ( F ( S ))

1 Mixture exp 2 Log-Modulars 3 Metropolis Accept with probability Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 13 The M 3 chain → M 3 = Mixture of Log-Modulars Metropolis ◦ Target p ( S ) ∝ exp ( F ( S )) ◦ Proposal q ( S, T )

13 Discrete Sampling using Semigradient-based Product Mixtures 1 Mixture exp 2 Log-Modulars 3 Metropolis Alkis Gotovos The M 3 chain → M 3 = Mixture of Log-Modulars Metropolis ◦ Target p ( S ) ∝ exp ( F ( S )) ◦ Proposal q ( S, T ) { } 1 , p ( T ) q ( T,S ) ◦ Accept with probability min p ( S ) q ( S,T )

13 Discrete Sampling using Semigradient-based Product Mixtures 1 Mixture exp 2 Log-Modulars 3 Metropolis Alkis Gotovos The M 3 chain → M 3 = Mixture of Log-Modulars Metropolis ◦ Target p ( S ) ∝ exp ( F ( S )) ◦ Proposal q ( S, T ) { } 1 , p ( T ) q ( T,S ) ◦ Accept with probability min p ( S ) q ( S,T )

13 2 Alkis Gotovos Discrete Sampling using Semigradient-based Product Mixtures Metropolis Log-Modulars 3 1 Mixture The M 3 chain → M 3 = Mixture of Log-Modulars Metropolis r q ( S, T ) = 1 ∑ w i exp ( m i ( T )) Z q i =1 ◦ Target p ( S ) ∝ exp ( F ( S )) ◦ Proposal q ( S, T ) { } 1 , p ( T ) q ( T,S ) ◦ Accept with probability min p ( S ) q ( S,T )

13 2 Alkis Gotovos Discrete Sampling using Semigradient-based Product Mixtures Metropolis Log-Modulars 3 1 Mixture The M 3 chain → M 3 = Mixture of Log-Modulars Metropolis r q ( S, T ) = 1 ∑ w i exp ( m i ( T )) Z q i =1 ◦ Target p ( S ) ∝ exp ( F ( S )) ◦ Proposal q ( S, T ) { } 1 , p ( T ) q ( T,S ) ◦ Accept with probability min p ( S ) q ( S,T )

13 2 1 Mixture Alkis Gotovos Discrete Sampling using Semigradient-based Product Mixtures Metropolis 3 Log-Modulars The M 3 chain → M 3 = Mixture of Log-Modulars Metropolis r q ( S, T ) = 1 ∑ w i exp ( m i ( T )) Z q i =1 ∑ m i ( T ) = m iv v ∈ T ◦ Target p ( S ) ∝ exp ( F ( S )) ◦ Proposal q ( S, T ) { } 1 , p ( T ) q ( T,S ) ◦ Accept with probability min p ( S ) q ( S,T )

13 2 1 Mixture Alkis Gotovos Discrete Sampling using Semigradient-based Product Mixtures Metropolis 3 Log-Modulars The M 3 chain → M 3 = Mixture of Log-Modulars Metropolis r q ( T ) = 1 ∑ w i exp ( m i ( T )) Z q i =1 ∑ m i ( T ) = m iv v ∈ T ◦ Target p ( S ) ∝ exp ( F ( S )) ◦ Proposal q ( T ) { } 1 , p ( T ) q ( S ) ◦ Accept with probability min p ( S ) q ( T )

14 Mixture Alkis Gotovos Discrete Sampling using Semigradient-based Product Mixtures ) number of components May need an exponential (in BUT arbitrarily well. can approximate any distribution Proposition 1 time in Can sample from The M 3 chain r Proposal q ( T ) = 1 ∑ w i exp ( m i ( T )) Z q i =1

14 can approximate any distribution Alkis Gotovos Discrete Sampling using Semigradient-based Product Mixtures ) number of components May need an exponential (in BUT arbitrarily well. Mixture Proposition 1 The M 3 chain r Proposal q ( T ) = 1 ∑ w i exp ( m i ( T )) Z q i =1 → Can sample from q in O ( n ) time

BUT Proposition 1 May need an exponential (in ) number of components Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 14 The M 3 chain r Proposal q ( T ) = 1 ∑ w i exp ( m i ( T )) Z q i =1 → Can sample from q in O ( n ) time Mixture q can approximate any distribution p arbitrarily well.

BUT Proposition 1 Discrete Sampling using Semigradient-based Product Mixtures Alkis Gotovos 14 The M 3 chain r Proposal q ( T ) = 1 ∑ w i exp ( m i ( T )) Z q i =1 → Can sample from q in O ( n ) time Mixture q can approximate any distribution p arbitrarily well. May need an exponential (in n ) number of components r