Advanced Algorithms (XIV) Shanghai Jiao Tong University Chihao - PowerPoint PPT Presentation

Advanced Algorithms (XIV) Shanghai Jiao Tong University Chihao Zhang June 8, 2020

Mixing Time via Coupling

Mixing Time via Coupling The state space Ω

Mixing Time via Coupling The state space Ω Transition matrix P ∈ ℝ Ω×Ω

Mixing Time via Coupling The state space Ω Transition matrix P ∈ ℝ Ω×Ω Two chains and ( X 0 , X 1 , …) ( Y 0 , Y 1 , …)

Mixing Time via Coupling The state space Ω Transition matrix P ∈ ℝ Ω×Ω Two chains and ( X 0 , X 1 , …) ( Y 0 , Y 1 , …) A distance d : Ω × Ω → ℝ ≥ 0

Mixing Time via Coupling The state space Ω Transition matrix P ∈ ℝ Ω×Ω Two chains and ( X 0 , X 1 , …) ( Y 0 , Y 1 , …) A distance d : Ω × Ω → ℝ ≥ 0 Two chains are “coupled” so that:

Mixing Time via Coupling The state space Ω Transition matrix P ∈ ℝ Ω×Ω Two chains and ( X 0 , X 1 , …) ( Y 0 , Y 1 , …) A distance d : Ω × Ω → ℝ ≥ 0 Two chains are “coupled” so that: E [ d ( X t +1 , Y t +1 ) ∣ ( X t , Y t )] ≤ (1 − α ) ⋅ d ( X t , Y t )

In other words, is a super martingale { d ( X t , Y t )} t ≥ 0

In other words, is a super martingale { d ( X t , Y t )} t ≥ 0 Recall the mixing time

In other words, is a super martingale { d ( X t , Y t )} t ≥ 0 Recall the mixing time

In other words, is a super martingale { d ( X t , Y t )} t ≥ 0 Recall the mixing time By coupling lemma

In other words, is a super martingale { d ( X t , Y t )} t ≥ 0 Recall the mixing time By coupling lemma d TV ( X t , Y t ) ≤ Pr [ X t ≠ Y t ] = Pr [ d ( X t , Y t ) > 0]

In other words, is a super martingale { d ( X t , Y t )} t ≥ 0 Recall the mixing time By coupling lemma d TV ( X t , Y t ) ≤ Pr [ X t ≠ Y t ] = Pr [ d ( X t , Y t ) > 0] For finite , we assume WLOG that Ω x , y ∈Ω : x ≠ y d ( x , y ) = 1 min

In other words, is a super martingale { d ( X t , Y t )} t ≥ 0 Recall the mixing time By coupling lemma d TV ( X t , Y t ) ≤ Pr [ X t ≠ Y t ] = Pr [ d ( X t , Y t ) > 0] For finite , we assume WLOG that Ω x , y ∈Ω : x ≠ y d ( x , y ) = 1 min Pr [ d ( X t , Y t ) > 0] = Pr [ d ( X t , Y t ) ≥ 1] ≤ E [ d ( X t , Y t )] ≤ (1 − α ) t ⋅ d ( X 0 , Y 0 )

Sampling Proper Colorings

Sampling Proper Colorings

Sampling Proper Colorings

Sampling Proper Colorings

Sampling Proper Colorings

Sampling Proper Colorings - the number of proper colorings q

Sampling Proper Colorings - the number of proper colorings q - a graph of maximum degree Δ G

Sampling Proper Colorings - the number of proper colorings q - a graph of maximum degree Δ G Is colorable using colors? G q

The problem is NP -hard in general

The problem is NP -hard in general We consider the case when q > Δ

The problem is NP -hard in general We consider the case when q > Δ Consider the chain obtained via the “Metropolis Rule”

The problem is NP -hard in general We consider the case when q > Δ Consider the chain obtained via the “Metropolis Rule” •Pick and u.a.r. v ∈ V c ∈ [ q ] •Recolor with if possible v c

The problem is NP -hard in general We consider the case when q > Δ Consider the chain obtained via the “Metropolis Rule” •Pick and u.a.r. v ∈ V c ∈ [ q ] •Recolor with if possible v c The chain is irreducible when q ≥ Δ + 2

The Coupling

The Coupling Two chains choose the same and v c

The Coupling Two chains choose the same and v c Good Move X t Y t c = v v

The Coupling Two chains choose the same and v c Good Move d ( X t +1 , Y t +1 ) = d ( X t , Y t ) − 1 X t Y t c = v v

The Coupling Two chains choose the same and v c Good Move d ( X t +1 , Y t +1 ) = d ( X t , Y t ) − 1 X t Y t c = Pr [ ⋅ ] ≥ d ( X t , Y t ) ⋅ q − 2( Δ − 1) v v N q

The Coupling Two chains choose the same and v c Good Move d ( X t +1 , Y t +1 ) = d ( X t , Y t ) − 1 X t Y t c = Pr [ ⋅ ] ≥ d ( X t , Y t ) ⋅ q − 2( Δ − 1) v v N q Bad Move X t Y t c = v v

The Coupling Two chains choose the same and v c Good Move d ( X t +1 , Y t +1 ) = d ( X t , Y t ) − 1 X t Y t c = Pr [ ⋅ ] ≥ d ( X t , Y t ) ⋅ q − 2( Δ − 1) v v N q Bad Move d ( X t +1 , Y t +1 ) = d ( X t , Y t ) + 1 X t Y t c = v v

The Coupling Two chains choose the same and v c Good Move d ( X t +1 , Y t +1 ) = d ( X t , Y t ) − 1 X t Y t c = Pr [ ⋅ ] ≥ d ( X t , Y t ) ⋅ q − 2( Δ − 1) v v N q Bad Move d ( X t +1 , Y t +1 ) = d ( X t , Y t ) + 1 X t Y t c = Pr [ ⋅ ] ≤ 2 d ( X t , Y t ) Δ v v Nq

E [ d ( X t +1 , Y t +1 ) ∣ ( X t , Y t )] ≤ d ( X t , Y t ) ⋅ ( 1 + 2 Δ − ( q − 2 Δ + 2)) ) qN = d ( X t , Y t ) ⋅ ( 1 − q − 4 Δ + 2 ) qN

E [ d ( X t +1 , Y t +1 ) ∣ ( X t , Y t )] ≤ d ( X t , Y t ) ⋅ ( 1 + 2 Δ − ( q − 2 Δ + 2)) ) qN = d ( X t , Y t ) ⋅ ( 1 − q − 4 Δ + 2 ) qN So if , we have q ≥ 4 Δ − 1

E [ d ( X t +1 , Y t +1 ) ∣ ( X t , Y t )] ≤ d ( X t , Y t ) ⋅ ( 1 + 2 Δ − ( q − 2 Δ + 2)) ) qN = d ( X t , Y t ) ⋅ ( 1 − q − 4 Δ + 2 ) qN So if , we have q ≥ 4 Δ − 1 E [ d ( X t +1 , Y t +1 ) ∣ ( X t , Y t )] ≤ ( 1 − 1 qN ) d ( X t , Y t )

E [ d ( X t +1 , Y t +1 ) ∣ ( X t , Y t )] ≤ d ( X t , Y t ) ⋅ ( 1 + 2 Δ − ( q − 2 Δ + 2)) ) qN = d ( X t , Y t ) ⋅ ( 1 − q − 4 Δ + 2 ) qN So if , we have q ≥ 4 Δ − 1 E [ d ( X t +1 , Y t +1 ) ∣ ( X t , Y t )] ≤ ( 1 − 1 qN ) d ( X t , Y t )

E [ d ( X t +1 , Y t +1 ) ∣ ( X t , Y t )] ≤ d ( X t , Y t ) ⋅ ( 1 + 2 Δ − ( q − 2 Δ + 2)) ) qN = d ( X t , Y t ) ⋅ ( 1 − q − 4 Δ + 2 ) qN So if , we have q ≥ 4 Δ − 1 E [ d ( X t +1 , Y t +1 ) ∣ ( X t , Y t )] ≤ ( 1 − 1 qN ) d ( X t , Y t ) d TV ( X t , Y t ) ≤ ( 1 − 1 t qN ) ⋅ N ≤ ε

E [ d ( X t +1 , Y t +1 ) ∣ ( X t , Y t )] ≤ d ( X t , Y t ) ⋅ ( 1 + 2 Δ − ( q − 2 Δ + 2)) ) qN = d ( X t , Y t ) ⋅ ( 1 − q − 4 Δ + 2 ) qN So if , we have q ≥ 4 Δ − 1 E [ d ( X t +1 , Y t +1 ) ∣ ( X t , Y t )] ≤ ( 1 − 1 qN ) d ( X t , Y t ) d TV ( X t , Y t ) ≤ ( 1 − 1 t qN ) ⋅ N ≤ ε ⟹ τ mix ( ε ) ≤ qN ( log N + log ε − 1 )

Geometric View of Mixing

Geometric View of Mixing A Markov chain is a random walk on the state space

Geometric View of Mixing A Markov chain is a random walk on the state space

Geometric View of Mixing A Markov chain is a random walk on the state space Which random walk mixes faster?

Geometric View of Mixing A Markov chain is a random walk on the state space Which random walk mixes faster? We will develop tools to formalize the intuition

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