Slow and Fast Mixing of Tempering and Swapping for the Potts Model
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Slow and Fast Mixing of Tempering and Swapping for the
Potts Model
Nayantara Bhatnagar, UC BerkeleyDana Randall, Georgia Tech
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lim Pr[Xt = Y | X0] = π(Y)
t → ∞
Markov Chains
K = (Ω, P)
Theorem: If K is connected and “aperiodic”, the Markov chain X0,X1,... converges in the limit to a unique stationary distribution π over Ω.
P(X,Y)
P(Y,X)
If P(X,Y) = P(Y,X), π is uniform over Ω.
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Matchings Independent Sets
Partition functions of Ising, Potts models
Volume of a convex body
Broder’s Markov chain Glauber dynamics
Glauber dynamicsBall walk, Lattice walk
δ
Markov Chains
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Introduction:Markov Chain Monte Carlo
Markov Chains:
• Matchings – Broder’s Markov chain
• Colorings – Glauber dynamics
• Independent Sets – Glauber dynamics
• Ising, Potts model – Glauber dynamics
• Volume – Ball walk, Lattice walk
Mixing Time, T: time to get within 1/4 in variation distance to π.
Rapid mixing (polynomial), slowly mixing (exponential).
Techniques for proving rapid mixing:
Coupling, Spectral Gap, Conductance and isoperimetry, Multicommodity flows, Decomposition, Comparison ...
What if natural Markov chain is slowly mixing?
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The q-state Potts Model
q-state Ferromagnetic Potts Model: Underlying graph: G(V,E)
Configurations Ω = { x : x [q]n}Inverse temperature β > 0,
πβ(x) e β( H(x)) H(x) = Σ δxi = x
j
Glauber dynamics Markov Chain• Choose (v, ct+1(v)) R V x [q].
• Update ct(v) to ct+1(v) with Metropolis probabilities.
(i,j)
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Why Simulated Tempering
πβ(x)
H(x)
Glauber dynamics mixes slowly for the q-state
Potts for Kn for q ≥ 2, at large enough β.
ΦS = P[ Xt+1 S | Xt ~ π(S)]
SSc
Theorem : T c1
Φ
c2
Φ2
Φ = min ΦS
S: π(S)
½
Conductance: [Jerrum-Sinclair ’89, Lawler-Sokal ’88]
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Simulated Tempering[Marinari-Parisi ’92]
Define inverse temperatures 0 = β0βM
=βand distributions π0π1πM = πβ on Ω.
i = M· i
M
……
πM
π(x,i) = ˆ 1
M+1πi (x)
Tempering Markov Chain:
From (x,i),
• W.p. ½, Glauber dynamics at βi
• W.p. ½, randomly move to (x,i
±1)
π0
Ω̂ = Ω × [M+1],
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Swapping[Geyer ’91]
Define inverse temperatures 0 = β0βM
=βand distributions π0π1πM = πβ on Ω.
i = M· i
M
……
πM
π(x) = Π ˆ πi (xi)
Swapping Markov Chain:
From x, choose random i
• W.p. ½, Glauber dynamics at βi
• W.p. ½, move to x(i,i+1)
π0
Ω̂ = Ω
[M+1],
i
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Theoretical Results
• Madras-Zheng ’99:
∙ Tempering mixes rapidly at all temperatures for the ferromagnetic Ising model (Potts model, q = 2) on Kn.
∙ Rapid mixing for symmetric bimodal exponential distribution on an interval.
• Zheng ’99: ∙ Rapid mixing of swapping implies tempering mixes
rapidly.
• B-Randall ’04:
∙Simulated Tempering mixes slowly for 3 state ferromagnetic Potts model on Kn.
∙Modified swapping algorithm is rapidly mixing for mean-field Ising model with an external field.
• Woodard, Schmidler, Huber ’08:
∙ Sufficient conditions for rapid mixing of tempering and swapping.
∙ Sufficient conditions for torpid mixing of tempering and swapping.
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In This Talk:
B-Randall ’04:
Tempering and swapping for the mean-field Potts model. Slow Mixing.
Tempering can be slowly mixing for any choice of temperatures.
Rapid Mixing Alternative tempered distributions for rapid mixing.
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Tempering for Potts Model
Theorem [BR]: There exists βcrit> 0, such that
tempering for Potts model on Kn at βcrit mixes slowly.
(0,0,n)
Proof idea: Bound conductance on Ω = Ω × [M+1].
• Cut depends on number of vertices of each color.
• Induces the same cut on Ω at each βi
The space Ω partitioned into equivalence classes σ:
ˆ
(n/2, 0, n/2)
(n,0,0)
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Stationary Distribution of Tempering Chain
At βcrit
At β0
…
At 0 < βi < βcrit
disordered mode
ordered mode
πi (σ) n
σR σB σGe β
i( )(σR)2 + (σB)2 + (σG)2
…
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Tempering Fails to Converge
βcrit
β0
…
0 < βi < βcrit
…
At βcrittempering mixes
slowly for any set of intermediate temperatures.
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Swapping and Tempering for Assymetric Distributions – Rapid Mixing
Assymetric exponential
Ising Model with an external Field
Potts model on KR, the line σB = σG n/3
01n- 2n
πβ(x) e β( H(x))
H(x) = Σ δxi = x
j + B Σ δx
i=+
(i,j) i
π(x) C |x| , x [-n1,n2 ]
n1 > n2
0 n
3
n
2
n
3
2n
πβ(x) e β( H(x))
H(x) = Σ δxi = x
j
n
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Decomposition of Swapping Chain
πi(x) C |x|
i
M
Madras-Randall ’02
Decomposition for Markov chains
1. Mixing of restricted chains R0,i and R1,i at each temperature.
2. Mixing of the projection chain P.
Tswap C min TRb,i x TP
b {0,1},
i M
01n- 2n
…
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Decomposition of Swapping Chain
πi(x) C |x|
i
M
011010 010110
011010 011011
Projection for Swapping chain
01n- 2n
…
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Decomposition of Swapping Chain
Projection for Swapping chain Weighted Cube (WC)
011010 010110
011010 011011
011010 010010
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Decomposition of Swapping Chain
Projection for Swapping chain Weighted Cube (WC)
Upto polynomials, πi(0) Cn1 i / M /Zi and πi(1) Cn2 i / M /Zi
Lemma: If for i > j,
πi(1) πj(0) p(n)πi(0) πj(1),
then TP q(n) TWC.
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• Modify more than just temperature
• Define π’M … π’0 so cut is not preserved.
……
Flat-Swap: Fast Mixing for Mean-Field Models
πi (σ) n
σR σB σGe β
i( )(σR)2 + (σB)2 + (σG)2
3
n
2
n
3
2n n
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• Modify more than just temperature
• Define π’M … π’0 so cut is not preserved.
Flat-Swap: Fast Mixing for Mean-Field Models
π’i (σ) n
σR σB σGe β
i( )(σR)2 + (σB)2 + (σG)2
……
i
M
π’i (σ) = πi (σ) fi(σ) = πi (σ) n
σR σB σG
i-M
M
3
n
2
n
3
2n n
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• Modify more than just temperature
• Define π’M … π’0 so cut is not preserved.
Flat Swap for Mean-Field Models
Theorem [B-Randall]:
• Flat swap for the 3-state Potts model onb KR using
the distributions π’M … π’0 mixes rapidly at every
temperature.
• Flat swap mixes rapidly for the mean field Ising model at every temperature and for any external field B.
Lemma: For i > j, π’i(0) π’j(1) p(n)π’i(1) π’j(0)
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Summary and Open problems
• Simulated tempering algorithms for other problems?
• Relative complexity of swapping and tempering
Open Problems
Summary
• Insight into why tempering can fail to converge.
• Designing more robust tempering algorithms.
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……
0
SM > crit
Tempering vs. Fixed Temperature
3
n
2
n n3
2n
Theorem[BR]: On the line KR, σG = σB ≤ n/3, Tempering
mixes slower than Metropolis at M > crit by an
exponential factor.
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