Simulated Tempering Mixes Slowly on the Meanfield Potts Model

1
Simulated Tempering Mixes Slowly on the
Mean-field Potts Model

• Nayantara Bhatnagar
• Dana Randall
• College of Computing
• Georgia Tech

2
Outline of Talk

• Ising and Potts models and sampling
• Glauber (local) dynamics Markov Chain
• Simulated Tempering
• Examples when Tempering fails
• Speeding up Tempering

3
Ising and Potts Model
Ising Model (2 colorings)
G Kn (n,E)
ß
3-state Potts Model (3 colorings)
G Kn (n,E)
Goal
Rapidly mixing MC to sample
from p at large ß.
4
Glauber (local) Dynamics MC
s1 s2 if d(s1, s2) ? 1
(i.e. color differs at most
at 1 vx)
sR
do nothing
sB
sR
Theorem CDFR, BCFKTVV Glauber dynamics for
the Ising and Potts on Kn mixes slowly for large
ß.
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Glauber Dynamics Mixes Slowly
(large ß)
Conductance
Jerrum-Sinclair 88
S
Sc
F min FS S p(S) ? ½
Conductance
Let .
Theorem
? Mixing time ?
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Tempering Marinari-Parisi 92
Define inverse temperatures 0 b0 lt b1 lt ... lt
bM and distributions p0, p1, ... ,pM on O.
p(bM)


Theorem Madras,Zheng 99 Tempering mixes
rapidly at all temperatures for the Ising model
on Kn.
p(b0)
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Tempering for Potts Model?
Theorem BR There exists b bcrit such that
tempering for Potts model on Kn at b mixes slowly.

Proof idea Bound conductance on O O M1.
ly
lx
O U O(r,s,t)
( , , )
r s tn
(0,0,n)
lx
Sufficient to bound conductance on ly M1.
8
Proof
?
?
FS
Ftemp
bM bcrit
S
Fglauber
pb crit( )

?
pb crit( )
O(e-cn)

b0

• Theorem BR There exists ß gt ßcrit at which
  Tempering slows Glauber MC by exponential factor.

• NO temperature based interpolants will help.

9
Speeding Up Tempering?

• Modify more than just temperature
• Define pM p0 so cut is not preserved.

eb(EREBEG)
eb(EREBEG)
?i(s)
pi (s)
f i (s)
Z
Z
eb(EREBEG)
? i (O(r,s,t))
Z
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Speeding up Tempering
eb(EREBEG)
?i (O(r,s,t))
Z
Thm 2 BR Tempering with ?i mixes rapidly for
the Potts model.

• Proof
• Decomposition Theorem
• Comparison Theorem
• Proof techniques generalize MZ to certain
  asymmetric distributions for usual tempering.

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Conclusions

• Tempering fails when phase transition is
  discontinuous
• Tempering with other interpolants could be
  better.
• Defining fi (s) in general ?
• other models (independent sets)
• other underlying graphs (Cartesian lattice)