Slow and Fast Mixing of Tempering and Swapping for the

Potts Model

Nayantara Bhatnagar, UC BerkeleyDana Randall, Georgia Tech

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.

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

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?

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)

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]

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],

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

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.

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.

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)

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

…

Tempering Fails to Converge

βcrit

β0

…

0 < βi < βcrit

…

At βcrittempering mixes

slowly for any set of intermediate temperatures.

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

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

…

Decomposition of Swapping Chain

πi(x) C |x|

i

M

011010 010110

011010 011011

Projection for Swapping chain

01n- 2n

…

Decomposition of Swapping Chain

Projection for Swapping chain Weighted Cube (WC)

011010 010110

011010 011011

011010 010010

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.

• 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

• 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

• 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)

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.

……

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.