Slow and Fast Mixing of Tempering and Swapping for the Potts Model Nayantara Bhatnagar, UC Berkeley Dana Randall, Georgia Tech

Introduction: Approximate Counting Problems MatchingsIndependent Sets Partition functions of Ising, Potts models Volume of a convex body # P-hard : We dont expect efficient exact algorithms.

Introduction: Approximate Counting and Sampling Theorem: Jerrum-Valiant-Vazirani '86 For "self-reducible" problems, Approximately Uniform Generation Approximate Counting Approximate Sampling Approximate Counting Matchings Colorings Independent Sets Volume of a convex body

lim Pr[X t = Y | X 0 ] = π(Y) t Markov Chains K = (Ω, P) Theorem: If K is connected and aperiodic, the Markov chain X 0,X 1,... 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.

MatchingsIndependent Sets Partition functions of Ising, Potts models Volume of a convex body Broders Markov chainGlauber dynamics Ball walk, Lattice walk Markov Chains

Introduction: Markov Chain Monte Carlo Markov Chains: Matchings – Broders 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, π β (x) e β( H(x)) H(x) = Σ δ x i = x j Glauber dynamics Markov Chain Choose (v, c t+1 (v)) R V x [q]. Update c t (v) to c t+1 (v) with Metropolis probabilities. (i,j)

Why Simulated Tempering π β (x) H(x) Glauber dynamics mixes slowly for the q-state Potts for K n for q 2, at large enough β. Φ S = P[ X t+1 S | X t ~ π(S)] S ScSc Theorem : T c 1 Φ c 2 Φ2 Φ2 Φ = min Φ S S: π(S) ½ Conductance: [Jerrum-Sinclair 89, Lawler-Sokal 88]

Simulated Tempering [Marinari-Parisi 92] Define inverse temperatures 0 = β 0 β M = β and distributions π 0 π 1 π M = π β on Ω. i = M · i M … … π M π (x,i) = ˆ 1 M+1 π i (x) Tempering Markov Chain: From ( x,i ), W.p. ½, Glauber dynamics at β i W.p. ½, randomly move to (x,i ±1) π 0 ˆ Ω = Ω × [M+1],

Swapping [Geyer 91] Define inverse temperatures 0 = β 0 β M = β and distributions π 0 π 1 π M = π β on Ω. i = M · i M … … π M π (x) = Π ˆ π i (x i ) Swapping Markov Chain: From x, choose random i W.p. ½, Glauber dynamics at β i W.p. ½, move to x (i,i+1) π 0 ˆ Ω = 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 K n. 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 K n. 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 β crit > 0, such that tempering for Potts model on K n at β crit 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 β i The space Ω partitioned into equivalence classes σ: ˆ (n/2, 0, n/2) (n,0,0)

Stationary Distribution of Tempering Chain At β 0 At β i > 0 π i (σ) n σ R σ B σ G e β i ( ) (σ R ) 2 + (σ B ) 2 + (σ G ) 2 π i (σ) n σ R σ B σ G

Stationary Distribution of Tempering Chain At β crit At β 0 … At 0 < β i < β crit disordered mode ordered mode π i (σ) n σ R σ B σ G e β i ( ) (σ R ) 2 + (σ B ) 2 + (σ G ) 2 …

Tempering Fails to Converge β crit β0β0 … 0 < β i < β crit … At β crit tempering 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 K R, the line σ B = σ G n/3 π β (x) e β( H(x)) H(x) = Σ δ x i = x j + B Σ δ x i =+ (i,j) i π(x) C |x|, x [-n 1,n 2 ] n 1 > n 2 n π β (x) e β( H(x)) H(x) = Σ δ x i = x j

Decomposition of Swapping Chain π i (x) C |x| i M Madras-Randall 02 Decomposition for Markov chains 1.Mixing of restricted chains R 0,i and R 1,i at each temperature. 2.Mixing of the projection chain P. T swap C min T R b,i x T P b {0,1}, i M …

Decomposition of Swapping Chain π i (x) C |x| i M Projection for Swapping chain …

Decomposition of Swapping Chain Projection for Swapping chain Weighted Cube (WC)

Decomposition of Swapping Chain Projection for Swapping chain Weighted Cube (WC) Upto polynomials, π i (0) C n 1 i / M /Z i and π i (1) C n 2 i / M /Z i Lemma: If for i > j, π i (1) π j (0) p(n)π i (0) π j (1), then T P q(n) T WC.

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 σ R σ B σ G e β i ( ) (σ R ) 2 + (σ B ) 2 + (σ G ) 2

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 σ R σ B σ G e β i ( ) (σ R ) 2 + (σ B ) 2 + (σ G ) 2 … … i M π i (σ) = π i (σ) f i (σ) = π i (σ) n σ R σ B σ G i-M M

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 K R 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 S M > crit Tempering vs. Fixed Temperature n Theorem[BR]: On the line K R, σ G = σ B n/3, Tempering mixes slower than Metropolis at M > crit by an exponential factor.