Mixing

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Mixing
A tutorial on Markov chains

• Dana Randall
• Georgia Tech

( Slides at www.math.gatech.edu/randall )
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Outline

• Fundamentals for designing a Markov chain
• Bounding running times (convergence rates)
• Connections to statistical physics

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Markov chains for sampling
Given A large set (matchings, colorings,

independent sets,)
Main Q What do typical elements
look like?
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Markov chains
Andrei Andreyevich Markov 1856-1922
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Sampling using Markov chains
State space ?
( ? cn )
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Sampling using Markov chains
State space ?
( ? cn )

• Step 1. Connect the state space.

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Basics of Markov chains
x
y
H
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The stationary distribution p
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Sampling from non-uniform distributions
Q What if we want to sample from some other
distribution?

• Step 2. Carefully define the
• transition probabilities.

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The Metropolis Algorithm
(MRRTT 53)
Propose a move from x to y as before, but accept
with probability
min (1, p(y)/p(x))
(with remaining probability stay at x).
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Basics continued

• Step 1. Connect the state space.

• Step 2. Carefully define the
• transition probabilities.

Starting at any state x0, take a random walk for
some number of steps . . . and output the final
state (from p?).
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The mixing rate
Defn The total variation distance is
Pt,p max __ ? Pt(x,y) - p(x).
1
2
xÎ ?
yÎ ?
A Markov chain is rapidly mixing if t(e)
is poly (n, log(e-1)).
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Spectral gap
Let 1 l1 gt l2 l? be
the eigenvalues of P.
Defn Gap(P) 1-l2 is the
spectral gap.
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Outline

• Fundamentals for designing a Markov chain
• Bounding running times (convergence rates)
• Connections to statistical physics

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Outline for rest of talk

• Techniques
• Coupling
• Flows and paths
• Indirect methods

• Problems
• Walk on the hypercube
• Colorings
• Matchings
• Independent sets

• Connections with statistical physics
• - problems
• - algorithms
• - physical insights

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Coupling
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Coupling
Simulate 2 processes
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Coupling
Defn A coupling is a MC on ? x ?

• Each process Xt, Yt is a faithful copy of
  the original MC,
• If Xt Yt, then Xt1 Yt1.

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Ex1 Walk on the hypercube

• MCCUBE
• Start at v0(0,0,,0).
• Repeat
• - Pick i Î n, b Î 0,1.
• - Set vi b.

Symmetric, ergodic p is uniform.

Mixing time? Use coupling
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Outline

• Techniques
• Coupling
• - path coupling
• Flows and paths
• Indirect methods

• Problems
• Walk on the hypercube
• Colorings
• Matchings
• Independent sets

• Connections with statistical physics
• - problems
• - algorithms
• - physical insights

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Ex 2 Colorings
Given A graph G (max deg d), k gt 1. Goal Find
a random k-coloring of G.

• MCCOL (Single point replacement)
• Starting at some k-coloring C0
• Repeat
• - With prob 1/2 do nothing.
• - Pick v Î V, c Î k
• - Recolor v with c, if possible.

The lazy chain
If k d 2, then the state
space is connected.
(Therefore p is uniform.)
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Path Coupling
Bubley,Dyer,Greenhill97-8
Coupling Show for all x,y Î W, E D
(dist(x,y)) lt 0.
-
Path coupling Show for all u,v s.t.
dist(u,v)1, that E D (dist(u,v)) lt
0.
-
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Path coupling for MCCOL
Thm MCCOL is rapidly mixing if k 3d.
(Jerrum 95)
Pf Use path coupling dist(x,y) 1.

• o.w. ?dist 0.

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Summary Coupling
Pros Can yield very easy proofs
Cons Demands a lot from the chain

• Extensions
• Careful coupling (k 2d) (Jerrum95)
• Change the MC (Luby-R-Sinclair95)
• Macromoves
• - burn in (Dyer-Frieze01, Molloy02)
• - non-Markovian couplings
• (Hayes-Vigoda03)

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Outline

• Techniques
• Coupling
• Flows and paths
• Indirect methods

• Problems
• Walk on the hypercube
• Colorings
• Matchings
• Independent sets

• Connections with statistical physics
• - problems
• - algorithms
• - physical insights

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Conductance and flows
(Jerrum-Sinclair88)
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Min cut Max flow

(Sinclair92)

• G Make ?2 canonical

paths gxy from xÎ?, to yÎ?, x ? y,
carrying p(x)p(y) units of flow.
?
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Ex 3 Back to the hypercube
- Define a canonical path from s to t.
- Bound the number of paths through (u,v) Î E.
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Outline

• Techniques
• Coupling
• Flows and paths
• Indirect methods

• Problems
• Walk on the hypercube
• Colorings
• Matchings
• Independent sets

• Connections with statistical physics
• - problems
• - algorithms
• - physical insights

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Ex 4 Sampling matchings
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Ex 4 Sampling matchings

• MCMATCH
• Starting at M0, repeat
• Pick e (u,v) Î E

- If e Î M, remove e - If u and v unmatched in
M, add e - If u matched (by e) and v
unmatched (or vice versa), add e and remove
e - Otherwise do nothing.
Thm Coupling wont work!
(Kumar-Ramesh99)
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Mixing time of MCMATCH
s
t
Å
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Outline

• Techniques
• Coupling
• Flows and paths
• Indirect methods

• Problems
• Walk on the hypercube
• Colorings
• Matchings
• Independent sets

• Connections with statistical physics
• - problems
• - algorithms
• - physical insights

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Ex 5 Independent Sets
Goal Given l, sample ind. set I
with prob p(I) lI/Z, Z ?J
lJ.
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Slow mixing of MCIND (large l)
(Even)
(Odd)
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Summary Flows
Pros Offers a combinatorial approach to
mixing especially useful for proving slow
mixing.
Cons Requires global knowledge of the chain
to spread out paths.
Extensions Balanced flows
(Morris-Sinclair99) MCMC -- Major
highlights - The permanent
(Jerrum-Sinclair-Vigoda02) -
Volume of a convex polytope
(Dyer-Frieze-Kannan89, )
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Outline

• Techniques
• Coupling
• Flows and paths
• Indirect methods
• - Comparison
• - Decomposition

• Problems
• Walk on the hypercube
• Colorings
• Matchings
• Independent sets

• Connections with statistical physics
• - problems
• - algorithms
• - physical insights

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Comparison
(Diaconis,Saloff-Coste93)
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Comparison
x
y
known P
_
_

(x,y) Î P gx,y (using P) G(z,w) is the
set of paths gx,y using (z,w)
w
unknown P
z
_
1
Thm Gap(P) Gap(P).
A
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Comparison, aka . . .
Adjacency . . . The Matrix Reloaded
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Disjoint decomposition
(Madras-R.96, Martin-R.00)
A2
A1
P
A5
A3
A4
A6
?
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Ex 6 MCIND on small ind. sets
For G(V,E)
Let ? ind. sets of G ?k ind.
sets of size k.
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Ind. sets w/bounded size (cont.)
Thm MCIND is rapidly mixing on
K
Ç
?k , where KV/2(?1).
k 1
MCSWAP
?0 ?1 ?2 . . . ?K-1 ?K
Projection
Restrictions
a0 a1 a2 . . .aK-1 aK
?k
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The Restrictions of MCswap
Projection
Restrictions

• .

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Summary Indirect methods
Pros Offer a top down approach allow hybrid
methods to be used..
Cons Can increase the complexity.
Extensions Comparison thm for log-Sobolev
(Diaconis-Saloff-Coste96)
Comparison for Glauber dynamics
(R.-Tetali 98)
Decomposition for log-Sobolev
(Jerrum-Son-Tetali-Vigoda 02)
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Outline

• Techniques
• Coupling
• Flows and paths
• Hybrid methods

• Problems
• Walk on the hypercube
• Colorings
• Matchings
• Independent sets

• Connections with statistical physics
• - problems
• - algorithms
• - physical insights

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Why Statistical Physics?

• They have a need for sampling
• Use many interesting heuristics
• Great intuition
• Experts on large data sets
• Microscopic Macroscopic
• details behavior
• (i.e., phase transitions)

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Models from statistical physics
Hardcore model
Potts model
(3-colorings) (Independent
sets) (Matchings)
(Min cut)
-
-
-
-

Dimer model
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Models (cont.)
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Models (The physics perspective)
Given A physical system ? s Define A
Gibbs measure as follows
H(s) (the Hamiltonian),
b 1/kT (inverse temperature),
p(s) e-bH(s)/ Z,
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Physics perspective (cont.)
Q What about on the infinite lattice?
Use conditional probabilities
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Phase transitions Ind. sets
Low temperature long range effects
High temperature ? effects die out
TC indicates a phase transition.
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Slow mixing of MCIND revisited
R/B
8
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Group by of fault lines
Fault lines are vacant paths of width 2 from top
to bottom (or left to
right).
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Peierls Argument
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Peierls Argument cont.
2n/2 3l
S1
SB
( l - n/2 more points)
(and similarly for S2, S3, ) x
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Conclusions
Techniques
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Conclusions
Open problems

• Sampling 4,5,6-colorings on the grid.

• Sampling perfect matchings on
• non-bipartite graphs.

• Sampling acyclic orientations in a graph.

• Sampling configurations of the Potts
• model (a generalization
• of Ising, but with more colors).

• How can we further exploit phase
• transitions? Other physical intuition?

...
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