why is it useful to walk randomly? lászló lovász mathematical institute eötvös loránd...
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Why is it useful to walk randomly?
László Lovász
Mathematical InstituteEötvös Loránd University
October 2012
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Random walk on a graph
October 2012
Graph G=(V,E)
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Random walk on a graph
October 2012
t(v): probability of being
at node v after t steps
1
( )
1( ) ( )
deg( )t t
j N i
i jj
1deg( ) deg( )( ) ( )
2 2t ti i
i i i im m
Stationary distribution
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Hitting time
H(s,t) = hitting time from s to t
= expected # of steps, starting at s,
before hitting t
k(s,t) = commute time between s and t
= H(s,t) + H(t,s)
October 2012
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( )
1( )
ha
( ).deg(
rmonic in
)
:
j N i
f i f ji
f i V
( )
1( )
ha
( )deg(
s pole i
)
n :
j N i
f i V
f i f ji
Every nonconstant function has at least 2 poles.
Harmonic functions
Every function defined on SV (S) has a unique extension harmonic on V \ S.
G=(V,E) graph,
f: V
October 2012 5
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S
2
3
1 f(v)= E(f(Zv))
Zv: (random) point where
random walk from v hits Sv
0v
1
f(v)= P(random walk from
v hits t before s)
s t
Harmonic functions and random walks
October 2012 6
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0v
1f(v)=electrical potentials t
Harmonic functions and electrical networks
October 2012 7
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f(v) = position of nodes
0 1
Harmonic functions and rubber bands
October 2012 8
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2( , ) 2 ( , )
force acting on
mu v mR u v
u
Commute time and resistance
October 2012 9
effective resistence between u and v
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( )
11 ( , )
d )( )
eg(,
i N s
H i ts
H s t
( )
deg( ) ( ( , ) ( , )) 0i N s
s H i t H s t
Distance from s to t = H(s,t).
t
weight=degree
strength=1
Hitting time and rubber bands
October 2012 10
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1{
7
1{
5
}1
}12
7
} 1
7
9
3
5
Hitting time and rubber bands
October 2012 11
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Random maze
October 2012
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Random maze
October 2012
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14October 2012
We obtain every mazewith the same probability!
Random maze
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Random spanning tree
October 2012
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- card shuffling
- statistics
- simulation
- counting
- numerical integration
- optimization
- …
Sampling: a general algorithmic task
October 2012
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{ : ( ) ( , )}L x y xA y
polynomial time algorithm
certificate
October 2012
L: a „language” (a family of graphs, numbers,...)
Sampling: a general algorithmic task
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{ : ( ) ( , )}L x y xA y
Find: - a certificate
Given: x
- an optimal certificate
- the number of certificates
- a random certificate
(uniform, or given distribution)
October 2012
L: a „language” (a family of graphs, numbers,...)
Sampling: a general algorithmic task
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One general method for sampling: Random walks
(+rejection sampling, lifting,…)
Construct regular graph with node set V
Want: sample uniformly from V
Simulate (run) random walk for T steps
Output the final node ????????????
mixing timeOctober 2012
Sampling by random walk
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Given: convex body K n
Want: volume of K
Not possible in polynomial time, even if an errorof nn/10 is allowed.
Elekes, Bárány, Füredi
Volume computation
October 2012 20
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Dyer-Frieze-Kannan 1989
But if we allow randomization:
There is a polynomial time randomized algorithmthat computes the volume of a convex body
with high probability with arbitrarily small relative error
Volume computation
October 2012 21
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B
K
Why not just....
***
*
*
*
*
*
**
* *
**
*
*
*
*
S
| |vol( ) vol( )
| |
S KK B
S
Need exponential size S
to get nonzero!
Volume computation by plain Monte-Carlo
October 2012 22
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i iK K B
0B1B
2B
mB
1 10
1 2 0
vol( ) vol( ) vol( )vol( ) vol( )
vol( ) vol( ) vol( )m m
m m
K K KK K
K K K
mK K
0 0K B
1/1 2 n
i iB B
Volume computation by multiphase Monte-Carlo
October 2012 23
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1vol( )1 2
vol( )i
i
K
K
Can use Monte-Carlo!
But...Now we have to generate random points from Ki+1.
Need sampling to computethe volume
Volume computation by multiphase Monte-Carlo
October 2012 24
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Do sufficiently longrandom walk on centersof cubes in K
Construct sufficiently dense lattice
Pick random point p from little cube
If p is outside K, abort;else return p
Dyer-Frieze-Kannan 1989
Sampling by random walk on lattice
October 2012 25
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Sampling by ball walk
October 2012 26
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Sampling by hit-and-run walk
October 2012 27
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steplength can be large!
Sampling by reflecting walk
October 2012 28
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- Stepsize
- Where to start the walk?
- How long to walk?
- How close will be the returned point to random?
Issues with all these walks
October 2012 29
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bottleneck
1S 2S1 1 2( ) ( , ' )PS x S x S
1 21
1 2
( , ' )( )
( ) ( ' )
P
P P
x S x SS
x S x S
isoperimetric quantity
inf { ( ) : }S S K
: uniform random point inx K
' : one step fromx x
x
'x
Conductance
October 2012 30
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Dyer-Frieze-Kannan 1989 ** 27 * 32 23( ) (( ) )O nO n O n
Polynomial time!
Cost of volume computation
(number of oracle calls)Amortized cost
of sample point
Cost ofsample point
Time bounds
October 2012 31
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Dyer-Frieze-Kannan 1989
Lovász-Simonovits 1990
Applegate-Kannan 1990
Lovász 1991
Dyer-Frieze 1991
Lovász-Simonovits 1992,93
Kannan-Lovász-Simonovits 1997
** 27 * 32 23( ) (( ) )O nO n O n
Lovász 1999
** 16 * 41 13( ) (( ) )O nO n O n** 10 * 87(( ) ( ))O n O nO n** 10 * 87(( ) ( ))O n O nO n
* 8 ** 6 7( ) ( )( )O nO n O n* 7 ** 5 6( ) ( )( )O nO n O n* 5 ** 3 4( ) ( )( )O nO n O n
Kannan-Lovász 1999
Lovász-Vempala 2002 * 3( )O n
Lovász-Vempala 2003 * 4( )O n
* 3( )O n
* 3( )O n
Time bounds
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- The Slicing Conjecture
- Reflecting walk
Possibilities for further improvement
October 2012 33
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Reflecting random walk in K
v
u
steplength h large
How fast does this mix?
Stationary distribution: uniform
Chain is time-reversible
(e.g. exponentially distributedwith expectation = diam(K))
October 2012 34
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Smallest bisecting surface
F H
Smallest bisecting hyperplane
1 1vol ( ) vol ( )n nH F
??
The Slicing Conjecture
October 2012 35