Volume computation László Lovász Microsoft Research [email protected].

Volume computation

László Lovász

Microsoft Research

[email protected]

Volume computation

nK Given: , convex

Want: volume of K

by a membership oracle;

(0,1) (0, )B K B R

with relative error ε

Not possible in polynomial time, even if ε=ncn.

in n, 1/ε and log R

Elekes, Bárány, Füredi

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

B

K

Why not just....

***

*

*

*

*

*

**

* *

**

*

*

*

*

S

| |vol( ) vol( )

| |

S KK B

S

Need exponential size Sbefore nonzero!

i iK K B 0B

1B2B

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

1vol( )1 2

vol( )i

i

K

K

Can use Monte-Carlo!

But...

Now we have to generate random points from Ki+1.

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

- How dense should be the lattice?

- Where to start the walk?

- How long to walk?

- How many trials will be aborted?

- How close will be the returned point to random?

“warm start”: use the points you already have

infinitely...

mixing time > bottleneck > isoperimetric inequality

“rounding”: preprocessing by affine transformation

mixing time + small isolated parts

Issues

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

Conductance

: uniform random point inx K

' : one step fromx x

x

'x

0 1 2, , ,... : random walkv v v

General mixing time bound

: :stationary distribution, distribution after stepsk k 0 ( )

sup( )A

AM

A

starting density

2 /2| ( ) ( ) |k kP v A A M e

Jerrum - Sinclair

Mixing time is >1/φ but <(log M)/φ2.

- make boundary small (sandwiching)

bottleneck

isolated cubeThe problem with the boundary

- make boundary smoother

- re-define conductance byexcluding small sets

- walk on all points

- separate global andlocal conductance

- start far from trouble

Dyer-Frieze-Kannan 1989 ** 27 * 32 23( ) (( ) )O nO n O n

multi-Phase Monte-Carlo (product estimator) Markov chain samplingisoperimetric inequalities

Polynomial time!

Cost of volume computation(number of oracle calls) Amortized cost

of sample point

Cost ofsample point


Lovász-Simonovits 1990

** 27 * 32 23( ) (( ) )O nO n O n** 16 * 41 13( ) (( ) )O nO n O n

isoperimetric inequalities via Localization Lemma,exceptional small sets,warm start: start from random point from a distribution already close to uniform

> start far from trouble> avoid start penalty

Bootstrapping: re-using points from previousphase as starting points

3S

1S

3S

1 23

2 vol( ) vol( )vol( )

vol( )

S SS

D K

Isoperimetric Inequality

diam( )D K1 2( , )d S S

, : nf g 0, 0n n

f g

11

0

11

0

, , : [0,1]

( (1 ) ) ( ) 0,

( (1 ) ) ( ) 0,

linearn

n

n

a b

f ta t b t dt

g ta t b t dt

infinitesimally narrow truncated cone

0, 0T T

f g

Localization Lemma



Applegate-Kannan 1990

** 27 * 32 23( ) (( ) )O nO n O n** 16 * 41 13( ) (( ) )O nO n O n** 10 * 87(( ) ( ))O n O nO n

integration of logconcave functions,isoperimetric inequality for logconcave functions,Metropolis algorithm, better sandwiching

The Metropolis algorithm

Given: time-reversible Markov chain M on V with stationary distribution ;

Want: Sample from distribution with density proportional to F.

: , 0.F V F

Modified Markov chain M’:

- generate step ij

- if F(j)F(i), make step;

- if F(j)≤F(i), make step with probability F(j)/F(i), else stay where you are.

M’ is time-reversible, and its density is proportional to F.




Lovász 1991

** 27 * 32 23( ) (( ) )O nO n O n** 16 * 41 13( ) (( ) )O nO n O n** 10 * 87(( ) ( ))O n O nO n** 10 * 87(( ) ( ))O n O nO n

ball walk




Lovász 1991

Dyer-Frieze 1991


* 8 ** 6 7( ) ( )( )O nO n O n

independence of errors




Lovász 1991

Dyer-Frieze 1991

Lovász-Simonovits 1992,93


* 8 ** 6 7( ) ( )( )O nO n O n* 7 ** 5 6( ) ( )( )O nO n O n

integration of smoother functionsrandomized preprocessinggeneralization of multi-phase Monte-Carlo to simulated annealing scheme

Want: (1 )K

Mf f Random walk on K randomX K

1

1( )

N

iiK

f f XN

?N

2

2 2

( )1 1

( ( ))

Var

E

ff XM

f X A

2Need

MN

“Simulated annealing” for integration

2

2

(log1

)2 need samplesY

M

/ log,k

K

kmA m Mf

0 1, mA A A 1 2

0 1 1

... m

m

AA AA

A A A

( 1) / /1/1

/ /

k m k mmk

k m k mk

kd

fA fd f d

A f f

1/( ) mY f XX: sample from k,

1 ( )Ek

k

AY

A




Lovász 1991

Dyer-Frieze 1991


Kannan-Lovász-Simonovits 1997


* 8 ** 6 7( ) ( )( )O nO n O n* 7 ** 5 6( ) ( )( )O nO n O n* 5 ** 3 4( ) ( )( )O nO n O n

isotropic posititionlocal and global obstructions (speedy walk)bootstrapping preprocessing and sampling




Lovász 1991

Dyer-Frieze 1991



** 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


* 3( )O n

analysis of the hit-and-run algorithm

Smith 1984Hit-and-run walk




Lovász 1991

Dyer-Frieze 1991



** 27 * 32 23( ) (( ) )O nO n O n

Lovász 1999



Kannan-Lovász 1999 * 3( )O n

average conductance,log-Cheeger inequality

* 3( )O n




Lovász 1991

Dyer-Frieze 1991



** 27 * 32 23( ) (( ) )O nO n O n

Lovász 1999



Kannan-Lovász 1999

Lovász-Vempala 2002 * 3( )O n

sampling from general logconcave distributions,ball walk and hit-and-run walk

* 3( )O n * 3( )O n




Lovász 1991

Dyer-Frieze 1991



** 27 * 32 23( ) (( ) )O nO n O n

Lovász 1999



Kannan-Lovász 1999

Lovász-Vempala 2002 * 3( )O n

A.Kalai-Lovász-Vempala 2003 * 4( )O n

Simulated annealing

* 3( )O n

* 3( )O n

The pencil construction

1

2vol( )

nZ K

0

'

( )K

axZ a e dx

K

'K

0x0

n

/21

(2 )22

n

Z nenn

(0,1) (0, ), lnB K B R R n n

2R

0 1

12 ...

2ma n a a

n

1

2vol( )

nZ K

0

'

( )K

axZ a e dx/2

1(2 )

22

n

Z nenn

0

0

'

: prob distribution with densityi

K

i

i

a x

a xe

e dx

0 00 0

0 0

1

1

1 1( () )( )

( )k

k

k

k kk k k k

k k

a x xa x a x

x x

aa a

a a

eZ a ee e

Z a e e

dxdx d

dx dx

1 10

1 2 0

( ) ( ) ( )vol( ) ( ) ... ( )

( ) ( ) ( )m m

m

m m

Z a Z a Z aK Z a Z a

Z a Z a Z a

/2 , 0,1,..., 2 log(2 )k nka ne k m n n

0 1

12 ...

2ma n a a

n

random point fromk kX 01( )( )k

k k ka a XY e

01 1( )( )( )

( )Ek

k k

k

k ka xaZ ae Y

Z ad

2( ) 2 ( )Var Ek kY Y

Two possibilities for further improvement:

- The Slicing Conjecture

- Reflecting walk

The Slicing Conjecture

Smallest bisecting surface

F H

Smallest bisecting hyperplane

1 1vol ( ) vol ( )n nH F ??

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).

Volume computation László Lovász Microsoft Research [email protected].

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Transcript of Volume computation László Lovász Microsoft Research [email protected].