the poincar é constant of a random walk in high-dimensional convex bodies

The Poincaré Constant ofa Random Walk in High-

Dimensional Convex Bodies

Ivona Bezáková

Thesis Advisor: Prof. Eric Vigoda

Goal:

Efficient algorithm for sampling points from high-dimensional convex bodies

Approach: Random walks

Overview of the talk

• Motivation & History

• Introduction to random walks, Markov chains

• Definition and basic properties of ball walks

• Overview of the algorithm for uniform sampling

• Analysis of the Poincaré constant

Sampling points from convex bodies efficiently

• Computation of volume

• Sampling contingency tables [Morris]

• Universal portfolios [Kalai & Vempala]

• Convex programs [Bertsimas & Vempala]

Motivation

Why is computing volume difficult?

Straightforward approach:

- find bounding box

- sample sufficiently many points from the box

- compute ratio: # points in the body vs. total # points

This algorithm correctly approximates volume.

Where is the catch?

Ratio of the volume of the body vs. volume of the box decreases exponentially with dimension!

This results in exponential running time.

Note: For small dimension (2D, 3D) this algorithms works well.

Goal: Find algorithm running in time polynomial in dimension.

How efficient sampling helps with volume?

• intersect with balls doubling in volume

• this defines sequence of convex bodies

• sample points from i-th body, compute ratio: i-th vs. (i-1)-st body

• return product of ratios

Why is this better than the bounding box?

- Volume of bodies max. doubles

Result: volume of the original body vs. volume of the smallest body (a ball)

History

• negative results for volume computation:

- cannot be approximated by deterministic polytime algorithm

• randomized approximation:

- Dyer, Frieze, Kannan ’89

- improvements: combinations of [Applegate, Dyer, Frieze, Kannan, Lovász, Simonovits]

- Kannan, Lovász, Simonovits ‘97 * 5( )O n

* 23( )O n

- #P-complete

[Elekes, Bárány & Füredi, Dyer & Frieze, Khachiyan, Lawrence, Lovász & Simonovits]

Notation

• convex body of diameter

(given by membership oracle)

nK R D

• step-size • ball ( , )B x

D

K

x

( , )B x 0

unit-b

all

Ball Walks

Kx

Speedy Walk – next point ( , )y B x K

y

Metropolis Walk – next point

if then

( , )y B x y K :y x

Problem: How to implement?

Markov Chains

State space

Transition distribution

( , )P x

(likelihood of going from to )x y

For speedy walk we have

K state space = convex body

( , ) 1 ( ( , ) )P x y vol B x K if || ||x y

0 otherwise

x y

Markov Chains 2

Stationary distribution = limiting distribution

P (fix-point) ( ) ( ) ( , )x y P y x dy

i.e.

Mixing time

For given mixing time is the expected number of steps needed to get close to the stationary distribution.

0

Want: rapid mixing, i.e. time polynomial in and 1 n

Comparison KLS vs. this work

Kannan, Lovász, Simonovits study so-called conductance for bounding mixing time.

* 3 2( )O n D

? poly-logarithmic in 1

We bound so-called Poincaré constant (generalization of conductance) and get mixing time

* 3 2( )O n D

cubic in 1

spectr

al gap

New ideas in KLS

• separated analysis of speedy walk (fast mixing in principle) and Metropolis walk (efficient implementation)

• for volume computation: introduced isotropic position to reduce diameter of the body

Why Poincaré constant?

• generalization might lead to better analysis through other quantities (log-Sobolev, [Frieze & Kannan, Jerrum & Son])

• the same difficulty

our focus: survey

Poincaré constant

where

:

( , )inf

( )f

f f

Var f

R

2( ) ( )( ( ) ( ))Var f x f x E f dx

( , ) ( ) ( )f f x h x dx

Well-studied Quantities

21( ) ( , )( ( ) ( ))

2h x P x y f x f y dy

and Dirichlet form

21( ) ( ) ( )( ( ) ( ))

2Var f x y f x f y dxdy

21( , ) ( ) ( , )( ( ) ( ))

2f f x P x y f x f y dxdy

(local variance) mea

sure

s de

cayi

ng o

f va

rianc

e

Well-studied Quantities 2

(Properties of Poincaré)

Thm: For (lazy reversible) Markov Chain

2 0 2|| || (1 / 2) || ||tt

For Markov chains defined on finite state spaces the Poincaré constant equals the spectral gap.

twhere is the distribution after stepst

with probability ½ stay at the same statecorresponds to symmetric chains

and is the stationary distribution

Thoughts about Poincaré constant

If then ( , ) ( )f f Var f ( , ) ( )P x y y

Thus, in this case and the chain mixes (very) rapidly.

Intuitively, this corresponds to a complete graph, where we can get from any point to any other point.

1

21( , ) ( ) ( , )( ( ) ( ))

2f f x P x y f x f y dxdy

21( ) ( ) ( )( ( ) ( ))

2Var f x y f x f y dxdy

Well-studied Quantities 3

10 ( ) 1/ 2inf Pr( | )t t

SX S X S

: {0,1}

( , )inf

( )f

f f

Var f

Conductance

equals Poincaré over indicator functions

trivially

Cheeger-type inequality by Jerrum and Sinclair, ‘89

2

2

Properties of Ball Walks

Local conductance( ( , ) )

( )( ( , ))

vol B x Kl x

vol B x

Ball walks:

• stationary distribution( )

( )l x

xL

( )K

L l x dxwhere

• reversible( ) 1

( ) ( , )( ( , ) )

l xx P x y

L vol B x K

From Speedy Walk to Uniform Sampling

(Overview)

• bound Poincaré constant for speedy walk

• mixing time for speedy walk

• running time of Metropolis walk (assuming good starting distribution)

• obtain a good starting distribution

• from a sample point from the speedy distribution obtain a sample point close to the uniform distribution


Poincaré inequality (for speedy walk):

/c D n If then2

2

c

D n

for some dimension-independent constant c

Mixing time of speedy walk:

For given 0 distribution after1 1( ln )t O

steps within from speedy distribution

(assuming reasonable starting distribution )0

Thm: For (lazy reversible) Markov Chain

2 0 2|| || (1 / 2) || ||tt

twhere is the distribution after stepstand is the stationary distribution


(Overview)






From Speedy Walk to Uniform Sampling 2

From speedy to Metropolis walk

Run M. walk until speedy steps

Mixing time of Metropolis walk:

If 0 0 2: || || 1m then we expect the

1 1( ln )t O

total number of steps (speedy + Metropolis)

(with exception )20mto be at most

20

3

(1 )

t

m

where is the average local conductance:

( ) ( )K

l x dx vol K


(Overview)







Obtaining a good starting distribution

Let (0,1) (0, )B K B D and /(0, (1 ) )i n

iB B

i iK B K for where0,...,i b

Algo: • Sample from according to

• For obtain :

Run Metropolis in starting at

1logb n D

0x 0K 0

1,...,i b

1ix iKix


Good starting distribution for Metropolis walk

Thm: For sufficiently small and0 2 n

the distribution of is within of .ix i

Expected total number of oracle calls

(with exception ) is less than3b

3 2 3( (1 ) ln ln(1 ))O n D D


(Overview)







From speedy distribution to the uniform distribution

Algo: • Shrink ' (2 1 2 )K n n K K• Sample from until 'x K

• Return : (2 2 1)y n n x

Thm: If and sufficiently small

1 8 ln( / )n n 0 then the distribution of is y 10

away from the uniform distribution.

Expected number of samples needed .2


(Overview)






Proof of the Poincaré Inequality

Restricted variance, Dirichlet form, expected value

' '[ ( ) ( ')] ( )K K

f x K f x dx 2

' ''( ) ( )( ( ) )K KK

Var f x f x f dx ' '( , ) ( ) ( )K Kf f x h x dx

Poincaré inequality (for speedy walk):

/c D n If then for any function2

2

c

D n

for some dimension-independent constant c

where

:f K R

( , ) ( )f f Var f

Idea of the proof:

• For a sufficiently small set such that does not vary much within

• Assuming Poincaré does not hold, we find a set contradicting the above

'K K'K

( )l x

' '( , ) ( ) 1 [2 ( 1)]K K lf f Var f c e

( , ) ( )K Kf f Var f

Find needle-like s.t.

1 1( , ) ( )K Kf f Var f

1K K

wlog ( ) 0KE f f

and10Kf

Chop to obtain desired set1K 0K

Needle-like Body

Eliminate dimensions one by one (inductively)

( , ) ( )i iK Kf f Var f 0

iKf • assume has fat dimensions and iK 1i

while

• projection of onto two fat dim.J iK

• there exists a point s.t. any lineJthrough cuts into appx. half

• take hyperplane s.t. 0i iK H K H

f f H

( , ) ( )i iK H K H

f f Var f

• at least one of these must be true

( , ) ( )i iK H K H

f f Var f

or

iK

J H

Shrinking Last Dimension1K

'D

Goal: find s.t. last dim. of is0 1K K 0K'

:c D

D n

and0 0 0( , ) ( )K Kf f Var f c

where is a constant (dependent on )0c c

How? Chop into1K 0 1,..., mS S

Ideally

1 1 0( , ) ( , ) ( ) ( )i iS K K Sf f f f Var f c Var f

?

But

1 1

2( ) ( ) ( )( )

i iK S i S KVar f Var f S f f

Assumption1 1

2( ) 2 ( )( )

iK i S KVar f S f f Idea: relate to ( )

iUVar f where 1:i i iU S S

We get

1 ,0

1( ) ( ) ( )ki j

i j mK UK Var f Var fa

where

, : 2 1j

i j i j kk i

a w w w

and : ( )k kw S

Next goal: bound ,0

i ji j m

a

Chopping of 1K

What do we need?

• does not vary much within , i.e. for any ( )l x

, ix y S let ( ) / ( ) 2l x l y

• width of is at mostiS / 2

iS

We will show that this chopping allows us to bound

,0

i ji j m

a appropriately

Properties of local conductance

• is concave over 1/( ) nl x K

From Brunn-Minkowski Thm:

• is Lipschitz over : For anyln ( )l x K ,x y K

| ln ( ) ln ( ) | || ||n

l x l y x y

Implications for the iS

• The width of increases, then it is (full width)

iS

and then it decreases

/ 2

• For sufficiently small the width of any is

at least

(2 )n

iS

From Dinghas’ Thm:

For the middle section is convex.11/ ,...,1/l rw w

Now we can split into several sums and estimate them separately

,i ja

, , , , ,0 ,

...i j i j i j i j i ji j m i j l i l j r i l r j l i j r

a a a a a

Thus 2, 1( ( ))i j

l i j r

a O M K

where is the number of slabs in the middle sectionM

What to do outside the middle section?

, 1( )i j li j l

a O lw

In the left section, the increase exponentiallyiw

This allows us to bound

We obtain similar bounds for other parts of the sum, putting them together we get

22

, 1 12 2(( ) ( )) ( ( ))i j

D na O l M m r K O K

c

We wanted 0

, 1( )i jl i j r

ca K

Thus we proved the Poincaré inequality.

THANK YOU

the poincar é constant of a random walk in high-dimensional convex bodies

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