Near Optimal Deterministic Algorithms for Volume Computatio - PowerPoint Presentation

Slide1

Near Optimal Deterministic Algorithms for Volume Computation via M-ellipsoids

Daniel

Dadush (CWI)

Joint

with Santosh

VempalaSlide2

Volume Estimation

Given convex body

and factor

, compute

such that .

given by a membership oracle.

 Slide3

Volume Estimation

Barany,

Furedi

86+88:

Any deterministic approximation uses at least membership queries.Dyer, Frieze,

Kannan 91: Randomized polynomial time

approximation algorithm. 

 Slide4

Volume Estimation

Barany,

Furedi

86+88:

Any deterministic approximation uses at least membership queries.Dyer, Frieze,

Kannan 91: Randomized polynomial time

approximation algorithm.Open Problem:Deterministic PTAS for explicit polytopes?  Slide5

Volume Estimation

Theorem

[D-

Vempala 12, D 14]: Deterministic approximation using time and

space.

Nearly matches optimal dependence on .Main Ingredients:Algorithmic construction of M-ellipsoid from convex geometry [Milman 86].Reduction to counting points in “well-calibrated” lattice. Slide6

Volume Estimation

Assumption:

is symmetric about the origin

  asymmetric

symmetricSlide7

Volume Estimation

First Goal:

Approximate

to within

.Already very challenging.First key step for general algorithm. Slide8

Ellipsoidal Approximation

Approximate

using an

ellipsoid

 Slide9

Ellipsoidal Approximation

How well can we approximate volume

using sandwiching ellipsoids?

 Slide10

Ellipsoidal Approximation

max volume ellipsoid contained in

.

min volume ellipsoid containing .  Unfortunately can have

.

  = 

 Slide11

Ellipsoidal Approximation

Unfortunately can have

.

=  

 Slide12

Milman’s

Ellipsoid

Covering Numbers:

For sets

, let

is the minimum number of translates of

needed to cover

.

 Slide13

Milman’s

Ellipsoid

Lemma:

For symmetric convex bodies

 Slide14

Milman’s

Ellipsoid

is an

M-Ellipsoid

of

:

 Slide15

Milman’s

Ellipsoid

is an

M-ellipsoid

of

:

 Slide16

Milman’s

Ellipsoid

is an

M-Ellipsoid

of :

 Slide17

Milman’s

Ellipsoid

is an

M-Ellipsoid

of

:

Covering properties give

.

 Slide18

M-ellipsoid Constructions

Milman `86:

Use sequence of “surgeries” to transform

into an ellipsoid

while maintaining

Klartag `06: 1. Sample uniformly.2. Compute covariance matrix of , .3. Return ellipsoid

.

Slicing conjecture

Can choose

above.

 Slide19

M-ellipsoid Constructions

Milman `86:

Use sequence of “surgeries” to transform

into an ellipsoid

while maintaining

Remainder of talk:Reduction to lattice point counting.Outline of Milman’s construction. Slide20

Volume via Counting

Asymptotically

volume is

 Slide21

Volume via Counting

For which

does this yield a good approximation?

volume is

 Slide22

Volume via Counting

Lemma:

If

(

covers space w.r.t.

),

then for any

:

volume is

 Slide23

Volume via Counting

1. Because

covers space w.r.t.

, can find

which tiles space w.r.t.

.

 Slide24

Volume via Counting

2. Since

is tiling

.

Centers of this tiling live in

.

 Slide25

Volume via Counting

Hence

.

 Slide26

Volume via Counting

Issues:

How do we build lattice

such that

and

?

(Thin lattice covering)How do we enumerate lattice points in efficiently?Must solve both problems simultaneously.

 Slide27

Volume via Counting

Observation:

and are a

good pair.

 Slide28

Volume via Counting

[DV `12]

Approach:

Partial Surgery + EnumerationTransform via surgeries to such that

and there exists ellipsoid

such that2. Let be generated by the axes of scaled by .Return

by enumeration.

Yields

approximation in time

.

 Slide29

Building a good ellipsoid

Input:

Symmetric convex body

Want:

Ellipsoid optimal solution to max s.t.

.

Unfortunately, no known tractable formulation for the above program. Slide30

Many ellipsoids do have optimization characterizations.

John Ellipsoid `48:

max

s.t. Problem: Too restrictive. Only gives

approximation.

 Building a good ellipsoidSlide31

How about relaxing strict containment a “little” bit?

-

Ellipsoid [

Tomzcak-Jaegermann

, Figiel `79]: max s.t.

Has a nearly-equivalent convex formulation!

Milman’s idea: Iteratively use -Ellipsoid to reduce distance of to an Ellipsoid.  Building a good ellipsoidSlide32

Milman’s Iteration

For

Compute

-Ellipsoid

for

. .

.

Return

M-Ellipsoid

.

 Slide33

-Ellipsoid: Covering Estimates

Assume

can be sandwiched within

factor by some ellipsoid.Let be the -Ellipsoid of , then for any

Requires deep theorem of

Pisier `80

(K-convexity).Sudakov inequalities for covering numbers.

 Slide34

“Slowed down” Milman’s

Iteration

For

Compute -Ellipsoid

for

.

.

.

Return

symmetrized body

.

 Slide35

-norm:

-Ellipsoid

 Slide36

-Ellipsoid

Semidefinite

Program:

s.t. P.S.D.

-Ellipsoid :

for optimal [TjF `79] Claim: Computes largest volume ellipsoid that is “half-contained” in .

Largest volume

Half contained in

 Slide37

Half Containment

Assume

(

wlog

).

is standard

gaussian

in

.

Markov Inequality

 Slide38

Assume

(

wlog

).

is standard

gaussian

in .  

concentrated

at

Concentration

of

sphere in

.

Markov Inequality

Half Containment

 Slide39

Assume

(

wlog

).

is standard

gaussian

in .  

of

sphere in

of

in

.

Half Containment

 Slide40

s.t.

P.S.D.

Want to use

Ellipsoid Algorithm from Convex Opt.Requirements:Objective function is “nice”. (yes) Feasible region for T is “well-rounded”. (easy)Membership oracle for feasible region. (???) Slide41

s.t.

P.S.D.

Sufficient to build a

deterministic, convex, and “efficient” estimator satisfying

 Slide42

: gaussian measure

 Slide43

Theorem [D-

Vemp

. 11,

Meka

12]:

The simple estimator satisfiesFurthermore is convex and can be computed in deterministic time using

space.

 Slide44

Conclusions

1. Nearly optimal deterministic algorithm for estimating the volume of a convex body.

2. Algorithmic construction of the M-ellipsoid.

Open Problems

1. Can one construct the M-ellipsoid of an explicit polytope in deterministic polytime?2. Can one approximate

in deterministic

polytime given an extended formulation of ? Slide45

 Slide46

Lattice Problems

Shortest Vector Problem (SVP):

For lattice

and norm

, find shortest non-zero vector in .[D, Peikert, Vempala 11]: Given an ellipsoid as advice, the SVP for

and can be solved in time

,where unit ball of ). Useful for other lattice problems as well (CVP, IP, …).  Slide47

Idea:

Cover

with translates of an ellipsoid

.

Enumerate in these and retain points in

.[D, Peikert, Vempala 11] 

Lattice Enumeration: M-EllipsoidSlide48

Lattice Enumeration: M-Ellipsoid

A lattice

is the integer span of a basis

.

Task:

Compute . 

 Slide49

Idea:

Cover

with translates of an ellipsoid

.

Enumerate in these and retain points in

.[D, Peikert, Vempala 11] 

Lattice Enumeration: M-EllipsoidSlide50

Want:

Not too many translates of

needed to cover

.

is not too much bigger than . 

Lattice Enumeration: M-Ellipsoid