Volume computation László Lovász Microsoft Research

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Presentation transcript:

Volume computation László Lovász Microsoft Research

Volume computation Given:, convex Want: volume of K by a membership oracle; with relative error ε Not possible in polynomial time, even if ε=n cn. 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 algorithm that computes the volume of a convex body with high probability with arbitrarily small relative error

Why not just.... * * * * * * * * * * ** * * * * * * S Need exponential size S before nonzero!

Can use Monte-Carlo! But... Now we have to generate random points from K i+1.

Do sufficiently long random walk on centers of 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 isoperimetric quantity Conductance

General mixing time bound starting density Jerrum - Sinclair Mixing time is >1/φ but < (log M)/φ 2.

- make boundary small (sandwiching) bottleneck isolated cube The problem with the boundary - make boundary smoother - re-define conductance by excluding small sets - walk on all points - separate global and local conductance - start far from trouble

Dyer-Frieze-Kannan 1989 multi-Phase Monte-Carlo (product estimator) Markov chain sampling isoperimetric inequalities Polynomial time! Cost of volume computation (number of oracle calls) Amortized cost of sample point Cost of sample point

Dyer-Frieze-Kannan 1989 Lovász-Simonovits 1990 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 previous phase as starting points

Isoperimetric Inequality

infinitesimally narrow truncated cone Localization Lemma

Dyer-Frieze-Kannan 1989 Lovász-Simonovits 1990 Applegate-Kannan 1990 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. Modified Markov chain M’: - generate step i  j - 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.

Dyer-Frieze-Kannan 1989 Lovász-Simonovits 1990 Applegate-Kannan 1990 Lovász 1991 ball walk

Dyer-Frieze-Kannan 1989 Lovász-Simonovits 1990 Applegate-Kannan 1990 Lovász 1991 Dyer-Frieze 1991 independence of errors

Dyer-Frieze-Kannan 1989 Lovász-Simonovits 1990 Applegate-Kannan 1990 Lovász 1991 Dyer-Frieze 1991 Lovász-Simonovits 1992,93 integration of smoother functions randomized preprocessing generalization of multi-phase Monte-Carlo to simulated annealing scheme

Want: Random walk on K “Simulated annealing” for integration

X : sample from  k,

Dyer-Frieze-Kannan 1989 Lovász-Simonovits 1990 Applegate-Kannan 1990 Lovász 1991 Dyer-Frieze 1991 Lovász-Simonovits 1992,93 Kannan-Lovász-Simonovits 1997 isotropic positition local and global obstructions (speedy walk) bootstrapping preprocessing and sampling

Dyer-Frieze-Kannan 1989 Lovász-Simonovits 1990 Applegate-Kannan 1990 Lovász 1991 Dyer-Frieze 1991 Lovász-Simonovits 1992,93 Kannan-Lovász-Simonovits 1997 Lovász 1999 analysis of the hit-and-run algorithm

Smith 1984 Hit-and-run walk

Dyer-Frieze-Kannan 1989 Lovász-Simonovits 1990 Applegate-Kannan 1990 Lovász 1991 Dyer-Frieze 1991 Lovász-Simonovits 1992,93 Kannan-Lovász-Simonovits 1997 Lovász 1999 Kannan-Lovász 1999 average conductance, log-Cheeger inequality

Dyer-Frieze-Kannan 1989 Lovász-Simonovits 1990 Applegate-Kannan 1990 Lovász 1991 Dyer-Frieze 1991 Lovász-Simonovits 1992,93 Kannan-Lovász-Simonovits 1997 Lovász 1999 Kannan-Lovász 1999 Lovász-Vempala 2002 sampling from general logconcave distributions, ball walk and hit-and-run walk

Dyer-Frieze-Kannan 1989 Lovász-Simonovits 1990 Applegate-Kannan 1990 Lovász 1991 Dyer-Frieze 1991 Lovász-Simonovits 1992,93 Kannan-Lovász-Simonovits 1997 Lovász 1999 Kannan-Lovász 1999 Lovász-Vempala 2002 A.Kalai-Lovász-Vempala 2003 Simulated annealing

The pencil construction 0 2R2R

Two possibilities for further improvement: - The Slicing Conjecture - Reflecting walk

The Slicing Conjecture Smallest bisecting surface F H Smallest bisecting hyperplane ? ?

Reflecting random walk in K steplength h large How fast does this mix? Stationary distribution: uniform Chain is time-reversible e.g. exponentially distributed with expectation  diam(K).