The Lovasz-Vempala algorithm for computing the volume of a convex body in O*(n^4) - Theory and Implementation

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1 The Lovasz-Vempala algorithm for computing the volume of a convex body in O*(n^4) - Theory and Implementation Mittagsseminar, 20. Dec Christian L. Müller, MOSAIC group, Institute of Theoretical Computer Science, ETH Zürich Dec 20, 2011:ETHZ

2 Papers Journal of Computer and System Sciences 72 (2006) Simulated annealing in convex bodies and an O (n 4 ) volume algorithm László Lovász a, Santosh Vempala b,,1 a Microsoft Research, One Microsoft Way, Redmond, WA 98052, USA b Department of Mathematics, MIT, Cambridge, MA 02139, USA Received 21 April 2004; received in revised form 27 July 2005 Available online 9 November 2005 European Journal of Operational Research 216 (2012) Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: Stochastics and Statistics Computational results of an O (n 4 ) volume algorithm q L. Lovász a, I. Deák b, a Department of Computer Science, Eötvös University, 1117 Budapest, Pázmány P. Sétány 1/c, Hungary b Corvinus University of Budapest, Hungary 2

3 Problem statement Given a convex body oracle and K R n B(0, 1) K B(0,R) accessible via a membership K Compute the volume vol(k) with relative error 3

4 Problem statement Given a convex body oracle and K R n B(0, 1) K B(0,R) accessible via a membership K Compute the volume vol(k) with relative error Not possible in polynomial time (in n, 1/ε, log R) for deterministic algorithms! 3

5 Problem statement Given a convex body oracle and K R n B(0, 1) K B(0,R) accessible via a membership K Compute the volume vol(k) with relative error Theorem 1. (See [4].) For every deterministic algorithm that uses at most n a membership queries and given a convex body K with B n K nb n outputs two numbers A,B such that A vol(k) B, thereexistsabodyk for which the ratio B/A is at least Not possible in polynomial time (in n, 1/ε, log R) for deterministic algorithms! where c is an absolute constant. ( cn a log n ) n 3 plexity of an algorith Bárány and Füredi [ e the volume to with

6 The break-through: Randomized algorithms There is a polynomial time randomized algorithm that computes the volume with high probability 1 δ with arbitrarily small relative error. Break-through paper by Dyer, Frieze, and Kannan in J.ACM, 1991describes a randomized algorithm with O*(n 23 ) complexity 4

7 The break-through: Randomized algorithms There is a polynomial time randomized algorithm that computes the volume with high probability 1 δ with arbitrarily small relative error. Break-through paper by Dyer, Frieze, and Kannan in J.ACM, 1991describes a randomized algorithm with O*(n 23 ) complexity Authors Walk Sandwiching Sampling Remark #S Type method Dyer-Frieze grid d = n n Break- -Kannan [36] +Minkowski sum through n 23 Lovász- grid Localization -Sim.[78] Lemma n 16 Applegate grid two cubes Metropolis -Kannan [8] d = n n 10 Lovász Ball [76] walk n 8 Dyer-Frieze grid two cubes [35] n 8 Lovász ball randomized, d = cn Metropolis -Sim.[80] walk R n e, Cost: n 7 Kannan-Lovász ball Isotropic, interlaced c 1 n 3 d 2 + Theorems n 10 -Sim. [60] walk Cost: n 5 c 2 Nn 2 d 2 12 n 5 Lovász [77] Hit-and Warm start Warm -Run see Remark 11 start n 5 Lovász Simulated See -Vempala [84] Annealing Appendix II n 4 4

8 The break-through: Randomized algorithms There is a polynomial time randomized algorithm that computes the volume with high probability 1 δ with arbitrarily small relative error. Break-through paper by Dyer, Frieze, and Kannan in J.ACM, 1991describes a randomized algorithm with O*(n 23 ) complexity Theorem 1.1. The volume of a convex body K, given by a membership oracle, and a parameter R such that B K RB, can be approximated to within a relative error of ε with probability 1 δ using ( n 4 O n ε 2 log9 εδ + n4 log 8 n ) δ log R = O (n 4 ) oracle calls. Lovasz and Vempala 4

9 Talk overview Part 1: Tools for volume computation - Membership oracle and boundary oracles - MCMC methods and the Hit-and-Run sampler - Classical volume computation - Simulated Annealing and log-concave functions 5

10 Talk overview Part 1: Tools for volume computation - Membership oracle and boundary oracles - MCMC methods and the Hit-and-Run sampler - Classical volume computation - Simulated Annealing and log-concave functions Part 2: The Lovasz-Vempala algorithm - Key algorithmic steps - The pencil construction - Multi-phase Monte Carlo sampling - Computational difficulties and solutions - Numerical results for hypercubes 5

11 Membership oracle Input x R n Output x 1 x 2 Yes if x K x i x n Convex body K No otherwise 6

12 Boundary oracles for line samplers Boundary oracle: Given a interior point in the body and an arbitrary direction. Return the upper and lower bounds of the body along the direction Boundary oracles are known for linear constraints, ellipsoidal constraints, and semidefinite constraints (Polyak et al. 2006) Boundary oracles can be constructed from membership oracles using bisection Ax b x T Ax 1 A 0 + x 1 A x n A n

13 MCMC sampling General objective: Draw unbiased samples from a continuous target probability distribution that is given in black-box form (and most often only up to a normalizing constant) Adaptive Stochastic methods x AuGust 25, 2011::UNIL

14 MCMC sampling General objective: Draw unbiased samples from a continuous target probability distribution that is given in black-box form (and most often only up to a normalizing constant) Black-box sampling... Adaptive Stochastic methods x AuGust 25, 2011::UNIL

15 Adaptive Stochastic methods AuGust 25, 2011::UNIL MCMC sampling Evaluate density at new sample and calculate the Metropolis criterion: α(x (g), x (g+1) )=min Draw a random number u from U(0,1) and accept 1, f(x(g+1) ) f(x (g) ) x (g+1) if u < α

16 MCMC sampling Markov Chain methods that iteratively sample from a fixed proposal distribution (often Gaussian distribution) and evaluate the target density there Random Walk Metropolis algorithm (Metropolis et al., 1953) Adaptive Stochastic methods AuGust 25, 2011::UNIL 08/24/11 5

17 MCMC sampling Markov Chain methods that iteratively sample from a fixed proposal distribution (often Gaussian distribution) and evaluate the target density there Random Walk Metropolis algorithm (Metropolis et al., 1953) x (g) x (g-1) Adaptive Stochastic methods AuGust 25, 2011::UNIL 08/24/11 5

18 MCMC and Hit-and-Run sampler K (R. L. Smith, 1984) 11

19 MCMC and Hit-and-Run sampler K (R. L. Smith, 1984) 11

20 MCMC and Hit-and-Run sampler K (R. L. Smith, 1984) 11

21 MCMC and Hit-and-Run sampler K (R. L. Smith, 1984) 11

22 MCMC and Hit-and-Run sampler K (R. L. Smith, 1984) 11

23 MCMC and Hit-and-Run sampler K (R. L. Smith, 1984) 11

24 MCMC and Hit-and-Run sampler K (R. L. Smith, 1984) 11

25 MCMC and Hit-and-Run sampler K (R. L. Smith, 1984) Lovasz shows in 1999 that Hit-and-Run mixes in O (n 2 R 2 ) for the uniform distribution. He also shows that the bound is best possible in terms of R and n. 11

26 Naive volume algorithm vol(k) = S K S vol(b) * * * * * * * * Needs exponential size S * * * * * * * * * * S 12

27 Classical volume algorithm B i =2 i/n B 0,i=1,...,m K 0 = B 0 K i = K B i K m = K 13

28 Classical volume algorithm B i =2 i/n B 0,i=1,...,m K 0 = B 0 K i = K B i K m = K vol(k) = vol(k m) vol(k m 1 ) vol(k m 1 ) vol(k m 2 )...vol(k 1) vol(k 0 ) vol(k 0) 13

29 Classical volume algorithm It can be shown that the number of phases must be and m =Ω(n) samples are needed per phase. m =Ω(n) The body must be sandwiched by the unit ball and an outer ball with radius R = O ( n). This can be achieved in O (n 4 ). Using Hit-and-Run to generate samples from the convex body results in an algorithm. O (n 5 ) This is best possible for this type of Multi-Phase Monte Carlo scheme. 14

30 Simulated annealing Simulated annealing has been introduced by Kirkpatrick, Gelatt, Vecchi in 1983 and Cerny in General-purpose heuristic optimizer for discrete and continuous problems f(x). Identical to the Metropolis algorithm but the acceptance ratio α is parametrized by a temperature Ti that is continuously lowered (annealed) during optimization. 15

31 Log-concave functions For log-concave functions the following relationship holds: f : R n R + f(θx +(1 θ)y) f(x) θ f(y) 1 θ Examples are the indicator function and the Gaussian function Hit-and-run also mixes fast for log-concave densities. 16

32 Further reading Math. Program., Ser. B 97: (2003) Digital Object Identifier (DOI) /s x Miklós Simonovits How to compute the volume in high dimension? Received: January 28, 2003 / Accepted: April 29, 2003 Published online: May 28, 2003 Springer-Verlag 2003 Abstract. In some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones. Approximating the volume of a convex n-dimensional body, given by an oracle is one of the areas where this difference can be proved. In general, if we use a deterministic algorithm to approximate the volume, it requires exponentially many oracle questions in terms of n as n.dyer,friezeandkannangave a randomized polynomial approximation algorithm for the volume of a convex body K R n, given by a membership oracle. The DKF algorithm was improved in a sequence of papers. The area is full of deep and interesting problems and results. This paper is an introduction to this field and also a survey. Combinatorial and Computational Geometry MSRI Publications Volume 52, 2005 Geometric Random Walks: A Survey SANTOSH VEMPALA Abstract. The developing theory of geometric random walks is outlined here. Three aspects general methods for estimating convergence (the mixing rate), isoperimetric inequalities in R n and their intimate connection to random walks, and algorithms for fundamental problems (volume computation and convex optimization) that are based on sampling by random walks are discussed. 17

33 Part 1: Tools for volume computation - Membership oracle and boundary oracles - MCMC methods and the Hit-and-Run sampler - Classical volume computation - Simulated Annealing and log-concave functions Part 2: The Lovasz-Vempala algorithm - Key algorithmic steps - The pencil construction - Multi-phase Monte Carlo integration - Computational difficulties and solutions - Numerical results for hypercubes 18

34 Key algorithmic steps Preprocessing: The body must be in isotropic position and it is well-rounded (Rudelson, 1998). We do the so-called pencil construction in n+1 dimension. We define a sequence of log-concave densities over the pencil inspired by inverse simulated annealing. We generate random points from the densities using Hit-and- Run sampling. We approximate the volume by a product of ratios of integrals of consecutive densities. 19

35 The pencil construction C = {x x R n+1,x 0 0, K = C [(0, 2R) K] n i=1 x 2 i 2x 2 0} 20

36 The pencil construction C = {x x R n+1,x 0 0, K = C [(0, 2R) K] n i=1 x 2 i 2x 2 0} 0 2R 20

37 A parametrized integral over the pencil Log-concave function f i (x) =exp( a i x 0 ), i =1,...,m x =(x 0,...,x n ) K 21

38 A parametrized integral over the pencil Log-concave function f i (x) =exp( a i x 0 ), i =1,...,m x =(x 0,...,x n ) K a i 1/T i a 0 a 1... a m (inverse annealing) 21

39 A parametrized integral over the pencil Log-concave function f i (x) =exp( a i x 0 ), i =1,...,m x =(x 0,...,x n ) K a i 1/T i a 0 a 1... a m (inverse annealing) a 0 6n a i = a 0 (1 1 n ) i,i=1,...,m 21

40 A parametrized integral over the pencil Log-concave function f i (x) =exp( a i x 0 ), i =1,...,m x =(x 0,...,x n ) K a i 1/T i a 0 a 1... a m (inverse annealing) a 0 6n a i = a 0 (1 1 n ) i,i=1,...,m Parametrized integral 21

41 A parametrized integral over the pencil Log-concave function f i (x) =exp( a i x 0 ), i =1,...,m x =(x 0,...,x n ) K a i 1/T i a 0 a 1... a m (inverse annealing) a 0 6n a i = a 0 (1 1 n ) i,i=1,...,m Parametrized integral Z(a) = exp ( ax 0 )dx K 21

42 A parametrized integral over the pencil Log-concave function f i (x) =exp( a i x 0 ), i =1,...,m x =(x 0,...,x n ) K a i 1/T i a 0 a 1... a m (inverse annealing) a 0 6n a i = a 0 (1 1 n ) i,i=1,...,m Parametrized integral Z(a) = exp ( ax 0 )dx K Z(a 0 )= K f 0 (x)dx C 21 Z(a m )= exp ( a 0 x 0 )dx = n!π 0 a (n+1) 0 K f m (x)dx

43 A parametrized integral over the pencil f i (x) =exp( a i x 0 ), i =1,...,m x =(x 0,...,x n ) K 0 1 2R 22

44 A parametrized integral over the pencil f i (x) =exp( a i x 0 ), i =1,...,m x =(x 0,...,x n ) K 0 f 0 1 2R 22

45 A parametrized integral over the pencil f i (x) =exp( a i x 0 ), i =1,...,m x =(x 0,...,x n ) K 0 f 0 1 2R 22

46 A parametrized integral over the pencil f i (x) =exp( a i x 0 ), i =1,...,m x =(x 0,...,x n ) K 0 f 0 1 f m 2R 22

47 Computing the pencil volume in each phase Z(a) = K exp ( ax 0 )dx Product of ratios Z(a m )=Z(a 0 ) m 1 i=0 Z(a i+1 ) Z(a i ) 23

48 Computing the pencil volume in each phase Z(a) = K exp ( ax 0 )dx Product of ratios Z(a m )=Z(a 0 ) m 1 i=0 Z(a i+1 ) Z(a i ) Monte Carlo approximation Wi 23

49 Computing the pencil volume in each phase Z(a) = K exp ( ax 0 )dx Product of ratios Z(a m )=Z(a 0 ) m 1 i=0 Z(a i+1 ) Z(a i ) Monte Carlo approximation Wi W i = 1 k k j=1 exp (a i a i+1 )x (j) 0 k: Number of threads 23

50 Computing the pencil volume in each phase Z(a) = K exp ( ax 0 )dx Product of ratios Z(a m )=Z(a 0 ) m 1 i=0 Z(a i+1 ) Z(a i ) Monte Carlo approximation Wi W i = 1 k k j=1 exp (a i a i+1 )x (j) 0 Final acceptance/rejection step to get the volume V from V k: Number of threads 23

51 The algorithm in a nutshell ± 2 Volume algorithm: V1. Set m = 2 n ln n 512 ( ) i ε, k = ε n ln n 2 ε, δ = ε 2 n 10 and a i = 2n 1 1 n for i = 1,...,m. V2. For i = 1,...,m, do the following. Run the sampler k times for convex body T i K, with vector (a i /γ i, 0,...,0) (i.e., exponential function e a ix 0 /γ i ), error parameter δ, and (for i > 0) starting points T i Xi 1 1,...,T ixi 1 k 1. Apply Ti to the resulting points to get points Xi 1,...,Xk i. Using these points, compute W i = 1 k k j=1 e (a i a i+1 )(X j i ) 0. V3. Return Z = n!π n (2n) (n+1) W 1...W m as the estimate of the volume of K. 24

52 The algorithm in a nutshell ± 2 Volume algorithm: V1. Set m = 2 6n n ln n 512 ( ε, k = ε n ln n 2 ε, δ = ε 2 n 10 and a i = 2n 1 1 for i = 1,...,m. V2. For i = 1,...,m, do the following. n ) i Run the sampler k times for convex body T i K, with vector (a i /γ i, 0,...,0) (i.e., exponential function e a ix 0 /γ i ), error parameter δ, and (for i > 0) starting points T i Xi 1 1,...,T ixi 1 k 1. Apply Ti to the resulting points to get points Xi 1,...,Xk i. Using these points, compute W i = 1 k k j=1 e (a i a i+1 )(X j i ) 0. V3. Return 6n Z = n!π n (2n) (n+1) W 1...W m as the estimate of the volume of K. 24

53 Computational issues The computational paper gives all details for implementation. Code in FORTRAN is also available from the second author The algorithm is numerically is extremely unstable. Large size differences between the first and the last volume. The ratios are initially large and then abruptly approach 1. Pre-factor in dimension dependence of the sampling size not clear for Hit-and-Run (when does it reach stationarity). Sample size in each phase is constant but should maybe be variable 25

54 Computational enhancements Special treatment of the first integral using stratified sampling Extra sampling between phases to reach stationarity faster Double-point estimator as variance reduction technique in integral computation 2n+1 ortho-normalized points around a random point is used 26

55 Numerical results on hypercubes in n=2,9 Table 1 Estimated values of the ratio R i, its standard deviation, and partial errors d i = Z 0 ZD(W i )/W i, for examples C 2, C 9. The sample size is s in each thread and in each phase, the number of threads is k thr = 25. i Dim. n + 1 = 3 Dim. n + 1 = 10 Sample size s = 100 Sample size s = 1000 Z 0 = , Z = 2.27E +3 Z 0 = 0.56E 11, Z 4.0E + 14 W i D(W i ) Z 0 ZD(W i )/W i W i D(W i ) Z 0 ZD(W i )/W i E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 1 27

56 Numerical results on hypercubes in n=2,9 Table 5 Results of three cube-pencils P 3, P 5, P 9. C 2 C 5 C 8 n k thr V e V E E E+2 r V Exact V e V 0.66E E E+2 Time (sec)

57 Concluding remarks Theoretically, it might be possible to further bound the mixing time of Hit-and-Run There is room for improvement in almost all aspects of the implementation. Sequential Monte Carlo as alternative paradigm for integral approximation Parallelization is possible (independent threads). I try to implement the algorithm next year. Any help, suggestions are welcome... 29

58 Further reading 30

59 Thanks and Merry christmas and happy holidays 31

60 Figures used in the presentation I took the pencil construction image and the sandwiching balls from L. Lovasz 2004 slides on volume computation ( 32