Benjamin Doerr

Max-Planck-Institut für Informatik Saarbrücken

Component-by-Component Construction of Low-Discrepancy Point Sets

joint work with Michael Gnewuch (Kiel), Peter Kritzer (Salzburg), and Friedrich Pillichshammer (Linz)

Benjamin Doerr

Reminder: Star Discrepancy Definition:

– s ∈ NN “dimension” (Austrian notation)

– P = {p0, p1, ..., pN-1} multi-set of N points in [0,1)s

– Discrepancy function: For x ∈ [0,1]s, Δ(x,P) := λ([0,x)) – #{i : pi ∈ [0,x)} / N “(how many points should be in [0,x) – how many

actually are there) normalized by N”

– Star discrepancy: D*(P) := sup{ |Δ(x,P)| : x ∈ [0,1]s}

Measure of how evenly P is distributed in [0,1)s

Benjamin Doerr

Reminder: Star Discrepancy Application: Numerical Integration

– Given f : [0,1]s → R

– Compute/estimate ∫[0,1]s f(x) dx !

Hope: ∫[0,1]s f(x) dx ≈ (1/N) ∑i f(pi)

Koksma-Hlawka inequality:

| ∫[0,1]s f(x) dx - (1/N) ∑i f(pi) | ≤ V(f)

D*(P)– V(f): Variation in the sense of Hardy and Krause

Low star discrepancy = good integration

Benjamin Doerr

Reminder: Star Discrepancy How good?

Benjamin Doerr

Reminder: Star Discrepancy Very good! There are N-point sets P with

D*(P) ≤ Cs (log N)s-1 / N

“More points = drastically reduced integration error” Really?

Note: All constants ‘C’ may be different. They never depend on N. If they depend on s, I call them ‘Cs’.

Benjamin Doerr

Reminder: Star Discrepancy Very good! – There are N-point sets P with

D*(P) ≤ Cs (log N)s-1 / N

“More points = drastically reduced integration error” Really? No!

Benjamin Doerr

Problem: Only good for many points!

– Increasing for N ≤ e10 (more points = worse integration?)– ≥ 1 (trivial bound), if N ≤ 1010

– ≥ D*(random point set), if N ≤ 102∙10

Need for “small” low-discrepancy point sets!

Benjamin Doerr

Motivation, Outline

Previous slides: – O((log N)s-1/N) bounds only good for many points

many: at least exponential in dimension.

– Otherwise: Random points have better guarantees.

Plan for this talk: – Be inspired by random points– ...and use this to construct better point sets

Note: Almost all ugly details omitted in this talk!– For many technicalities, the sharpest bounds and

more results see the full paper (MCMAppl, to appear).

Benjamin Doerr

Previous Work (1)

Heinrich, Novak, Wasilkowski, Woźniakowski (Acta Arith., 2002): – There are point sets with D*(P) ≤ C (s/N)1/2

randomized construction Talagrand inequality

– Good bounds for N polynomial in s– Existential result only, implicit constants not

known

Benjamin Doerr

Previous Work (2)

We build on previous results by D., Gnewuch, Srivastav (JoC ‘05, MCQMC ’06): – D*(P) ≤ C (s/N)1/2 (log N)1/2, C small– via randomized rounding:

discrepancy guarantee holds with high probability

– derandomization: deterministic construction of P in run-time (CN)s+2

computes the exact star discrepancy on the way– wait for Michael’s talk (next talk) to see how difficult

computing the star discrepancy can be...

Benjamin Doerr

Rounding Approach

Task: Put N = 16 points in the unit cube nicely

Benjamin Doerr

Rounding Approach

Task: Put N = 16 points in the unit cube nicely Partition the cube into small rectangles (“boxes”)

Benjamin Doerr

Rounding Approach

Task: Put N = 16 points in the unit cube nicely Partition the cube into small rectangles (“boxes”) Compute the ‘fair’ number xB of points for each box B: xB = N

vol(B)

3.0625

1.96875

1.96875

1.2656..

1.09375

0.875

Benjamin Doerr

Rounding Approach

Task: Put N = 16 points in the unit cube nicely Partition the cube into small rectangles (“boxes”) Compute the ‘fair’ number xB of points for each box B: xB = N

vol(B) Round these numbers to integers yB ...

3.0625

1.96875

1.96875

1.2656..

1.09375

0.875

3 2

1

1

1

2

0.7031..

0.5625

1

0 1

1

1

1 1

0

0

0

0.31..

Benjamin Doerr

Rounding Approach

Task: Put N = 16 points in the unit cube nicely Partition the cube into small rectangles (“boxes”) Compute the ‘fair’ number xB of points for each box B: xB = N vol(B)

Round these numbers to integers yB such that for all aligned corners C,yC := ∑B⊆CyB is close to xC := ∑B⊆CxB.

3.0625

1.96875

1.96875

1.2656..

1.09375

0.875

3 2

1

1

1

2

0.7031..

0.5625

1

0 1

1

1

1 1

0

0

0

0.31..

xC=12.25 yC=12

Benjamin Doerr

Rounding Approach

Task: Put N = 16 points in the unit cube nicely Partition the cube into small rectangles (“boxes”) Compute the ‘fair’ number xB of points for each box B: xB = N vol(B)

Round these numbers to integers yB such that for all aligned corners C,yC := ∑B⊆CyB is close to xC := ∑B⊆CxB. Then put yB points in B arbitrarily.

3.0625

1.96875

1.96875

1.2656..

1.09375

0.875

3 2

1

1

1

2

0.7031..

0.5625

1

0 1

1

1

1 1

0

0

0

0.31..

Benjamin Doerr

Classical Rounding Theory Let x1, ..., xn be numbers, N:=||x||1 and I1, ..., Im ⊆ {1, ...,n}.

Randomized Rounding: – If xi is an integer, yi := xi

– If not, then yi := ⌈xi⌉ with probability equal to the fractional part of xi and yi := ⌊xi⌋ otherwise

Theorem: With probability 1-ε, we have for all 1 ≤ k ≤ m

(*) | ∑i ∈ Ik yi - ∑i ∈ Ik xi | ≤ (0.5 N log(2m/ε))1/2

Derandomization: A rounding (yi) satisfying (*) with ε=1 can be computed deterministically in time O(mn).

Experiment: Derandomization yields smaller rounding errors.

Benjamin Doerr

Rounding Approach (continued)

Task: Put N = 16 points in the unit cube nicely Partition the cube into small rectangles (“boxes”) Compute the ‘fair’ number xB of points for each box B: xB = N vol(B)

Round these numbers to integers yB such that for all aligned corners C,| ∑B⊆CyB - ∑B⊆CxB | ≤ (0.5 N log(2 #boxes))1/2 ...

3.0625

1.96875

1.96875

1.2656..

1.09375

0.875

3 2

1

1

1

2

0.7031..

0.5625

1

0 1

1

1

1 1

0

0

0

0.31..

Benjamin Doerr

New Result: The same can be done in a component-by-component

way: – Compoment-by-compoment: Given an (s-1)-dimensional low-

discrepancy point set, add an sth component to each point.– Adjust the randomized-rounding approach accordingly.

Advantage: – Fewer variables to be rounded in each iteration.– Total run-time (over all s iterations) roughly N(s+3)/2 instead of Ns+2.

Surprise: Discrepancy increases only by a factor of s.– Roughly C s3/2 N-1/2 log(N)1/2 instead of C s1/2 N-1/2 log(N)1/2

That’s OK, because we can now compute N2 points in roughly the same time as needed for N points before.

Benjamin Doerr

Summary and Conclusion Result: Component-by-Component derandomized

construction is much faster and yields only slightly higher discrepancies compared to “all at once”.

Outlook: Could also be useful if components are of different importance. E.g., do the expensive derandomization only for few components.

Open problem: Come up with something really efficient.... (instead of NCs).

Merci/Thanks!