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Discrepancy and SDPs

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Discrepancy and SDPs

Nikhil Bansal (TU Eindhoven)

Discrepancy: definitions and applications

Basic results: upper/lower bounds

Partial Coloring method (non-constructive)

SDPs: basic method

Algorithmic Spencer’s Result

Lovett-Meka result

Lower bounds via SDP duality (Matousek)

Classic: Geometric Discrepancy by J. Matousek

Papers:

Bansal. Constructive algorithms for discrepancy minimization, FOCS 2010

Matousek. The determinant lower bound is almost tight

Lovett, Meka. Discrepancy minimization by walking on the edges

Survey with fewer technical details:

Bansal. …

Study of gaps in approximating the continuous by the discrete.

Original motivation: Numerical Integration/ Sampling

Problem: How well can you approximate a region by discrete points

Discrepancy:

Max over intervals I

|(# points in I) – (length of I)|

Study of gaps in approximating the continuous by the discrete.

Problem: How uniformly can you distribute points in a grid.

“Uniform” : For every axis-parallel rectangle R

| (# points in R) - (Area of R) | should be low.

Discrepancy:

Max over rectangles R

|(# points in R) – (Area of R)|

n1/2

n1/2

Problem: How uniformly can you distribute points in a grid.

“Uniform” : For every axis-parallel rectangle R

| (# points in R) - (Area of R) | should be low.

n= 64

points

Van der Corput Set

Uniform

Random

n1/2 discrepancy

n1/2 (loglog n)1/2

O(log n) discrepancy!

With N random samples: Error \prop 1/\sqrt{n}

Quasi-Monte Carlo Methods: \prop Disc/n

Can discrepancy be O(1) for 2d grid?

No. \Omega(log n) [Schmidt …]

d-dimensions: O(log^{d-1} n) [Halton-Hammersely ]

\Omega(log^{(d-1)/2} n) [Roth ]

\Omega(log^{(d-1)/2 + \eta} n [Bilyk,Lacey,Vagharshakyan’08]

Input: n points placed arbitrarily in a grid.

Color them red/blue such that each rectangle is colored as evenly as possible

Discrepancy: max over rect. R ( | # red in R - # blue in R | )

Continuous: Color each element

1/2 red and 1/2 blue (0 discrepancy)

Discrete:

Random has about O(n1/2 log1/2 n)

Can achieve O(log2.5 n)

S3

S4

S1

S2

Universe: U= [1,…,n]

Subsets: S1,S2,…,Sm

Color elements red/blue so each

set is colored as evenly as possible.

Find : [n] ! {-1,+1} to

Minimize |(S)|1 = maxS | i 2 S(i) |

If A is m \times n incidence matrix.

Disc(A) = min_{x \in {-1,1}^n} |Ax|_\infty

CS: Computational Geometry, Comb. Optimization, Monte-Carlo simulation, Machine learning, Complexity, Pseudo-Randomness, …

Math: Dynamical Systems, Combinatorics, Mathematical Finance,

Number Theory, Ramsey Theory, Algebra, Measure Theory, …

Lovasz-Spencer-Vesztermgombi’86

Given any matrix A, and x \in R^n

can round x to \tilde{x} \in Z^n s.t.

|Ax – A\tilde{x}|_\infty < Herdisc(A)

Proof: Round the bits one by one.

Can we find it efficiently?

Nothing known until recently.

Thm [B’10]. Can efficiently round so that

Error \leq O(\sqrt{log m log n}) Herdisc(A)

Bin Packing

Refined further by Rothvoss(Entropy rounding method)

N points in a 2-d region.

Weights update over time.

Query: Given an axis-parallel rectangle R, determine the total weight on points in R.

Preprocess:

- Low query time
- Low update time (upon weight change)

Line:

Query = O(n) Update = 1

Query = 1 Update = O(n^2)

Query = 2 Update = O(n)

Query = O(log n) Update = O(log n)

Recursively can get for 2-d.

Query

Circles arbitrary rectangles aligned triangle

Turns out t_q t_u \geq n^{1/2}/log^2 n ?

Larsen-Green: t_q t_u \geq disc(S)^n/log^2 n

A good data structure implies

D = A P

A = row sparse P = Column sparse

(low query time) (low update time)

Random: Color each element i independently as

x(i) = +1 or -1 with probability ½ each.

Thm: Discrepancy = O (n log n)1/2

Pf: For each set, expect O(n1/2) discrepancy

Standard tail bounds: Pr[ | i 2 S x(i) | ¸c n1/2 ] ¼e-c2

Union bound + Choose c ¼ (log n)1/2

Analysis tight: Random actually incurs ((n log n)1/2).

[Spencer 85]: (Six standard deviations suffice)

Always exists coloring with discrepancy ·6n1/2

(In general for arbitrary m, discrepancy = O(n1/2log(m/n)1/2)

Tight: For m=n, cannot beat 0.5 n1/2 (Hadamard Matrix, “orthogonal” sets)

Inherently non-constructive proof

(pigeonhole principle on exponentially large universe)

Challenge: Can we find it algorithmically ?

Certain algorithms do not work [Spencer]

Conjecture[Alon-Spencer]: May not be possible.

S3

S4

S1

S2

U = [1,…,n] Sets: S1,S2,…,Sm

Suppose each element lies in at most t sets (t << n).

[Beck Fiala’ 81]: Discrepancy 2t -1.

(elegant linear algebraic argument, algorithmic result)

Beck Fiala Conjecture: O(t1/2) discrepancy possible

Other results: O( t1/2 log t log n ) [Beck]

O( t1/2 log n ) [Srinivasan]

O( t1/2 log1/2 n ) [Banaszczyk]

Non-constructive

1 2 … n

1’ 2’ … n’

S1

S2

…

S’1

S’2

…

Question: If a set system has low discrepancy (say << n1/2)

Can we find a good discrepancy coloring ?

[Charikar, Newman, Nikolov 11]:

Even 0 vs. O (n1/2) is NP-Hard

(Matousek): What if system has low Hereditary discrepancy?

herdisc (U,S) = maxU’ ½ U disc (U’, S|U’)

Robust measure of discrepancy (often same as discrepancy)

Widely used: TU set systems, Geomety, …

Thm 1: Can get Spencer’s bound constructively.

That is, O(n1/2) discrepancy for m=n sets.

Thm 2: If each element lies in at most t sets, get bound of O(t1/2 log n) constructively (Srinivasan’s bound)

Thm 3: For any set system, can find

Discrepancy ·O(log (mn))Hereditary discrepancy.

Other Problems: Constructive bounds (matching current best)

k-permutation problem [Spencer, Srinivasan,Tetali]

Geometric problems , …

Not clear how to use.

Linear Program is useless. Can color each element ½ red and ½ blue. Discrepancy of each set = 0!

SDPs(LP on vi¢ vj, cannot control dimension of v’s)

| i 2 S vi |2· n 8 S

|vi|2 = 1

Intended solution vi = (+1,0,…,0) or (-1,0,…,0).

Trivially feasible: vi = ei (all vi’s orthogonal)

Yet, SDPs will be a major tool.

SDP very helpful if “tighter” bounds needed for some sets.

|i 2 S vi |2· 2 n

| i 2 S’ vi|2· n/log n

|vi|2· 1

Not apriori clear why one can do this.

Entropy Method.

Algorithm will construct coloring over time and

use several SDPs in the process.

Tighter bound for S’

Introduction

The Method

Low Hereditary discrepancy -> Good coloring

Additional Ideas

Spencer’s O(n1/2) bound

-n

n

Can be improved to O(\sqrt{n})/2^n

If you pick a random {-1,1} coloring s

w.p. say >= ½ |a \cdot s| \leq c \sqrt{n}

2^{n-1} colorings s, with |a\cdot s| \leq c \sqrt{n}

Easy: 1/poly(n) (How?)

Answer: Pick any poly(n) colorings.

[Karmarkar-Karp’81]: \approx 1/n^log n

Huge gap: Major open question

Remark: {-1,+1} not enough. Really need color 0 also.

E.g. a_1 = 1, a_2=…=a_n = 1/(2n)

There is a {-1,0,1} coloring with at least

n/2 {-1,1}’s s.t. \sum_i a_i s_i \leq n/2^{n/5}

Make buckets of size 2n/2^{n/5}

At least 2^{4n/5} sums fall in same bucket

Claim: Some two s’ and s’’ in same bucket and differ in at least n/2 coordinates

Again consider s = (s’-s’’)/2

Claim: Any set of 2^{4n/5} vertices of the

boolean cube has

[Kleitman’66] Isoperimetry for cube.

Hamming ball B(v,r) has the smallest diameter for a given number of vertices.

|B(v,n/4)| < 2^{4n/5}

start

finish

Each dimension: An Element

Each vertex: A Coloring

Cube: {-1,+1}n

Algorithm: “Sticky” random walk

Each step generated by rounding a suitable SDP Move in various dimensions correlated, e.g. t1 + t2¼ 0

Analysis: Few steps to reach a vertex (walk has high variance)

Disc(Si) does a random walk (with low variance)

Hereditary disc. ) the following SDP is feasible

SDP:

Low discrepancy: |i 2 Sj vi |2 ·2

|vi|2 = 1

Obtain vi2 Rn

Rounding:

Pick random Gaussian g = (g1,g2,…,gn)

each coordinate gi is iid N(0,1)

For each i, consider i = g¢ vi

Lemma: If g 2 Rn is random Gaussian. For any v 2 Rn,

g ¢ v is distributed as N(0, |v|2)

Pf: N(0,a2) + N(0,b2) = N(0,a2+b2) g¢ v = i v(i) gi» N(0, i v(i)2)

Recall: i = g ¢ vi

- Each i» N(0,1)
- For each set S,
- i 2 Si = g ¢ (i2 S vi) » N(0, ·2)
- (std deviation ·)

SDP:

|vi|2 = 1

|i2S vi|2·2

’s mimics a low discrepancy coloring (but is not {-1,+1})

+1

time

-1

Construct coloring iteratively.

Initially: Start with coloring x0 = (0,0,0, …,0) at t = 0.

At Time t: Update coloring as xt = xt-1 + (t1,…,tn)

( tiny: 1/n suffices)

xt(i) = (1i + 2i + … + ti)

Color of element i: Does random walk

over time with step size ¼ N(0,1)

x(i)

Fixed if reaches -1 or +1.

Set S: xt(S) = i 2 S xt(i) does a random walk w/ step N(0,·2)

Consider time T = O(1/2)

Claim 1: With prob. ½, at least n/2 elements reach -1 or +1.

Pf: Each element doing random walk with size ¼.

Recall: Random walk with step 1, is ¼ O(t1/2) away in t steps.

A Trouble: Various element updates are correlated

Consider basic walk x(t+1) = x(t) 1 with prob ½

Define Energy (t) = x(t)2

E[(t+1)] = ½ (x(t)+1)2 + ½ (x(t)-1)2 = x(t)2 + 1 = (t)+1

Expected energy = n at t= n.

Claim 2: Each set has O() discrepancy in expectation.

Pf: For each S, xt(S) doing random walk with step size ¼

Consider time T = O(1/2)

Claim 1: With prob. ½, at least n/2 variables reach -1 or +1.

) Everything colored in O(log n) rounds.

Claim 2: Each set has O() discrepancy in expectation per round.

) Expected discrepancy of a set at end = O( log n)

Thm: Obtain a coloring with discrepancy O( log (mn))

Pf: By Chernoff, Prob. that disc(S) >= 2 Expectation + O( log m)

= O( log (mn))

is tiny (poly(1/m)).

At each step of walk, formulate SDP on unfixed variables.

Use some (existential) property to argue SDP is feasible

Rounding SDP solution -> Step of walk

Properties of walk:

High Variance -> Quick convergence

Low variance for discrepancy on sets -> Low discrepancy

Spencer’s six std deviations result:

Goal: Obtain O(n1/2) discrepancy for any set system on m = O(n) sets.

Random coloring has n1/2(log n)1/2 discrepancy

Previous approach seems useless:

Expected discrepancy for a set O(n1/2),

but some random walks will deviate by up to (log n)1/2 factor

Need an additional idea to prevent this.

Partial Coloring Lemma: For any system with m sets, there exists a coloring on ¸ n/2 elements with discrepancy O(n1/2 log1/2 (2m/n))

[For m=n, disc = O(n1/2)]

Algorithm for total coloring:

Repeatedly apply partial coloring lemma

Total discrepancy

O( n1/2 log1/2 2 ) [Phase 1]

+ O( (n/2)1/2 log1/2 4 ) [Phase 2]

+ O((n/4)1/2 log1/2 8 ) [Phase 3]

+ … = O(n1/2)

X1 = ( 1,-1, 1 , …,1,-1,-1)

X2 = (-1,-1,-1, …,1, 1, 1)

X = ( 1, 0, 1 , …,0,-1,-1)

Beautiful Counting argument (entropy method + pigeonhole)

Idea: Too many colorings (2n), but few “discrepancy profiles”

Key Lemma: There exist k=24n/5 colorings X1,…,Xk such that

every two Xi, Xj are “similar” for every set S1,…,Sn.

Some X1,X2 differ on ¸ n/2 positions

Consider X = (X1 – X2)/2

Pf: X(S) = (X1(S) – X2(S))/2 2 [-10 n1/2 , 10 n1/2]

There exists a partial coloring with non-uniform discrepancy bound S for set S

Even if S = ( n1/2) in some average sense

Suppose there exists partial coloring X:

1. On ¸ n/2 elements

2. Each set S has |X(S)| ·S

SDP:

Low discrepancy: |i 2 Sj vi |2·S2

Many colors:i |vi|2¸ n/2

|vi|2· 1

Pick random Gaussian g = (g1,g2,…,gn)

each coordinate gi is iid N(0,1)

For each i, consider i = g ¢ vi

Obtain vi2 Rn

Initially write SDP with S = c n1/2

Each set S does random walk and expects to reach

discrepancy of O(DS) = O(n1/2)

Some sets will become problematic.

Reduce their S on the fly.

Not many problematic sets, and entropy penalty low.

Danger 3 …

Danger 1

Danger 2

…

35n1/2

0

30n1/2

20n1/2

Construct coloring over time by solving sequence of SDPs (guided by existence results)

Works quite generally

Can be derandomized[Bansal-Spencer]

(use entropy method itself for derandomizing + usual tech.)

E.g. Deterministic six standard deviations can be viewed as a way to derandomize something stronger than Chernoff bounds.

- How to generate i with required properties.
- How to update S over time.
Show n1/2 (log log log n)1/2 bound.

- Often algorithms rely on continuous relaxations.
- Linear Program is useless. Can color each element ½ red and ½ blue.

- Improved results of Spencer, Beck, Srinivasan, … based on clever counting (entropy method).
- Pigeonhole Principle on exponentially large systems (seems inherently non-constructive)

Suppose we have discrepancy bound S for set S.

Consider 2n possible colorings

Signature of a coloring X: (b(S1), b(S2),…, b(Sm))

Want partial coloring with signature (0,0,0,…,0)

Energy increases at each step:

E(t) = \sum_i x_i(t)^2

Initially energy =0, can be at most n.

Expected value of E(t) = E(t-1) + \sum_i \gamma_i(t)^2

Markov’s inequality.

- How to generate the \eta_i
- How to update \Delta_S over time

X1 = (1,-1, 1 , …, 1,-1,-1)

X2 = (-1,-1,-1, …, 1,1, 1)

If exist two colorings X1,X2

1. Same signature (b1,b2,…,bm)

2. Differ in at least n/2 positions.

Consider X = (X1 –X2)/2

- -1 or 1 on at least n/2 positions, i.e. partial coloring
- Has signature (0,0,0,…,0)
X(S) = (X1(S) – X2(S)) / 2, so |X(S)| ·S for all S.

Can show that there are 24n/5 colorings with same signature.

So, some two will differ on > n/2 positions. (Pigeon Hole)

Partial Coloring Lemma: For any system with m sets,

there exists a coloring on ¸ n/2 elements with discrepancy O(n1/2 log1/2 (2m/n))

[For m=n, disc = O(n1/2)]

Algorithm for total coloring:

Repeatedly apply partial coloring lemma

Total discrepancy

O( n1/2 log1/2 2 ) [Phase 1]

+ O( (n/2)1/2 log1/2 4 ) [Phase 2]

+ O((n/4)1/2 log1/2 8 ) [Phase 3]

+ … = O(n1/2)

Let us prove the lemma for m = n

Ent(b1) · 1/5

-30 n1/2

-10 n1/2

10 n1/2

30 n1/2

-2

-1

0

1

2

Pf: Associate with coloring X, signature = (b1,b2,…,bn)

(bi = bucket in which X(Si) lies )

Wish to show: There exist 24n/5 colorings with same signature

Choose X randomly: Induces distribution on signatures.

Entropy () · n/5 implies some signature has prob. ¸ 2-n/5.

Entropy ( ) ·i Entropy( bi) [Subadditivity of Entropy]

bi = 0 w.p. ¼ 1- 2 e-50,

= 1 w.p. ¼ e-50

= 2 w.p. ¼ e-450

….

For each set S, consider the “bucketing”

-2

-1

2

0

1

S

-3S

-S

3S

5S

Bucket of n1/2/100

has penalty ¼ ln(100)

Partial coloring with non-uniform discrepancy S for set S

Suffices to have s Ent (bs) · n/5

Or, if S = s n1/2 , then s g(s) · n/5

g() ¼ e-2/2 > 1

¼ln(1/) < 1

Partial Coloring:S¼ 10 n1/2 gives low entropy

) 24n/5 colorings exist with same signature.

) some X1,X2 with large hamming distance.

(X1 – X2) /2 gives the desired partial coloring.

Trouble: 24n/5/2n is an exponentially small fraction.

Only if we could find the partial coloring efficiently…