Discrepancy Minimization by Walking

on the EdgesRaghu Meka (IAS & DIMACS)

Shachar Lovett (IAS)

1 2 3 4 5

Discrepancy• Subsets • Color with or - to minimize imbalance

1 * 1 1 *

* 1 1 * 1

1 1 1 1 1

* * * 1 1

1 * 1 * 1

1 2 3 4 51 * 1 1 *

* 1 1 * 1

1 1 1 1 1

* * * 1 1

1 * 1 * 1

3

1

1

0

1

Discrepancy Examples• Fundamental combinatorial

conceptArithmetic Progressions

Roth 64: Matousek, Spencer 96:

Discrepancy Examples• Fundamental combinatorial

conceptHalfspaces

Alexander 90: Matousek 95:

Discrepancy Examples• Fundamental combinatorial

conceptAxis-aligned boxes

Beck 81: Srinivasan 97:

Why Discrepancy?

Complexity theory

Communication Complexity

Computational Geometry

PseudorandomnessMany more!

Spencer’s Six Sigma Theorem

• Central result in discrepancy theory.

• Beats random:• Tight: Hadamard.

Spencer 85: System with n sets has discrepancy at most .

“Six standard deviations suffice”

Conjecture (Alon, Spencer): No efficient algorithm can find one.

Bansal 10: Can efficiently get discrepancy .

A Conjecture and a Disproof

• Non-constructive pigeon-hole proof

Spencer 85: System with n sets has discrepancy at most .

This Work

• Truly constructive• Algorithmic partial coloring lemma• Extends to other settings

Main: Can efficiently find a coloring with discrepancy

New elemantary constructive proof of Spencer’s result

EDGE-WALK: New algorithmic tool

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Beck-Fiala Setting

1 * 1 * *

* 1 1 * 1

* 1 * 1 *

* * * 1 *

1 * * * 1

Beck-Fiala 80: Srinivasan 97: (log n)) (non-algorithmic)Banszczyk 98: (non-algorithmic)

Each element occurs in at most sets

Beck-Fiala SettingEach element occurs in at most

setsThm: Can efficiently find a coloring with discrepancy

Matches Bansal 10

Outline

1. Partial coloring Method

2. EDGE-WALK: Geometric picture

3. Analysis of algorithm

Partial Coloring Method

1 * 1 1 *

* 1 1 * 1

1 1 1 1 1

* * * 1 1

1 * 1 * 1

1 * 1 1 *

* 1 1 * 1

1 1 1 1 1

* * * 1 1

1 * 1 * 1

• Beck 80: find partial assignment with zeros

1 -1 1 1 -1

1 * 1 1 *

* 1 1 * 1

1 1 1 1 1

* * * 1 1

1 * 1 * 1

1 -1 0 0 0

1 * 1 1 *

* 1 1 * 1

1 1 1 1 1

* * * 1 1

1 * 1 * 1

1 * 1 1 *

* 1 1 * 1

1 1 1 1 1

* * * 1 1

1 * 1 * 1

1 1 0-1

• Focus on m = n case.Lemma: Can do this in randomized

time.

Partial Coloring Method

Input:

Output:

Outline

1. Partial coloring Method

2. EDGE-WALK: Geometric picture

3. Analysis of algorithm

1 * 1 1 ** 1 1 * 11 1 1 1 1* * * 1 11 * 1 * 1

Discrepancy: Geometric View• Subsets

• Color with or - to minimize imbalance

1-111-1

3

1

1

0

1

3

1

1

0

1

1 2 3 4 5

1 * 1 1 ** 1 1 * 11 1 1 1 1* * * 1 11 * 1 * 1

Discrepancy: Geometric View

1-111-1

31101

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• Vectors • Want

Discrepancy: Geometric View• Vectors

• Want

Goal: Find non-zero lattice point inside

Polytope view used earlier by Gluskin’ 88.

Claim: Will find good partial coloring.

Edge-Walk

• Start at origin• Gaussian walk

until you hit a face• Gaussian walk

within the face

Goal: Find non-zero lattice point in

Edge-Walk: Algorithm

Gaussian random walk in subspaces

• Subspace V, rate • Gaussian walk in V

Standard normal in V:Orthonormal basis

change

Edge-Walk Algorithm

Discretization issues: hitting faces

• Might not hit face• Slack: face hit if

close to it.

1. For

2. Cube faces nearly hit by .

Disc. faces nearly hit by .

Subspace orthogonal to

Edge-Walk: Algorithm• Input: Vectors • Parameters:

Pr [𝑊𝑎𝑙𝑘h𝑖𝑡𝑠𝑎𝑑𝑖𝑠𝑐 . 𝑓𝑎𝑐𝑒 ]≪ Pr [𝑊𝑎𝑙𝑘h𝑖𝑡𝑠𝑎𝑐𝑢𝑏𝑒′ 𝑠 ]

Edge-Walk: Intuition

1

100 Hit cube more often!

Discrepancy faces much farther than cube’s

Outline

1. Partial coloring Method

2. EDGE-WALK: Geometric picture

3. Analysis of algorithm

Edge-Walk: Analysis

Lem: For

with prob 0.1 and

Edge-Walk Analysis

• Claim 1: Never cross polytope.

• Claim 2: Number of disc. faces hit .–

• Win-Win: Hit many cube faces or grow–

Edge-Walk Analysis

Claim 1: Never cross polytope

• Must make a big jump

• Unlikely:

𝛿

Edge-Walk Analysis

• Claim 2: Number of disc. faces hit

• Small progress each step

• Martingale tail bound:– .

• Linearity of expectation

100

Edge-Walk Analysis

• Claim 3: Hit many cube faces -

• norm grows dimension of subspace

• Final dimension small: ∝dim (𝑉 𝑡)

Main Partial Coloring Lemma

Algorithmic partial coloring lemma

Th: Given thresholds

Can find with 1. 2.

Summary

1. Edge-Walk: Algorithmic partial coloring lemma

2. Recurse on unfixed variables

Spencer’s Theorem

Beck-Fiala setting similar

Open Problems

Q: Other applications?General IP’s, Minkowski’s theorem?

• Some promise: our PCL “stronger” than Beck’s

Q: Beck-Fiala Conjecture 81: Discrepancy for degree t.

• Constructive version of Banszczyk’s bound?

Thank you