Beating the Union Bound by Geometric Techniques

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Beating the Union Bound by Geometric Techniques. Raghu Meka (IAS &amp; DIMACS). Union Bound. Popularized by Erdos. “ When you have eliminated the impossible, whatever remains, however improbable, must be the truth ” . Probabilistic Method 101. Ramsey graphs Erdos Coding theory Shannon

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### Beating the Union Bound by Geometric Techniques

Raghu Meka (IAS & DIMACS)

Union Bound

Popularized by Erdos

“When you have eliminated the impossible, whatever remains,

however improbable, must be the truth”

Probabilistic Method 101
• Ramsey graphs
• Erdos
• Coding theory
• Shannon
• Metric embeddings
• Johnson-Lindenstrauss
Beating the Union Bound
• Not always enough
• Constructive: Beck’91, …, Moser’09, …

Lovasz Local Lemma:

, dependent.

Beating the Union Bound
• Optimal, explicit -nets for Gaussians
• Kanter’s lemma, convex geometry
• Constructive Discrepancy Minimization
• EdgeWalk: New LP rounding method

Geometric techniques

“Truly” constructive

Outline
• Optimal, explicit -nets for Gaussians
• Kanter’s lemma, convex geometry
• Constructive Discrepancy Minimization
• EdgeWalk: New LP rounding method
Epsilon Nets
• Discrete approximations
• Applications: integration, comp. geometry, …
Epsilon Nets for Gaussians

Discrete approximations of Gaussian

Explicit

Even existence not clear!

Nets in Gaussian space

Thm: Explicit -net of size .

• Optimal: Matching lower bound
• Union bound:
Gaussian Processes (GPs)

Multivariate Gaussian Distribution

Supremum of Gaussian Processes (GPs)

Given want to study

• Supremum is natural: eg., balls and bins
Supremum of Gaussian Processes (GPs)

Given want to study

• Union bound: .
• Covariance matrix
• More intuitive

Random

Gaussian

When is the supremum smaller?

Why Gaussian Processes?

Stochastic Processes

Functional analysis

Convex Geometry

Machine Learning

Many more!

Cover times of Graphs

Aldous-Fill 94: Compute cover time deterministically?

Fundamental graph parameter

Eg:

• KKLV00: approximation
• Feige-Zeitouni’09: FPTAS for trees
Cover Times and GPs

Transfer to GPs

Compute supremum of GP

Thm (Ding, Lee, Peres 10): O(1) det. poly. time approximation for cover time.

Computing the Supremum

Question (Lee10, Ding11): PTAS for computing the supremum of GPs?

• Ding, Lee, Peres 10: approximation
• Can’t beat : Talagrand’s majorizing measures
Main Result

Thm: PTAS for computing the supremum of Gaussian processes.

Thm: PTAS for computing cover time of bounded degree graphs.

Heart of PTAS: Epsilon net

(Dimension reduction ala JL, use exp. size net)

Construction of -net

Simplest possible: univariate to multivariate

1. How fine a net?

Naïve: . Union bound!

2. How big a net?

Construction of -net

Simplest possible: univariate to multivariate

Lem: Granularity enough.

Key point that beats union bound

Construction of -net

This talk: Analyze ‘step-wise’ approximator

Even out mass in interval .

-

Construction of -net

Take univariate net and lift to multivariate

Lem: Granularity enough.

-

Dimension Free Error Bounds

Thm: For , a norm,

• Proof by “sandwiching”
• Exploit convexity critically

-

Analysis of Error

Def: Sym. (less peaked), if sym. convex sets K

• Why interesting? For any norm,
Analysisfor Univarate Case

Fact:

Proof:

-

Analysis for Univariate Case

Fact:

Proof: For inward push compensates earlier spreading.

Def: scaled down

, , pdf of .

Push mass towards origin.

Analysis for Univariate Case

Combining upper and lower:

Lifting to Multivariate Case

Key for univariate: “peakedness”

Dimension free!

Kanter’s Lemma(77): and unimodal,

Lifting to Multivariate Case

Dimension free: key point that beats union bound!

Summary of Net Construction

Optimal -net

Granularity enough

Cut points outside -ball

Outline
• Optimal, explicit -nets for Gaussians
• Kanter’s lemma, convex geometry
• Constructive Discrepancy Minimization
• EdgeWalk: New LP rounding method
Discrepancy
• Subsets
• Color with or -to minimize imbalance

1 2 3 4 5

123 45

3

1

1

0

1

Discrepancy Examples
• Fundamental combinatorial concept
• Arithmetic Progressions

Roth 64:

Matousek, Spencer 96:

Discrepancy Examples
• Fundamental combinatorial concept
• Halfspaces

Alexander 90:

Matousek 95:

Why Discrepancy?

Complexity theory

Communication Complexity

Computational Geometry

Pseudorandomness

Many more!

Spencer’s Six Sigma Theorem

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

“Six standard deviations suffice”

• Central result in discrepancy theory.
• Beats union bound:
A Conjecture and a Disproof

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

Bansal 10: Can efficiently get discrepancy .

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

• Non-constructive pigeon-hole proof
Six Sigma Theorem

Main: Can efficiently find a coloring with discrepancy

New elementary geometric proof of Spencer’s result

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

EDGE-WALK: New LP rounding method

Outline of Algorithm

Partial coloring method

EDGE-WALK: geometric picture

Partial Coloring Method
• Beck 80: find partial assignment with zeros

1-111-1

1-10 0 0

11 0

-1

Partial Coloring Method

Lemma: Can do this in randomized

time.

Input:

Output:

Outline of Algorithm

Partial coloring Method

EDGE-WALK: Geometric picture

Discrepancy: Geometric View
• Subsets
• Color with or -to minimize imbalance

123 45

3

1

1

0

1

Discrepancy: Geometric View
• Vectors
• Want

123 45

Discrepancy: Geometric View
• Vectors
• Want

Gluskin 88: Polytopes, Kanter’s lemma, ... !

Goal: Find non-zero lattice point inside

Edge-Walk

Claim: Will find good partial coloring.

• Start at origin
• Brownian motion till you hit a face
• Brownian motion 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.
Edge-Walk: Algorithm
• Input: Vectors
• Parameters:

For

Cube faces nearly hit by .

Disc. faces nearly hit by .

Subspace orthogonal to

Edgewalk: Partial Coloring

Lem: For

with prob 0.1 and

Edgewalk: Analysis

Discrepancy faces much farther than cube’s

Hit cube more often!

Key point that beats union bound

100

1

Six Suffice

Spencer’s Theorem

Edge-Walk: Algorithmic partial coloring

Recurseon unfixed variables

Summary
• Optimal, explicit -nets for Gaussians
• Kanter’s lemma, convex geometry
• Constructive Discrepancy Minimization
• EdgeWalk: New LP rounding method

Geometric techniques

Others: Invariance principle for polytopes

(Harsha, Klivans, M.’10), …

Open Problems
• FPTAS for computing supremum?
• Beck-Fiala conjecture 81?
• Discrepancy for degree .
• Applications of Edgewalk rounding?

Rothvoss’13: Improvements for

bin-packing!

EdgewalkRounding

Th: Given thresholds

Can find with

1.

2.