beating the union bound by geometric techniques n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Beating the Union Bound by Geometric Techniques PowerPoint Presentation
Download Presentation
Beating the Union Bound by Geometric Techniques

Loading in 2 Seconds...

play fullscreen
1 / 57

Beating the Union Bound by Geometric Techniques - PowerPoint PPT Presentation


  • 93 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Beating the Union Bound by Geometric Techniques' - udell


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
union bound
Union Bound

Popularized by Erdos

“When you have eliminated the impossible, whatever remains,

however improbable, must be the truth”

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

Lovasz Local Lemma:

, dependent.

beating the union bound1
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
Outline
  • Optimal, explicit -nets for Gaussians
    • Kanter’s lemma, convex geometry
  • Constructive Discrepancy Minimization
    • EdgeWalk: New LP rounding method
epsilon nets
Epsilon Nets
  • Discrete approximations
  • Applications: integration, comp. geometry, …
epsilon nets for gaussians
Epsilon Nets for Gaussians

Discrete approximations of Gaussian

Explicit

Even existence not clear!

nets in gaussian space
Nets in Gaussian space

Thm: Explicit -net of size .

  • Optimal: Matching lower bound
  • Union bound:
  • Dadusch-Vempala’12:
gaussian processes gps
Gaussian Processes (GPs)

Multivariate Gaussian Distribution

supremum of gaussian processes gps
Supremum of Gaussian Processes (GPs)

Given want to study

  • Supremum is natural: eg., balls and bins
supremum of gaussian processes gps1
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
Why Gaussian Processes?

Stochastic Processes

Functional analysis

Convex Geometry

Machine Learning

Many more!

cover times of graphs
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
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
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
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 net1
Construction of -net

Simplest possible: univariate to multivariate

1. How fine a net?

Naïve: . Union bound!

2. How big a net?

construction of net2
Construction of -net

Simplest possible: univariate to multivariate

Lem: Granularity enough.

Key point that beats union bound

construction of net3
Construction of -net

This talk: Analyze ‘step-wise’ approximator

Even out mass in interval .

-

construction of net4
Construction of -net

Take univariate net and lift to multivariate

Lem: Granularity enough.

-

dimension free error bounds
Dimension Free Error Bounds

Thm: For , a norm,

  • Proof by “sandwiching”
  • Exploit convexity critically

-

analysis of error
Analysis of Error

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

  • Why interesting? For any norm,
analysis for univarate case
Analysisfor Univarate Case

Fact:

Proof:

Spreading away from origin!

-

analysis for univariate case
Analysis for Univariate Case

Fact:

Proof: For inward push compensates earlier spreading.

Def: scaled down

, , pdf of .

Push mass towards origin.

analysis for univariate case1
Analysis for Univariate Case

Combining upper and lower:

lifting to multivariate case
Lifting to Multivariate Case

Key for univariate: “peakedness”

Dimension free!

Kanter’s Lemma(77): and unimodal,

lifting to multivariate case1
Lifting to Multivariate Case

Dimension free: key point that beats union bound!

summary of net construction
Summary of Net Construction

Optimal -net

Granularity enough

Cut points outside -ball

outline1
Outline
  • Optimal, explicit -nets for Gaussians
    • Kanter’s lemma, convex geometry
  • Constructive Discrepancy Minimization
    • EdgeWalk: New LP rounding method
discrepancy
Discrepancy
  • Subsets
  • Color with or -to minimize imbalance

1 2 3 4 5

123 45

3

1

1

0

1

discrepancy examples
Discrepancy Examples
  • Fundamental combinatorial concept
  • Arithmetic Progressions

Roth 64:

Matousek, Spencer 96:

discrepancy examples1
Discrepancy Examples
  • Fundamental combinatorial concept
  • Halfspaces

Alexander 90:

Matousek 95:

why discrepancy
Why Discrepancy?

Complexity theory

Communication Complexity

Computational Geometry

Pseudorandomness

Many more!

spencer s six sigma theorem
Spencer’s Six Sigma Theorem

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

“Six standard deviations suffice”

  • Central result in discrepancy theory.
  • Tight: Hadamard
  • Beats union bound:
a conjecture and a disproof
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
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
Outline of Algorithm

Partial coloring method

EDGE-WALK: geometric picture

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

1-111-1

1-10 0 0

11 0

-1

partial coloring method1
Partial Coloring Method

Lemma: Can do this in randomized

time.

Input:

Output:

outline of algorithm1
Outline of Algorithm

Partial coloring Method

EDGE-WALK: Geometric picture

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

123 45

3

1

1

0

1

discrepancy geometric view1
Discrepancy: Geometric View
  • Vectors
  • Want

123 45

discrepancy geometric view2
Discrepancy: Geometric View
  • Vectors
  • Want

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

Goal: Find non-zero lattice point inside

edge walk
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
Edge-Walk: Algorithm

Gaussian random walk in subspaces

  • Subspace V, rate
  • Gaussian walk in V

Standard normal in V:

Orthonormal basis change

edge walk algorithm1
Edge-Walk Algorithm

Discretization issues: hitting faces

  • Might not hit face
  • Slack: face hit if close to it.
edge walk algorithm2
Edge-Walk: Algorithm
  • Input: Vectors
  • Parameters:

For

Cube faces nearly hit by .

Disc. faces nearly hit by .

Subspace orthogonal to

edgewalk partial coloring
Edgewalk: Partial Coloring

Lem: For

with prob 0.1 and

edgewalk analysis
Edgewalk: Analysis

Discrepancy faces much farther than cube’s

Hit cube more often!

Key point that beats union bound

100

1

six suffice
Six Suffice

Spencer’s Theorem

Edge-Walk: Algorithmic partial coloring

Recurseon unfixed variables

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

Rothvoss’13: Improvements for

bin-packing!

edgewalk rounding
EdgewalkRounding

Th: Given thresholds

Can find with

1.

2.