Beating the Union Bound by Geometric Techniques Raghu Meka (IAS & DIMACS)

“ When you have eliminated the impossible, whatever remains, however improbable, must be the truth ” Union Bound Popularized by Erdos

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, …

Beating the Union Bound Geometric techniques “Truly” constructive

Outline

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

First: Application to Gaussian Processes and Cover Times 10

Gaussian Processes (GPs) Multivariate Gaussian Distribution

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

When is the supremum smaller? Supremum of Gaussian Processes (GPs) Random Gaussian Covariance matrix More intuitive

Why Gaussian Processes? Stochastic Processes Functional analysis Convex Geometry Machine Learning Many more!

Aldous-Fill 94: Compute cover time deterministically? Cover times of Graphs

Cover Times and GPs Thm (Ding, Lee, Peres 10): O(1) det. poly. time approximation for cover time. Transfer to GPs Compute supremum of GP

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

Main Result Thm: PTAS for computing the supremum of Gaussian processes. Heart of PTAS: Epsilon net (Dimension reduction ala JL, use exp. size net) Thm: PTAS for computing cover time of bounded degree graphs.

Construction of Net 19

Simplest possible: univariate to multivariate 1. How fine a net? 2. How big a net?

Simplest possible: univariate to multivariate Key point that beats union bound

This talk: Analyze ‘step-wise’ approximator

Take univariate net and lift to multivariate

Dimension Free Error Bounds Proof by “sandwiching” Exploit convexity critically

Analysis of Error Why interesting? For any norm,

Analysis for Univarate Case Spreading away from origin!

Analysis for Univariate Case Push mass towards origin.

Analysis for Univariate Case Combining upper and lower:

Kanter’s Lemma(77): and unimodal, Lifting to Multivariate Case Key for univariate: “peakedness” Dimension free!

Lifting to Multivariate Case Dimension free: key point that beats union bound!

Summary of Net Construction

Outline

Discrepancy 1*11* *11* ***11 1*1* *11* *11* ***11 1*1*

Discrepancy Examples Fundamental combinatorial concept Arithmetic Progressions

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 Central result in discrepancy theory. Tight: Hadamard Beats union bound: 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.

Six Sigma Theorem Truly constructive Algorithmic partial coloring lemma Extends to other settings New elementary geometric proof of Spencer’s result EDGE-WALK: New LP rounding method

Outline of Algorithm 1.Partial coloring method 2.EDGE-WALK: geometric picture

Partial Coloring Method 1*11* *11* ***11 1*1*1 1*11* *11* ***11 1*1* *11* *11* ***11 1*1* *11* *11* ***11 1*1*1 1*11* *11* ***11 1*1*

Lemma: Can do this in randomized time. Partial Coloring Method Input: Output:

Outline of Algorithm 1.Partial coloring Method 2.EDGE-WALK: Geometric picture

1*11* *11* ***11 1*1*1 Discrepancy: Geometric View

1*11* *11* ***11 1*1*1 Discrepancy: Geometric View

Discrepancy: Geometric View Goal: Find non-zero lattice point inside Gluskin 88: Polytopes, Kanter’s lemma,... !

Claim: Will find good partial coloring. Edge-Walk 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 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

Edgewalk: Partial Coloring Lem: For with prob 0.1 and

Edgewalk: Analysis Hit cube more often! Discrepancy faces much farther than cube’s Key point that beats union bound

Six Suffice 1.Edge-Walk: Algorithmic partial coloring 2.Recurse on unfixed variables Spencer’s Theorem

Summary Geometric techniques Others: Invariance principle for polytopes (Harsha, Klivans, M.’10), …

Open Problems Rothvoss’13: Improvements for bin-packing!

Thank you

Edgewalk Rounding