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Beating the Union Bound by Geometric Techniques

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

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Beating the Union Bound by Geometric Techniques

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

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

  3. Probabilistic Method 101 • Ramsey graphs • Erdos • Coding theory • Shannon • Metric embeddings • Johnson-Lindenstrauss • …

  4. Beating the Union Bound • Not always enough • Constructive: Beck’91, …, Moser’09, … Lovasz Local Lemma: , dependent.

  5. 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

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

  7. Epsilon Nets • Discrete approximations • Applications: integration, comp. geometry, …

  8. Epsilon Nets for Gaussians Discrete approximations of Gaussian Explicit Even existence not clear!

  9. Nets in Gaussian space Thm: Explicit -net of size . • Optimal: Matching lower bound • Union bound: • Dadusch-Vempala’12:

  10. First: Application to Gaussian Processes and Cover Times

  11. Gaussian Processes (GPs) Multivariate Gaussian Distribution

  12. Supremum of Gaussian Processes (GPs) Given want to study • Supremum is natural: eg., balls and bins

  13. Supremum of Gaussian Processes (GPs) Given want to study • Union bound: . • Covariance matrix • More intuitive Random Gaussian When is the supremum smaller?

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

  15. Cover times of Graphs Aldous-Fill 94: Compute cover time deterministically? Fundamental graph parameter Eg: • KKLV00: approximation • Feige-Zeitouni’09: FPTAS for trees

  16. 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.

  17. 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

  18. 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)

  19. Construction of Net

  20. Construction of -net Simplest possible: univariate to multivariate 1. How fine a net? Naïve: . Union bound! 2. How big a net?

  21. Construction of -net Simplest possible: univariate to multivariate Lem: Granularity enough. Key point that beats union bound

  22. Construction of -net This talk: Analyze ‘step-wise’ approximator Even out mass in interval . -

  23. Construction of -net Take univariate net and lift to multivariate Lem: Granularity enough. -

  24. Dimension Free Error Bounds Thm: For , a norm, • Proof by “sandwiching” • Exploit convexity critically -

  25. Analysis of Error Def: Sym. (less peaked), if sym. convex sets K • Why interesting? For any norm,

  26. Analysisfor Univarate Case Fact: Proof: Spreading away from origin! -

  27. Analysis for Univariate Case Fact: Proof: For inward push compensates earlier spreading. Def: scaled down , , pdf of . Push mass towards origin.

  28. Analysis for Univariate Case Combining upper and lower:

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

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

  31. Summary of Net Construction Optimal -net Granularity enough Cut points outside -ball

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

  33. Discrepancy • Subsets • Color with or -to minimize imbalance 1 2 3 4 5 123 45 3 1 1 0 1

  34. Discrepancy Examples • Fundamental combinatorial concept • Arithmetic Progressions Roth 64: Matousek, Spencer 96:

  35. Discrepancy Examples • Fundamental combinatorial concept • Halfspaces Alexander 90: Matousek 95:

  36. Why Discrepancy? Complexity theory Communication Complexity Computational Geometry Pseudorandomness Many more!

  37. 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:

  38. 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

  39. 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

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

  41. Partial Coloring Method • Beck 80: find partial assignment with zeros 1-111-1 1-10 0 0 11 0 -1

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

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

  44. Discrepancy: Geometric View • Subsets • Color with or -to minimize imbalance 123 45 3 1 1 0 1

  45. Discrepancy: Geometric View • Vectors • Want 123 45

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

  47. 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

  48. Edge-Walk: Algorithm Gaussian random walk in subspaces • Subspace V, rate • Gaussian walk in V Standard normal in V: Orthonormal basis change

  49. Edge-Walk Algorithm Discretization issues: hitting faces • Might not hit face • Slack: face hit if close to it.

  50. Edge-Walk: Algorithm • Input: Vectors • Parameters: For Cube faces nearly hit by . Disc. faces nearly hit by . Subspace orthogonal to

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