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Beating the Union Bound by Geometric Techniques

Explore the use of geometric techniques to surpass the limitations of the union bound in probabilistic methods. Topics include Ramsey graphs, coding theory, Johnson-Lindenstrauss lemma, and more.

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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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