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Preconditioning in Expectation

Preconditioning in Expectation. Richard Peng. MIT. Joint with Michael Cohen (MIT), Rasmus Kyng (Yale), Jakub Pachocki (CMU), and Anup Rao (Yale). CMU theory seminar, April 5, 2014. Random Sampling. Collection of many objects Pick a small subset of them. Goals of Sampling.

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Preconditioning in Expectation

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  1. Preconditioning in Expectation Richard Peng MIT Joint with Michael Cohen (MIT), RasmusKyng (Yale), JakubPachocki (CMU), and AnupRao (Yale) CMU theory seminar, April 5, 2014

  2. Random Sampling • Collection of many objects • Pick a small subset of them

  3. Goals of Sampling • Estimate quantities • Approximate higher dimensional objects • Use in algorithms

  4. Sample to Approximate • ε- nets / cuttings • Sketches • Graphs • Gradients This talk: matrices

  5. Numerical Linear Algebra • Linear system in n x n matrix • Inverse is dense • [Concus-Golub-O'Leary `76]: incomplete Cholesky, drop entries

  6. How To Analyze? • Show sample is good • Concentration bounds • Scalar: [Bernstein `24][Chernoff`52] • Matrices: [AW`02][RV`07][Tropp `12]

  7. This talk • Directly show algorithm using samples runs well • Better bounds • Simpler analysis

  8. Outline • Random matrices • Iterative methods • Randomized preconditioning • Expected inverse moments

  9. How To Drop Entries? • Entry based representation hard • Group entries together • Symmetric with positive entries  adjacency matrix of a graph

  10. Sample with Guarantees • Sample edges in graphs • Goal: preserve size of all cuts • [BK`96] graph sparsification • Generalization of expanders

  11. Dropping Entries/edges • L: graph Laplacian • 0-1 x : |x|L2 = size of cut between 0s-and-1s Unit weight case: |x|L2 = Σuv (xu – xv)2 Matrix norm: |x|P2 = xTPx

  12. Decomposing a Matrix u Σuv (xu – xv)2 P.S.D. multi-variate version of positive L = Σuv v • Sample based on positive representations • P = Σi Pi, with each Pi P.S.D • Graphs: one Pi per edge u v

  13. Matrix Chernoff Bounds ≼ : Loewner’s partial ordering, A ≼B B – A positive semi definite P = Σi Pi, with each Pi P.S.D Can sample Q with O(nlognε-2) rescaled Piss.t.P≼ Q ≼ (1 +ε) P

  14. Can we do better? • Yes, [BSS `12]: O(nε-2) is possible • Iterative, cubic time construction • [BDM `11]: extends to general matrices

  15. Direct Application Find Qvery close toP Solve problem on Q Return answer For ε accuracy, need P≼ Q≼(1 +ε) P Size of Q depends inversely on ε ε-1 is best that we can hope for

  16. Use inside iterative methods Find Qsomewhat similar to P Solve problem on P using Q as a guide • [AB `11]: crude samples give good answers • [LMP `12]: extensions to row sampling

  17. Algorithmic View • Crude approximations are ok • But need to be efficient • Can we use [BSS `12]?

  18. Speed up [BSS `12] • Expander graphs, and more • ‘i.i.d. sampling’ variant related to the Kadison-Singer problem

  19. Motivation • One dimensional sampling: • moment estimation, • pseudorandom generators • Rarely need w.h.p. • Dimensions should be disjoint

  20. Motivation • Randomized coordinate descent for electrical flows [KOSZ`13,LS`13] • ACDM from [LS `13] improves various numerical routines

  21. Randomized coordinate descent • Related to stochastic optimization • Known analyses when Q= Pj • [KOSZ`13][LS`13] can be viewed as ways of changing bases

  22. Our Result For numerical routines, random Q gives same performances as [BSS`12], in expectation

  23. Implications • Similar bounds to ACDM from [LS `13] • Recursive Chebyshev iteration ([KMP`11]) runs faster • Laplacian solvers in ~ mlog1/2n time

  24. Outline • Random matrices • Iterative methods • Randomized preconditioning • Expected inverse moments

  25. Iterative Methods Find Qs.t.P≼ Q≼10 P Use Q as guide to solve problem on P • [Gauss, 1823] Gauss-Siedel iteration • [Jacobi, 1845] Jacobi Iteration • [Hestnes-Stiefel `52] conjugate gradient

  26. [Richardson `1910] x(t + 1) = x(t) + (b – Px(t)) • Fixed point: b – Px(t) = 0 • Each step: one matrix-vector multiplication

  27. Iterative Methods • Multiplication is easier than division, especially for matrices • Use verifier to solve problem

  28. 1D case Know: 1/2 ≤ p ≤ 1  1 ≤ 1/p ≤ 2 • 1 is a ‘good’ estimate • Bad when p is far from 1 • Estimate of error: 1 - p

  29. Iterative Methods • 1 + (1 – p) = 2 – p is more accurate • Two terms of Taylor expansion • Can take more terms

  30. Iterative Methods 1/p = 1 + (1 – p) + (1 – p)2 + (1 – p)3… Generalizes to matrix settings: P-1 = I + (I – P) + (I – P)2 + …

  31. [Richardson `1910] x(0) = Ib X(1)= (I + (I – P))b x(2)= (I + (I – P) (I + (I – P)))b … x(t + 1) = b + (I – P) x(t) • Error of x(t) = (I – P)t b • Geometric decrease if P is close to I

  32. Optimization view Residue: r(t) = x(t ) – P-1b Error: |r(t)|22 • Quadratic potential function • Goal: walk down to the bottom • Direction given by gradient

  33. Descent Steps x(t) x(t+1) x(t) x(t+1) • Step may overshoot • Need smooth function

  34. Measure of Smoothness x(t + 1) = b + (I – P) x(t) Note: b = PP-1b r(t + 1) = (I – P) r(t) |r(t + 1)|2 ≤|I – P|2 |x(t)|2

  35. Measure of Smoothness • |I – P|2 :smoothness of |r(t)|22 • Distance between P and I • Related to eigenvalues of P 1 / 2 I ≼ P≼ I  |I – P|2 ≤ 1/2

  36. More general • Convex functions • Smoothness / strong convexity This talk: only quadratics

  37. Outline • Random matrices • Iterative methods • Randomized preconditioning • Expected inverse moments

  38. Ill Posed Problems • Smoothness of directions differ • Progress limited by steeper parts

  39. Preconditioning P Q P • Solve similar problem Q • Transfer steps across

  40. Preconditioned Richardson P Q • Optimal step down energy function of Q given by Q-1 • Equivalent to solving Q-1Px = Q-1b

  41. Preconditioned Richardson x(t + 1) = b + (I – Q-1P) x(t) Residue: r(t + 1) = (I – Q-1P) r(t) |r(t + 1)|P = |(I– Q-1P)r(t)|P

  42. Convergence P Q Improvements depend on |I– P1/2Q-1P1/2|2 • If P≼ Q≼10 P, error halves in O(1) iterations • How to find a good Q?

  43. Matrix Chernoff P = ΣiPi Q = ΣisiPi s has small support • Take O(nlogn) (rescaled) Pis with probability ~ trace(PiP-1) • Matrix Chernoff ([AW`02],[RV`07]): w.h.p. P≼ Q≼ 2P Note: Σitrace(PiP-1) = n

  44. Why These Probabilities? • trace(PiP-1): • Matrix ‘dot product’ • If P is diagonal • 1 for all i • Need all entries Overhead of concentration: union bound on dimensions

  45. Is Chernoff necessary? • P: diagonal matrix • Missing one entry: unbounded approximation factor

  46. Better Convergence? • [Kaczmarz `37]: random projections onto small subspaces can work • Better (expected) behavior than what matrix concentration gives!

  47. How? Q1 P ≠ • Will still progress in good directions • Can have (finite) badness if they are orthogonal to goal

  48. Quantify Degeneracies P D • Have some D≼P ‘for free’ • D = λmin (P)I (min eigenvalue) • D = tree when P is a graph • D = crude approximation/ rank certificate

  49. Removing Degeneracies P D • ‘Padding’ to remove degeneracy • If D≼P and 0.5 P ≼ Q≼P, 0.5P≼ D + Q≼ 2P

  50. Role of D P D • Implicit in proofs of matrix Chernoff, as well as [BSS`12] • Splitting of P in numerical analysis • D and P can be very different

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