Randomized Kaczmarz Nick Freris EPFL
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1 Randomized Kaczmarz Nick Freris EPFL (Joint work with A. Zouzias)
2 Outline Randomized Kaczmarz algorithm for linear systems Consistent (noiseless) Inconsistent (noisy) Optimal de-noising Convergence analysis and simulations Application in sensor networks Distributed consensus algorithm for synchronization Faster convergence and energy savings - Faster for sparse systems - Consensus design method 1 / 17
3 Applications Computer science Parallel and distributed algorithms Random projections Sensor networks Optimization & Control Distributed estimation Consensus Signal processing Sampling Compressed Sensing Linear Inverse problems Imaging (ART) Tomography Acoustics and more.. Convergence lab (CSL, Univ. Illinois) SmartSense, EPFL 2 / 17
4 Kaczmarz algorithm Iterative algorithm for solving Ax = b also known as ART in image reconstruction / tomography Round-Robin row selection Projection to the solution space of selected row Convergece: alternating projections performance depends on row order 3 / 17
5 Randomized Kaczmarz Iterative algorithm for solving Ax = b Randomized selection of row Projection to the solution space of selected row Exponential convergence in m.s. (SV 09, FZ 12) Rate of convergence: apple 2 F := A 2 F 2 min performance depends on row scaling 4 / 17
6 Noisy measurements: Oscillatory behavior Noisy case Asymptotically constrained in a ball (N 10, FZ 12) Under-relaxation (RKU) Convergence to a point in the ball slower Least-squares: Bad idea (squaring the condition number) 5 / 17
7 Optimal de-noising LS for inconsistent system: Solution: projection to the range space of A Ax = b R(A) Randomized selection of column Projection to the orthogonal complement of the selected column same rate of convergence 6 / 17
8 Putting the pieces together RK and de-noising: Randomized orthogonal projection Randomized Kaczmarz Termination criteria 7 / 17
9 Analysis of REK Rate of convergence (ZF 13): Ekx (k) x LS k 2 apple (1 1 apple 2 F (A))k [kx LS k 2 + ckb R(A) k 2 k] same exponent, no delay Expected number of arithmetic operations: proportional to sparsity squared condition number Designed for sparse wellconditioned systems 8 / 17
10 Implementation in C REK-C Implementation REK-BLAS (level-1 BLAS routines + Blendenpik) Comparison Matlab backslash \ LAPACK DGELSY (QR factorization) DGELSD (SVD) LSNR Blendenpik 9 / 17
11 Experiments Excellent performance for sparse systems 10 / 17
12 A sensor network problem Relative measurements For two neighbors: Network problem: y ij = x i x j + w ij Jacobi algorithm for LSE Local averaging (distributed) Synchronous: Exponential convergence (GK 06) Asynchronous: Exponential convergence (FZ 13) Applications Clock synchronization (smoothing time differences) Localization (smoothing distance/angular differences) 11 / 17
13 Smoothing via RK Randomized sampling Distributed averaging Asynchronous implementation Exponential clocks 12 / 17
14 An extension Over-smoothing (RKO) Faster convergence in absolute time vs More messages / iteration 13 / 17
15 Convergence analysis Algorithm Convergence Reference Jacobi O( 2 (G) m ) GK 06 (FZ 12) OSE Faster than Jacobi BDE 06 RKS FZ 12 O( 2 (G) m ) 2 RKLS FZ 12 O( 2(G) m 2 ) RKU FZ 12 O( 2 (G) m ) RKO Faster than RKS FZ 12 Cheeger s inequality: depends on network connectivity 14 / 17
16 Simulations Faster convergence Energy savings 15 / 17
17 Conclusions Randomized Kaczmarz (RK) algorithm Exponential convergence in the mean-square Same rate regardless of noise Distributed asynchronous smoothing Experiments Linear systems: Gains for sparse systems Sensor networks: Faster convergence and energy savings Efficient sparse linear system solver 16 / 17
18 Ongoing work Distributed implementation of REK Range projection matrix pre-conditioning termination criteria Stochastic approximation convergence to the true values slower (gradient method) improved convergence Numerical analysis is not dead! 17 / 17
19 References 1. N. Freris and A. Zouzias, Fast distributed smoothing of relative measurements," 51 st IEEE Conference on Decision and Control (CDC), pp , Dec Anastasios Zouzias and Nikolaos Freris, Randomized Extended Kaczmarz for Solving Least Squares. SIAM Journal on Matrix Analysis and Applications, 34(2), , T. Strohmer and R. Vershynin, A Randomized Kaczmarz Algorithm with Exponential Convergence, Journal of Fourier Analysis and Applications, vol. 15, no. 1, pp , D. Needell. Randomized Kaczmarz Solver for Noisy Linear Systems. Bit Numerical Mathematics, 50(2): , 2010.
20 Thank you
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