NDLI: Learning Theory of Randomized Sparse Kaczmarz Method

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Zhou, Ding-Xuan Lei, Yunwen
Copyright Year	2018
Abstract	In this paper we propose an online learning algorithm, a general randomized sparse Kaczmarz method, for generating sparse approximate solutions to linear systems and present learning theory analysis for its convergence. Under a mild assumption covering the case of noisy random measurements in the sampling process or nonlinear regression function, we show that the algorithm converges in expectation if and only if the step size sequence $\{\eta_t\}_{t\in\mathbb{N}}$ satisfies $\lim_{t\to\infty}\eta_t=0$ and $\sum_{t=1}^{\infty}\eta_t=\infty$. Convergence rates are also obtained and linear convergence is shown to be impossible under the assumption of positive variance of the sampling process. A sufficient condition for almost sure convergence is derived with an additional restriction $\sum_{t=1}^{\infty}\eta_t^2 <\infty$. Our novel analysis is performed by interpreting the randomized sparse Kaczmarz method as a special online mirror descent algorithm with a nondifferentiable mirror map and using the Bregman distance. The sufficient and necessary conditions are derived by establishing a restricted variant of strong convexity for the involved generalization error and using the special structures of the soft-thresholding operator.
Sponsorship	Research Grants Council, University Grants Committee. National Natural Science Foundation of China
Starting Page	547
Ending Page	574
Page Count	28
File Format	PDF
DOI	10.1137/17M1136225
e-ISSN	19364954
Journal	SIAM Journal on Imaging Sciences (SJISBI)
Issue Number	1
Volume Number	11
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2018-02-20
Access Restriction	Subscribed
Subject Keyword	linearized Bregman iteration randomized sparse Kaczmarz algorithm Stochastic learning and adaptive control Computational learning theory Bregman distance learning theory online learning
Content Type	Text
Resource Type	Article

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