Approximate message passing for nonconvex sparse regularization with stability and asymptotic analysis

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Journal Title
Journal ISSN
Volume Title
A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
Date
2018-03-13
Major/Subject
Mcode
Degree programme
Language
en
Pages
1-35
Series
Journal of Statistical Mechanics: Theory and Experiment, Volume 2018, issue 3
Abstract
We analyse a linear regression problem with nonconvex regularization called smoothly clipped absolute deviation (SCAD) under an overcomplete Gaussian basis for Gaussian random data. We propose an approximate message passing (AMP) algorithm considering nonconvex regularization, namely SCAD-AMP, and analytically show that the stability condition corresponds to the de Almeida-Thouless condition in spin glass literature. Through asymptotic analysis, we show the correspondence between the density evolution of SCAD-AMP and the replica symmetric (RS) solution. Numerical experiments confirm that for a sufficiently large system size, SCADAMP achieves the optimal performance predicted by the replica method. Through replica analysis, a phase transition between replica symmetric and replica symmetry breaking (RSB) region is found in the parameter space of SCAD. The appearance of the RS region for a nonconvex penalty is a significant advantage that indicates the region of smooth landscape of the optimization problem. Furthermore, we analytically show that the statistical representation performance of the SCAD penalty is better than that of ℓ1-based methods, and the minimum representation error under RS assumption is obtained at the edge of the RS/RSB phase. The correspondence between the convergence of the existing coordinate descent algorithm and RS/RSB transition is also indicated.
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Keywords
analysis of algorithms, learning theory, message-passing algorithms
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Citation
Sakata , A & Xu , Y 2018 , ' Approximate message passing for nonconvex sparse regularization with stability and asymptotic analysis ' , Journal of Statistical Mechanics: Theory and Experiment , vol. 2018 , no. 3 , 033404 , pp. 1-35 . https://doi.org/10.1088/1742-5468/aab051