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Journal of Data Science


Accelerating Fixed-Point Algorithms in Statistics and Data Science: A State-of-Art Review
Volume 21, Issue 1 (2023), pp. 1–26
Pub. online: 19 July 2022      Type: Data Science Reviews      Open accessOpen Access
Received
7 March 2022
Accepted
25 May 2022
Published
19 July 2022

Abstract

Fixed-point algorithms are popular in statistics and data science due to their simplicity, guaranteed convergence, and applicability to high-dimensional problems. Well-known examples include the expectation-maximization (EM) algorithm, majorization-minimization (MM), and gradient-based algorithms like gradient descent (GD) and proximal gradient descent. A characteristic weakness of these algorithms is their slow convergence. We discuss several state-of-art techniques for accelerating their convergence. We demonstrate and evaluate these techniques in terms of their efficiency and robustness in six distinct applications. Among the acceleration schemes, SQUAREM shows robust acceleration with a mean 18-fold speedup. DAAREM and restarted-Nesterov schemes also demonstrate consistently impressive accelerations. Thus, it is possible to accelerate the original fixed-point algorithm by using one of SQUAREM, DAAREM, or restarted-Nesterov acceleration schemes. We describe implementation details and software packages to facilitate the application of the acceleration schemes. We also discuss strategies for selecting a particular acceleration scheme for a given problem.

Supplementary material

 Supplementary Material
1. We provide an R package AccelBenchmark, available from Github that can be used to reproduce the experiments in this paper. Please check the vignette and the demo.R file under vignette folder for reference. 2. We provide an additional pdf file that includes a) some basic properties of fixed point iterations; b) visualization of the convergence for all the experiments and c) additional analysis for the Sinkhorn scaling problem.

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Copyright
2023 The Author(s). Published by the School of Statistics and the Center for Applied Statistics, Renmin University of China.
Open access article under the CC BY license.

Keywords
convergence acceleration EM high dimensional models MM proximal gradient

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