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• W3009716613 abstract "First-order operator splitting methods are ubiquitous among many fields through science and engineering, such as inverse problems, signal/image processing, statistics, data science and machine learning, to name a few. In this paper, we study a geometric property of first-order methods when applying to solve non-smooth optimization problems. With the tool of partial smoothness, we design a framework to analyze the trajectory of the fixed-point sequence generated by first-order methods and show that locally, the fixed-point sequence settles onto a regular trajectory such as a straight line or a spiral. Based on this finding, we discuss the limitation of current widely used inertial acceleration technique, and propose a trajectory following adaptive acceleration algorithm. Global convergence is established for the proposed acceleration scheme based on the perturbation of fixed-point iteration. Locally, we first build connections between the acceleration scheme and the well-studied vector extrapolation technique in the field of numerical analysis, and then discuss local acceleration guarantees of the proposed acceleration scheme. Moreover, our result provides a geometric interpretation of these vector extrapolation techniques. Numerical experiments on various first-order methods are provided to demonstrate the advantage of the proposed adaptive acceleration scheme." @default.
• W3009716613 created "2020-03-13" @default.
• W3009716613 creator A5004350216 @default.
• W3009716613 creator A5066789410 @default.
• W3009716613 date "2020-03-08" @default.
• W3009716613 modified "2023-10-17" @default.
• W3009716613 title "Geometry of First-Order Methods and Adaptive Acceleration" @default.
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• W3009716613 doi "https://doi.org/10.48550/arxiv.2003.03910" @default.
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