Convergence Rate Analysis of Accelerated Forward-Backward Algorithm with Generalized Nesterov Momentum Scheme
Year: 2023
Author: Yizun Lin, Si Li, Yunzhong Zhang
International Journal of Numerical Analysis and Modeling, Vol. 20 (2023), Iss. 4 : pp. 518–537
Abstract
Nesterov’s accelerated forward-backward algorithm (AFBA) is an efficient algorithm for solving a class of two-term convex optimization models consisting of a differentiable function with a Lipschitz continuous gradient plus a nondifferentiable function with a closed form of its proximity operator. It has been shown that the iterative sequence generated by AFBA with a modified Nesterov’s momentum scheme converges to a minimizer of the objective function with an $o (\frac{1}{k^2})$ convergence rate in terms of the function value (FV-convergence rate) and an $o(\frac{1}{k})$ convergence rate in terms of the distance between consecutive iterates (DCI-convergence rate). In this paper, we propose a more general momentum scheme with an introduced power parameter $ω ∈ (0, 1]$ and show that AFBA with the proposed momentum scheme converges to a minimizer of the objective function with an $o ( \frac{1}{ k^{2ω}} )$ FV-convergence rate and an $o (\frac{1}{k^ω})$ DCI-convergence rate. The generality of the proposed momentum scheme provides us a variety of parameter selections for different scenarios, which makes the resulting algorithm more flexible to achieve better performance. We then employ AFBA with the proposed momentum scheme to solve the smoothed hinge loss $ℓ_1$-support vector machine model. Numerical results demonstrate that the proposed generalized momentum scheme outperforms two existing momentum schemes.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ijnam2023-1022
International Journal of Numerical Analysis and Modeling, Vol. 20 (2023), Iss. 4 : pp. 518–537
Published online: 2023-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 20
Keywords: Nesterov’s momentum forward-backward algorithm convergence rate support vector machine.