The Novel Adaptive Fractional Order Gradient Decent Algorithms Design via Robust Control

• URL: http://arxiv.org/abs/2303.04328v1
• Date: Wed, 8 Mar 2023 02:03:30 GMT
• Title: The Novel Adaptive Fractional Order Gradient Decent Algorithms Design via Robust Control
• Authors: Jiaxu Liu and Song Chen and Shengze Cai and Chao Xu
• Abstract summary: The vanilla fractional order descent may oscillatively converge to a region around the global minimum instead of converging to the exact minimum point, or even diverge, in the case where the objective function is strongly convex. To address this problem, a novel adaptive fractional order descent (AFDOG) and a novel fractional descent (AFOAGD) method are proposed in this paper.
• Score: 5.5491171448159715
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The vanilla fractional order gradient descent may oscillatively converge to a region around the global minimum instead of converging to the exact minimum point, or even diverge, in the case where the objective function is strongly convex. To address this problem, a novel adaptive fractional order gradient descent (AFOGD) method and a novel adaptive fractional order accelerated gradient descent (AFOAGD) method are proposed in this paper. Inspired by the quadratic constraints and Lyapunov stability analysis from robust control theory, we establish a linear matrix inequality to analyse the convergence of our proposed algorithms. We prove that the proposed algorithms can achieve R-linear convergence when the objective function is $\textbf{L-}$smooth and $\textbf{m-}$strongly-convex. Several numerical simulations are demonstrated to verify the effectiveness and superiority of our proposed algorithms.

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