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

Jiaxu Liu; Song Chen; Shengze Cai; Chao Xu

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arXiv · 2303.04328

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

Jiaxu Liu·Song Chen·Shengze Cai·Chao Xu

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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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Jiaxu Liu, Song Chen, Shengze Cai, Chao Xu. 2023-03-08. The Novel Adaptive Fractional Order Gradient Decent Algorithms Design via Robust Control. https://arxiv.org/abs/2303.04328

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