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

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.