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

arXiv: https://arxiv.org/abs/2303.04328 · HTML: https://mlsanity.com/paper/2303.04328/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 23 references that are in the map.
- First posted on arXiv 8 Mar 2023.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints](https://mlsanity.com/paper/1408.3595/) (2014)
- [First-order optimization algorithms via inertial systems with Hessian driven damping](https://mlsanity.com/paper/1907.10536/) (2019)
- [The connections between Lyapunov functions for some optimization algorithms and differential equations](https://mlsanity.com/paper/2009.00673/) (2020)
- [Lyapunov functions for nabla discrete fractional order systems](https://mlsanity.com/paper/1812.11368/) (2018)
- [Neural Observer with Lyapunov Stability Guarantee for Uncertain Nonlinear Systems](https://mlsanity.com/paper/2208.13006/) (2022)

## Built on by (most cited first, 2 of 2)

- [Applications of fractional calculus in learned optimization](https://mlsanity.com/paper/2411.14855/) (2024)
- [Enhancing Fractional Gradient Descent with Learned Optimizers](https://mlsanity.com/paper/2510.18783/) (2025)

## Abstract (arXiv)

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 L-smooth and m-strongly-convex. Several numerical simulations are demonstrated to verify the effectiveness and superiority of our proposed algorithms.