| 引用本文: | 刘佳旭,陈嵩,蔡声泽,许超,褚健.基于鲁棒控制的自适应分数阶梯度优化算法设计(英文)[J].控制理论与应用,2024,41(7):1187~1196.[点击复制] |
| LIU Jia-xu,CHEN Song,CAI Sheng-ze,XU Chao,CHU Jian.The novel adaptive fractional order gradient decent algorithms design via robust control[J].Control Theory and Technology,2024,41(7):1187~1196.[点击复制] |
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| 基于鲁棒控制的自适应分数阶梯度优化算法设计(英文) |
| The novel adaptive fractional order gradient decent algorithms design via robust control |
| 摘要点击 111 全文点击 38 投稿时间：2023-08-08 修订日期：2024-04-10 |
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| DOI编号 DOI: 10.7641/CTA.2024.30534 |
| 2024,41(7):1187-1196 |
| 中文关键词 梯度下降法 自适应算法 鲁棒控制 分数阶微积分 加速算法 |
| 英文关键词 gradient descent adaptive algorithm robust control fractional order calculus accelerated algorithm |
| 基金项目 科技创新2030新一代人工智能重大项目(2018AAA0100902),国家重点研发计划(2019YFB1705800)，国家自然科学基金(61973270) |
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| 中文摘要 |
| 当目标函数是强凸函数时, 一般的分数阶梯度下降法不能够使函数收敛到最小值点, 只能收敛到一个包含最小值点的区域内或者是发散的. 为了解决这个问题, 本文提出了自适应分数阶梯度下降法(AFOGD)和自适应分数阶加速梯度下降法(AFOAGD)两种新的优化算法. 受到鲁棒控制理论中二次约束和李雅普诺夫稳定性理论的启发, 建立了一个线性矩阵不等式去分析所提出的算法的收敛性. 当目标函数是L-光滑且m-强凸时, 算法可以达到R线性收敛. 最后几个数值仿真证明了算法的有效性和优越性. |
| 英文摘要 |
| The vanilla fractional order gradient descent may 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 our 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 |