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

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

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