基于鲁棒控制的自适应分数阶梯度优化算法设计(英文)

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.

The novel adaptive fractional order gradient decent algorithms design via robust control

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