Convergence of the Momentum Method for Semialgebraic Functions with Locally Lipschitz Gradients
提出了一个新的长度公式来研究动量方法在最小化可微半代数函数时的迭代行为，证明了局部和全局收敛性，无需全局利普希茨梯度等假设，首次保证了从任意初始点出发对矩阵分解、矩阵感知和线性神经网络应用的收敛性。
We propose a new length formula that governs the iterates of the momentum method when minimizing differentiable semialgebraic functions with locally Lipschitz gradients. It enables us to establish local convergence, global convergence, and convergence to local minimizers without assuming global Lipschitz continuity of the gradient, coercivity, and a global growth condition, as is done in the literature. As a result, we provide the first convergence guarantee of the momentum method starting from arbitrary initial points when applied to matrix factorization, matrix sensing, and linear neural networks.