发表机构
Xi’an Jiaotong University; Hong Kong Polytechnic University; Huazhong University of Science and Technology; National University of Singapore; A*STAR（西安交通大学; 香港理工大学; 华中科技大学; 新加坡国立大学; 新加坡科技研究局）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究重球方法在强凸目标上的初始化问题，发现保证收敛的邻域随动量增大而缩小，并基于梯度推导出预热启动规则，确保加速收敛。
AI 中文摘要
重球方法可以加速接近极小点的收敛，但在一般强凸目标上也可能进入持续振荡。应如何初始化以可靠地实现这种局部优势？我们研究了仅利用曲率和正则性界即可保证收敛的初始化邻域。该邻域随着动量趋近于1而缩小；对于Lipschitz连续的Hessian，其半径与阻尼同阶。匹配的上界和下界表明，这种初始化要求不能在函数类上统一放宽。然后，我们基于当前梯度推导出一个启动规则：有限的梯度下降预热在切换到重球之前进入保证区域。对于强凸二次型最优的参数，后续迭代满足加速的局部渐近速率界。代码可在https://this URL获取。
英文摘要
The heavy-ball method can accelerate convergence near the minimizer, but it may also enter persistent oscillations on general strongly convex objectives. How should it be initialized to reliably realize this local advantage? We study the initialization neighborhood in which convergence can be guaranteed using only curvature and regularity bounds. This neighborhood shrinks as momentum approaches one; for Lipschitz-continuous Hessians, its radius is of the same order as damping. Matching upper and lower bounds show that this initialization requirement cannot be relaxed uniformly over the function class. We then derive a startup rule based on the current gradient: a finite gradient-descent warm-up enters the guaranteed region before switching to heavy-ball. With parameters optimal for strongly convex quadratics, the subsequent iterates satisfy an accelerated local asymptotic rate bound. Code is available at https://anonymous.4open.science/r/HeavyBall-811D/.
