发表机构
Yale University（耶鲁大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究将Litman提出的边缘耦合扩展到重球和Nesterov动量，证明其临界点可表征梯度下降的不动点与两点轨道，海森矩阵可表征稳定性，且其与对称Verlet作用等同，确立了稳定性边缘与离散力学的联系。
AI 中文摘要
当离散时间优化器在稳定性边缘运行时，会呈现接近双周期的行为。这种振荡动力学让人联想到保守系统，比如辛积分器生成的动力学。然而，离散时间优化器在稳定性边缘与离散力学之间联系的精确公式仍未得到充分探索。最近，Litman引入了“边缘耦合”：一种关于连续梯度下降迭代的泛函，其临界点编码梯度下降动力学的不动点和两点轨道。在此，我们将边缘耦合扩展到重球（heavy-ball）和Nesterov动量。我们表明，其临界点表征不动点和两点轨道，其海森矩阵（Hessian）表征它们的稳定性。我们还表明，边缘耦合可与对称Verlet作用等同，正式确立了稳定性边缘与离散力学之间的联系。
英文摘要
When discrete-time optimizers operate at the edge of stability, they exhibit near-two-periodic behavior. These oscillatory dynamics are reminiscent of conservative systems, such as the dynamics generated by symplectic integrators. However, a precise formulation of the connection between discrete-time optimizers at the edge of stability and discrete mechanics remains underexplored. Recently, Litman introduced the "edge coupling": a functional on consecutive gradient descent iterates whose critical points encode the fixed points and two-point orbits of the gradient descent dynamics. Here we extend the edge coupling to heavy-ball and Nesterov momentum. We show that its critical points characterize the fixed points and two-point orbits, with its Hessian characterizing their stability. We also show that the edge coupling can be identified with the symmetric Verlet action, formalizing the connection between the edge of stability and discrete mechanics.
