Variational Structure at the Edge of Stability

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.