Acceleration via Symplectic Discretization of High-Resolution Differential Equations
This work addresses the challenge of efficient optimization in machine learning by providing a novel discretization approach, though it is incremental as it builds on existing ODE frameworks.
The paper tackles the problem of designing accelerated first-order optimization methods by discretizing high-resolution ODEs, showing that a symplectic scheme achieves an accelerated rate for smooth strongly convex functions, while other schemes fail or are impractical.
We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: an explicit Euler scheme, an implicit Euler scheme, and a symplectic scheme. We show that the optimization algorithm generated by applying the symplectic scheme to a high-resolution ODE proposed by Shi et al. [2018] achieves an accelerated rate for minimizing smooth strongly convex functions. On the other hand, the resulting algorithm either fails to achieve acceleration or is impractical when the scheme is implicit, the ODE is low-resolution, or the scheme is explicit.