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Acceleration via Symplectic Discretization of High-Resolution Differential Equations
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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.
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On the stability of optimization algorithms given by discretizations of the Euler-Lagrange ODE
The iteration-stability bound claimed for the explicit-implicit Euler discretization of the Euler-Lagrange acceleration ODE is not proven and is false for some parameter values permitted by the theorem.
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