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We formulate two classes of first-order algorithms more general thanpreviously studied for minimizing smooth and strongly convex or, respectively,smooth and convex functions. We establish sufficient conditions, via newdiscrete Lyapunov analyses, for achieving accelerated convergence rates whichmatch Nesterov's methods in the strongly and general convex settings. Next, westudy the convergence of limiting ordinary differential equations (ODEs) andpoint out currently notable gaps between the convergence properties of thecorresponding algorithms and ODEs. Finally, we propose a novel class ofdiscrete algorithms, called the Hamiltonian assisted gradient method, directlybased on a Hamiltonian function and several interpretable operations, and thendemonstrate meaningful and unified interpretations of our accelerationconditions.

Understanding Accelerated Gradient Methods: Lyapunov Analyses and Hamiltonian Assisted Interpretations