  We formulate two classes of first-order algorithms more general than
previously studied for minimizing smooth and strongly convex or, respectively,
smooth and convex functions. We establish sufficient conditions, via new
discrete Lyapunov analyses, for achieving accelerated convergence rates which
match Nesterov's methods in the strongly and general convex settings. Next, we
study the convergence of limiting ordinary differential equations (ODEs) and
point out currently notable gaps between the convergence properties of the
corresponding algorithms and ODEs. Finally, we propose a novel class of
discrete algorithms, called the Hamiltonian assisted gradient method, directly
based on a Hamiltonian function and several interpretable operations, and then
demonstrate meaningful and unified interpretations of our acceleration
conditions.
