  Motivated by understanding the behavior of the Alternating Mirror Descent
(AMD) algorithm for bilinear zero-sum games, we study the discretization of
continuous-time Hamiltonian flow via the symplectic Euler method. We provide a
framework for analysis using results from Hamiltonian dynamics, Lie algebra,
and symplectic numerical integrators, with an emphasis on the existence and
properties of a conserved quantity, the modified Hamiltonian (MH), for the
symplectic Euler method. We compute the MH in closed-form when the original
Hamiltonian is a quadratic function, and show that it generally differs from
the other conserved quantity known previously in that case. We derive new error
bounds on the MH when truncated at orders in the stepsize in terms of the
number of iterations, $K$, and use these bounds to show an improved
$\mathcal{O}(K^{1/5})$ total regret bound and an $\mathcal{O}(K^{-4/5})$
duality gap of the average iterates for AMD. Finally, we propose a conjecture
which, if true, would imply that the total regret for AMD scales as
$\mathcal{O}\left(K^{\varepsilon}\right)$ and the duality gap of the average
iterates as $\mathcal{O}\left(K^{-1+\varepsilon}\right)$ for any
$\varepsilon>0$, and we can take $\varepsilon=0$ upon certain convergence
conditions for the MH.
