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.

翻译：受理解交替镜像下降算法在双线性零和博弈中行为的启发，我们研究了通过辛欧拉方法对连续时间哈密顿流进行离散化的问题。我们利用哈密顿动力学、李代数和辛数值积分器的结果建立了一个分析框架，重点研究了辛欧拉方法所保持的守恒量——修正哈密顿量的存在性与性质。当原始哈密顿量为二次函数时，我们以闭式形式计算了修正哈密顿量，并证明它通常不同于该情况下先前已知的其他守恒量。我们推导了修正哈密顿量在步长阶数截断时关于迭代次数 $K$ 的新误差界，并利用这些界证明了交替镜像下降算法改进的 $\mathcal{O}(K^{1/5})$ 总遗憾界以及平均迭代点的 $\mathcal{O}(K^{-4/5})$ 对偶间隙。最后，我们提出了一个猜想：若该猜想成立，则对于任意 $\varepsilon>0$，交替镜像下降的总遗憾可缩放到 $\mathcal{O}\left(K^{\varepsilon}\right)$，平均迭代点的对偶间隙可缩放到 $\mathcal{O}\left(K^{-1+\varepsilon}\right)$；并且在修正哈密顿量满足特定收敛条件时，可取 $\varepsilon=0$。