交替镜像下降的辛分析
A Symplectic Analysis of Alternating Mirror Descent
AI总结：
本文针对双线性零和博弈中的AMD算法，基于辛欧拉方法结合哈密顿动力学等理论分析其离散化特性，推导得到更优的遗憾与对偶间隙界，并提出关于收敛量级的猜想。
摘要：
为厘清双线性零和博弈中交替镜像下降（Alternating Mirror Descent，AMD）算法的行为特性，本文研究了采用辛欧拉方法对连续时间哈密顿流进行离散化的过程。我们基于哈密顿动力学、李代数与辛数值积分器的相关结论构建分析框架，重点探究辛欧拉方法下修正哈密顿量（modified Hamiltonian，MH）这一守恒量的存在性与性质。当原始哈密顿量为二次函数时，我们推导得到了MH的闭式表达式，证明其通常与该场景下此前已知的其他守恒量存在差异。我们推导得到了按步长阶数截断时MH的全新误差界，该界以迭代次数K为变量；利用这些界，我们证明AMD可达到更优的O(K^{1/5})总遗憾界，以及平均迭代点O(K^{-4/5})的对偶间隙。最后我们提出一个猜想：若该猜想成立，则对任意ε>0，AMD的总遗憾量级为O(K^ε)，平均迭代点的对偶间隙量级为O(K^{-1+ε})；当MH满足特定收敛条件时，可取ε=0。
英文摘要：
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.
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