量子零和博弈中乐观矩阵镜像近端的平均迭代与最后迭代下界
Average-and Last-Iterate Lower Bounds for Optimistic Matrix Mirror-Prox in Quantum Zero-Sum Games
- University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究通过显式量子零和博弈，证明了乐观矩阵镜像近端算法在平均迭代和最后迭代收敛上的下界，并揭示了乐观梯度下降-上升和乐观矩阵乘法权重更新的收敛速率。
AI中文摘要：
乐观矩阵镜像近端（OMMP）算法在量子零和博弈中计算ε近似纳什均衡，并具有O(1/ε)的平均迭代保证[arXiv:2311.10859]。我们研究这种对精度的依赖是否紧密，以及是否能保证几何的最后迭代收敛。我们通过每个玩家一个量子比特的显式博弈来研究这些问题。首先，我们证明了均匀平均输出的Ω(1/ε)下界，该下界包括最大混合初始状态，且与正则化器和步长无关。其次，我们构造了一个固定博弈，在该博弈上，从最大混合状态初始化的乐观梯度下降-上升（OGDA）算法，对于每个足够小的固定步长，其最后迭代到均衡的Frobenius距离为Θ(1/t)，对偶间隙为Θ(1/t^3)。另一个固定博弈展示了在一系列初始状态中，将初始误差减少常数因子时可能出现任意长的延迟。最后，我们给出一个具有唯一严格互补均衡的固定博弈，在该博弈上，乐观矩阵乘法权重更新（OMMWU）从最大混合状态出发，对于每个固定的正步长，仅以多项式速度收敛。从均衡到迭代的最后迭代Frobenius距离和量子相对熵以Θ(1/t)衰减，而对偶间隙以Θ(1/t^2)衰减。
英文摘要：
Optimistic matrix mirror-prox (OMMP) computes $ε$-approximate Nash equilibria in quantum zero-sum games with an $O(1/\varepsilon)$ average-iterate guarantee [arXiv:2311.10859]. We investigate whether this dependence on accuracy is tight and whether geometric last-iterate convergence can be guaranteed. We study these questions through explicit games with one qubit per player. First, we prove an $Ω(1/\varepsilon)$ lower bound for the uniform-average output that includes the maximally mixed initial state, independently of the regularizer and step size. Second, we construct a fixed game on which optimistic gradient descent-ascent (OGDA), initialized at the maximally mixed state, has last-iterate Frobenius distance to equilibrium $Θ(1/t)$ and duality gap $Θ(1/t^3)$ for every sufficiently small fixed step size. A separate fixed game exhibits arbitrarily long delays in reducing the initial error by a constant factor across a family of initial states. Finally, we give a fixed game with a unique, strictly complementary equilibrium on which optimistic matrix multiplicative weights updates (OMMWU) converge only polynomially from the maximally mixed state for every fixed positive step size. The last-iterate Frobenius distance and quantum relative entropy from the equilibrium to the iterates decay as $Θ(1/t)$, while the duality gap decays as $Θ(1/t^2)$.
