AI 中文总结
该研究针对紧致状态空间上的测度值变分不等式，提出KL镜像近端算法，结合正则化方法，在单调性假设下证明收敛性与误差界，可处理势型及非势型代价算子。
AI 中文摘要
我们研究紧致状态空间上静态平均场均衡的计算，将均衡条件表述为概率测度上的变分不等式。我们提出了Korpelevich额外梯度算法的熵变体——Kullback-Leibler镜像近端（KL Mirror-Prox）方法，其中欧氏投影被替换为相对熵近端步。因此，每半步都是当前测度在有限状态空间离散化上的显式指数重加权。在Lasry-Lions单调性和连续性假设下，我们证明了网格细化遍历平均的收敛性，并得到了有限迭代的Minty残差和近似均衡界，该界联合量化了迭代与离散化误差。在强单调性下，我们推导了最后、最优及平均迭代的度量收敛速率。我们还开发了KL型Tikhonov正则化，用于选择相对于参考测度最小化相对熵的均衡。该框架适用于势型和非势型代价算子，且不要求个体状态下代价的可微性或凸性。
英文摘要
We study the computation of static mean-field equilibria on a compact state space by formulating the equilibrium condition as a variational inequality over probability measures. We propose an entropic variant of Korpelevich's extragradient algorithm---the Kullback--Leibler Mirror-Prox method---in which Euclidean projections are replaced by relative-entropy proximal steps. Each half-step is therefore an explicit exponential reweighting of the current measure, implemented on a finite state-space discretization. Under Lasry--Lions monotonicity and continuity assumptions, we prove convergence of mesh-refined ergodic averages and obtain finite-iteration Minty-residual and approximate-equilibrium bounds that jointly quantify iteration and discretization errors. Under strong monotonicity, we derive metric convergence rates for the last, best, and averaged iterates. We also develop a KL-type Tikhonov regularization that selects the equilibrium minimizing relative entropy with respect to a reference measure. The framework applies to potential and nonpotential cost operators and does not require differentiability or convexity of the cost in the individual state.
Comments45 pages, 7 figures
