弱单调离散时间有限时域平均场博弈的单调包含方法
Monotone Inclusion Approach to Weakly Monotone Discrete-Time Finite-Horizon Mean-Field Games
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中文总结 AI 辅助
该研究针对离散时间有限时域单调平均场博弈的均衡计算问题，将其转化为占用测度空间的优化及约束Lipschitz单调包含问题，提出无正则化的锚定近端梯度下降算法，实现O(1/√T)收敛速率且不依赖均衡唯一性。
中文摘要 AI 辅助
我们重新研究了离散时间、单调、有限时域平均场博弈（MFGs）中平均场均衡（MFEs）的计算问题。研究表明，当转移核与状态测度项无关，且奖励函数满足常规弱单调性条件并为Lipschitz连续时，可采用锚定近端梯度下降方法计算单调MFE，同时我们还建立了这些方法的最后迭代收敛结果。我们的方法核心是将计算问题转化为占用测度空间上的优化问题，通过该表述证明该问题等价于一类带约束的Lipschitz单调包含问题，随后将迭代方法应用于该单调包含表述，推导出一种易处理的算法。所得算法在T次迭代后达到O(1/√T)的收敛速率，且无需任何正则化；即使在对应MFE不满足唯一性假设的情况下，该收敛速率依然成立。
英文摘要
We revisit the problem of computing mean-field equilibria (MFEs) in discrete-time, monotone, finite-horizon mean-field games (MFGs). We show that, when the transition kernel is independent of the state-measure term and the reward function satisfies the usual weak monotonicity condition and is Lipschitz continuous, anchored proximal gradient descent methods can be used to compute a monotone MFE. We also establish last-iterate convergence results for these methods. Our approach relies on formulating the computation problem as an optimization problem over the space of occupation measures. Using this formulation, we show that the problem is equivalent to a class of constrained Lipschitz monotone inclusion problems. We then apply iterative methods for this monotone inclusion formulation to derive a tractable algorithm. The resulting algorithm achieves a convergence rate of \(O(1/\sqrt{T})\) after \(T\) iterations, without requiring any regularization. This rate holds even in the absence of a uniqueness assumption for the corresponding MFE.
