Motivated by applications of multi-agent learning in noisy environments, this paper studies the robustness of gradient-based learning dynamics with respect to disturbances. While disturbances injected along a coordinate corresponding to any individual player's actions can always affect the overall learning dynamics, a subset of players can be disturbance decoupled---i.e., such players' actions are completely unaffected by the injected disturbance. We provide necessary and sufficient conditions to guarantee this property for games with quadratic cost functions, which encompass quadratic one-shot continuous games, finite-horizon linear quadratic (LQ) dynamic games, and bilinear games. Specifically, disturbance decoupling is characterized by both algebraic and graph-theoretic conditions on the learning dynamics, the latter is obtained by constructing a game graph based on gradients of players' costs. For LQ games, we show that disturbance decoupling imposes constraints on the controllable and unobservable subspaces of players. For two player bilinear games, we show that disturbance decoupling within a player's action coordinates imposes constraints on the payoff matrices. Illustrative numerical examples are provided.

中文翻译：

---

具有二次成本的基于梯度的多智能体学习的干扰解耦

受多智能体学习在嘈杂环境中的应用的启发，本文研究了基于梯度的学习动态对干扰的鲁棒性。虽然沿着对应于任何个体玩家动作的坐标注入的干扰总是会影响整体学习动态，但一部分玩家可以被干扰解耦——即，这些玩家的动作完全不受注入干扰的影响。我们为具有二次成本函数的博弈提供了充分必要条件来保证这种性质，这些博弈包括二次单次连续博弈、有限水平线性二次（LQ）动态博弈和双线性博弈。具体来说，干扰解耦的特点是学习动力学的代数和图论条件，后者是通过构建基于玩家成本梯度的博弈图来获得的。对于 LQ 游戏，我们表明干扰解耦对玩家的可控和不可观察的子空间施加了约束。对于两个玩家的双线性游戏，我们展示了玩家动作坐标内的干扰解耦对支付矩阵施加了约束。提供了说明性的数字示例。