  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.
