Continual Learning as Asymmetric Nash Bargaining with Absent Players

Gradient-based multi-task learning methods assume that all task gradients are available when an update is selected. Continual learning violates this assumption: past tasks are absent and can only be represented through imperfect proxies, such as replay buffers, regularization statistics, or frozen models. Unlike ordinary minibatch noise, these proxies may introduce systematic, non-vanishing errors in the gradient and curvature information that defines each task utility. We formulate task-incremental continual learning as an asymmetric $n$-player Nash bargaining problem in which absent players are represented by uncertain proxies, utilities are second order, and disagreement points are tolerable-forgetting budgets. Taking the worst case over bounded proxy errors yields an exact robust bargaining objective whose feasible set is nonempty and bounded without a trust region. We first characterize how proxy errors perturb the bargaining solution, deriving a sensitivity bound governed by the proxy error bounds, bargaining slack, and curvature conditioning. We then connect this perturbation to catastrophic forgetting through a bound that separates the ideal bargaining contribution, proxy-induced update error, and a third-order curvature remainder. Finally, we derive a damage certificate that is computable at the returned update and involves no oracle quantity. Together, these results identify bargaining slack as a central source of fragility in continual learning and provide a principled basis for bargaining with imperfect representations of past tasks. Our source code is available at https://anonymous.4open.science/r/CL-Nash-bargaining-8584/