Sequential Stability of the Value Function and the Solution Mapping in Berge's Maximum Theorem via Variational Convergence

What the math gives to ML

The paper develops constructive stability criteria for value functions and argmax solution mappings under simultaneous perturbations of objectives and feasible sets. Its most transferable asset is the distinction between value stability and optimizer-set stability: convergent objective values do not guarantee that selected solutions remain stable, and inner convergence can fail even under continuous convergence. This directly applies to hard top-k routing, attention sparsification, and discrete action heads. A practical adaptation is to train routers with perturbation consistency and explicit selection margins, then test graph stability through route agreement under logit and mask perturbations.