Adversarial Water-Filling: Theory, Algorithms, and a Domain-Specific Wireless Foundation Model

-cross Abstract: Competitive resource allocation problems over frequency and space can be formulated as minimax interaction between transmit power and worst-case interference. This formulation naturally arises in multi-operator low Earth orbit (LEO) satellite spectrum sharing, where transmissions from competing constellations interfere in real-time. Under Gaussian channels, the corresponding power-allocation problem admits a convex-concave formulation with a unique saddle point. Discrete constellations yield generally nonconvex mercury/water-filling formulations. In this paper we propose the adversarial water-filling (AWF) problem with corresponding theory and algorithms for these settings. In addition, we develop a domain-specific wireless foundation model for AWF to learn the AWF search dynamics. The architecture incorporates permutation-invariant channel representations, a constraint-aware graph neural network (GNN) with sparse message passing, and global latent variables capturing the low-dimensional water level implied by the AWF optimality. Through learned projected extragradient iterations, the model approximates stationary solutions of the constrained minimax problem arising under mercury/water-filling. We further establish projected-stationarity/Karush-Kuhn-Tucker consistency and conditional local convergence of the learned AWF dynamics under local regularity and stability conditions. Experiments demonstrate empirical generalization across unseen problem sizes, constraint structures, discrete constellations, and channel-weighted objectives, while achieving a median speedup exceeding one order of magnitude over Mirror-Prox on matched instances at comparable first-order solution quality.The related code can be found at https://github.com/convexsoft/AWF.