Convergence Rates for Variational Inequality Projection Neural Networks with a State-Dependent Metric

What the math gives to ML

The paper develops projected dynamical systems with a state-dependent positive-definite matrix metric, extending beyond fixed Euclidean metrics and Hessian-generated metrics. The transferable asset is a stability framework linking strong monotonicity of an operator, spectral bounds on the preconditioner, and Lipschitz control of metric variation to exponential convergence. It also identifies a concrete failure mode: sufficiently variable non-Hessian metrics can create periodic orbits even for strongly monotone linear operators. A neural optimizer can therefore use adaptive preconditioning only after imposing eigenvalue clipping and explicit per-step variation limits, especially when training under convex parameter constraints.