Nonlinear Forward-Backward Algorithm for Solving Non-monotone+Lipschitz Inclusions with Applications to Adjoint Mismatch Problems

What the math gives to ML

The paper develops a forward-backward splitting framework that remains convergent when one forward operator is Lipschitz but non-monotone, a regime that includes approximate adjoints and imperfect gradient-like operators. The transferable asset is the warped resolvent: a variable metric or nonlinear preconditioner can absorb part of the operator mismatch, while an explicit semimonotonicity-dependent lower bound controls instability. A promising neural-network adaptation is a safeguarded proximal optimizer for nonconvex training, using curvature or minibatch-gradient mismatch estimates to choose the metric scale and step size rather than assuming ordinary monotonicity of the loss gradient.