When Rates Are Geometric: Rate-Certificate Transfer for Contact Splittings in Optimization

What the math gives to ML

The paper supplies a structure-preserving route for turning continuous-time contact-Hamiltonian convergence certificates into discrete optimization guarantees. The transferable asset is not merely heavy-ball dynamics, but the fact that contact splittings preserve a modified contact Hamiltonian and inherit an intrinsic decay law with controllable finite-step error. A practical neural-network optimizer can add an auxiliary contact coordinate and replace an ad hoc momentum update with exact subflows for kinetic motion, objective gradients, and damping. The method should be evaluated both by training loss and by whether the discrete conformal-factor residual scales as the predicted O(h^r).