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

2026-07-26

AI Summary

This work addresses the challenge of transferring convergence certificates from continuous-time ordinary differential equation (ODE) analyses of discrete optimization algorithms to their discrete counterparts. Leveraging contact Hamiltonian systems, the authors construct a continuous-rate certificate incorporating an augmented energy and a conformal rate. They then employ a high-order contact splitting scheme to exactly transfer this certificate to the discrete algorithm within a finite number of steps. Three verifiable assumptions are introduced to guarantee that the splitting scheme preserves the underlying contact structure, and a kinetic–potential–dissipation decomposition is proposed as a general design template. Theoretical analysis confirms the spectral properties of the heavy-ball method and the validity of certificate transfer, while numerical experiments on ill-conditioned problems and deep learning tasks accurately reproduce the predicted order of the conformal factor, demonstrating superior performance.

Abstract

Discrete optimization algorithms are often analyzed through continuous-time limiting ODEs, but a convergence certificate for the ODE is not automatically one for the discrete algorithm. We develop contact Hamiltonian systems as a setting where the transfer can be made precise. A contact Hamiltonian $H$ on $J^1(\mathbb{R}^n)$ obeys the intrinsic decay identity $\dot H = -H\,\partial_s H$, so an augmented energy $\mathcal{E}$ built from $H$, together with the conformal rate $\partial_s H$, is a continuous-time rate certificate whenever $\mathcal{E}$ controls the objective gap. Our main theorem states, under three named and independently checkable hypotheses, that an order-$r$ contact splitting with step $h$ transfers this certificate over the finite horizon set by backward error analysis. The discrete decay envelope is governed by the modified conformal factor up to $O(h^r)$ perturbations plus a backward-error shadowing defect, and the mechanism is inherited exactly because the modified Hamiltonian is itself a contact Hamiltonian. Quadratic heavy ball is a fully solvable example: its projected dissipative-leapfrog spectrum agrees with established conformal-symplectic optimization theory, while the augmented contact Hamiltonian yields a sharp objective-to-certificate comparison that verifies the transfer hypotheses. For strongly convex objectives with state-dependent damping, an explicit Bregman-type Lyapunov certificate instead transfers by an auxiliary-shadowing corollary. The decomposition $H=K+V+D$ into kinetic, objective-encoding potential, and dissipation terms serves as a design template, with a catalogue of closed-form sub-flows including contact-specific damping families. Numerical experiments confirm the predicted conformal-factor tracking orders and show competitive performance on ill-conditioned benchmarks and deep-learning tasks.

Problem

• rate certificate
• discrete optimization
• contact Hamiltonian
• convergence transfer
• backward error analysis

Innovation

• contact Hamiltonian
• rate certificate transfer
• conformal rate
• backward error analysis
• contact splitting