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

arXiv: https://arxiv.org/abs/2607.23642 · HTML: https://mlsanity.com/paper/2607.23642/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 43 references that are in the map.
- First posted on arXiv 26 Jul 2026.

## Builds on (most cited first, 10 of 10)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Stochastic Gradient Hamiltonian Monte Carlo](https://mlsanity.com/paper/1402.4102/) (2014)
- [A Differential Equation for Modeling Nesterov's Accelerated Gradient Method: Theory and Insights](https://mlsanity.com/paper/1503.01243/) (2015)
- [A Variational Perspective on Accelerated Methods in Optimization](https://mlsanity.com/paper/1603.04245/) (2016)
- [Understanding the Acceleration Phenomenon via High-Resolution Differential Equations](https://mlsanity.com/paper/1810.08907/) (2018)
- [A Dynamical Systems Perspective on Nesterov Acceleration](https://mlsanity.com/paper/1905.07436/) (2019)
- [Conformal Symplectic and Relativistic Optimization](https://mlsanity.com/paper/1903.04100/) (2019)
- [On dissipative symplectic integration with applications to gradient-based optimization](https://mlsanity.com/paper/2004.06840/) (2020)
- [Dissipativity Theory for Nesterov's Accelerated Method](https://mlsanity.com/paper/1706.04381/) (2017)
- [Bregman dynamics, contact transformations and convex optimization](https://mlsanity.com/paper/1912.02928/) (2019)

## Abstract (arXiv)

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 J1(ℝn) obeys the intrinsic decay identity dot H = -H,partials H, so an augmented energy E built from H, together with the conformal rate partials H, is a continuous-time rate certificate whenever 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(hr) 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.