这篇论文把连续优化理论的收敛证书精确传递到离散算法，用接触哈密顿系统给出通用框架，二次重球例子很透彻。想理解优化加速机制可以读。
本文开发了接触哈密顿系统作为优化算法中速率证书精确传递的框架。作者提出的主要定理在三个可独立验证的假设下，证明阶为r的接触分裂（步长h）能在有限时间区间内将连续时间速率证书传递给离散算法。离散衰减包络由修正共形因子控制，误差为O(h^r)加上后向误差影缺陷。以二次重球为例，其投影耗散-跳跃谱与已知共形辛优化理论一致。数值实验验证了共形因子跟踪阶，并在病态基准和深度学习任务中展现了竞争性能。
When Rates Are Geometric: Rate-Certificate Transfer for Contact Splittings in Optimization
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.