发表机构
Johns Hopkins University（约翰斯·霍普金斯大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究离散优化算法通过接触哈密顿系统转移速率证书问题，在特定假设下，\(r\)阶接触分裂可在有限时间范围转移证书，离散衰减受修正共形因子等控制，二次重球等为例，数值实验验证了理论及性能。
AI 中文摘要
离散优化算法通常通过连续时间极限常微分方程（ODE）进行分析，但ODE的收敛证书并非自动适用于离散算法。我们开发了接触哈密顿系统作为一种能使转移精确化的设置。在\(J^1(\mathbb{R}^n)\)上的接触哈密顿量\(H\)遵循内在衰减恒等式\(\dot H = -H\,\partial_s H\)，所以由\(H\)构建的增强能量\(\mathcal{E}\)，连同共形速率\(\partial_s H\)，在\(\mathcal{E}\)控制目标差距时是连续时间速率证书。我们的主要定理表明，在三个可命名且可独立检验的假设下，步长为\(h\)的\(r\)阶接触分裂在由向后误差分析设定的有限时间范围内转移此证书。离散衰减包络由修正共形因子控制，误差为\(O(h^r)\)扰动加上向后误差阴影缺陷，并且该机制被精确继承，因为修正哈密顿量本身就是接触哈密顿量。二次重球是一个完全可解的例子：其投影耗散蛙跳谱与已建立的共形辛优化理论一致，而增强接触哈密顿量产生了目标与证书的精确比较，验证了转移假设。对于具有状态依赖阻尼的强凸目标，一个明确的布雷格曼型李雅普诺夫证书通过辅助阴影推论进行转移。分解\(H = K + V + D\)为动能、目标编码势和耗散项，作为设计模板，有一系列封闭形式的子流，包括特定接触阻尼族。数值实验证实了预测的共形因子跟踪阶数，并在病态基准和深度学习任务中显示出有竞争力的性能。
英文摘要
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.
