Recently, continuous-time dynamical systems have proved useful in providing conceptual and quantitative insights into gradient-based optimization, widely used in modern machine learning and statistics. An important question that arises in this line of work is how to discretize the system in such a way that its stability and rates of convergence are preserved. In this paper we propose a geometric framework in which such discretizations can be realized systematically, enabling the derivation of ‘rate-matching’ algorithms without the need for a discrete convergence analysis. More specifically, we show that a generalization of symplectic integrators to non-conservative and in particular dissipative Hamiltonian systems is able to preserve rates of convergence up to a controlled error. Moreover, such methods preserve a shadow Hamiltonian despite the absence of a conservation law, extending key results of symplectic integrators to non-conservative cases. Our arguments rely on a combination of backward error analysis with fundamental results from symplectic geometry. We stress that although the original motivation for this work was the application to optimization, where dissipative systems play a natural role, they are fully general and not only provide a differential geometric framework for dissipative Hamiltonian systems but also substantially extend the theory of structure-preserving integration.

最近，连续时间动态系统已被证明在为基于梯度的优化提供概念和定量见解方面很有用，广泛用于现代机器学习和统计。在这一系列工作中出现的一个重要问题是如何以保持其稳定性和收敛速度的方式对系统进行离散化。在本文中，我们提出了一个几何框架，其中可以系统地实现此类离散化，从而无需离散收敛分析即可推导出“速率匹配”算法。更具体地说，我们表明辛积分器对非保守尤其是耗散哈密顿系统的推广能够保持收敛速度达到受控误差。而且，尽管没有守恒律，但这些方法保留了影子哈密顿量，将辛积分器的关键结果扩展到非守恒情况。我们的论点依赖于后向误差分析与辛几何的基本结果的结合。我们强调，虽然这项工作的最初动机是应用于优化，其中耗散系统发挥着自然的作用，但它们是完全通用的，不仅为耗散哈密顿系统提供了微分几何框架，而且大大扩展了结构保持理论一体化。