On dissipative symplectic integration with applications to gradient-based optimization
| Autor: | França, Guilherme, Jordan, Michael I., Vidal, René |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | J. Stat. Mech. (2021) 043402 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1088/1742-5468/abf5d4 |
| Popis: | 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 nonconservative 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 nonconservative 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. Comment: matches published version |
| Databáze: | arXiv |
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