On dissipative symplectic integration with applications to gradient-based optimization

• URL: http://arxiv.org/abs/2004.06840v4
• Date: Wed, 28 Apr 2021 17:24:58 GMT
• Title: On dissipative symplectic integration with applications to gradient-based optimization
• Authors: Guilherme Fran\c{c}a, Michael I. Jordan, Ren\'e Vidal
• Abstract summary: We propose a geometric framework in which discretizations can be realized systematically. We show that a generalization of symplectic to nonconservative and in particular dissipative Hamiltonian systems is able to preserve rates of convergence up to a controlled error.
• Score: 77.34726150561087
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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