Figure 1 - uploaded by Valentin Duruisseaux
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Error vs. Iterations number as the standard PolyHTVI and ExpoHTVI algorithms and their restarted versions (Function (F), Gradient (G) and Velocity (V)) are applied to Problem 1 (left), Problem 2 (middle), and Problem 3 (right).
Source publication
Geometric numerical integration has recently been exploited to design symplectic accelerated optimization algorithms by simulating the Lagrangian and Hamiltonian systems from the variational framework introduced in Wibisono et al. In this paper, we discuss practical considerations which can significantly boost the computational performance of these...
Citations
... We will also study the convergence properties of the discrete-time algorithms, and try to better understand how to reconcile the Nesterov barrier theorem with the convergence properties of the continuous Bregman flows. It would also be useful to study the extent to which the practical considerations recently presented in [69], which significantly improved the computational performance of the symplectic optimization algorithms in the normed vector space setting, extend to the Riemannian manifold and Lie group settings with the Lagrangian Riemannian and Lie group variational integrators. ...
A variational framework for accelerated optimization was recently introduced on normed vector spaces and Riemannian manifolds in [1] and [2]. It was observed that a careful combination of time-adaptivity and symplecticity in the numerical integration can result in a significant gain in computational efficiency. It is however well known that symplectic integrators lose their near-energy preservation properties when variable time-steps are used. The most common approach to circumvent this problem involves the Poincaré transformation on the Hamiltonian side, and was used in [3] to construct efficient explicit algorithms for symplectic accelerated optimization. However, the current formulations of Hamiltonian variational integrators do not make intrinsic sense on more general spaces such as Riemannian manifolds and Lie groups. In contrast, Lagrangian variational integrators are well-defined on manifolds, so we develop here a framework for time-adaptivity in Lagrangian variational integrators and use the resulting geometric integrators to solve optimization problems on vector spaces and Lie groups.
