论文标题
通过惯性动力学结合粘性和Hessian驱动阻尼与时间重新缩放的快速凸优化
Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling
论文作者
论文摘要
在希尔伯特的环境中，我们开发了凸出不受约束优化的快速方法。我们依赖于将几何阻尼与时间缩放结合的惯性系统的渐近行为。凸功能最小化通过其梯度进入动态。动态包括三个系数随时间而变化，一个是粘性阻尼系数，第二个系数连接到Hessian驱动的阻尼，第三个是时间缩放系数。我们研究涉及阻尼和时间尺度系数的一般条件下值的收敛速率。获得的结果基于新的Lyapunov分析，它们涵盖了该主题的已知结果。我们特别注意渐近消失的粘性阻尼，这与Nesterov的加速梯度方法直接相关。 Hessian驱动的阻尼大大降低了振荡方面。作为主要结果，我们在不假定目标函数的强凸度的情况下获得了指数级的汇率。这些动力学的时间离散化为大量惯性优化算法打开了门。
In a Hilbert setting, we develop fast methods for convex unconstrained optimization. We rely on the asymptotic behavior of an inertial system combining geometric damping with temporal scaling. The convex function to minimize enters the dynamic via its gradient. The dynamic includes three coefficients varying with time, one is a viscous damping coefficient, the second is attached to the Hessian-driven damping, the third is a time scaling coefficient. We study the convergence rate of the values under general conditions involving the damping and the time scale coefficients. The obtained results are based on a new Lyapunov analysis and they encompass known results on the subject. We pay particular attention to the case of an asymptotically vanishing viscous damping, which is directly related to the accelerated gradient method of Nesterov. The Hessian-driven damping significantly reduces the oscillatory aspects. As a main result, we obtain an exponential rate of convergence of values without assuming the strong convexity of the objective function. The temporal discretization of these dynamics opens the gate to a large class of inertial optimization algorithms.
