论文标题
在几何假设下使用Hessian驱动阻尼的惯性动力学的快速收敛
Fast convergence of inertial dynamics with Hessian-driven damping under geometry assumptions
论文作者
论文摘要
一阶优化算法可以视为普通微分方程（ODES）的离散化\ cite {su2014differential}。从这个角度来看，研究相应轨迹的特性可能会导致收敛结果，这些结果可以转移到数值方案中。在本文中，我们分析了Attouch等人引入的以下ODE。在\ cite {attouch2016fast}中：\ begin {qore*} \ forall t \ geqslant t_0，〜\ ddot {x}（x}（t）+\fracα{t} {t} \ dot {x} \ dot {x}（x}（x}（x）） f（x（t））= 0，\ end {equation*}其中$α> 0 $，$β> 0 $和$ h_f $表示$ f $的hessian。该颂歌可以得出来构建不需要$ f $的数值方案，以使其两倍可减小，如\ cite {attouch2020first，attouch2021convergence}中所示。我们在$ \ | \ nabla f（x（t））上的错误$ f（x（t）） - f（x（t）） - f（x（t）） - f（x（t））\ | $在$ f $上的某些几何假设（例如围绕二次增长）的某些几何假设下，提供了强大的收敛结果。特别是，我们表明，强凸功能的误差的衰减率为$ o（t^{ - α-\ varepsilon}）$对于任何$ \ varepsilon> 0 $。这些结果在论文的末尾简要说明。
First-order optimization algorithms can be considered as a discretization of ordinary differential equations (ODEs) \cite{su2014differential}. In this perspective, studying the properties of the corresponding trajectories may lead to convergence results which can be transfered to the numerical scheme. In this paper we analyse the following ODE introduced by Attouch et al. in \cite{attouch2016fast}: \begin{equation*} \forall t\geqslant t_0,~\ddot{x}(t)+\fracα{t}\dot{x}(t)+βH_F(x(t))\dot{x}(t)+\nabla F(x(t))=0,\end{equation*} where $α>0$, $β>0$ and $H_F$ denotes the Hessian of $F$. This ODE can be derived to build numerical schemes which do not require $F$ to be twice differentiable as shown in \cite{attouch2020first,attouch2021convergence}. We provide strong convergence results on the error $F(x(t))-F^*$ and integrability properties on $\|\nabla F(x(t))\|$ under some geometry assumptions on $F$ such as quadratic growth around the set of minimizers. In particular, we show that the decay rate of the error for a strongly convex function is $O(t^{-α-\varepsilon})$ for any $\varepsilon>0$. These results are briefly illustrated at the end of the paper.
