SIAM Journal on Optimization, Volume 30, Issue 3, Page 1850-1877, January 2020.
In this article a family of second-order ODEs associated with the inertial gradient descent is studied. These ODEs are widely used to build trajectories converging to a minimizer $x^*$ of a function $F$, possibly convex. This family includes the continuous version of the Nesterov inertial scheme and the continuous heavy ball method. Several damping parameters, not necessarily vanishing, and a perturbation term $g$ are thus considered. The damping parameter is linked to the inertia of the associated inertial scheme and the perturbation term $g$ is linked to the error that can be made on the gradient of the function $F$. This article presents new asymptotic bounds on $F(x(t))-F(x^*)$, where $x$ is a solution of the ODE, when $F$ is convex and satisfies local geometrical properties such as Łojasiewicz properties and under integrability conditions on $g$. Even if geometrical properties and perturbations were already studied for most ODEs of these families, it is the first time they are jointly studied. All these results give an insight on the behavior of these inertial and perturbed algorithms if $F$ satisfies some Łojasiewicz properties especially in the setting of stochastic algorithms.

中文翻译：

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几何条件和摄动条件下阻尼惯性动力学的收敛速度

SIAM优化杂志，第30卷，第3期，第1850-1877页，2020年1月。
在本文中，研究了与惯性梯度下降相关的二阶ODE族。这些ODE被广泛用于构建轨迹，收敛到函数$ F $的极小化子$ x ^ * $，可能是凸的。该族包括Nesterov惯性方案的连续版本和连续重球方法。因此考虑了几个阻尼参数，不一定消失，以及一个扰动项$ g $。阻尼参数链接到相关惯性方案的惯性，而扰动项$ g $链接到可以对函数$ F $的梯度进行的误差。本文介绍了$ F（x（t））-F（x ^ *）$的新渐近界线，其中$ x $是ODE的解，当$ F $凸且满足局部几何特性（例如Łojasiewicz特性）并且在可积条件下满足$ g $时。即使已经对这些族的大多数ODE进行了几何性质和微扰的研究，但这仍是首次对其进行联合研究。如果$ F $满足某些Łojasiewicz属性，尤其是在随机算法的设置中，所有这些结果将提供对这些惯性和扰动算法的行为的了解。