鞍点动力学下鞍点的渐近稳定性

A. Cherukuri, J. Cortés
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引用次数: 12

摘要

本文研究具有最小-最大鞍点的两个向量变量的连续可微函数。研究了相关鞍点动力学的渐近收敛性质(第一个变量为梯度下降,第二个变量为梯度上升)。在鞍点动力学下,我们确定了鞍点集渐近稳定的一组互补条件。我们的第一组结果是基于定义鞍点动力学的函数的凹凸性来建立收敛保证。对于不具有此特征的函数,我们的第二组结果依赖于动力学线性化的性质以及沿鞍集的近法线的函数。我们还提供了渐近收敛结果的全局版本。不同的例子说明了我们的讨论。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Asymptotic stability of saddle points under the saddle-point dynamics
This paper considers continuously differentiable functions of two vector variables that have (possibly a continuum of) min-max saddle points. We study the asymptotic convergence properties of the associated saddle-point dynamics (gradient-descent in the first variable and gradient-ascent in the second one). We identify a suite of complementary conditions under which the set of saddle points is asymptotically stable under the saddle-point dynamics. Our first set of results is based on the convexity-concavity of the function defining the saddle-point dynamics to establish the convergence guarantees. For functions that do not enjoy this feature, our second set of results relies on properties of the linearization of the dynamics and the function along the proximal normals to the saddle set. We also provide global versions of the asymptotic convergence results. Various examples illustrate our discussion.
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