论文标题
关于功能的一阶近似质量
On the Quality of First-Order Approximation of Functions with Hölder Continuous Gradient
论文作者
论文摘要
我们表明，梯度的Hölder连续性不仅是足够的条件，而且是在一阶Taylor近似误差上存在全局上限的必要条件。我们还将这种全球上限与梯度的Hölder常数联系起来。该关系表示为间隔，具体取决于Hölder常数，其中保证一阶泰勒近似值的误差为。我们表明，对于Lipschitz的连续情况，无法减少间隔。提出了二次形式规范的应用，这使我们能够得出欧几里得规范的新颖表征。
We show that Hölder continuity of the gradient is not only a sufficient condition, but also a necessary condition for the existence of a global upper bound on the error of the first-order Taylor approximation. We also relate this global upper bound to the Hölder constant of the gradient. This relation is expressed as an interval, depending on the Hölder constant, in which the error of the first-order Taylor approximation is guaranteed to be. We show that, for the Lipschitz continuous case, the interval cannot be reduced. An application to the norms of quadratic forms is proposed, which allows us to derive a novel characterization of Euclidean norms.
