一种简单形导数浮点误差分析的一般框架
A general framework for floating point error analysis of simplex derivatives
AI总结：
本文提出了一种用于分析简单形导数浮点误差的一般框架，讨论了简单形梯度和中心简单形梯度的精度，并给出了样本集近似直径的最小选择建议。
摘要：
梯度近似是一种在数值优化中至关重要的数值近似技术。在无梯度优化中，大多数梯度近似，包括简单形梯度、中心简单形梯度和适应中心简单形梯度，均以简单形导数形式出现。由于机器精度，任何数值近似技术的近似精度都受浮点误差的影响。本文提供了一种简单形导数的浮点误差分析的一般框架。该框架不依赖于所选简单形导数，只要其满足一般形式。我们回顾了广义简单形梯度和广义中心简单形梯度的定义和近似精度。我们定义并分析了广义适应中心简单形梯度的精度。作为例子，我们将该框架应用于广义简单形梯度、广义中心简单形梯度和广义适应中心简单形梯度。基于结果，我们给出了样本集近似直径的最小选择建议。
英文摘要：
Gradient approximations are a class of numerical approximation techniques that are of central importance in numerical optimization. In derivative-free optimization, most of the gradient approximations, including the simplex gradient, centred simplex gradient, and adapted centred simplex gradient, are in the form of simplex derivatives. Owing to machine precision, the approximation accuracy of any numerical approximation technique is subject to the influence of floating point errors. In this paper, we provide a general framework for floating point error analysis of simplex derivatives. Our framework is independent of the choice of the simplex derivative as long as it satisfies a general form. We review the definition and approximation accuracy of the generalized simplex gradient and generalized centred simplex gradient. We define and analyze the accuracy of a generalized version of the adapted centred simplex gradient. As examples, we apply our framework to the generalized simplex gradient, generalized centred simplex gradient, and generalized adapted centred simplex gradient. Based on the results, we give suggestions on the minimal choice of approximate diameter of the sample set.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
