In this paper, we analyze several methods for approximating gradients of noisy functions using only function values. These methods include finite differences, linear interpolation, Gaussian smoothing, and smoothing on a sphere. The methods differ in the number of functions sampled, the choice of the sample points, and the way in which the gradient approximations are derived. For each method, we derive bounds on the number of samples and the sampling radius which guarantee favorable convergence properties for a line search or fixed step size descent method. To this end, we use the results in Berahas et al. (Global convergence rate analysis of a generic line search algorithm with noise, arXiv:1910.04055, 2019) and show how each method can satisfy the sufficient conditions, possibly only with some sufficiently large probability at each iteration, as happens to be the case with Gaussian smoothing and smoothing on a sphere. Finally, we present numerical results evaluating the quality of the gradient approximations as well as their performance in conjunction with a line search derivative-free optimization algorithm.

在本文中，我们分析了仅使用函数值来逼近噪声函数梯度的几种方法。这些方法包括有限差分，线性插值，高斯平滑和球体平滑。这些方法的不同之处在于采样的函数数量，采样点的选择以及得出梯度近似值的方式。对于每种方法，我们得出样本数量和采样半径的界限，这些界限保证了线搜索或固定步长下降方法的有利收敛性。为此，我们使用Berahas等人的结果。（带有噪声的通用线搜索算法的全局收敛速度分析，arXiv：1910.04055，2019）并展示了每种方法如何满足足够的条件，可能仅在每次迭代时才具有足够大的概率，就像高斯平滑和球面上的平滑一样。最后，我们结合线搜索无导数优化算法，给出了评估梯度近似质量及其性能的数值结果。