SIAM Journal on Optimization, Volume 31, Issue 2, Page 1489-1518, January 2021.
In this paper, we develop convergence analysis of a modified line search method for objective functions whose value is computed with noise and whose gradient estimates are inexact and possibly random. The noise is assumed to be bounded in absolute value without any additional assumptions. We extend the framework based on stochastic methods from [C. Cartis and K. Scheinberg, Math. Program., 169 (2018), pp. 337--375] which was developed to provide analysis of a standard line search method with exact function values and random gradients to the case of noisy functions. We introduce two alternative conditions on the gradient which, when satisfied with some sufficiently large probability at each iteration, guarantees convergence properties of the line search method. We derive expected complexity bounds to reach a near optimal neighborhood for convex, strongly convex and nonconvex functions. The exact dependence of the convergence neighborhood on the noise is specified.

中文翻译：

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一种带噪声的通用线搜索算法的全局收敛率分析

SIAM Journal on Optimization，第 31 卷，第 2 期，第 1489-1518 页，2021 年 1 月。
在本文中，我们针对目标函数开发了一种改进的线搜索方法的收敛性分析，目标函数的值是用噪声计算的，其梯度估计是不精确的，可能是随机的。假设噪声的绝对值是有界的，没有任何额外的假设。我们基于来自 [C. Cartis 和 K. Scheinberg，数学。Program., 169 (2018), pp. 337--375] 其开发目的是为噪声函数的情况提供具有精确函数值和随机梯度的标准线搜索方法的分析。我们在梯度上引入了两个替代条件，当每次迭代满足足够大的概率时，保证线搜索方法的收敛性。我们推导出预期的复杂度界限，以达到凸面的近最优邻域，强凸函数和非凸函数。指定了收敛邻域对噪声的确切依赖性。