In this paper, we study the well-known "Heavy Ball" method for convex and nonconvex optimization introduced by Polyak in 1964, and establish its convergence under a variety of situations. Traditionally, most algorithms use "full-coordinate update," that is, at each step, every component of the argument is updated. However, when the dimension of the argument is very high, it is more efficient to update some but not all components of the argument at each iteration. We refer to this as "batch updating" in this paper. When gradient-based algorithms are used together with batch updating, in principle it is sufficient to compute only those components of the gradient for which the argument is to be updated. However, if a method such as backpropagation is used to compute these components, computing only some components of gradient does not offer much savings over computing the entire gradient. Therefore, to achieve a noticeable reduction in CPU usage at each step, one can use first-order differences to approximate the gradient. The resulting estimates are biased, and also have unbounded variance. Thus some delicate analysis is required to ensure that the HB algorithm converge when batch updating is used instead of full-coordinate updating, and/or approximate gradients are used instead of true gradients. In this paper, we establish the almost sure convergence of the iterations to the stationary point(s) of the objective function under suitable conditions; in addition, we also derive upper bounds on the rate of convergence. To the best of our knowledge, there is no other paper that combines all of these features. This paper is dedicated to the memory of Boris Teodorovich Polyak

翻译：本文研究了由Polyak于1964年提出的著名“重球”方法在凸与非凸优化问题中的应用，并建立了其在多种情境下的收敛性。传统上，大多数算法采用“全坐标更新”，即每一步更新自变量的所有分量。然而，当自变量维数极高时，每次迭代仅更新部分分量而非全部分量更为高效。本文将此称为“批量更新”。当梯度类算法与批量更新结合使用时，原则上只需计算待更新自变量分量对应的梯度分量。然而，若采用反向传播等方法计算这些分量，仅计算部分梯度分量相比计算完整梯度节省的计算资源有限。因此，为显著降低每一步的CPU使用量，可利用一阶差分来近似梯度。由此产生的估计量存在偏差且方差无界，因此需要精细的分析以确保重球算法在使用批量更新替代全坐标更新，和/或使用近似梯度替代真实梯度时仍能收敛。本文在适当条件下证明了迭代序列几乎必然收敛至目标函数的稳定点，并进一步推导了收敛速率的上界。据我们所知，尚无其他文献综合涵盖所有这些特性。本文谨献给Boris Teodorovich Polyak的纪念。