Convergence of Momentum-Based Heavy Ball Method with Batch Updating and/or Approximate Gradients

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