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In this paper, we present a unified and general framework for analyzing thebatch updating approach to nonlinear, high-dimensional optimization. Theframework encompasses all the currently used batch updating approaches, and isapplicable to nonconvex as well as convex functions. Moreover, the frameworkpermits the use of noise-corrupted gradients, as well as first-orderapproximations to the gradient (sometimes referred to as "gradient-free"approaches). By viewing the analysis of the iterations as a problem in theconvergence of stochastic processes, we are able to establish a very generaltheorem, which includes most known convergence results for zeroth-order andfirst-order methods. The analysis of "second-order" or momentum-based methodsis not a part of this paper, and will be studied elsewhere. However, numericalexperiments indicate that momentum-based methods can fail if the true gradientis replaced by its first-order approximation. This requires further theoreticalanalysis.

Convergence of Batch Updating Methods with Approximate Gradients and/or Noisy Measurements: Theory and Computational Results