  In this contribution, we present a full overview of the continuous stochastic
gradient (CSG) method, including convergence results, step size rules and
algorithmic insights. We consider optimization problems in which the objective
function requires some form of integration, e.g., expected values. Since
approximating the integration by a fixed quadrature rule can introduce
artificial local solutions into the problem while simultaneously raising the
computational effort, stochastic optimization schemes have become increasingly
popular in such contexts. However, known stochastic gradient type methods are
typically limited to expected risk functions and inherently require many
iterations. The latter is particularly problematic, if the evaluation of the
cost function involves solving multiple state equations, given, e.g., in form
of partial differential equations. To overcome these drawbacks, a recent
article introduced the CSG method, which reuses old gradient sample information
via the calculation of design dependent integration weights to obtain a better
approximation to the full gradient. While in the original CSG paper convergence
of a subsequence was established for a diminishing step size, here, we provide
a complete convergence analysis of CSG for constant step sizes and an
Armijo-type line search. Moreover, new methods to obtain the integration
weights are presented, extending the application range of CSG to problems
involving higher dimensional integrals and distributed data.
