  The main aim of this paper is to provide an analysis of gradient descent (GD)
algorithms with gradient errors that do not necessarily vanish, asymptotically.
In particular, sufficient conditions are presented for both stability (almost
sure boundedness of the iterates) and convergence of GD with bounded,
(possibly) non-diminishing gradient errors. In addition to ensuring stability,
such an algorithm is shown to converge to a small neighborhood of the minimum
set, which depends on the gradient errors. It is worth noting that the main
result of this paper can be used to show that GD with asymptotically vanishing
errors indeed converges to the minimum set. The results presented herein are
not only more general when compared to previous results, but our analysis of GD
with errors is new to the literature to the best of our knowledge. Our work
extends the contributions of Mangasarian & Solodov, Bertsekas & Tsitsiklis and
Tadic & Doucet. Using our framework, a simple yet effective implementation of
GD using simultaneous perturbation stochastic approximations (SP SA), with
constant sensitivity parameters, is presented. Another important improvement
over many previous results is that there are no `additional' restrictions
imposed on the step-sizes. In machine learning applications where step-sizes
are related to learning rates, our assumptions, unlike those of other papers,
do not affect these learning rates. Finally, we present experimental results to
validate our theory.
