Gradient convergence in gradient methods with errors

We consider the gradient method x(t+1) = x(t) + gamma(t)(s(t) + w(t)), where s(t) is a descent direction of a function f : R-n --> R and w(t) is a deterministic or stochastic error. We assume that del f is Lipschitz continuous, that the stepsize gamma(t) diminishes to 0, and that s(t) and w(t) satisfy standard conditions. We show that either f(x(t)) --> -infinity or f(x(t)) converges to a finite value and del f(x(t)) --> 0 (with probability 1 in the stochastic case), and in doing so, we remove various boundedness conditions that are assumed in existing results, such as boundedness from below of f, boundedness of del f(x(t)), or boundedness of x(t).