Gradient Convergence in Gradient methods with Errors

We consider the gradient method xₓ+₁=xₜ+ₜ (sₜ+wₜ), where sₜ is a descent direction of a function f: and wₜ is a deterministic or stochastic error. We assume that f is Lipschitz continuous, that the stepsize ₜ diminishes to 0, and that sₜ and wₜ satisfy standard conditions. We show that either f (xₜ) - or f (xₜ) converges to a finite value and f (xₜ) 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 f (xₜ), or boundedness of xt.