Gradient Convergence in Gradient methods with Errors

We consider the gradient method xt+1 = xt + γt(st + wt), where st is a descent direction of a function f : �n → � and wt is a deterministic or stochastic error. We assume that ∇f is Lipschitz continuous, that the stepsize γt 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.