Gradient convergence in gradient methods with errors

We consider the gradient method cursive Greek chi_t+1 = cursive Greek chi_t + γ_t(s_t + w_t), where s_t is a descent direction of a function f : ℛⁿ → ℛ and w_t is a deterministic or stochastic error. We assume that ∇f is Lipschitz continuous, that the stepsize γ_t diminishes to 0, and that s_t and w_t satisfy standard conditions. We show that either f(cursive Greek chi_t) → -∞ or f(cursive Greek chi_t) converges to a finite value and ∇ f(cursive Greek chi_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 ∇ f(cursive Greek chi_t), or boundedness of cursive Greek chi_t.