Convergence of Gradient Methods with Deterministic and Bounded Noise

Abstract

In this paper, we analyse the effects of noise on the gradient methods for solving a convex unconstraint optimization problem. Assuming that the objective function is with Lipschitz continuous gradients, we analyse the convergence properties of the gradient method when the noise is deterministic and bounded. Our theoretical results show that the gradient algorithm converges to the related optimality within some tolerance, where the tolerance depends on the underlying noise, step size, and the gradient Lipschitz continuity constant of the underlying objective function. Moreover, we consider an application of distributed optimization, where the objective function is a sum of two strongly convex functions. Then the related convergences are discussed based on dual decomposition together with gradient methods, where the associated noise is considered as a consequence of quantization errors. Finally, the theoretical results are verified using numerical experiments. Keywords: The gradient method; deterministic and bounded noise; distributed optimization; dual decomposition