Performance of noisy Nesterov's accelerated method for strongly convex optimization problems

We study the performance of noisy gradient descent and Nesterov's accelerated methods for strongly convex objective functions with Lipschitz continuous gradients. The steady-state second-order moment of the error in the iterates is analyzed when the gradient is perturbed by an additive white noise with zero mean and identity covariance. For any given condition number $\kappa$, we derive explicit upper bounds on noise amplification that only depend on $\kappa$ and the problem size. We use quadratic objective functions to derive lower bounds and to demonstrate that the upper bounds are tight up to a constant factor. The established upper bound for Nesterov's accelerated method is larger than the upper bound for gradient descent by a factor of $\sqrt{\kappa}$. This gap identifies a fundamental tradeoff that comes with acceleration in the presence of stochastic uncertainties in the gradient evaluation.