We study the robustness of accelerated first-order algorithms to stochastic\nuncertainties in gradient evaluation. Specifically, for unconstrained, smooth,\nstrongly convex optimization problems, we examine the mean-squared error in the\noptimization variable when the iterates are perturbed by additive white noise.\nThis type of uncertainty may arise in situations where an approximation of the\ngradient is sought through measurements of a real system or in a distributed\ncomputation over a network. Even though the underlying dynamics of first-order\nalgorithms for this class of problems are nonlinear, we establish upper bounds\non the mean-squared deviation from the optimal solution that are tight up to\nconstant factors. Our analysis quantifies fundamental trade-offs between noise\namplification and convergence rates obtained via any acceleration scheme\nsimilar to Nesterov's or heavy-ball methods. To gain additional analytical\ninsight, for strongly convex quadratic problems, we explicitly evaluate the\nsteady-state variance of the optimization variable in terms of the eigenvalues\nof the Hessian of the objective function. We demonstrate that the entire\nspectrum of the Hessian, rather than just the extreme eigenvalues, influence\nrobustness of noisy algorithms. We specialize this result to the problem of\ndistributed averaging over undirected networks and examine the role of network\nsize and topology on the robustness of noisy accelerated algorithms.\n