On the noise amplification of primal-dual gradient flow dynamics based on proximal augmented Lagrangian

In this paper, we examine amplification of additive stochastic disturbances to primal-dual gradient flow dynamics based on proximal augmented Lagrangian. These dynamics can be used to solve a class of non-smooth composite optimization problems and are convenient for distributed implementation. We utilize the theory of integral quadratic constraints to show that the upper bound on noise amplification is inversely proportional to the strong-convexity module of the smooth part of the objective function. Furthermore, to demonstrate tightness of these upper bounds, we exploit the structure of quadratic optimization problems and derive analytical expressions in terms of the eigenvalues of the corresponding dynamical generators. We further specialize our results to a distributed optimization framework and discuss the impact of network topology on the noise amplification.