On the probability distribution of distributed optimization strategies

We study the steady-state probability distribution of diffusion and consensus strategies that employ constant step-sizes to enable continuous adaptation and learning. We show that, in the small step-size regime, the estimation error at each agent approaches a Gaussian distribution. More importantly, the covariance matrix of this distribution is shown to coincide with the error covariance matrix that would result from a centralized stochastic-gradient strategy. The results hold regardless of the connected topology and help clarify the convergence and learning behavior of distributed strategies in an interesting way.