The impact of stochastic noisy feedback on distributed network utility maximization

The implementation of distributed network utility maximization (NUM) algorithms hinges heavily on information feedback through message passing among network elements. In practical systems the feedback is often obtained using errorprone measurement mechanisms and suffers from random errors. There has been little work in this direction, and by and large the impact of noisy feedback remains unclear. A main objective of this study is to fill this void and to obtain a rigorous and systematic understanding of the impact of stochastic noisy feedback. In this paper, we consider distributed NUM in multi-hop wireless networks, and focus on the impact of noisy feedback on the distributed algorithms based on the Lagrangian dual method. These algorithms can in general be regarded as some form of gradient (or sub-gradient) based methods. Assuming strong duality, we study both cases when the stochastic gradients are unbiased or biased, and develop a general theory on the stochastic stability of these algorithms in the presence of noisy feedback. When the gradient estimator is unbiased, we establish, via a combination of the stochastic Lyapunov Stability Theorem and local analysis, that the iterates generated by distributed NUM algorithms converge with probability one to the optimal point, under standard technical conditions. In contrast, when the gradient estimator is biased, we show that the iterates converge to a contraction region around the optimal point, provided that the biased terms are asymptotically bounded by a scaled version of the true gradients. We also investigate the rate of convergence for the unbiased case, and find that, in general, the limit process of the interpolated process corresponding to the normalized iterate sequence is a stationary reflected linear diffusion process, not necessarily a Gaussian diffusion process. We also apply the above general theory to investigate stability of cross-layer rate control for joint congestion control and random access. Our numerical examples corroborate the theoretic findings well.