Section: New Results

Stochastic optimization

A key feature of modern data networks is their distributed nature and the stochasticity surrounding users and their possible actions. To account for these issues in a general optimization context, we proposed in [4] a distributed, asynchronous algorithm for stochastic semidefinite programming which is a stochastic approximation of the continous-time matrix exponential scheme derived in [9]. This algorithm converges almost surely to an $ϵ$-approximation of an optimal solution requiring only an unbiased estimate of the gradient of the problem's stochastic objective. When applied to throughput maximization in wireless multiple-input and multiple-output (MIMO) systems, the proposed algorithm retains its convergence properties under a wide array of mobility impediments such as user update asynchronicities, random delays and/or ergodically changing channels.

More generally, in view of solving convex optimization problems with noisy gradient input, we also analyzed in [43] the asymptotic behavior of gradient-like flows that are subject to stochastic disturbances. For concreteness, we focused on the widely studied class of mirror descent methods for constrained convex programming and we examined the dynamics' convergence and concentration properties in the presence of noise. In the small noise limit, we showed that the dynamics converge to the solution set of the underlying problem with probability 1. Otherwise, in the case of persistent noise, we estimated the measure of the dynamics' long-run concentration around interior solutions and their convergence to boundary solutions that are sufficiently “robust”. Finally, we showed that a rectified variant of the method with a decreasing sensitivity parameter converges irrespective of the magnitude of the noise or the structure of the underlying convex program, and we derived an explicit estimate for its rate of convergence.