A stochastic approximation algorithm for stochastic semidefinite programming

TL;DR

This paper introduces a distributed stochastic approximation algorithm for stochastic semidefinite programming, specifically tailored for multi-antenna wireless networks, demonstrating convergence and robustness under various practical conditions.

Contribution

It presents a novel asynchronous, distributed algorithm based on a matrix exponential scheme with entropy regularization for stochastic semidefinite problems, applicable to wireless MIMO systems.

Findings

Algorithm converges almost surely to an ε-approximate solution.

Maintains convergence under user asynchrony, delays, and changing channels.

Numerical simulations confirm robustness and scalability.

Abstract

Motivated by applications to multi-antenna wireless networks, we propose a distributed and asynchronous algorithm for stochastic semidefinite programming. This algorithm is a stochastic approximation of a continous- time matrix exponential scheme regularized by the addition of an entropy-like term to the problem's objective function. We show that the resulting algorithm converges almost surely to an $\varepsilon$-approximation of the 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. Our theoretical analysis is complemented by extensive numerical simulations which illustrate the robustness and scalability of the proposed method in realistic network conditions.