A Converse Result on Convergence Time for Opportunistic Wireless Scheduling

Abstract

This paper proves an impossibility result for stochastic network utility maximization for multi-user wireless systems, including multi-access and broadcast systems. Every time slot an access point observes the current channel states and opportunistically selects a vector of transmission rates. Channel state vectors are assumed to be independent and identically distributed with an unknown probability distribution. The goal is to learn to make decisions over time that maximize a concave utility function of the running time average transmission rate of each user. Recently it was shown that a stochastic Frank-Wolfe algorithm converges to utility-optimality with an error of O(log(T)/T), where T is the time the algorithm has been running. An existing Ω(1/T) converse is known. The current paper improves the converse to Ω(log(T)/T), which matches the known achievability result. The proof uses a reduction from the opportunistic scheduling problem to a Bernoulli estimation problem. Along the way, it refines a result on Bernoulli estimation.