A Converse Result on Convergence Time for Opportunistic Wireless Scheduling

This paper proves an impossibility result for stochastic network utility maximization for multi-user wireless systems, including multiple access and broadcast systems. Every time slot an access point observes the current channel states for each user 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 $\Omega(1/T)$ converse is known. The current paper improves the converse to $\Omega(\log(T)/T)$, which matches the known achievability result. It does this by constructing a particular (simple) system for which no algorithm can achieve a better performance. The proof uses a novel reduction of the opportunistic scheduling problem to a problem of estimating a Bernoulli probability $p$ from independent and identically distributed samples. Along the way we refine a regret bound for Bernoulli estimation to show that, for any sequence of estimators, the set of values $p \in [0,1]$ under which the estimators perform poorly has measure at least $1/8$.