Optimal Convergence and Adaptation for Utility Optimal Opportunistic Scheduling

This paper considers the fundamental convergence time for opportunistic scheduling over time-varying channels. The channel state probabilities are unknown and algorithms must perform some type of estimation and learning while they make decisions to optimize network utility. Existing schemes can achieve a utility within $\epsilon$ of optimality, for any desired $\epsilon>0$, with convergence and adaptation times of $O(1/\epsilon^2)$. This paper shows that if the utility function is concave and smooth, then $O(\log(1/\epsilon)/\epsilon)$ convergence time is possible via an existing stochastic variation on the Frank-Wolfe algorithm, called the RUN algorithm. Next, a converse result is proven to show it is impossible for any algorithm to have convergence time better than $O(1/\epsilon)$, provided the algorithm has no a-priori knowledge of channel state probabilities. Hence, RUN is within a logarithmic factor of convergence time optimality. However, RUN has a vanishing stepsize and hence has an infinite adaptation time. Using stochastic Frank-Wolfe with a fixed stepsize yields improved $O(1/\epsilon^2)$ adaptation time, but convergence time increases to $O(1/\epsilon^2)$, similar to existing drift-plus-penalty based algorithms. This raises important open questions regarding optimal adaptation.