Optimal convergence and adaptation for utility optimal opportunistic scheduling

Abstract

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 ∊ of optimality, for any desired ∊ > 0, with convergence and adaptation times of O(1/∊2). This paper shows that if the utility function is concave and smooth, then O(log(1/∊)/∊) 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/∊), 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/∊2) adaptation time, but convergence time increases to O(1/∊2), similar to existing drift-plus-penalty based algorithms. This raises important open questions regarding optimal adaptation.