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.Comment: Preprint of Allerton 2017 conference paper. 14 pages, 2 figure

Primal-Dual Frank-Wolfe for Constrained Stochastic Programs with Convex and Non-convex Objectives

We study constrained stochastic programs where the decision vector at each time slot cannot be chosen freely but is tied to the realization of an underlying random state vector. The goal is to minimize a general objective function subject to linear constraints. A typical scenario where such programs appear is opportunistic scheduling over a network of time-varying channels, where the random state vector is the channel state observed, and the control vector is the transmission decision which depends on the current channel state. We consider a primal-dual type Frank-Wolfe algorithm that has a low complexity update during each slot and that learns to make efficient decisions without prior knowledge of the probability distribution of the random state vector. We establish convergence time guarantees for the case of both convex and non-convex objective functions. We also emphasize application of the algorithm to non-convex opportunistic scheduling and distributed non-convex stochastic optimization over a connected graph.Comment: 25 pages, 1 figur