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Wireless network optimization via stochastic sub-gradient descent: Rate analysis

This paper considers a general stochastic resource allocation problem that arises widely in wireless networks, cognitive radio networks, smart-grid communications, and cross-layer design. The problem formulation involves expectations with respect to a collection of random variables with unknown distributions, representing exogenous quantities such as channel gain, user density, or spectrum occupancy. The problem is solved in dual domain using a constant step-size stochastic dual subgradient descent (SDSD) method. This results in a primal resource allocation subproblem at each time instant. The goal here is to characterize the non-asymptotic behavior of such stochastic resource allocations in an almost sure sense. This paper establishes a convergence rate result for the SDSD algorithm that precisely characterizes the trade-off between the rate of convergence and the choice of constant step size e. Towards this end, a novel stochastic bound on the gap between the objective function and the optimum is developed. The asymptotic behavior of the stochastic term is characterized in an almost sure sense, thereby generalizing the existing results for the stochastic subgradient methods. As an application, the power and user-allocation problem in device-to-device networks is formulated and solved using the SDSD algorithm. Further intuition on the rate results is obtained from the verification of the regularity conditions and accompanying simulation results.