A dual stochastic DFP algorithm for optimal resource allocation in wireless systems

Aryan Mokhtari, Alejandro Ribeiro

Abstract: A stochastic implementation of the Davidon-Fletcher-Powell (DFP) quasi-Newton method to minimize dual functions of optimal resource allocation problems in wireless systems is introduced. While the use of dual stochastic gradient descent algorithms is widespread, they suffer from slow convergence rate. Application of second order methods, on the other hand, is impracticable because computation of dual Hessian inverses incurs excessive cost. The proposed method utilizes stochastic gradients in lieu of deterministic gradients for both, the determination of descent directions and the approximation of the dual function's curvature. Since stochastic gradients can be computed at manageable computational cost stochastic DFP is realizable and retains the convergence rate advantages of its deterministic counterparts. Convergence results show that lower and upper bounds on the instantaneous form of the dual Hessian are sufficient to guarantee convergence to a small neighborhood of optimality. Numerical experiments illustrate that for ill conditioned dual functions stochastic DFP outperforms stochastic gradient descent by an order of magnitude.