Search

To efficiently trade off system sum-rate and link fairness, this paper is dedicated to maximizing the sum of \alpha-fair utility in spectrum-sharing networks, where multiple interfering links share one channel. In the literature, three special cases, including \alpha=\mbox{0} (sum-rate maximization), \alpha=\mbox{1} (proportional fairness), and \alpha=\infty (max-min fairness), have been investigated; the complexity for cases \mbox{1} < \alpha < \infty and \mbox{0} < \alpha < \mbox{1} is still unknown. In this paper, we prove that the problem is convex when \mbox1 < \alpha < \infty and is NP-hard when \mbox0 < \alpha < \mbox1. To deal with the latter case, we transform the objective function and represent it by the difference of two concave functions (D.C.). Then, a power allocation algorithm is proposed with fast convergence to a local optimal point. Simulation results show that the proposed algorithm can obtain global optimality in two-link cases when \mbox0 < \alpha < \mbox1. In addition, we can get a flexible tradeoff between sum-rate and fairness in terms of Jain's index by adjusting $\alpha$. [ABSTRACT FROM AUTHOR]

In this paper, we study Stochastic Composite Optimization (SCO) for sparse learning that aims to learn a sparse solution from a composite function. Most of the recent SCO algorithms have already reached the optimal expected convergence rate \mathcal O(1/\lambda T) , but they often fail to deliver sparse solutions at the end either due to the limited sparsity regularization during stochastic optimization (SO) or due to the limitation in online-to-batch conversion. Even when the objective function is strongly convex, their high probability bounds can only attain \mathcal O(\sqrt\log (1/\delta)/T) with $\delta$ is the failure probability, which is much worse than the expected convergence rate. To address these limitations, we propose a simple yet effective two-phase Stochastic Composite Optimization scheme by adding a novel powerful sparse online-to-batch conversion to the general Stochastic Optimization algorithms. We further develop three concrete algorithms, OptimalSL, LastSL and AverageSL, directly under our scheme to prove the effectiveness of the proposed scheme. Both the theoretical analysis and the experiment results show that our methods can really outperform the existing methods at the ability of sparse learning and at the meantime we can improve the high probability bound to approximately \mathcal {O}(\log (\log (T)/\delta)/\lambda T) . [ABSTRACT FROM AUTHOR]