Successive Convex Approximation Algorithms for Sparse Signal Estimation with Nonconvex Regularizations

In this paper, we propose a successive convex approximation framework for sparse optimization where the nonsmooth regularization function in the objective function is nonconvex and it can be written as the difference of two convex functions. The proposed framework is based on a nontrivial combination of the majorization-minimization framework and the successive convex approximation framework proposed in literature for a convex regularization function. The proposed framework has several attractive features, namely, i) flexibility, as different choices of the approximate function lead to different type of algorithms; ii) fast convergence, as the problem structure can be better exploited by a proper choice of the approximate function and the stepsize is calculated by the line search; iii) low complexity, as the approximate function is convex and the line search scheme is carried out over a differentiable function; iv) guaranteed convergence to a stationary point. We demonstrate these features by two example applications in subspace learning, namely, the network anomaly detection problem and the sparse subspace clustering problem. Customizing the proposed framework by adopting the best-response type approximation, we obtain soft-thresholding with exact line search algorithms for which all elements of the unknown parameter are updated in parallel according to closed-form expressions. The attractive features of the proposed algorithms are illustrated numerically.Comment: submitted to IEEE Journal of Selected Topics in Signal Processing, special issue in Robust Subspace Learnin

Expander Graph and Communication-Efficient Decentralized Optimization

In this paper, we discuss how to design the graph topology to reduce the communication complexity of certain algorithms for decentralized optimization. Our goal is to minimize the total communication needed to achieve a prescribed accuracy. We discover that the so-called expander graphs are near-optimal choices. We propose three approaches to construct expander graphs for different numbers of nodes and node degrees. Our numerical results show that the performance of decentralized optimization is significantly better on expander graphs than other regular graphs.Comment: 2016 IEEE Asilomar Conference on Signals, Systems, and Computer