Relaxed Sparse Eigenvalue Conditions for Sparse Estimation via Non-convex Regularized Regression
This work addresses sparse estimation for statistical learning, providing incremental theoretical improvements for non-convex regularization methods.
The paper tackles the problem of sparse estimation by analyzing conditions for non-convex regularizers, showing that they require weaker sparse eigenvalue conditions than L1-regularization for parameter and sparseness estimation, with experiments demonstrating performance improvements.
Non-convex regularizers usually improve the performance of sparse estimation in practice. To prove this fact, we study the conditions of sparse estimations for the sharp concave regularizers which are a general family of non-convex regularizers including many existing regularizers. For the global solutions of the regularized regression, our sparse eigenvalue based conditions are weaker than that of L1-regularization for parameter estimation and sparseness estimation. For the approximate global and approximate stationary (AGAS) solutions, almost the same conditions are also enough. We show that the desired AGAS solutions can be obtained by coordinate descent (CD) based methods. Finally, we perform some experiments to show the performance of CD methods on giving AGAS solutions and the degree of weakness of the estimation conditions required by the sharp concave regularizers.