Tree-guided group lasso for multi-response regression with structured sparsity, with an application to eQTL mapping

We consider the problem of estimating a sparse multi-response regression function, with an application to expression quantitative trait locus (eQTL) mapping, where the goal is to discover genetic variations that influence gene-expression levels. In particular, we investigate a shrinkage technique capable of capturing a given hierarchical structure over the responses, such as a hierarchical clustering tree with leaf nodes for responses and internal nodes for clusters of related responses at multiple granularity, and we seek to leverage this structure to recover covariates relevant to each hierarchically-defined cluster of responses. We propose a tree-guided group lasso, or tree lasso, for estimating such structured sparsity under multi-response regression by employing a novel penalty function constructed from the tree. We describe a systematic weighting scheme for the overlapping groups in the tree-penalty such that each regression coefficient is penalized in a balanced manner despite the inhomogeneous multiplicity of group memberships of the regression coefficients due to overlaps among groups. For efficient optimization, we employ a smoothing proximal gradient method that was originally developed for a general class of structured-sparsity-inducing penalties. Using simulated and yeast data sets, we demonstrate that our method shows a superior performance in terms of both prediction errors and recovery of true sparsity patterns, compared to other methods for learning a multivariate-response regression.Comment: Published in at http://dx.doi.org/10.1214/12-AOAS549 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

Convex optimization over intersection of simple sets: improved convergence rate guarantees via an exact penalty approach

We consider the problem of minimizing a convex function over the intersection of finitely many simple sets which are easy to project onto. This is an important problem arising in various domains such as machine learning. The main difficulty lies in finding the projection of a point in the intersection of many sets. Existing approaches yield an infeasible point with an iteration-complexity of $O(1/\varepsilon^2)$ for nonsmooth problems with no guarantees on the in-feasibility. By reformulating the problem through exact penalty functions, we derive first-order algorithms which not only guarantees that the distance to the intersection is small but also improve the complexity to $O(1/\varepsilon)$ and $O(1/\sqrt{\varepsilon})$ for smooth functions. For composite and smooth problems, this is achieved through a saddle-point reformulation where the proximal operators required by the primal-dual algorithms can be computed in closed form. We illustrate the benefits of our approach on a graph transduction problem and on graph matching

A clustering heuristic to improve a derivative-free algorithm for nonsmooth optimization

In this paper we propose an heuristic to improve the performances of the recently proposed derivative-free method for nonsmooth optimization CS-DFN. The heuristic is based on a clustering-type technique to compute an estimate of Clarke’s generalized gradient of the objective function, obtained via calculation of the (approximate) directional derivative along a certain set of directions. A search direction is then calculated by applying a nonsmooth Newton-type approach. As such, this direction (as it is shown by the numerical experiments) is a good descent direction for the objective function. We report some numerical results and comparison with the original CS-DFN method to show the utility of the proposed improvement on a set of well-known test problems

Fast Low-Rank Matrix Learning with Nonconvex Regularization

Low-rank modeling has a lot of important applications in machine learning, computer vision and social network analysis. While the matrix rank is often approximated by the convex nuclear norm, the use of nonconvex low-rank regularizers has demonstrated better recovery performance. However, the resultant optimization problem is much more challenging. A very recent state-of-the-art is based on the proximal gradient algorithm. However, it requires an expensive full SVD in each proximal step. In this paper, we show that for many commonly-used nonconvex low-rank regularizers, a cutoff can be derived to automatically threshold the singular values obtained from the proximal operator. This allows the use of power method to approximate the SVD efficiently. Besides, the proximal operator can be reduced to that of a much smaller matrix projected onto this leading subspace. Convergence, with a rate of O(1/T) where T is the number of iterations, can be guaranteed. Extensive experiments are performed on matrix completion and robust principal component analysis. The proposed method achieves significant speedup over the state-of-the-art. Moreover, the matrix solution obtained is more accurate and has a lower rank than that of the traditional nuclear norm regularizer.Comment: Long version of conference paper appeared ICDM 201