Improving Sparsity and Scalability in Regularized Nonconvex Truncated-Loss Learning Problems

The truncated regular L 1 -loss support vector machine can eliminate the excessive number of support vectors (SVs); thus, it has significant advantages in robustness and scalability. However, in this paper, we discover that the associated state-of-the-art solvers, such as difference convex algorithm and concave-convex procedure, not only have limited sparsity promoting property for general truncated losses especially the L 2 -loss but also have poor scalability for large-scale problems. To circumvent these drawbacks, we present a general multistage scheme with explicit interpretation regarding SVs as well as outliers. In particular, we solve the general nonconvex truncated loss minimization through a sequence of associated convex subproblems, in which the outliers are removed in advance. The proposed algorithm can be regarded as a structural optimization attempt carefully considering sparsity imposed by the nonconvex truncated losses. We show that this general multistage algorithm offers sufficient sparsity especially for the truncated L 2 -loss. To further improve the scalability, we propose a linear multistep algorithm by employing a single iteration of coordinate descent to monotonically decrease the objective function at each stage and a kernel algorithm by using the Karush-Kuhn-Tucker conditions to cheaply find most part of the outliers for the next stage. Comparison experiments demonstrate that our methods have superiority in sparsity as well as efficiency in scalability.