DC approximation approaches for sparse optimization (1407.0286v2)

Abstract: Sparse optimization refers to an optimization problem involving the zero-norm in objective or constraints. In this paper, nonconvex approximation approaches for sparse optimization have been studied with a unifying point of view in DC (Difference of Convex functions) programming framework. Considering a common DC approximation of the zero-norm including all standard sparse inducing penalty functions, we studied the consistency between global minimums (resp. local minimums) of approximate and original problems. We showed that, in several cases, some global minimizers (resp. local minimizers) of the approximate problem are also those of the original problem. Using exact penalty techniques in DC programming, we proved stronger results for some particular approximations, namely, the approximate problem, with suitable parameters, is equivalent to the original problem. The efficiency of several sparse inducing penalty functions have been fully analyzed. Four DCA (DC Algorithm) schemes were developed that cover all standard algorithms in nonconvex sparse approximation approaches as special versions. They can be viewed as, an $\ell _{1}$-perturbed algorithm / reweighted-$\ell _{1}$ algorithm / reweighted-$\ell _{1}$ algorithm. We offer a unifying nonconvex approximation approach, with solid theoretical tools as well as efficient algorithms based on DC programming and DCA, to tackle the zero-norm and sparse optimization. As an application, we implemented our methods for the feature selection in SVM (Support Vector Machine) problem and performed empirical comparative numerical experiments on the proposed algorithms with various approximation functions.

• The paper demonstrates that under proper parameter settings, the global minimizers of approximate problems align with those of the original sparse optimization problem.
• It rigorously compares sparsity-inducing penalty functions like Capped-l1 and SCAD, identifying optimal ranges to effectively approximate the zero-norm.
• The study develops four DCA schemes, applying them to SVM feature selection to achieve higher classification accuracy with fewer features.

Overview of DC Approximation Approaches for Sparse Optimization

The paper "DC Approximation Approaches for Sparse Optimization" by LE THI Hoai An, PHAM DINH Tao, LE Hoai Minh, and VO Xuan Thanh explores the intricate domain of sparse optimization, specifically concentrating on nonconvex approximation methods within the Difference of Convex (DC) programming framework. Sparse optimization, characterized by the involvement of the zero-norm in objective functions or constraints, presents significant challenges due to the discontinuity of the zero-norm at the origin. This paper provides a comprehensive examination of DC approximation functions and their application to sparse optimization tasks, offering promising results for both theoretical exploration and practical algorithm design.

Key Contributions

1. Consistency and Equivalence Results: The authors rigorously prove the consistency between the minima of approximate and original problems. They demonstrate that global (and local) minimizers of the approximate problem align with those of the original problem under specific conditions. Remarkably, they provide evidence that with appropriate parameters, approximate problems can be equivalent to the original sparse optimization problem, especially in cases where the feasible set is a bounded polyhedral convex set.
2. Analysis of Sparsity-Inducing Penalty Functions: A detailed comparative analysis of various penalty functions—including Capped-l1 and SCAD functions—highlights their advantages and effectiveness in approximating the zero-norm. The paper identifies optimal parameter ranges for these approximations, thereby guiding future research on parameter selection.
3. Development of DCA Schemes: Four distinct DC Algorithm (DCA) schemes are developed, each capable of solving different formulations of sparse optimization problems. These include l1-perturbed and reweighted-l1 algorithms, which are presented as special cases under the unifying framework provided by DC programming.
4. Feature Selection in SVM: The proposed DC approximation methods are applied to feature selection in Support Vector Machines (SVMs). The authors conducted empirical experiments demonstrating the superiority of their approaches over standard algorithms, particularly in achieving higher classification accuracy with fewer features.

Numerical Results and Claims

The empirical results exhibit the robustness of the proposed DCA schemes in feature selection tasks within SVMs. When applied to several datasets, the DC approximation approaches consistently showed improved classification performance and reduced dimensionality of the feature space by selecting fewer yet more relevant features.

Implications and Future Directions

The potential implications of this research span both theoretical and practical spheres. Theoretically, the work enriches the foundation of DC programming by illuminating the conditions and parameters under which nonconvex approximations achieve optimality comparable to or surpassing their convex counterparts. Practically, these findings empower machine learning practitioners with methods for constructing more interpretable and efficient sparse models, particularly in high-dimensional data such as genomic applications and image recognition.

Future research could explore the scalability of the DC programming approach to even larger datasets and continue the development of more efficient algorithms, possibly incorporating advances in parallel computing. Furthermore, the application of these methodologies could extend beyond feature selection to broader machine learning tasks, reinforcing the versatility of DC decomposition strategies in dealing with real-world sparse optimization challenges.

This paper stands as a substantive contribution to the field, offering a robust and scalable approach to tackling complex sparse optimization problems through the lens of DC programming and its associated algorithms. Its implications for both theory and application are profound, providing valuable insights and tools for future advancements in optimization and machine learning.

Overall, the research presented embodies a significant stride in the understanding and application of nonconvex optimization methods, with the DC framework proving to be an effective and comprehensive approach to sparse optimization challenges.