Best subset selection is considered the `gold standard' for many sparse learning problems. A variety of optimization techniques have been proposed to attack this non-smooth non-convex problem. In this paper, we investigate the dual forms of a family of $\ell_0$-regularized problems. An efficient primal-dual algorithm is developed based on the primal and dual problem structures. By leveraging the dual range estimation along with the incremental strategy, our algorithm potentially reduces redundant computation and improves the solutions of best subset selection. Theoretical analysis and experiments on synthetic and real-world datasets validate the efficiency and statistical properties of the proposed solutions.

翻译：最优子集选择被视作众多稀疏学习问题的“金标准”。针对这一非光滑非凸优化问题，已有多种优化方法被提出。本文研究了一系列$\ell_0$正则化问题的对偶形式，基于原始问题与对偶问题的结构特性，提出了一种高效的原始-对偶算法。通过结合对偶范围估计与增量策略，该算法能够有效减少冗余计算并改进最优子集选择的求解结果。在合成数据集与真实数据集上的理论分析与实验验证了所提方法在计算效率与统计性能上的优越性。