DC-LiGME: An Efficient Algorithm for Improved Convex Sparse Regularization

Convex sparse regularization techniques such as l
1
-regularization enjoy robustness to random initialization and computational efficiency, but often underestimate the true solution. To mitigate the estimation bias, the generalized minimax concave (GMC) penalty (Selesnick 2017) and its linearly involved extension LiGME (Abe et al. 2020) have been proposed recently to yield improved convex sparse regularization. However, empirical results indicate that existing proximal splitting algorithms for GMC/LiGME do not necessarily achieve the convergence speed desired in some real-world applications (e.g., stream data processing).To resolve this situation, we propose a fast and well scalable algorithm (termed DC-LiGME) based on DC programming, which allows us to solve the difficult original problem via solution to a sequence of simpler subproblems. We establish convergence guarantee of the DC-LiGME algorithm and present an intuitive explanation of its fast convergence. Numerical experiments demonstrate superior convergence speed of the DC-LiGME algorithm, and suggest that mere two iterations of the algorithm suffice to stably produce satisfactory solutions (at the cost of solving three standard convex regularization models), which is a favourable feature in practice.