Scalable Algorithms for the Sparse Ridge Regression

Sparse regression and variable selection for large-scale data have been rapidly developed in the past decades. This work focuses on sparse ridge regression, which enforces the sparsity by use of the L0 norm. We pave out a theoretical foundation to understand why existing approaches may not work well for this problem, in particular on large-scale datasets. By reformulating the problem as a chance-constrained program, we derive a novel mixed integer second order conic (MISOC) reformulation using perspective formulation and prove that its continuous relaxation is equivalent to that of the convex integer formulation proposed in recent work. Based upon these two formulations (i.e., the proposed MISOC formulation and an existing convex integer formulation), we develop two new scalable algorithms, the greedy and randomized algorithms, for sparse ridge regression with desirable theoretical properties. The proposed algorithms are proved to yield near-optimal solutions under mild conditions. We further propose to integrate the greedy algorithm with the randomized algorithm, which can greedily search the features from the nonzero subset identified by the continuous relaxation of the MISOC formulation. The merits of the proposed methods are illustrated through numerical examples in comparison with several existing ones.