Sparsity-Constraint Optimization via Splicing Iteration

Abstract

Sparsity-constraint optimization has wide applicability in signal processing, statistics, and machine learning. Existing fast algorithms must burdensomely tune parameters, such as the step size or the implementation of precise stop criteria, which may be challenging to determine in practice. To address this issue, we develop an algorithm named Sparsity-Constraint Optimization via sPlicing itEration (SCOPE) to optimize nonlinear differential objective functions with strong convexity and smoothness in low dimensional subspaces. Algorithmically, the SCOPE algorithm converges effectively without tuning parameters. Theoretically, SCOPE has a linear convergence rate and converges to a solution that recovers the true support set when it correctly specifies the sparsity. We also develop parallel theoretical results without restricted-isometry-property-type conditions. We apply SCOPE's versatility and power to solve sparse quadratic optimization, learn sparse classifiers, and recover sparse Markov networks for binary variables. The numerical results on these specific tasks reveal that SCOPE perfectly identifies the true support set with a 10--1000 speedup over the standard exact solver, confirming SCOPE's algorithmic and theoretical merits. Our open-source Python package skscope based on C++ implementation is publicly available on GitHub, reaching a ten-fold speedup on the competing convex relaxation methods implemented by the cvxpy library.