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We consider the projected gradient algorithm for the nonconvex best subsetselection problem that minimizes a given empirical loss function under an$\ell_0$-norm constraint. Through decomposing the feasible set of the givensparsity constraint as a finite union of linear subspaces, we present twoacceleration schemes with global convergence guarantees, one by same-spaceextrapolation and the other by subspace identification. The former fullyutilizes the problem structure to greatly accelerate the optimization speedwith only negligible additional cost. The latter leads to a two-stagemeta-algorithm that first uses classical projected gradient iterations toidentify the correct subspace containing an optimal solution, and then switchesto a highly-efficient smooth optimization method in the identified subspace toattain superlinear convergence. Experiments demonstrate that the proposedaccelerated algorithms are magnitudes faster than their non-acceleratedcounterparts as well as the state of the art.

Accelerated projected gradient algorithms for sparsity constrained optimization problems