Greedy Sparsity-Constrained Optimization
This work addresses a gap in sparsity-constrained optimization for nonlinear and non-quadratic cases, which is incremental as it generalizes existing results from linear models.
The paper tackles the problem of sparsity-constrained optimization for nonlinear models or non-quadratic cost functions, proposing the Gradient Support Pursuit (GraSP) algorithm, which is guaranteed to approximate sparse minima under certain conditions and demonstrated through synthetic data simulations for sparse logistic regression.
Sparsity-constrained optimization has wide applicability in machine learning, statistics, and signal processing problems such as feature selection and compressive Sensing. A vast body of work has studied the sparsity-constrained optimization from theoretical, algorithmic, and application aspects in the context of sparse estimation in linear models where the fidelity of the estimate is measured by the squared error. In contrast, relatively less effort has been made in the study of sparsity-constrained optimization in cases where nonlinear models are involved or the cost function is not quadratic. In this paper we propose a greedy algorithm, Gradient Support Pursuit (GraSP), to approximate sparse minima of cost functions of arbitrary form. Should a cost function have a Stable Restricted Hessian (SRH) or a Stable Restricted Linearization (SRL), both of which are introduced in this paper, our algorithm is guaranteed to produce a sparse vector within a bounded distance from the true sparse optimum. Our approach generalizes known results for quadratic cost functions that arise in sparse linear regression and Compressive Sensing. We also evaluate the performance of GraSP through numerical simulations on synthetic data, where the algorithm is employed for sparse logistic regression with and without $\ell_2$-regularization.