Greedy Sparsity-Constrained Optimization (1203.5483v3)

Abstract: 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.

• The paper introduces the GraSP algorithm, extending sparse optimization to nonlinear models with a novel gradient support pursuit strategy.
• It iteratively improves sparse estimates by merging gradient directions with projection steps, supported by Stable Restricted Hessian and Linearization properties.
• Empirical results demonstrate GraSP outperforms traditional methods like LASSO, particularly in high-dimensional and complex cost function settings.

Greedy Sparsity-Constrained Optimization: An Analysis of GraSP Algorithm

The paper Greedy Sparsity-Constrained Optimization introduces the Gradient Support Pursuit (GraSP) algorithm, aimed at addressing sparsity-constrained optimization in nonlinear models. While the field has extensively explored sparsity within linear models using quadratic costs, GraSP extends these explorations to complex models where standard approaches like $\ell_1$-norm regularization may fall short.

Overview and Algorithmic Approach

GraSP is a greedy algorithm designed to approximate sparse minima of generic, possibly nonlinear cost functions. It draws inspiration from CoSaMP, a well-known algorithm in the context of Compressed Sensing for sparse recovery in linear models with squared-error loss function. However, GraSP adapts this framework for a broader class of problems while retaining computational tractability.

The algorithm operates iteratively, improving an estimate of the sparse solution at each step. It focuses on significant gradient directions to identify potential adjustments needed in the sparse support, followed by minimization over this merged support and subsequent projection onto a sparse subspace. A notable distinction in GraSP is not relying solely on the $\ell_1$-norm for imposing sparsity, which is common in convex relaxation approaches.

Theoretical Contributions

The innovations in GraSP are underpinned by two theoretical constructs, Stable Restricted Hessian (SRH) and Stable Restricted Linearization (SRL), which provide the necessary conditions for algorithmic accuracy. Unlike the RIP in Compressed Sensing, SRH and SRL are generalized properties applicable to broader cost function landscapes:

• SRH measures constraints on the Hessian over sparse subspaces, generalizing the quadratic cost structures.
• SRL extends this concept to non-smooth objectives by evaluating restricted Bregman divergence, ensuring reliable and robust estimations in nonsmooth scenarios.

The paper delivers rigorous convergence guarantees under these properties, hypothetically providing an algorithmic error margin that converges as iterations proceed. The bounds are adaptable depending on how SRH/SRL constants are conditioned relative to the cost function’s characteristics.

Numerical Simulations and Results

Empirical validation via synthetic datasets shows GraSP's effectiveness compared to existing methods like LASSO, Logit-OMP, and elastic net regularization. The paper identifies areas where GraSP outperforms alternatives, specifically in scenarios where standard convex methods might fail to achieve minimal sparsity. Furthermore, variants of GraSP that incorporate debiasing and $\ell_2$ regularization exhibit improved stability, particularly in high dimensional contexts.

Practical Implications and Future Directions

GraSP’s flexibility suggests potential widespread applications in diverse fields where parameter estimation is crucial, such as bioinformatics and social network analysis. Its ability to adapt to both smooth and nonsmooth cost functions extends its applicability beyond traditional approaches constrained by linearity and convexity.

The theoretical constructs SRH and SRL can inspire further paper into sparsity-constrained optimization across varying domains of machine learning and signal processing. Future research might explore GraSP's synergy with randomized or hybrid algorithms to mitigate computational complexity in extremely high dimensional datasets.

Conclusion

Overall, the paper makes substantial strides in non-linear sparsity-constrained optimization, effectively bridging gaps left by traditional convex methods within the field of complex models. GraSP emerges as a pivotal tool, equipped with theoretical robustness and empirical efficacy that can ingratiate itself into modern analytical frameworks across diverse research disciplines.