Attribute-Efficient PAC Learning of Sparse Halfspaces with Constant Malicious Noise Rate

Attribute-efficient PAC learning of sparse halfspaces has been a fundamental problem in machine learning theory. In recent years, machine learning algorithms are faced with prevalent data corruptions or even malicious attacks. It is of central interest to design computationally-efficient algorithms that are robust to malicious corruptions. In this paper, we consider that there exists a constant amount of malicious noise in the data and the goal is to learn an underlying $s$-sparse halfspace $w^* \in \mathbb{R}^d$ with $\text{poly}(s,\log d)$ samples. Specifically, we follow a recent line of works and assume that the underlying distribution satisfies a certain concentration condition and a margin condition at the same time. Under such conditions, we show that attribute-efficiency can be achieved with simple variants to existing hinge loss minimization programs. Our key contribution includes: 1) an attribute-efficient PAC learning algorithm that works under a constant malicious noise rate; 2) a new gradient analysis that carefully handles the sparsity admitted constraints in hinge loss minimization program.