Learning General Halfspaces with General Massart Noise under the Gaussian Distribution

TL;DR

This paper develops new algorithms for learning halfspaces under Gaussian distribution with Massart noise, removing previous restrictions on the target halfspace's homogeneity and noise parameter, and establishes matching lower bounds for the problem.

Contribution

It introduces the first algorithms for general halfspaces with Massart noise without prior assumptions and proves matching lower bounds for their complexity.

Findings

Algorithms for general halfspaces with Massart noise for ta<1/2 and ta=1/2.

Sample and computational complexity bounds depending on ta and psilon.

Matching SQ lower bounds indicating inherent difficulty of the problem.

Abstract

We study the problem of PAC learning halfspaces on $\mathbb{R}^d$ with Massart noise under the Gaussian distribution. In the Massart model, an adversary is allowed to flip the label of each point $\mathbf{x}$ with unknown probability $\eta(\mathbf{x}) \leq \eta$, for some parameter $\eta \in [0,1/2]$. The goal is to find a hypothesis with misclassification error of $\mathrm{OPT} + \epsilon$, where $\mathrm{OPT}$ is the error of the target halfspace. This problem had been previously studied under two assumptions: (i) the target halfspace is homogeneous (i.e., the separating hyperplane goes through the origin), and (ii) the parameter $\eta$ is strictly smaller than $1/2$. Prior to this work, no nontrivial bounds were known when either of these assumptions is removed. We study the general problem and establish the following: For $\eta <1/2$, we give a learning algorithm for general halfspaces with sample and computational complexity $d^{O_{\eta}(\log(1/\gamma))}\mathrm{poly}(1/\epsilon)$, where $\gamma =\max\{\epsilon, \min\{\mathbf{Pr}[f(\mathbf{x}) = 1], \mathbf{Pr}[f(\mathbf{x}) = -1]\} \}$ is the bias of the target halfspace $f$. Prior efficient algorithms could only handle the special case of $\gamma = 1/2$. Interestingly, we establish a qualitatively matching lower bound of $d^{\Omega(\log(1/\gamma))}$ on the complexity of any Statistical Query (SQ) algorithm. For $\eta = 1/2$, we give a learning algorithm for general halfspaces with sample and computational complexity $O_\epsilon(1) d^{O(\log(1/\epsilon))}$. This result is new even for the subclass of homogeneous halfspaces; prior algorithms for homogeneous Massart halfspaces provide vacuous guarantees for $\eta=1/2$. We complement our upper bound with a nearly-matching SQ lower bound of $d^{\Omega(\log(1/\epsilon))}$, which holds even for the special case of homogeneous halfspaces.