Learning general halfspaces with general Massart noise under the Gaussian distribution

Ilias Diakonikolas, Daniel M. Kane, Vasilis Kontonis, Christos Tzamos, Nikos Zarifis

Abstract: We study the problem of PAC learning halfspaces on ℝd with Massart noise under the Gaussian distribution. In the Massart model, an adversary is allowed to flip the label of each point x with unknown probability η(x) ≤ η, for some parameter η ∈ [0,1/2]. The goal is to find a hypothesis with misclassification error of OPT + є, where 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 η 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: [leftmargin = *] For η <1/2, we give a learning algorithm for general halfspaces with sample and computational complexity dOη(log(1/γ))poly(1/є), where γ max{є, min{Pr[f(x) = 1], Pr[f(x) = −1]} } is the “bias” of the target halfspace f. Prior efficient algorithms could only handle the special case of γ = 1/2. Interestingly, we establish a qualitatively matching lower bound of dΩ(log(1/γ)) on the complexity of any Statistical Query (SQ) algorithm. For η = 1/2, we give a learning algorithm for general halfspaces with sample and computational complexity Oє(1)  dO(log(1/є)). This result is new even for the subclass of homogeneous halfspaces; prior algorithms for homogeneous Massart halfspaces provide vacuous guarantees for η=1/2. We complement our upper bound with a nearly-matching SQ lower bound of dΩ(log(1/є) ), which holds even for the special case of homogeneous halfspaces. Taken together, our results qualitatively characterize the complexity of learning general halfspaces with general Massart noise under Gaussian marginals. Our techniques rely on determining the existence (or non-existence) of low-degree polynomials whose expectations distinguish Massart halfspaces from random noise.