Learning General Halfspaces with General Massart Noise under the Gaussian Distribution

We study the problem of PAC learning halfspaces on R-d 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 eta(x) <= eta, for some parameter eta is an element of[0,1/2]. The goal is to find a hypothesis with misclassification error of OPT+epsilon, 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 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)))poly(1/epsilon), where gamma :=max{epsilon,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 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. 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.