Information-Computation Tradeoffs for Learning Margin Halfspaces with Random Classification Noise

We study the problem of PAC learning $γ$-margin halfspaces with Random Classification Noise. We establish an information-computation tradeoff suggesting an inherent gap between the sample complexity of the problem and the sample complexity of computationally efficient algorithms. Concretely, the sample complexity of the problem is $\widetildeΘ(1/(γ^2 ε))$. We start by giving a simple efficient algorithm with sample complexity $\widetilde{O}(1/(γ^2 ε^2))$. Our main result is a lower bound for Statistical Query (SQ) algorithms and low-degree polynomial tests suggesting that the quadratic dependence on $1/ε$ in the sample complexity is inherent for computationally efficient algorithms. Specifically, our results imply a lower bound of $\widetildeΩ(1/(γ^{1/2} ε^2))$ on the sample complexity of any efficient SQ learner or low-degree test.