Nearly Tight Bounds for Robust Proper Learning of Halfspaces with a Margin

We study the problem of em properly learning large margin halfspaces in the agnostic PAC model. In more detail, we study the complexity of properly learning d-dimensional halfspaces on the unit ball within misclassification error α · OPTγ + ε, where OPTγ is the optimal γ-margin error rate and α ≥ 1 is the approximation ratio. We give learning algorithms and computational hardness results for this problem, for all values of the approximation ratio α ≥ 1, that are nearly-matching for a range of parameters. Specifically, for the natural setting that α is any constant bigger than one, we provide an essentially tight complexity characterization. On the positive side, we give an α = 1.01-approximate proper learner that uses O(1/(ε2γ2)) samples (which is optimal) and runs in time poly(d/ε) · 2^tildeO(1/γ2). On the negative side, we show that em any constant factor approximate proper learner has runtime poly(d/ε) · 2^(1/γ)2-o(1), assuming the Exponential Time Hypothesis.