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 $alpha cdot mathrm{OPT}_{gamma} + epsilon$, where $mathrm{OPT}_{gamma}$ is the optimal $gamma$-margin error rate and $alpha geq 1$ is the approximation ratio. We give learning algorithms and computational hardness results for this problem, for all values of the approximation ratio $alpha geq 1$, that are nearly-matching for a range of parameters. Specifically, for the natural setting that $alpha$ is any constant bigger than one, we provide an essentially tight complexity characterization. On the positive side, we give an $alpha = 1.01$-approximate proper learner that uses $O(1/(epsilon^2gamma^2))$ samples (which is optimal) and runs in time $mathrm{poly}(d/epsilon) cdot 2^{tilde{O}(1/gamma^2)}$. On the negative side, we show that {em any} constant factor approximate proper learner has runtime $mathrm{poly}(d/epsilon) cdot 2^{(1/gamma)^{2-o(1)}}$, assuming the Exponential Time Hypothesis.
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