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 $\alpha \cdot \opt_{\gamma} + \eps$, where $\opt_{\gamma}$ 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 $O(1/(\eps^2\gamma^2))$ samples (which is optimal) and runs in time $\poly(d/\eps) \cdot 2^{\tilde{O}(1/\gamma^2)}$.

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 \opt_{\gamma} \eps, where \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.