  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 $O(1/(\epsilon^2\gamma^2))$ 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.
