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 \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.