Before revision:  OurO\big({} \big)  main techniques include attribute-efficient paradigms for instance reweighting and for empirical risk minimization, and a new analysis of uniform concentration for unbounded data ~ -- ~ all of them crucially take the structure of the underlying halfspace into account . To the best of our knowledge, this is the first near-optimal result in the setting. As a byproduct of our analysis, we show that by applying a result of Bienstock and Michalka (2014), a global optimum of a form of sparse principal component analysis can be found in polynomial time without any statistical assumption on the data, which has not been recognized before and might be of independent interest .
After revision:  OurO\big({} \big) main techniques include attribute-efficient paradigms for instance reweighting and for empirical risk minimization, and a new analysis of uniform concentration for unbounded data ~ -- ~ all of them crucially take the structure of the underlying halfspace into account .