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From average case complexity to improper learning complexity
The biggest challenge in proving complexity results is to establish hardness of {\em improper learning} (a. k. a.
More data speeds up training time in learning halfspaces over sparse vectors
That is, if more data is available, beyond the sample complexity limit, is it possible to use the extra examples to speed up the computation time required to perform the learning task?
The complexity of learning halfspaces using generalized linear methods
The best approximation ratio achievable by an efficient algorithm is $O\left(\frac{1/\gamma}{\sqrt{\log(1/\gamma)}}\right)$ and is achieved using an algorithm from the above class.
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