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Many popular learning algorithms (E.g. Regression, Fourier-Transform basedalgorithms, Kernel SVM and Kernel ridge regression) operate by reducing theproblem to a convex optimization problem over a vector space of functions.These methods offer the currently best approach to several central problemssuch as learning half spaces and learning DNF's. In addition they are widelyused in numerous application domains. Despite their importance, there are stillvery few proof techniques to show limits on the power of these algorithms.  We study the performance of this approach in the problem of (agnostically andimproperly) learning halfspaces with margin $\gamma$. Let $\mathcal{D}$ be adistribution over labeled examples. The $\gamma$-margin error of a hyperplane$h$ is the probability of an example to fall on the wrong side of $h$ or at adistance $\le\gamma$ from it. The $\gamma$-margin error of the best $h$ isdenoted $\mathrm{Err}_\gamma(\mathcal{D})$. An $\alpha(\gamma)$-approximationalgorithm receives $\gamma,\epsilon$ as input and, using i.i.d. samples of$\mathcal{D}$, outputs a classifier with error rate $\le\alpha(\gamma)\mathrm{Err}_\gamma(\mathcal{D}) + \epsilon$. Such an algorithmis efficient if it uses $\mathrm{poly}(\frac{1}{\gamma},\frac{1}{\epsilon})$samples and runs in time polynomial in the sample size.  The best approximation ratio achievable by an efficient algorithm is$O\left(\frac{1/\gamma}{\sqrt{\log(1/\gamma)}}\right)$ and is achieved using analgorithm from the above class. Our main result shows that the approximationratio of every efficient algorithm from this family must be $\ge\Omega\left(\frac{1/\gamma}{\mathrm{poly}\left(\log\left(1/\gamma\right)\right)}\right)$,essentially matching the best known upper bound.

The complexity of learning halfspaces using generalized linear methods