  Many popular learning algorithms (E.g. Regression, Fourier-Transform based
algorithms, Kernel SVM and Kernel ridge regression) operate by reducing the
problem to a convex optimization problem over a vector space of functions.
These methods offer the currently best approach to several central problems
such as learning half spaces and learning DNF's. In addition they are widely
used in numerous application domains. Despite their importance, there are still
very few proof techniques to show limits on the power of these algorithms.
  We study the performance of this approach in the problem of (agnostically and
improperly) learning halfspaces with margin $\gamma$. Let $\mathcal{D}$ be a
distribution 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 a
distance $\le\gamma$ from it. The $\gamma$-margin error of the best $h$ is
denoted $\mathrm{Err}_\gamma(\mathcal{D})$. An $\alpha(\gamma)$-approximation
algorithm 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 algorithm
is 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 an
algorithm from the above class. Our main result shows that the approximation
ratio 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.
