Proper Learning, Helly Number, and an Optimal SVM Bound

The classical PAC sample complexity bounds are stated for any Empirical Risk Minimizer (ERM) and contain an extra logarithmic factor $\log(1/{\epsilon})$ which is known to be necessary for ERM in general. It has been recently shown by Hanneke (2016) that the optimal sample complexity of PAC learning for any VC class C is achieved by a particular improper learning algorithm, which outputs a specific majority-vote of hypotheses in C. This leaves the question of when this bound can be achieved by proper learning algorithms, which are restricted to always output a hypothesis from C. In this paper we aim to characterize the classes for which the optimal sample complexity can be achieved by a proper learning algorithm. We identify that these classes can be characterized by the dual Helly number, which is a combinatorial parameter that arises in discrete geometry and abstract convexity. In particular, under general conditions on C, we show that the dual Helly number is bounded if and only if there is a proper learner that obtains the optimal joint dependence on $\epsilon$ and $\delta$. As further implications of our techniques we resolve a long-standing open problem posed by Vapnik and Chervonenkis (1974) on the performance of the Support Vector Machine by proving that the sample complexity of SVM in the realizable case is $\Theta((n/{\epsilon})+(1/{\epsilon})\log(1/{\delta}))$, where $n$ is the dimension. This gives the first optimal PAC bound for Halfspaces achieved by a proper learning algorithm, and moreover is computationally efficient.