  This paper studies the generalization performance of multi-class
classification algorithms, for which we obtain, for the first time, a
data-dependent generalization error bound with a logarithmic dependence on the
class size, substantially improving the state-of-the-art linear dependence in
the existing data-dependent generalization analysis. The theoretical analysis
motivates us to introduce a new multi-class classification machine based on
$\ell_p$-norm regularization, where the parameter $p$ controls the complexity
of the corresponding bounds. We derive an efficient optimization algorithm
based on Fenchel duality theory. Benchmarks on several real-world datasets show
that the proposed algorithm can achieve significant accuracy gains over the
state of the art.
