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

Multi-class SVMs: From Tighter Data-Dependent Generalization Bounds to Novel Algorithms