#### Authors

Nathan Srebro, Karthik Sridharan, Ambuj Tewari

#### Abstract

<p>We establish an excess risk bound of O(H R<em>n^2 + sqrt{H L*} R</em>n) for ERM with an H-smooth loss function and a hypothesis class with Rademacher complexity R<em>n, where L* is the best risk achievable by the hypothesis class. For typical hypothesis classes where R</em>n = sqrt{R/n}, this translates to a learning rate of ̃ O(RH/n) in the separable (L* = 0) case and O(RH/n + sqrt{L* RH/n}) more generally. We also provide similar guarantees for online and stochastic convex optimization of a smooth non-negative objective.</p>