Smooth approximation method for non-smooth empirical risk minimization based distance metric learning

Distance metric learning (DML) has become a very active research field in recent years. Bian and Tao (IEEE Trans. Neural Netw. Learn. Syst. 23(8) (2012) 1194–1205) presented a constrained empirical risk minimization (ERM) framework for DML. In this paper, we utilize smooth approximation method to make their algorithm applicable to the non-differentiable hinge loss function. We show that the objective function with hinge loss is equivalent to a non-smooth min–max representation, from which an approximate objective function is derived. Compared to the original objective function, the approximate one becomes differentiable with Lipschitz-continuous gradient. Consequently, Nesterov's optimal first-order method can be directly used. Finally, the effectiveness of our method is evaluated on various UCI datasets.