We proposed a doubly stochastic primal-dual coordinate optimization algorithm for regularized empirical risk minimization that can be formulated as a saddle-point problem using conjugate function. Different from existing coordinate methods, the proposed method randomly samples both primal and dual coordinates to update solutions, which is a desirable property when applied to data with both a high dimension and a large size. The convergence of our method is established not only in terms of the solution's distance to optimality but also in terms of the primal-dual objective gap. When applied to the data matrix factorized as a product of two smaller matrices, we show that the proposed method has a lower overall complexity than other coordinate methods, especially, when data size is large.
['Adams Wei Yu', 'Qihang Lin', 'Tianbao Yang']
Doubly Stochastic Primal-Dual Coordinate Method for Regularized Empirical Risk Minimization with Factorized Data