Doubly Stochastic Primal-Dual Coordinate Method for Bilinear Saddle-Point Problem
This work addresses computational efficiency for machine learning practitioners dealing with large-scale optimization problems, but it is incremental as it builds on existing coordinate methods with specific improvements.
The authors tackled the bilinear saddle-point problem in empirical risk minimization by proposing a doubly stochastic primal-dual coordinate optimization algorithm, which achieves linear convergence with lower overall complexity than existing methods in cases like factorized data matrices or expensive proximal mappings, as confirmed by empirical studies on applications such as multi-task large margin nearest neighbor.
We propose a doubly stochastic primal-dual coordinate optimization algorithm for empirical risk minimization, which can be formulated as a bilinear saddle-point problem. In each iteration, our method randomly samples a block of coordinates of the primal and dual solutions to update. The linear convergence of our method could be established in terms of 1) the distance from the current iterate to the optimal solution and 2) the primal-dual objective gap. We show that the proposed method has a lower overall complexity than existing coordinate methods when either the data matrix has a factorized structure or the proximal mapping on each block is computationally expensive, e.g., involving an eigenvalue decomposition. The efficiency of the proposed method is confirmed by empirical studies on several real applications, such as the multi-task large margin nearest neighbor problem.