Multi-Step Stochastic ADMM in High Dimensions: Applications to Sparse Optimization and Matrix Decomposition

In this paper, we consider a multi-step version of the stochastic ADMM method with efficient guarantees for high-dimensional problems. We first analyze the simple setting, where the optimization problem consists of a loss function and a single regularizer (e.g. sparse optimization), and then extend to the multi-block setting with multiple regularizers and multiple variables (e.g. matrix decomposition into sparse and low rank components). For the sparse optimization problem, our method achieves the minimax rate of $O(s\log d/T)$ for $s$-sparse problems in $d$ dimensions in $T$ steps, and is thus, unimprovable by any method up to constant factors. For the matrix decomposition problem with a general loss function, we analyze the multi-step ADMM with multiple blocks. We establish $O(1/T)$ rate and efficient scaling as the size of matrix grows. For natural noise models (e.g. independent noise), our convergence rate is minimax-optimal. Thus, we establish tight convergence guarantees for multi-block ADMM in high dimensions. Experiments show that for both sparse optimization and matrix decomposition problems, our algorithm outperforms the state-of-the-art methods.