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We propose an efficient ADMM method with guarantees for high-dimensionalproblems. We provide explicit bounds for the sparse optimization problem andthe noisy matrix decomposition problem. For sparse optimization, we establishthat the modified ADMM method has an optimal convergence rate of$\mathcal{O}(s\log d/T)$, where $s$ is the sparsity level, $d$ is the datadimension and $T$ is the number of steps. This matches with the minimax lowerbounds for sparse estimation. For matrix decomposition into sparse and low rankcomponents, we provide the first guarantees for any online method, and prove aconvergence rate of $\tilde{\mathcal{O}}((s+r)\beta^2(p) /T) +\mathcal{O}(1/p)$ for a $p\times p$ matrix, where $s$ is the sparsity level,$r$ is the rank and $\Theta(\sqrt{p})\leq \beta(p)\leq \Theta(p)$. Ourguarantees match the minimax lower bound with respect to $s,r$ and $T$. Inaddition, we match the minimax lower bound with respect to the matrix dimension$p$, i.e. $\beta(p)=\Theta(\sqrt{p})$, for many important statistical modelsincluding the independent noise model, the linear Bayesian network and thelatent Gaussian graphical model under some conditions. Our ADMM method is basedon epoch-based annealing and consists of inexpensive steps which involveprojections on to simple norm balls. Experiments show that for both sparseoptimization and matrix decomposition problems, our algorithm outperforms thestate-of-the-art methods. In particular, we reach higher accuracy with sametime complexity.

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