{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/multi-step-stochastic-admm-in-high-dimensions","title":"Multi-Step Stochastic ADMM in High Dimensions: Applications to Sparse Optimization and Noisy Matrix Decomposition","arxiv_id":"1402.5131","date":"2014-02-20","proceeding":"NeurIPS 2014","authors":["Hanie Sedghi","Anima Anandkumar","Edmond Jonckheere"],"abstract":"We propose an efficient ADMM method with guarantees for high-dimensional\nproblems. We provide explicit bounds for the sparse optimization problem and\nthe noisy matrix decomposition problem. For sparse optimization, we establish\nthat the modified ADMM method has an optimal convergence rate of\n$\\mathcal{O}(s\\log d/T)$, where $s$ is the sparsity level, $d$ is the data\ndimension and $T$ is the number of steps. This matches with the minimax lower\nbounds for sparse estimation. For matrix decomposition into sparse and low rank\ncomponents, we provide the first guarantees for any online method, and prove a\nconvergence rate of $\\tilde{\\mathcal{O}}((s+r)\\beta^2(p) /T) +\n\\mathcal{O}(1/p)$ for a $p\\times p$ matrix, where $s$ is the sparsity level,\n$r$ is the rank and $\\Theta(\\sqrt{p})\\leq \\beta(p)\\leq \\Theta(p)$. Our\nguarantees match the minimax lower bound with respect to $s,r$ and $T$. In\naddition, we match the minimax lower bound with respect to the matrix dimension\n$p$, i.e. $\\beta(p)=\\Theta(\\sqrt{p})$, for many important statistical models\nincluding the independent noise model, the linear Bayesian network and the\nlatent Gaussian graphical model under some conditions. Our ADMM method is based\non epoch-based annealing and consists of inexpensive steps which involve\nprojections on to simple norm balls. Experiments show that for both sparse\noptimization and matrix decomposition problems, our algorithm outperforms the\nstate-of-the-art methods. In particular, we reach higher accuracy with same\ntime complexity.","url_abs":"http://arxiv.org/abs/1402.5131v6","url_pdf":"http://arxiv.org/pdf/1402.5131v6.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"multi-step-stochastic-admm-in-high-dimensions","repo_url":"https://github.com/haniesedghi/REASON2","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"multi-step-stochastic-admm-in-high-dimensions","repo_url":"https://github.com/FanjieLUO/matlab","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":null}],"tasks":[],"methods":[{"method_slug":"admm","method_name":"ADMM"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}