Fast global convergence of gradient methods for high-dimensional statistical recovery
Many statistical M-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient and composite gradient methods for solving such problems, working within a high-di...
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| Format: | Article |
| Language: | en_US |
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Institute of Mathematical Statistics
2013
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| Online Access: | http://hdl.handle.net/1721.1/78602 |
| _version_ | 1826216416845496320 |
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| author | Agarwal, Alekh Negahban, Sahand N. Wainwright, Martin J. |
| author2 | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science |
| author_facet | Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Agarwal, Alekh Negahban, Sahand N. Wainwright, Martin J. |
| author_sort | Agarwal, Alekh |
| collection | MIT |
| description | Many statistical M-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient and composite gradient methods for solving such problems, working within a high-dimensional framework that allows the ambient dimension d to grow with (and possibly exceed) the sample size n. Our theory identifies conditions under which projected gradient descent enjoys globally linear convergence up to the statistical precision of the model, meaning the typical distance between the true unknown parameter θ[superscript ∗] and an optimal solution [ˆ over θ]. By establishing these conditions with high probability for numerous statistical models, our analysis applies to a wide range of M-estimators, including sparse linear regression using Lasso; group Lasso for block sparsity; log-linear models with regularization; low-rank matrix recovery using nuclear norm regularization; and matrix decomposition using a combination of the nuclear and ℓ[subscript 1] norms. Overall, our analysis reveals interesting connections between statistical and computational efficiency in high-dimensional estimation. |
| first_indexed | 2024-09-23T16:47:21Z |
| format | Article |
| id | mit-1721.1/78602 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T16:47:21Z |
| publishDate | 2013 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/786022022-10-03T08:17:07Z Fast global convergence of gradient methods for high-dimensional statistical recovery Agarwal, Alekh Negahban, Sahand N. Wainwright, Martin J. Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology. Laboratory for Information and Decision Systems Negahban, Sahand N. Many statistical M-estimators are based on convex optimization problems formed by the combination of a data-dependent loss function with a norm-based regularizer. We analyze the convergence rates of projected gradient and composite gradient methods for solving such problems, working within a high-dimensional framework that allows the ambient dimension d to grow with (and possibly exceed) the sample size n. Our theory identifies conditions under which projected gradient descent enjoys globally linear convergence up to the statistical precision of the model, meaning the typical distance between the true unknown parameter θ[superscript ∗] and an optimal solution [ˆ over θ]. By establishing these conditions with high probability for numerous statistical models, our analysis applies to a wide range of M-estimators, including sparse linear regression using Lasso; group Lasso for block sparsity; log-linear models with regularization; low-rank matrix recovery using nuclear norm regularization; and matrix decomposition using a combination of the nuclear and ℓ[subscript 1] norms. Overall, our analysis reveals interesting connections between statistical and computational efficiency in high-dimensional estimation. United States. Air Force Office of Scientific Research (Grant AFOSR-09NL184) National Science Foundation (U.S.) (NSF-CDI-0941742) 2013-04-25T16:38:29Z 2013-04-25T16:38:29Z 2012-10 Article http://purl.org/eprint/type/JournalArticle 0090-5364 http://hdl.handle.net/1721.1/78602 Agarwal, Alekh, Sahand Negahban, and Martin J. Wainwright. "Fast global convergence of gradient methods for high-dimensional statistical recovery." Annals of Statistics 40.5 (2012): 2452-2482. ©Institute of Mathematical Statistics en_US http://dx.doi.org/10.1214/12-AOS1032SUPP Annals of Statistics Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Institute of Mathematical Statistics Institute of Mathematical Statistics |
| spellingShingle | Agarwal, Alekh Negahban, Sahand N. Wainwright, Martin J. Fast global convergence of gradient methods for high-dimensional statistical recovery |
| title | Fast global convergence of gradient methods for high-dimensional statistical recovery |
| title_full | Fast global convergence of gradient methods for high-dimensional statistical recovery |
| title_fullStr | Fast global convergence of gradient methods for high-dimensional statistical recovery |
| title_full_unstemmed | Fast global convergence of gradient methods for high-dimensional statistical recovery |
| title_short | Fast global convergence of gradient methods for high-dimensional statistical recovery |
| title_sort | fast global convergence of gradient methods for high dimensional statistical recovery |
| url | http://hdl.handle.net/1721.1/78602 |
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