Sparse learning via Boolean relaxations

We introduce novel relaxations for cardinality-constrained learning problems, including least-squares regression as a special but important case. Our approach is based on reformulating a cardinality-constrained problem exactly as a Boolean program, to which standard convex relaxations such as the La...

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Published in:	Mathematical programming Vol. 151; no. 1; pp. 63 - 87
Main Authors:	Pilanci, Mert, Wainwright, Martin J., El Ghaoui, Laurent
Format:	Journal Article
Language:	English
Published:	Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2015 Springer Nature B.V
Subjects:	Artificial intelligence Boolean Calculus of Variations and Optimal Control; Optimization Combinatorics Control theory Electrical engineering Full Length Paper Machine learning Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Sparsity Studies Support vector machines Theoretical United States > US 68T05 Learning and adaptive systems 90C25 Convex programming 90C09 Boolean programming Sparsity 90C06 Large-scale problems Machine learning Convex relaxation Combinatorial optimization Regularization
ISSN:	0025-5610, 1436-4646
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Abstract	We introduce novel relaxations for cardinality-constrained learning problems, including least-squares regression as a special but important case. Our approach is based on reformulating a cardinality-constrained problem exactly as a Boolean program, to which standard convex relaxations such as the Lasserre and Sherali-Adams hierarchies can be applied. We analyze the first-order relaxation in detail, deriving necessary and sufficient conditions for exactness in a unified manner. In the special case of least-squares regression, we show that these conditions are satisfied with high probability for random ensembles satisfying suitable incoherence conditions, similar to results on ℓ 1 -relaxations. In contrast to known methods, our relaxations yield lower bounds on the objective, and it can be verified whether or not the relaxation is exact. If it is not, we show that randomization based on the relaxed solution offers a principled way to generate provably good feasible solutions. This property enables us to obtain high quality estimates even if incoherence conditions are not met, as might be expected in real datasets. We numerically illustrate the performance of the relaxation-randomization strategy in both synthetic and real high-dimensional datasets, revealing substantial improvements relative to ℓ 1 -based methods and greedy selection heuristics.
AbstractList	We introduce novel relaxations for cardinality-constrained learning problems, including least-squares regression as a special but important case. Our approach is based on reformulating a cardinality-constrained problem exactly as a Boolean program, to which standard convex relaxations such as the Lasserre and Sherali-Adams hierarchies can be applied. We analyze the first-order relaxation in detail, deriving necessary and sufficient conditions for exactness in a unified manner. In the special case of least-squares regression, we show that these conditions are satisfied with high probability for random ensembles satisfying suitable incoherence conditions, similar to results on ℓ 1 -relaxations. In contrast to known methods, our relaxations yield lower bounds on the objective, and it can be verified whether or not the relaxation is exact. If it is not, we show that randomization based on the relaxed solution offers a principled way to generate provably good feasible solutions. This property enables us to obtain high quality estimates even if incoherence conditions are not met, as might be expected in real datasets. We numerically illustrate the performance of the relaxation-randomization strategy in both synthetic and real high-dimensional datasets, revealing substantial improvements relative to ℓ 1 -based methods and greedy selection heuristics. (ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image) Issue Title: Special Issue: International Symposium on Mathematical Programming, Pittsburgh, July 2015 We introduce novel relaxations for cardinality-constrained learning problems, including least-squares regression as a special but important case. Our approach is based on reformulating a cardinality-constrained problem exactly as a Boolean program, to which standard convex relaxations such as the Lasserre and Sherali-Adams hierarchies can be applied. We analyze the first-order relaxation in detail, deriving necessary and sufficient conditions for exactness in a unified manner. In the special case of least-squares regression, we show that these conditions are satisfied with high probability for random ensembles satisfying suitable incoherence conditions, similar to results on ...-relaxations. In contrast to known methods, our relaxations yield lower bounds on the objective, and it can be verified whether or not the relaxation is exact. If it is not, we show that randomization based on the relaxed solution offers a principled way to generate provably good feasible solutions. This property enables us to obtain high quality estimates even if incoherence conditions are not met, as might be expected in real datasets. We numerically illustrate the performance of the relaxation-randomization strategy in both synthetic and real high-dimensional datasets, revealing substantial improvements relative to ...-based methods and greedy selection heuristics.
Author	Wainwright, Martin J. Pilanci, Mert El Ghaoui, Laurent
Author_xml	– sequence: 1 givenname: Mert surname: Pilanci fullname: Pilanci, Mert organization: Department of Electrical Engineering and Computer Sciences, University of California – sequence: 2 givenname: Martin J. surname: Wainwright fullname: Wainwright, Martin J. organization: Department of Electrical Engineering and Computer Sciences and Department of Statistics, University of California – sequence: 3 givenname: Laurent surname: El Ghaoui fullname: El Ghaoui, Laurent email: elghaoui@berkeley.edu organization: Department of Electrical Engineering and Computer Sciences, University of California
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References	Lovász, Schrijver (CR19) 1991; 1 Wainwright (CR33) 2009; 55 Laurent (CR17) 2003; 28 CR16 Oliveira (CR26) 2010; 15 CR37 Zhao, Yu (CR38) 2006; 7 Markowitz (CR20) 1959 CR35 Zou, Hastie (CR39) 2005; 67 CR31 Wainwright (CR34) 2014; 1 Candes, Tao (CR6) 2005; 51 Dekel, Singer (CR11) 2007; 19 Donoho (CR12) 2006; 52 Donoho, Elad, Temlyakov (CR13) 2006; 52 CR2 Davidson, Szarek (CR10) 2001 Bühlmann, van de Geer (CR5) 2011 CR8 Wasserman (CR36) 2000; 44 Boyd, Vandenberghe (CR4) 2004 Grant, Boyd, Blondel, Boyd, Kimura (CR15) 2008 CR28 Ahlswede, Winter (CR1) 2002; 48 CR9 CR27 McCullagh, Nelder (CR21) 1989 Ledoux (CR18) 2001 CR25 Bickel, Ritov, Tsybakov (CR3) 2009; 37 CR24 Wainwright (CR32) 2009; 55 Meinshausen, Bühlmann (CR22) 2006; 34 Tibshirani (CR30) 1996; 58 Fuchs (CR14) 2004; 2 Cristianini, Shawe-Taylor (CR7) 2000 Motwani, Raghavan (CR23) 1995 Sherali, Adams (CR29) 1990; 3 894_CR25 894_CR24 R Tibshirani (894_CR30) 1996; 58 JJ Fuchs (894_CR14) 2004; 2 RI Oliveira (894_CR26) 2010; 15 MJ Wainwright (894_CR33) 2009; 55 HD Sherali (894_CR29) 1990; 3 P Zhao (894_CR38) 2006; 7 HM Markowitz (894_CR20) 1959 M Laurent (894_CR17) 2003; 28 L Lovász (894_CR19) 1991; 1 894_CR9 R Motwani (894_CR23) 1995 MJ Wainwright (894_CR34) 2014; 1 894_CR8 O Dekel (894_CR11) 2007; 19 Larry Wasserman (894_CR36) 2000; 44 894_CR16 894_CR2 DL Donoho (894_CR13) 2006; 52 S Boyd (894_CR4) 2004 R Ahlswede (894_CR1) 2002; 48 P Bühlmann (894_CR5) 2011 EJ Candes (894_CR6) 2005; 51 DL Donoho (894_CR12) 2006; 52 894_CR31 894_CR37 894_CR35 H Zou (894_CR39) 2005; 67 N Cristianini (894_CR7) 2000 M Ledoux (894_CR18) 2001 KR Davidson (894_CR10) 2001 PJ Bickel (894_CR3) 2009; 37 M Grant (894_CR15) 2008 MJ Wainwright (894_CR32) 2009; 55 P McCullagh (894_CR21) 1989 N Meinshausen (894_CR22) 2006; 34 894_CR28 894_CR27
References_xml	– volume: 67 start-page: 301 issue: 2 year: 2005 end-page: 320 ident: CR39 article-title: Regularization and variable selection via the elastic net publication-title: J. R. Stat. Soc. Ser. B doi: 10.1111/j.1467-9868.2005.00503.x – volume: 7 start-page: 2541 year: 2006 end-page: 2567 ident: CR38 article-title: On model selection consistency of Lasso publication-title: J. Mach. Learn. Res. – year: 1989 ident: CR21 publication-title: Generalized Linear Models. Monographs on Statistics and Applied Probability 37 – volume: 34 start-page: 1436 year: 2006 end-page: 1462 ident: CR22 article-title: High-dimensional graphs and variable selection with the Lasso publication-title: Ann. Stat. doi: 10.1214/009053606000000281 – volume: 52 start-page: 1289 issue: 4 year: 2006 end-page: 1306 ident: CR12 publication-title: Compress. sensing. IEEE Trans. Info. Theory doi: 10.1109/TIT.2006.871582 – ident: CR2 – ident: CR16 – ident: CR37 – year: 1959 ident: CR20 publication-title: Portf. Sel. – volume: 55 start-page: 5728 year: 2009 end-page: 5741 ident: CR32 article-title: Information-theoretic bounds on sparsity recovery in the high-dimensional and noisy setting publication-title: IEEE Trans. Info. Theory doi: 10.1109/TIT.2009.2032816 – volume: 51 start-page: 4203 issue: 12 year: 2005 end-page: 4215 ident: CR6 article-title: Decoding by linear programming publication-title: IEEE Trans. Info. Theory doi: 10.1109/TIT.2005.858979 – year: 2000 ident: CR7 publication-title: An Introduction to Support Vector Machines (and Other Kernel Based Learning Methods) doi: 10.1017/CBO9780511801389 – volume: 52 start-page: 6 issue: 1 year: 2006 end-page: 18 ident: CR13 article-title: Stable recovery of sparse overcomplete representations in the presence of noise publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2005.860430 – year: 2001 ident: CR18 publication-title: The Concentration of Measure Phenomenon – volume: 58 start-page: 267 issue: 1 year: 1996 end-page: 288 ident: CR30 article-title: Regression shrinkage and selection via the Lasso publication-title: J. R. Stat. Soc. Ser. B – year: 2011 ident: CR5 publication-title: Statistics for High-Dimensional Data. Springer Series in Statistics doi: 10.1007/978-3-642-20192-9 – ident: CR35 – volume: 55 start-page: 2183 year: 2009 end-page: 2202 ident: CR33 article-title: Sharp thresholds for high-dimensional and noisy sparsity recovery using -constrained quadratic programming (Lasso) publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2009.2016018 – year: 2004 ident: CR4 publication-title: Convex Optimization doi: 10.1017/CBO9780511804441 – volume: 44 start-page: 92 issue: 1 year: 2000 end-page: 107 ident: CR36 article-title: Bayesian model selection and model averaging publication-title: J. Math. Psychol. doi: 10.1006/jmps.1999.1278 – ident: CR8 – volume: 19 start-page: 345 year: 2007 ident: CR11 article-title: Support vector machines on a budget publication-title: Adv. Neural Inf. Process. Syst. – ident: CR25 – volume: 1 start-page: 233 year: 2014 end-page: 253 ident: CR34 article-title: Structured regularizers: statistical and computational issues publication-title: Annu. Rev. Stat. Appl. – ident: CR27 – volume: 15 start-page: 203 year: 2010 end-page: 212 ident: CR26 article-title: Sums of random Hermitian matrices and an inequality by Rudelson publication-title: Elec. Comm. Prob. doi: 10.1214/ECP.v15-1544 – volume: 28 start-page: 470 year: 2003 end-page: 496 ident: CR17 article-title: A comparison of the Sherali-Adams, Lovász-Schrijver and Lasserre relaxations for 0–1 programming publication-title: Math. Oper. Res. doi: 10.1287/moor.28.3.470.16391 – volume: 2 start-page: 533 year: 2004 end-page: 536 ident: CR14 article-title: Recovery of exact sparse representations in the presence of noise publication-title: ICASSP – start-page: 317 year: 2001 end-page: 336 ident: CR10 article-title: Local operator theory, random matrices and Banach spaces publication-title: Handbook of Banach Spaces – ident: CR31 – year: 1995 ident: CR23 publication-title: Randomized Algorithms doi: 10.1017/CBO9780511814075 – ident: CR9 – volume: 1 start-page: 166 year: 1991 end-page: 190 ident: CR19 article-title: Cones of matrices and set-functions and 0–1 optimization publication-title: SIAM J. Optim. doi: 10.1137/0801013 – volume: 3 start-page: 411 year: 1990 end-page: 430 ident: CR29 article-title: A hierarchy of relaxations between the continuous and convex hull representations for zero-one programming problems publication-title: SIAM J. Discrete Math. doi: 10.1137/0403036 – ident: CR28 – volume: 37 start-page: 1705 issue: 4 year: 2009 end-page: 1732 ident: CR3 article-title: Simultaneous analysis of Lasso and Dantzig selector publication-title: Ann. Stat. doi: 10.1214/08-AOS620 – start-page: 95 year: 2008 end-page: 110 ident: CR15 article-title: Graph implementations for nonsmooth convex programs publication-title: Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences – ident: CR24 – volume: 48 start-page: 569 issue: 3 year: 2002 end-page: 579 ident: CR1 article-title: Strong converse for identification via quantum channels publication-title: IEEE Trans. Inf. Theory doi: 10.1109/18.985947 – volume: 7 start-page: 2541 year: 2006 ident: 894_CR38 publication-title: J. Mach. Learn. Res. – volume: 37 start-page: 1705 issue: 4 year: 2009 ident: 894_CR3 publication-title: Ann. Stat. doi: 10.1214/08-AOS620 – volume: 44 start-page: 92 issue: 1 year: 2000 ident: 894_CR36 publication-title: J. Math. Psychol. doi: 10.1006/jmps.1999.1278 – volume: 34 start-page: 1436 year: 2006 ident: 894_CR22 publication-title: Ann. Stat. doi: 10.1214/009053606000000281 – volume: 19 start-page: 345 year: 2007 ident: 894_CR11 publication-title: Adv. Neural Inf. Process. Syst. – volume: 15 start-page: 203 year: 2010 ident: 894_CR26 publication-title: Elec. Comm. Prob. doi: 10.1214/ECP.v15-1544 – ident: 894_CR28 – volume: 67 start-page: 301 issue: 2 year: 2005 ident: 894_CR39 publication-title: J. R. Stat. Soc. Ser. B doi: 10.1111/j.1467-9868.2005.00503.x – ident: 894_CR8 – volume: 52 start-page: 6 issue: 1 year: 2006 ident: 894_CR13 publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2005.860430 – volume-title: Statistics for High-Dimensional Data. Springer Series in Statistics year: 2011 ident: 894_CR5 doi: 10.1007/978-3-642-20192-9 – volume-title: An Introduction to Support Vector Machines (and Other Kernel Based Learning Methods) year: 2000 ident: 894_CR7 doi: 10.1017/CBO9780511801389 – volume: 3 start-page: 411 year: 1990 ident: 894_CR29 publication-title: SIAM J. Discrete Math. doi: 10.1137/0403036 – volume: 48 start-page: 569 issue: 3 year: 2002 ident: 894_CR1 publication-title: IEEE Trans. Inf. Theory doi: 10.1109/18.985947 – ident: 894_CR2 – volume-title: Convex Optimization year: 2004 ident: 894_CR4 doi: 10.1017/CBO9780511804441 – ident: 894_CR24 – volume: 1 start-page: 233 year: 2014 ident: 894_CR34 publication-title: Annu. Rev. Stat. Appl. doi: 10.1146/annurev-statistics-022513-115643 – ident: 894_CR25 doi: 10.2139/ssrn.912637 – ident: 894_CR31 – volume: 52 start-page: 1289 issue: 4 year: 2006 ident: 894_CR12 publication-title: Compress. sensing. IEEE Trans. Info. Theory doi: 10.1109/TIT.2006.871582 – ident: 894_CR27 – ident: 894_CR35 – start-page: 317 volume-title: Handbook of Banach Spaces year: 2001 ident: 894_CR10 – volume: 28 start-page: 470 year: 2003 ident: 894_CR17 publication-title: Math. Oper. Res. doi: 10.1287/moor.28.3.470.16391 – volume: 58 start-page: 267 issue: 1 year: 1996 ident: 894_CR30 publication-title: J. R. Stat. Soc. Ser. B doi: 10.1111/j.2517-6161.1996.tb02080.x – ident: 894_CR16 – start-page: 95 volume-title: Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences year: 2008 ident: 894_CR15 doi: 10.1007/978-1-84800-155-8_7 – volume: 55 start-page: 5728 year: 2009 ident: 894_CR32 publication-title: IEEE Trans. Info. Theory doi: 10.1109/TIT.2009.2032816 – volume-title: Generalized Linear Models. Monographs on Statistics and Applied Probability 37 year: 1989 ident: 894_CR21 – ident: 894_CR9 doi: 10.1007/s10107-010-0416-0 – ident: 894_CR37 – volume: 1 start-page: 166 year: 1991 ident: 894_CR19 publication-title: SIAM J. Optim. doi: 10.1137/0801013 – volume: 51 start-page: 4203 issue: 12 year: 2005 ident: 894_CR6 publication-title: IEEE Trans. Info. Theory doi: 10.1109/TIT.2005.858979 – volume: 55 start-page: 2183 year: 2009 ident: 894_CR33 publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.2009.2016018 – volume-title: The Concentration of Measure Phenomenon year: 2001 ident: 894_CR18 – volume: 2 start-page: 533 year: 2004 ident: 894_CR14 publication-title: ICASSP – volume-title: Randomized Algorithms year: 1995 ident: 894_CR23 doi: 10.1017/CBO9780511814075 – volume-title: Portf. Sel. year: 1959 ident: 894_CR20
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SubjectTerms	Artificial intelligence Boolean Calculus of Variations and Optimal Control; Optimization Combinatorics Control theory Electrical engineering Full Length Paper Machine learning Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Sparsity Studies Support vector machines Theoretical
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Title	Sparse learning via Boolean relaxations
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