Random Design Analysis of Ridge Regression
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This work gives a simultaneous analysis of both the ordinary least squares estimator and the ridge regression estimator in the random design setting under mild assumptions on the covariate/response distributions. In particular, the analysis provides sharp results on the “out-of-sample” prediction error, as opposed to the “in-sample” (fixed design) error. The analysis also reveals the effect of errors in the estimated covariance structure, as well as the effect of modeling errors, neither of which effects are present in the fixed design setting. The proofs of the main results are based on a simple decomposition lemma combined with concentration inequalities for random vectors and matrices.
- N. Ailon and B. Chazelle. Approximate nearest neighbors and the fast Johnson-Lindenstrauss transform. SIAM J. Comput., 39(1):302–322, 2009. CrossRef
- J.-Y. Audibert and O. Catoni. Linear regression through PAC-Bayesian truncation, 2010. arXiv:1010.0072.
- J.-Y. Audibert and O. Catoni. Robust linear least squares regression. The Annals of Statistics, 30(5):2766–2794, 2011. CrossRef
- A. Caponnetto and E. De Vito. Optimal rates for the regularized least-squares algorithm. Foundations of Computational Mathematics, 7(3):331–368, 2007. CrossRef
- O. Catoni. Statistical Learning Theory and Stochastic Optimization, Lectures on Probability and Statistics, Ecole d’Eté de Probabilitiés de Saint-Flour XXXI - 2001, volume 1851 of Lecture Notes in Mathematics. Springer, 2004.
- P. Drineas and M. W. Mahoney. Effective Resistances, Statistical Leverage, and Applications to Linear Equation Solving, 2010. arXiv:1005.3097.
- P. Drineas, M. W. Mahoney, S. Muthukrishnan, and T. Sarlós. Faster least squares approximation. Numerische Mathematik, 117(2):219–249, 2010. CrossRef
- L. Györfi, M. Kohler, A. Kryżak, and H. Walk. A Distribution-Free Theory of Nonparametric Regression. Springer, 2004.
- A. E. Hoerl. Application of ridge analysis to regression problems. Chemical Engineering Progress, 58:54–59, 1962.
- R. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, 1985.
- D. Hsu, S. M. Kakade, and T. Zhang. A tail inequality for quadratic forms of subgaussian random vectors, 2011. arXiv:1110.2842.
- D. Hsu, S. M. Kakade, and T. Zhang. Tail inequalities for sums of random matrices that depend on the intrinsic dimension. Electronic Communications in Probability, 17(14):1–13, 2012.
- D. Hsu and S. Sabato. Loss Minimization and Parameter Estimation with Heavy Tails, 2013. arXiv:1307.1827.
- V. Koltchinskii. Local Rademacher complexities and oracle inequalities in risk minimization. The Annals of Statistics, 34(6):2593–2656, 2006. CrossRef
- B. Laurent and P. Massart. Adaptive estimation of a quadratic functional by model selection. The Annals of Statistics, 28(5):1302–1338, 2000. CrossRef
- E. L. Lehmann and G. Casella. Theory of Point Estimation. Springer, second edition, 1998.
- M. Nussbaum. Minimax risk: Pinsker bound. In S. Kotz, editor, Encyclopedia of Statistical Sciences, Update Volume 3, pages 451–460. Wiley, New York, 1999.
- V. Rokhlin and M. Tygert. A fast randomized algorithm for overdetermined linear least-squares regression. Proc. Natl. Acad. Sci. USA, 105(36):13212–13217, 2008. CrossRef
- S. Smale and D.-X. Zhou. Learning theory estimates via integral operators and their approximations. Constructive Approximations, 26:153–172, 2007. CrossRef
- I. Steinwart, D. Hush, and C. Scovel. Optimal Rates for Regularized Least Squares Regression. In Proceedings of the 22nd Annual Conference on Learning Theory, pp. 79–93, 2009.
- G. W. Stewart and J.-G. Sun. Matrix Perturbation Theory. Academic Press, 1990.
- C. J. Stone. Optimal global rates of convergence for nonparametric regression. The Annals of Statistics, 10:1040–1053, 1982. CrossRef
- T. Zhang. Learning bounds for kernel regression using effective data dimensionality. Neural Computation, 17:2077–2098, 2005. CrossRef
- Random Design Analysis of Ridge Regression
Foundations of Computational Mathematics
Volume 14, Issue 3 , pp 569-600
- Cover Date
- Print ISSN
- Online ISSN
- Springer US
- Additional Links
- Linear regression
- Ordinary least squares
- Ridge regression
- Randomized approximation
- Primary 62J07
- Secondary 62J05
- Industry Sectors
- Author Affiliations
- 1. Department of Computer Science, Columbia University, 450 Computer Science Building, 1214 Amsterdam Avenue, Mailcode 0401, New York, NY, 10027-7003, USA
- 2. Microsoft Research, One Memorial Drive, Cambridge, MA, 02142, USA
- 3. Department of Statistics, Rutgers University, 501 Hill Center, 110 Frelinghuysen Road, Piscataway, NJ, 08854, USA