Foundations of Computational Mathematics

, Volume 14, Issue 3, pp 569-600

First online:

Random Design Analysis of Ridge Regression

  • Daniel HsuAffiliated withDepartment of Computer Science, Columbia University Email author 
  • , Sham M. KakadeAffiliated withMicrosoft Research
  • , Tong ZhangAffiliated withDepartment of Statistics, Rutgers University

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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.


Linear regression Ordinary least squares Ridge regression Randomized approximation

Mathematics Subject Classification

Primary 62J07 Secondary 62J05