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The degrees of freedom of partly smooth regularizers

Author

Listed:
  • Samuel Vaiter

    (CEREMADE, CNRS, Université Paris-Dauphine)

  • Charles Deledalle

    (IMB, CNRS, Université Bordeaux 1)

  • Jalal Fadili

    (Normandie Univ, ENSICAEN, CNRS, GREYC)

  • Gabriel Peyré

    (CEREMADE, CNRS, Université Paris-Dauphine)

  • Charles Dossal

    (IMB, CNRS, Université Bordeaux 1)

Abstract

We study regularized regression problems where the regularizer is a proper, lower-semicontinuous, convex and partly smooth function relative to a Riemannian submanifold. This encompasses several popular examples including the Lasso, the group Lasso, the max and nuclear norms, as well as their composition with linear operators (e.g., total variation or fused Lasso). Our main sensitivity analysis result shows that the predictor moves locally stably along the same active submanifold as the observations undergo small perturbations. This plays a pivotal role in getting a closed-form expression for the divergence of the predictor w.r.t. observations. We also show that, for many regularizers, including polyhedral ones or the analysis group Lasso, this divergence formula holds Lebesgue a.e. When the perturbation is random (with an appropriate continuous distribution), this allows us to derive an unbiased estimator of the degrees of freedom and the prediction risk. Our results unify and go beyond those already known in the literature.

Suggested Citation

  • Samuel Vaiter & Charles Deledalle & Jalal Fadili & Gabriel Peyré & Charles Dossal, 2017. "The degrees of freedom of partly smooth regularizers," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 69(4), pages 791-832, August.
    • Handle: RePEc:spr:aistmt:v:69:y:2017:i:4:d:10.1007_s10463-016-0563-z
    DOI: 10.1007/s10463-016-0563-z
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    References listed on IDEAS

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    1. Robert Tibshirani & Michael Saunders & Saharon Rosset & Ji Zhu & Keith Knight, 2005. "Sparsity and smoothness via the fused lasso," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(1), pages 91-108, February.
    2. Lukas Meier & Sara Van De Geer & Peter Bühlmann, 2008. "The group lasso for logistic regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 70(1), pages 53-71, February.
    3. Jérôme Bolte & Aris Daniilidis & Adrian S. Lewis, 2011. "Generic Optimality Conditions for Semialgebraic Convex Programs," Mathematics of Operations Research, INFORMS, vol. 36(1), pages 55-70, February.
    4. Kato, Kengo, 2009. "On the degrees of freedom in shrinkage estimation," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1338-1352, August.
    5. Ming Yuan & Yi Lin, 2007. "Model selection and estimation in the Gaussian graphical model," Biometrika, Biometrika Trust, vol. 94(1), pages 19-35.
    6. Ming Yuan & Yi Lin, 2006. "Model selection and estimation in regression with grouped variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 49-67, February.
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    Cited by:

    1. Adrian S. Lewis & Calvin Wylie, 2021. "Active‐Set Newton Methods and Partial Smoothness," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 712-725, May.
    2. Cesare Molinari & Jingwei Liang & Jalal Fadili, 2019. "Convergence Rates of Forward–Douglas–Rachford Splitting Method," Journal of Optimization Theory and Applications, Springer, vol. 182(2), pages 606-639, August.
    3. Tung Duy Luu & Jalal Fadili & Christophe Chesneau, 2020. "Sharp oracle inequalities for low-complexity priors," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(2), pages 353-397, April.

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