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PAC-Bayesian risk bounds for group-analysis sparse regression by exponential weighting

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  • Luu, Tung Duy
  • Fadili, Jalal
  • Chesneau, Christophe

Abstract

In this paper, we consider a high-dimensional nonparametric regression model with fixed design and iid random errors. We propose an estimator by exponential weighted aggregation with a group-analysis sparsity and a prior on the weights. We prove that our estimator satisfies a sharp group-analysis sparse oracle inequality with a small remainder term that ensures its good theoretical performance. We also propose a forward–backward proximal Langevin Monte Carlo algorithm to sample from the target distribution (which is neither smooth nor log-concave) and derive its convergence guarantees. In turn, this enables us to implement our estimator and validate it with numerical experiments.

Suggested Citation

  • Luu, Tung Duy & Fadili, Jalal & Chesneau, Christophe, 2019. "PAC-Bayesian risk bounds for group-analysis sparse regression by exponential weighting," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 209-233.
    • Handle: RePEc:eee:jmvana:v:171:y:2019:i:c:p:209-233
    DOI: 10.1016/j.jmva.2018.12.004
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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.
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    3. 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.
    4. Pradeep Ravikumar & John Lafferty & Han Liu & Larry Wasserman, 2009. "Sparse additive models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(5), pages 1009-1030, November.
    5. 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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