PAC-Bayesian Bounds for Sparse Regression Estimation with Exponential Weights

Abstract : We consider the sparse regression model where the number of parameters $p$ is larger than the sample size $n$. The difficulty when considering high-dimensional problems is to propose estimators achieving a good compromise between statistical and computational performances. The BIC estimator for instance performs well from the statistical point of view \cite{BTW07} but can be computed for values of $p$ of at most a few tens. The Lasso estimator is solution of a convex minimization problem. Hence it can be computed for large value of $p$. However stringent conditions on the design are required to establish the statistical properties of this estimator. Dalalyan and Tsybakov \cite{arnak} propose a method achieving a good compromise between the statistical and computational aspects of the problem. Their estimator can be computed for reasonably large $p$ and satisfies nice statistical properties under weak assumptions on the design. However, \cite{arnak} concerns only the empirical risk and proposes only results in expectation. In this paper, we propose an aggregation procedure similar to that of \cite{arnak} but with improved statistical performances. Our main result concerns the expected risk and is given in probability. We also propose a MCMC method to compute our estimator for reasonably large values of $p$.
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Preprints, Working Papers, ...
2010
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https://hal.archives-ouvertes.fr/hal-00465801
Contributor : Karim Lounici <>
Submitted on : Tuesday, September 14, 2010 - 6:01:31 PM
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Pierre Alquier, Karim Lounici. PAC-Bayesian Bounds for Sparse Regression Estimation with Exponential Weights. 2010. <hal-00465801v3>

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