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

International audienceIn this paper, we consider a high-dimensional non-parametric regression model with fixed design and i.i.d. random errors. We propose a powerful estimator by exponential weighted aggregation (EWA) with a group-analysis sparsity promoting prior on the weights. We prove that our estimator satisfies a sharp group-analysis sparse oracle inequality with a small remainder term ensuring its good theoretical performances. We also propose a forward-backward proximal Langevin Monte-Carlo algorithm to sample from the target distribution (which is not smooth nor log-concave) and derive its guarantees. In turn, this allows us to implement our estimator and validate it on some numerical experiments