In this paper, we propose proximal splitting-type algorithms for sampling from distributions whose densities are not necessarily smooth nor log-concave. Our approach brings together tools from, on the one hand, variational analysis and non-smooth optimization, and on the other hand, stochastic diffusion equations, and in particular the Langevin diffusion. We establish in particular consistency guarantees of our algorithms seen as discretization schemes in this context. These algorithms are then applied to compute the exponentially weighted aggregates for regression problems involving non-smooth penalties that are commonly used to promote some notion of simplicity/complexity. Some popular penalties are detailed and implemented on some numerical experiments.

In this paper, we consider a high-dimensional non-parametric regression model with fixed design and i.i.d. random errors. We propose an 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 convergence guarantees. In turn, this allows us to implement our estimator and validate it on some numerical experiments.