  We consider the problem of regression learning for deterministic design and
independent random errors. We start by proving a sharp PAC-Bayesian type bound
for the exponentially weighted aggregate (EWA) under the expected squared
empirical loss. For a broad class of noise distributions the presented bound is
valid whenever the temperature parameter $\beta$ of the EWA is larger than or
equal to $4\sigma^2$, where $\sigma^2$ is the noise variance. A remarkable
feature of this result is that it is valid even for unbounded regression
functions and the choice of the temperature parameter depends exclusively on
the noise level. Next, we apply this general bound to the problem of
aggregating the elements of a finite-dimensional linear space spanned by a
dictionary of functions $\phi_1,...,\phi_M$. We allow $M$ to be much larger
than the sample size $n$ but we assume that the true regression function can be
well approximated by a sparse linear combination of functions $\phi_j$. Under
this sparsity scenario, we propose an EWA with a heavy tailed prior and we show
that it satisfies a sparsity oracle inequality with leading constant one.
Finally, we propose several Langevin Monte-Carlo algorithms to approximately
compute such an EWA when the number $M$ of aggregated functions can be large.
We discuss in some detail the convergence of these algorithms and present
numerical experiments that confirm our theoretical findings.
