  We extend the correspondence between two-stage coding procedures in data
compression and penalized likelihood procedures in statistical estimation.
Traditionally, this had required restriction to countable parameter spaces. We
show how to extend this correspondence in the uncountable parameter case.
Leveraging the description length interpretations of penalized likelihood
procedures we devise new techniques to derive adaptive risk bounds of such
procedures. We show that the existence of certain countable coverings of the
parameter space implies adaptive risk bounds and thus our theory is quite
general. We apply our techniques to illustrate risk bounds for $\ell_1$ type
penalized procedures in canonical high dimensional statistical problems such as
linear regression and Gaussian graphical Models. In the linear regression
problem, we also demonstrate how the traditional $l_0$ penalty times
$\frac{\log(n)}{2}$ plus lower order terms has a two stage description length
interpretation and present risk bounds for this penalized likelihood procedure.
