  ``Localization'' has proven to be a valuable tool in the Statistical Learning
literature as it allows sharp risk bounds in terms of the problem geometry.
Localized bounds seem to be much less exploited in the Stochastic Optimization
literature. In addition, there is an obvious interest in both communities in
obtaining risk bounds that require weak moment assumptions or
``heavier-tails''. In this work we use a localization toolbox to derive risk
bounds in two specific applications. The first is in portfolio risk
minimization with conditional value-at-risk constraints. We consider a setting
where, among all assets with high returns, there is a portion of dimension $g$,
unknown to the investor, that has significant less risk than the other
remaining portion. Our rates for the SAA problem show that ``risk inflation'',
caused by a multiplicative factor, affects the statistical rate only via a term
proportional to $g$. As the ``normalized risk'' increases, the contribution in
the rate from the extrinsic dimension diminishes while the dependence on $g$ is
kept fixed. Localization is a key tool to show this property. As a second
application of our localization toolbox, we obtain sharp oracle inequalities
for least-squares estimators with a Lasso-type constraint under weak moment
assumptions. One main consequence of these inequalities is to obtain
\emph{persistence}, as posed by Greenshtein and Ritov, with covariates having
heavier tails. This gives improvements in prior work of Bartlett, Mendelson and
Neeman.
