  The minimization of the logistic loss is a popular approach to batch
supervised learning. Our paper starts from the surprising observation that,
when fitting linear (or kernelized) classifiers, the minimization of the
logistic loss is \textit{equivalent} to the minimization of an exponential
\textit{rado}-loss computed (i) over transformed data that we call Rademacher
observations (rados), and (ii) over the \textit{same} classifier as the one of
the logistic loss. Thus, a classifier learnt from rados can be
\textit{directly} used to classify \textit{observations}. We provide a learning
algorithm over rados with boosting-compliant convergence rates on the
\textit{logistic loss} (computed over examples). Experiments on domains with up
to millions of examples, backed up by theoretical arguments, display that
learning over a small set of random rados can challenge the state of the art
that learns over the \textit{complete} set of examples. We show that rados
comply with various privacy requirements that make them good candidates for
machine learning in a privacy framework. We give several algebraic, geometric
and computational hardness results on reconstructing examples from rados. We
also show how it is possible to craft, and efficiently learn from, rados in a
differential privacy framework. Tests reveal that learning from differentially
private rados can compete with learning from random rados, and hence with batch
learning from examples, achieving non-trivial privacy vs accuracy tradeoffs.
