  The optimality and sensitivity of the empirical risk minimization problem
with relative entropy regularization (ERM-RER) are investigated for the case in
which the reference is a sigma-finite measure instead of a probability measure.
This generalization allows for a larger degree of flexibility in the
incorporation of prior knowledge over the set of models. In this setting, the
interplay of the regularization parameter, the reference measure, the risk
function, and the empirical risk induced by the solution of the ERM-RER problem
is characterized. This characterization yields necessary and sufficient
conditions for the existence of a regularization parameter that achieves an
arbitrarily small empirical risk with arbitrarily high probability. The
sensitivity of the expected empirical risk to deviations from the solution of
the ERM-RER problem is studied. The sensitivity is then used to provide upper
and lower bounds on the expected empirical risk. Moreover, it is shown that the
expectation of the sensitivity is upper bounded, up to a constant factor, by
the square root of the lautum information between the models and the datasets.
