  The local Rademacher complexity framework is one of the most successful
general-purpose toolboxes for establishing sharp excess risk bounds for
statistical estimators based on the framework of empirical risk minimization.
Applying this toolbox typically requires using the Bernstein condition, which
often restricts applicability to convex and proper settings. Recent years have
witnessed several examples of problems where optimal statistical performance is
only achievable via non-convex and improper estimators originating from
aggregation theory, including the fundamental problem of model selection. These
examples are currently outside of the reach of the classical localization
theory.
  In this work, we build upon the recent approach to localization via offset
Rademacher complexities, for which a general high-probability theory has yet to
be established. Our main result is an exponential-tail excess risk bound
expressed in terms of the offset Rademacher complexity that yields results at
least as sharp as those obtainable via the classical theory. However, our bound
applies under an estimator-dependent geometric condition (the "offset
condition") instead of the estimator-independent (but, in general,
distribution-dependent) Bernstein condition on which the classical theory
relies. Our results apply to improper prediction regimes not directly covered
by the classical theory.
