  We formulate a uniform tail bound for empirical processes indexed by a class
of functions, in terms of the individual deviations of the functions rather
than the worst-case deviation in the considered class. The tail bound is
established by introducing an initial "deflation" step to the standard generic
chaining argument. The resulting tail bound is the sum of the complexity of the
"deflated function class" in terms of a generalization of Talagrand's $\gamma$
functional, and the deviation of the function instance, both of which are
formulated based on the natural seminorm induced by the corresponding
Cram\'{e}r functions. We also provide certain approximations for the mentioned
seminorm when the function class lies in a given (exponential type) Orlicz
space, that can be used to make the complexity term and the deviation term more
explicit.
