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.

翻译：我们针对由函数类索引的经验过程，提出了一种基于函数个体偏差（而非所考虑类中的最坏情况偏差）的一致尾部界。该尾部界通过向标准通用链论证引入一个初始“缩减”步骤而建立。所得尾部界是“缩减函数类”的复杂度（基于Talagrand γ泛函的推广）与函数实例偏差之和，两者均依据相应Cramér函数诱导的自然半范数进行表述。此外，我们为当函数类位于给定（指数型）Orlicz空间中的上述半范数提供了特定近似，这些近似可用于使复杂度项和偏差项更加显式化。