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：我们针对由一类函数索引的经验过程提出了一种一致尾部界，该界基于函数的个体偏差而非所考虑类中的最坏情况偏差。该尾部界通过在标准通用链论证中引入初始的"缩减"步骤建立。得到的尾部界是"缩减函数类"基于Talagrand γ泛函推广的复杂度与函数实例偏差之和，两者均根据相应Cramér函数诱导的自然半范数构造。当函数类处于给定（指数型）Orlicz空间时，我们还提供了所述半范数的某种近似，使复杂度项和偏差项更显式化。