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 has a main complexity component, a variant of Talagrand's $\gamma$ functional for the deflated function class, as well as an instance-dependent deviation term, measured by an appropriately scaled version of a suitable norm. Both of these terms are expressed using certain coefficients formulated based on the relevant cumulant generating functions. We also provide more explicit approximations for the mentioned coefficients, when the function class lies in a given (exponential type) Orlicz space.

翻译：我们针对由一类函数索引的实证过程，提出了一种以函数个体偏差而非所考虑类别中最坏情况偏差表示的一致尾界。该尾界通过在标准通用链论证中引入初始“收缩”步骤而建立。所得尾界包含两个核心组成部分：一是关于收缩函数类的Talagrand $\gamma$泛函变体这一主要复杂度成分，二是由适当缩放范数度量的依赖样本偏差项。这两项均基于相关累积量生成函数构建的特定系数进行表述。当函数类位于给定（指数型）Orlicz空间时，我们还为上述系数提供了更显式的近似表达式。