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. Leveraging another less demanding natural seminorm, we also show similar bounds, though with implicit dependence on the sample size, in the more general case where finite exponential moments cannot be assumed. We also provide approximations of the tail bounds in terms of the more prevalent Orlicz norms or their ``incomplete'' versions under suitable moment conditions.

翻译：我们针对一类函数索引的经验过程，提出了一种一致尾部界，该界基于各函数的个体偏差而非所考虑类别中的最坏情况偏差。通过向标准泛化链论证引入初始"收缩"步骤，建立了该尾部界。所得尾部界为"收缩函数类"的复杂度（以Talagrand $\gamma$泛函的推广形式表示）与函数实例偏差之和，两者均基于相应克拉默函数导出的自然半范数进行构建。利用另一个要求较低的自然半范数，我们还在更一般的情况下（无法假设有限指数矩时）证明了类似边界，尽管其与样本量的依赖关系是隐式的。在适当的矩条件下，我们还提供了基于更普遍的Orlicz范数或其"不完全"版本的尾部界近似。