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 $γ$ functional, and the deviation of the function instance, both of which are formulated based on the natural seminorm induced by the corresponding Cramér 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：我们针对由一类函数索引的经验过程，构建了一个均匀尾界，该尾界基于函数的个体偏差而非所考虑类别中的最坏情况偏差。通过向标准泛型链论证中引入一个初始的“收缩”步骤，建立了该尾界。所得尾界是“收缩函数类”复杂性的总和，其中复杂性以Talagrand γ泛函的推广形式度量，以及函数实例的偏差，两者均根据相应Cramér函数诱导的自然半范数定义。利用另一个要求较低的自然半范数，我们还展示了类似的界（尽管隐式依赖于样本量），适用于不能假设有限指数矩的更一般情况。此外，在合适的矩条件下，我们提供了用更常见的Orlicz范数或其“不完全”版本近似的尾界。