论文标题
实例依赖性统一的尾部范围用于经验过程
Instance-dependent uniform tail bounds for empirical processes
论文作者
论文摘要
我们根据函数的个体偏差而不是考虑类别中最严重的偏差，为通过一类函数索引的经验过程制定了统一的尾巴。通过将初始的``通缩''步骤引入标准通用链接参数来建立尾巴界。所得的尾巴结合是根据talagrand的$γ$函数的概括``缩放功能类别''的复杂性的总和，并且函数实例的偏差是基于相应的cramér函数诱导的自然中的eminorm制定的。利用另一个要求较少的天然半静电剂，我们也显示出相似的界限，尽管对样本量的隐含依赖性，在更一般的情况下，不能假定有限的指数力矩。我们还根据更普遍的ORLICZ规范或其``不完整''版本在适当的瞬间条件下提供了尾部边界的近似值。
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.
