论文标题
界定霍尔德类索引的经验过程的期望
Bounding the expectation of the supremum of empirical processes indexed by Hölder classes
论文作者
论文摘要
在本说明中，我们提供了上限，以任何平滑度的Hölder类索引的经验过程的期望以及在$ \ Mathbb r^d $中支持的任何分布中所索引的经验过程。当未知分布由基于$ n $独立的观测值估算出未知分布的经验对应物估算时，这些结果也可以被视为非反应风险界限，并且由积分概率指标（IPM）量化了估计误差。特别是，考虑了由Hölder类索引的IPM，并得出了相应的速率。这些结果在两个众所周知的极端情况之间插值：速率$ n^{ - 1/d} $对应于Wassertein-1距离（最小平滑的外壳）和对应于非常平滑函数的快速速率$ n^{ - 1/2} $（例如，由由有界的kernel定义的RKHS的功能）。
In this note, we provide upper bounds on the expectation of the supremum of empirical processes indexed by Hölder classes of any smoothness and for any distribution supported on a bounded set in $\mathbb R^d$. These results can be alternatively seen as non-asymptotic risk bounds, when the unknown distribution is estimated by its empirical counterpart, based on $n$ independent observations, and the error of estimation is quantified by the integral probability metrics (IPM). In particular, the IPM indexed by a Hölder class is considered and the corresponding rates are derived. These results interpolate between the two well-known extreme cases: the rate $n^{-1/d}$ corresponding to the Wassertein-1 distance (the least smooth case) and the fast rate $n^{-1/2}$ corresponding to very smooth functions (for instance, functions from an RKHS defined by a bounded kernel).
