A uniform Dvoretzky-Kiefer-Wolfowitz inequality
arxiv(2023)
摘要
We show that under minimal assumption on a class of functions $\mathcal{H}$
defined on a probability space $(\mathcal{X},\mu)$, there is a threshold
$\Delta_0$ satisfying the following: for every $\Delta\geq\Delta_0$, with
probability at least $1-2\exp(-c\Delta m)$ with respect to $\mu^{\otimes m}$,
\[ \sup_{h\in\mathcal{H}} \sup_{t\in\mathbb{R}} \left| \mathbb{P}(h(X)\leq t)
- \frac{1}{m}\sum_{i=1}^m 1_{(-\infty,t]}(h(X_i)) \right| \leq \sqrt{\Delta};\]
here $X$ is distributed according to $\mu$ and $(X_i)_{i=1}^m$ are
independent copies of $X$.
The value of $\Delta_0$ is determined by an unexpected complexity parameter
of the class $\mathcal{H}$ that captures the set's geometry (Talagrand's
$\gamma_1$-functional).
The bound, the probability estimate and the value of $\Delta_0$ are all
optimal up to a logarithmic factor.
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