凸函数的覆盖数
Covering Numbers for Convex Functions
摘要：
本文研究了多维凸一致有界函数空间的覆盖数。我们找到了$\C([a, b]^d, B)$的$ε$ -覆盖数的最优上界和下界，在$L_p$ -度量中，$1 \le p < \infty$，根据相关常数，其中$d \geq 1$，$a < b \in \mathbb{R}$，$B>0$，和$\C([a,b]^d, B)$表示$[a, b]^d$上的所有凸函数的集合，这些凸函数由$B$一致有界。我们总结了以前已知的结果覆盖数凸函数，也提供了一些已知结果的替代证明。我们的结果有直接的影响，在研究收敛速度的经验最小化程序，以及在众多的凸约束函数估计问题的最佳收敛速度。
英文摘要：
In this paper we study the covering numbers of the space of convex and uniformly bounded functions in multi-dimension. We find optimal upper and lower bounds for the $ε$-covering number of $\C([a, b]^d, B)$, in the $L_p$-metric, $1 \le p < \infty$, in terms of the relevant constants, where $d \geq 1$, $a < b \in \mathbb{R}$, $B>0$, and $\C([a,b]^d, B)$ denotes the set of all convex functions on $[a, b]^d$ that are uniformly bounded by $B$. We summarize previously known results on covering numbers for convex functions and also provide alternate proofs of some known results. Our results have direct implications in the study of rates of convergence of empirical minimization procedures as well as optimal convergence rates in the numerous convexity constrained function estimation problems.
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