AI 中文总结
研究通过子图表征复杂度的函数类，建立新伯恩斯坦型偏差不等式，用于核密度估计有自适应误差界，证明结合新原理与不等式，用于集合类时改进了经典不等式常数。
AI 中文摘要
我们为通过子图表征复杂度的函数类建立了一个新的伯恩斯坦型偏差不等式。该不等式是非渐近的，包含显式常数，并通过概率水平进行相对归一化。应用于核密度估计时，它在估计器和平滑密度之间产生了一个位置和带宽自适应误差界，在实线上所有点和所有正带宽上同时成立。证明是基本的，结合了一个纳入相对归一化的新对称化原理和极大次高斯不等式，既不需要集中论证也不需要熵积分论证。当专门针对集合类时，我们的技术改进了安东尼和肖 - 泰勒（1993）的经典瓦普尼克 - 切尔沃年基斯不等式中具有相对偏差的常数，将右尾不等式中的因子\(4S_A(2n)\)减少到\(S_A(2n)\)，左尾不等式中减少到\(3S_A(2n)\)。
英文摘要
We establish a new Bernstein-type deviation inequality for classes of functions whose complexity is characterized through subgraphs. The inequality is non-asymptotic, involves explicit constants, and features a relative normalization by the probability level. Applied to kernel density estimation, it produces a location and bandwidth-adaptive error bound between the estimator and the smoothed density, holding simultaneously over all points on the real line and all positive bandwidths. The proof is elementary, combining a new symmetrization principle, which incorporates the relative normalization, with the maximal sub-Gaussian inequality, and requires neither concentration nor entropy-integral arguments. When specialized to classes of sets, our technique improves the constants in the classical Vapnik-Chervonenkis inequality with relative deviation of Anthony and Shawe-Taylor (1993), reducing the factor 4 S A (2n) to S A (2n) in the right-tail inequality and to 3 S A (2n) in the left-tail inequality.
