We study distribution-free nonparametric regression following a notion of average smoothness initiated by Ashlagi et al. (2021), which measures the "effective" smoothness of a function with respect to an arbitrary unknown underlying distribution. While the recent work of Hanneke et al. (2023) established tight uniform convergence bounds for average-smooth functions in the realizable case and provided a computationally efficient realizable learning algorithm, both of these results currently lack analogs in the general agnostic (i.e. noisy) case. In this work, we fully close these gaps. First, we provide a distribution-free uniform convergence bound for average-smoothness classes in the agnostic setting. Second, we match the derived sample complexity with a computationally efficient agnostic learning algorithm. Our results, which are stated in terms of the intrinsic geometry of the data and hold over any totally bounded metric space, show that the guarantees recently obtained for realizable learning of average-smooth functions transfer to the agnostic setting. At the heart of our proof, we establish the uniform convergence rate of a function class in terms of its bracketing entropy, which may be of independent interest.

翻译：我们研究了基于Ashlagi等人（2021）提出的平均光滑性概念的无分布非参数回归，该概念衡量函数相对于任意未知潜在分布的“有效”光滑性。尽管Hanneke等人（2023）近期的工作在可实现情形下建立了平均光滑函数的紧致一致收敛界，并提供了计算高效的可实现学习算法，但这些结果在一般非依赖（即含噪声）情形下目前仍缺乏对应版本。在本工作中，我们完全填补了这些空白。首先，我们给出了非依赖设定下平均光滑函数类的无分布一致收敛界。其次，我们通过计算高效的非依赖学习算法匹配了导出的样本复杂度。我们的结果以数据的固有几何性质表述，并适用于任意全有界度量空间，表明近期针对平均光滑函数可实现学习所获得的保证可推广至非依赖设定。在证明的核心部分，我们基于函数的括号熵建立了其一致收敛速率，该结果可能具有独立研究价值。