Deep nonparametric regression, characterized by the utilization of deep neural networks to learn target functions, has emerged as a focal point of research attention in recent years. Despite considerable progress in understanding convergence rates, the absence of asymptotic properties hinders rigorous statistical inference. To address this gap, we propose a novel framework that transforms the deep estimation paradigm into a platform conducive to conditional mean estimation, leveraging the conditional diffusion model. Theoretically, we develop an end-to-end convergence rate for the conditional diffusion model and establish the asymptotic normality of the generated samples. Consequently, we are equipped to construct confidence regions, facilitating robust statistical inference. Furthermore, through numerical experiments, we empirically validate the efficacy of our proposed methodology.

利用深度神经网络学习目标函数的深度非参数回归近年来成为研究的焦点，尽管在理解收敛速度方面取得了相当的进展，但缺乏渐近性质阻碍了严格的统计推断。为填补这一空白，我们提出了一种新的框架，将深度估计范式转化为一种有利于条件均值估计的平台，利用条件扩散模型。在理论上，我们为条件扩散模型开发了端到端的收敛速度，并建立了生成样本的渐近正态性。因此，我们可以构建置信区间，便于进行鲁棒的统计推断。此外，通过数值实验，我们在实践中检验了我们提出的方法的有效性。

无模型预测及不确定性评估