提出最优置信带，实现核梯度流估计的统一推断。
Optimal Confidence Band for Kernel Gradient Flow Estimator
- 基于源条件构造连续与离散核梯度流的统一置信带。
- 置信带宽度逼近极小极大最优率，且收缩速度更快。
- 适用于需严格误差控制的核回归推断场景。
本文研究特定核回归方法——核梯度流的上确界范数泛化误差与统一推断问题。在核回归文献中广泛采用的容量-源条件框架下，我们建立了连续与离散核梯度流在源条件 $s > α_0$ 时的上确界范数泛化误差收敛速率，其中 $α_0 \in (0,1)$ 为核函数的嵌入指数。进一步证明这些速率达到极小极大最优。基于此结果，我们构建了连续与离散核梯度流的联合置信带，其宽度亦为最优：收缩速率大于但可任意接近极小极大最优速率。
原文摘要 · Abstract (English)
In this paper, we investigate the supremum-norm generalization error and the uniform inference for a specific class of kernel regression methods, namely the kernel gradient flows. Under the widely adopted capacity-source condition framework in the kernel regression literature, we first establish convergence rates for the supremum norm generalization error of both continuous and discrete kernel gradient flows under the source condition $s>α_0$, where $α_0\in(0,1)$ denotes the embedding index of the kernel function. Moreover, we show that these rates match the minimax optimal rates. Building on this result, we then construct simultaneous confidence bands for both continuous and discrete kernel gradient flows. Notably, the widths of the proposed confidence bands are also optimal, in the sense that their shrinkage rates are greater than, while can be arbitrarily close to, the minimax optimal rates.
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