We provide uniform confidence bands for kernel ridge regression (KRR), with finite sample guarantees. KRR is ubiquitous, yet--to our knowledge--this paper supplies the first exact, uniform confidence bands for KRR in the non-parametric regime where the regularization parameter $\lambda$ converges to 0, for general data distributions. Our proposed uniform confidence band is based on a new, symmetrized multiplier bootstrap procedure with a closed form solution, which allows for valid uncertainty quantification without assumptions on the bias. To justify the procedure, we derive non-asymptotic, uniform Gaussian and bootstrap couplings for partial sums in a reproducing kernel Hilbert space (RKHS) with bounded kernel. Our results imply strong approximation for empirical processes indexed by the RKHS unit ball, with sharp, logarithmic dependence on the covering number.

翻译：我们为核岭回归（KRR）提供了具有有限样本保证的一致置信带。KRR应用广泛，但据我们所知，本文首次在正则化参数 λ 趋于0的非参数框架下，针对一般数据分布给出了KRR的精确一致置信带。所提出的一致置信带基于一种新的、具有闭式解的对称化乘子自助法，该方法无需对偏差进行假设即可实现有效的不确定性量化。为证明该方法的合理性，我们推导了具有有界核的再生核希尔伯特空间（RKHS）中部分和的非渐近一致高斯耦合与自助法耦合。我们的结果表明，对于以RKHS单位球为索引的经验过程，实现了强逼近，且对覆盖数具有尖锐的对数依赖性。