Regularization plays a pivotal role in ill-posed machine learning and inverse problems. However, the fundamental comparative analysis of various regularization norms remains open. We establish a small noise analysis framework to assess the effects of norms in Tikhonov and RKHS regularizations, in the context of ill-posed linear inverse problems with Gaussian noise. This framework studies the convergence rates of regularized estimators in the small noise limit and reveals the potential instability of the conventional L2-regularizer. We solve such instability by proposing an innovative class of adaptive fractional RKHS regularizers, which covers the L2 Tikhonov and RKHS regularizations by adjusting the fractional smoothness parameter. A surprising insight is that over-smoothing via these fractional RKHSs consistently yields optimal convergence rates, but the optimal hyper-parameter may decay too fast to be selected in practice.

翻译：正则化在不适定机器学习与逆问题中扮演关键角色。然而，不同正则化范数的基础比较分析仍待解决。针对含高斯噪声的线性不适定逆问题，我们建立了一个小噪声分析框架，用于评估Tikhonov和RKHS正则化中范数的影响。该框架研究了小噪声极限下正则化估计量的收敛速率，并揭示了传统L2正则化潜在的失稳性。我们通过提出一类创新的自适应分数阶RKHS正则化器解决了该失稳问题——该类正则化器通过调节分数光滑度参数，同时涵盖L2 Tikhonov正则化与RKHS正则化。一个令人惊奇的发现是：采用这些分数阶RKHS进行过度光滑始终能获得最优收敛速率，但最优超参数可能因衰减过快而难以在实际中选取。