Without imposing light-tailed noise assumptions, we prove that Tikhonov regularization for Gaussian Empirical Gain Maximization (EGM) in a reproducing kernel Hilbert space is consistent and further establish its fast exponential type convergence rates. In the literature, Gaussian EGM was proposed in various contexts to tackle robust estimation problems and has been applied extensively in a great variety of real-world applications. A reproducing kernel Hilbert space is frequently chosen as the hypothesis space, and Tikhonov regularization plays a crucial role in model selection. Although Gaussian EGM has been studied theoretically in a series of papers recently and has been well-understood, theoretical understanding of its Tikhonov regularized variants in RKHS is still limited. Several fundamental challenges remain, especially when light-tailed noise assumptions are absent. To fill the gap and address these challenges, we conduct the present study and make the following contributions. First, under weak moment conditions, we establish a new comparison theorem that enables the investigation of the asymptotic mean calibration properties of regularized Gaussian EGM. Second, under the same weak moment conditions, we show that regularized Gaussian EGM estimators are consistent and further establish their fast exponential-type convergence rates. Our study justifies its feasibility in tackling robust regression problems and explains its robustness from a theoretical viewpoint. Moreover, new technical tools including probabilistic initial upper bounds, confined effective hypothesis spaces, and novel comparison theorems are introduced and developed, which can faciliate the analysis of general regularized empirical gain maximization schemes that fall into the same vein as regularized Gaussian EGM.

中文翻译：

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RKHS 中高斯经验增益最大化的 Tikhonov 正则化是一致的


在不施加轻尾噪声假设的情况下，我们证明了在再现核希尔伯特空间中高斯经验增益最大化 （EGM） 的 Tikhonov 正则化是一致的，并进一步建立了其快速指数型收敛率。在文献中，高斯 EGM 在各种环境中被提出来解决鲁棒估计问题，并已广泛应用于各种实际应用。再现核 Hilbert 空间经常被选为假设空间，而 Tikhonov 正则化在模型选择中起着至关重要的作用。尽管最近在一系列论文中对高斯 EGM 进行了理论研究并得到了很好的理解，但对其在 RKHS 中的 Tikhonov 正则化变体的理论理解仍然有限。仍然存在几个基本挑战，尤其是在没有轻尾噪声假设的情况下。为了填补空白并应对这些挑战，我们进行了本研究并做出了以下贡献。首先，在弱矩条件下，我们建立了一个新的比较定理，能够研究正则化高斯 EGM 的渐近平均校准特性。其次，在相同的弱矩条件下，我们证明正则化高斯 EGM 估计器是一致的，并进一步建立了它们的快速指数型收敛率。我们的研究证明了它解决稳健回归问题的可行性，并从理论角度解释了它的稳健性。 此外，引入和发展了新的技术工具，包括概率初始上限、受限有效假设空间和新颖的比较定理，这可以促进对与正则化高斯 EGM 属于同一脉络的一般正则化经验增益最大化方案的分析。