The Gaussian process (GP) algorithm is considered as a powerful nonparametric-learning approach, which can provide uncertainty measurements on the predictions. The standard GP requires clearly observed data, unexpected perturbations in the input may lead to learned regression model mismatching. Besides, GP also suffers from the lack of good generalization performance guarantees. To deal with data uncertainty and provide a numerical generalization performance guarantee on the unknown data distribution, this article proposes a novel robust noisy input GP (NIGP) algorithm based on the probably approximately correct (PAC) Bayes theory. Furthermore, to reduce the computational complexity, we develop a sparse NIGP algorithm, and then develop a sparse PAC-Bayes NIGP approach. Compared with NIGP algorithms, instead of maximizing the marginal log likelihood, one can optimize the PAC-Bayes bound to pursue a tighter generalization error upper bound. Experiments verify that the NIGP algorithms can attain greater accuracy. Besides, the PAC-NIGP algorithms proposed herein can achieve both robust performance and improved generalization error upper bound in the face of both uncertain input and output data.

中文翻译：

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具有输入不确定性的鲁棒高斯过程回归：PAC-贝叶斯视角


高斯过程（GP）算法被认为是一种强大的非参数学习方法，它可以提供预测的不确定性测量。标准 GP 需要清晰观察到的数据，输入中的意外扰动可能会导致学习到的回归模型不匹配。此外，GP还缺乏良好的泛化性能保证。为了处理数据不确定性并为未知数据分布提供数值泛化性能保证，本文提出了一种基于可能近似正确（PAC）贝叶斯理论的新型鲁棒噪声输入GP（NIGP）算法。此外，为了降低计算复杂度，我们开发了稀疏 NIGP 算法，然后开发了稀疏 PAC-Bayes NIGP 方法。与 NIGP 算法相比，我们可以优化 PAC-Bayes 界限以追求更严格的泛化误差上限，而不是最大化边际对数似然。实验验证NIGP算法可以获得更高的精度。此外，本文提出的PAC-NIGP算法在面对不确定的输入和输出数据时可以实现鲁棒性能和改进的泛化误差上限。