This paper provides a comprehensive error analysis of learning with
vector-valued random features (RF). The theory is developed for RF ridge
regression in a fully general infinite-dimensional input-output setting, but
nonetheless applies to and improves existing finite-dimensional analyses. In
contrast to comparable work in the literature, the approach proposed here
relies on a direct analysis of the underlying risk functional and completely
avoids the explicit RF ridge regression solution formula in terms of random
matrices. This removes the need for concentration results in random matrix
theory or their generalizations to random operators. The main results
established in this paper include strong consistency of vector-valued RF
estimators under model misspecification and minimax optimal convergence rates
in the well-specified setting. The parameter complexity (number of random
features) and sample complexity (number of labeled data) required to achieve
such rates are comparable with Monte Carlo intuition and free from logarithmic
factors.

本论文提供了关于向量值随机特征（RF）学习的全面误差分析，为 RF 岭回归在输入输出设置下建立了理论，该方法直接分析了风险函数，避免随机矩阵理论中的浓度结果，主要结果包括在模型未规范化情况下向量值 RF 估计量的强一致性和在规范化设置下的极小极小收敛速度。