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.

翻译：本文对向量值随机特征的学习进行了全面的误差分析。该理论在完全通用的无限维输入-输出框架下针对随机特征岭回归建立，但同样适用于并改进了现有的有限维分析。与文献中同类工作不同，本文提出的方法直接分析底层风险泛函，完全避免了基于随机矩阵的显式随机特征岭回归解公式。这消除了对随机矩阵理论中的集中结果或其向随机算子的推广的需求。本文建立的主要结果包括：模型设定错误下向量值随机特征估计量的强相合性，以及正确设定情形下的极小化最优收敛速度。实现这些速度所需的参数复杂度（随机特征数量）和样本复杂度（标注数据数量）与蒙特卡洛直觉相当，且不含对数因子。