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。

基于矢量值随机特征的学习误差界

Error Bounds for Learning with Vector-Valued Random Features

Gradient boosting is a state-of-the-art prediction technique that
sequentially produces a model in the form of linear combinations of simple
predictors---typically decision trees---by solving an infinite-dimensional
convex optimization problem. We provide in the present paper a thorough
analysis of two widespread versions of gradient boosting, and introduce a
general framework for studying these algorithms from the point of view of
functional optimization. We prove their convergence as the number of iterations
tends to infinity and highlight the importance of having a strongly convex risk
functional to minimize. We also present a reasonable statistical context
ensuring consistency properties of the boosting predictors as the sample size
grows. In our approach, the optimization procedures are run forever (that is,
without resorting to an early stopping strategy), and statistical
regularization is basically achieved via an appropriate $L^2$ penalization of
the loss and strong convexity arguments.

本文针对一种最先进的预测技术 —— 梯度提升方法，通过解决无限维凸优化问题，顺序地生成一个由简单预测器（通常为决策树）的线性组合构成的模型。我们对两个广泛使用的梯度提升版本进行了彻底的分析，并从函数优化的角度引入了一个通用框架来研究这些算法。证明了它们在迭代次数趋近于无穷时的收敛性，并强调了具有强凸风险函数的重要性。我们还提供了一个合理的统计环境，确保在样本大小增长时提高了预测器的一致性。在我们的方法中，优化程序是无限运行的（也就是说，没有采用早期停止策略），并且通过适当的 $L^2$ 损失惩罚和强凸性论证来实现统计正则化。