Randomized algorithms exploit stochasticity to reduce computational complexity. One important example is random feature regression (RFR) that accelerates Gaussian process regression (GPR). RFR approximates an unknown function with a random neural network whose hidden weights and biases are sampled from a probability distribution. Only the final output layer is fit to data. In randomized algorithms like RFR, the hyperparameters that characterize the sampling distribution greatly impact performance, yet are not directly accessible from samples. This makes optimization of hyperparameters via standard (gradient-based) optimization tools inapplicable. Inspired by Bayesian ideas from GPR, this paper introduces a random objective function that is tailored for hyperparameter tuning of vector-valued random features. The objective is minimized with ensemble Kalman inversion (EKI). EKI is a gradient-free particle-based optimizer that is scalable to high-dimensions and robust to randomness in objective functions. A numerical study showcases the new black-box methodology to learn hyperparameter distributions in several problems that are sensitive to the hyperparameter selection: two global sensitivity analyses, integrating a chaotic dynamical system, and solving a Bayesian inverse problem from atmospheric dynamics. The success of the proposed EKI-based algorithm for RFR suggests its potential for automated optimization of hyperparameters arising in other randomized algorithms.

翻译：随机算法利用随机性降低计算复杂度，随机特征回归是加速高斯过程回归的重要范例。该方法通过随机神经网络逼近未知函数，其隐藏层权重与偏置从概率分布中采样，仅对最终输出层进行数据拟合。在此类随机算法中，表征采样分布的超参数对性能影响显著，却无法直接从样本中获取，导致基于梯度的标准优化工具难以适用。受高斯过程回归中贝叶斯思想的启发，本文针对向量值随机特征的超参数调优，构建了专用的随机目标函数，并采用集成卡尔曼反演进行最小化。该无梯度粒子优化器具备高维可扩展性，且对目标函数随机性具有鲁棒性。数值研究通过四类对超参数选择敏感的问题，展示了这种新型黑箱方法学习超参数分布的能力：两项全局敏感性分析、混沌动力系统积分以及大气动力学贝叶斯反问题求解。基于集成卡尔曼反演的随机特征回归算法所取得的成功，表明该方法在其他随机算法的超参数自动化优化中具有应用潜力。