Reservoir computing approximation and generalization bounds are proved for a new concept class of input/output systems that extends the so-called generalized Barron functionals to a dynamic context. This new class is characterized by the readouts with a certain integral representation built on infinite-dimensional state-space systems. It is shown that this class is very rich and possesses useful features and universal approximation properties. The reservoir architectures used for the approximation and estimation of elements in the new class are randomly generated echo state networks with either linear or ReLU activation functions. Their readouts are built using randomly generated neural networks in which only the output layer is trained (extreme learning machines or random feature neural networks). The results in the paper yield a fully implementable recurrent neural network-based learning algorithm with provable convergence guarantees that do not suffer from the curse of dimensionality.

翻译：针对一类将广义巴伦泛函推广至动态情境的输入/输出系统新概念类，本文证明了储层计算的逼近与泛化界。该新类别通过构建在无限维状态空间系统上、具有特定积分表达式的读出层来刻画。研究表明，此类系统极为丰富，具备实用特性与通用逼近性质。用于逼近和估计该类元素的储层架构采用了随机生成的回声状态网络，其激活函数为线性函数或ReLU函数。其读出层由随机生成的神经网络构建，仅训练输出层（极端学习机或随机特征神经网络）。本文结果提供了一种完全可实现的、基于递归神经网络的学习算法，该算法具有可证明的收敛保证，且不受维度灾难影响。