基于梯度下降学习的过参数化深度神经网络，从相依数据中估计回归函数
Estimation of a regression function from dependent data by over-parametrized deep neural networks learned by gradient descent
中文总结 AI 辅助
本文针对指数β混合相依数据，提出用梯度下降学习的带logistic激活函数的过参数化深度神经网络估计回归函数，其收敛速率在设计集中于低维流形时依赖于流形维度而非设计维度。
中文摘要 AI 辅助
本文研究从指数β混合相依数据中估计回归函数的问题，采用关于设计分布积分的L₂误差作为误差准则。定义了采用logistic激活函数的深度神经网络估计量，所有参数均通过梯度下降学习得到。针对(p,C)-光滑回归函数，分析了期望L₂误差的收敛速率；在设计集中于d*维流形的特殊情形下，证明该估计量的期望L₂误差收敛速率依赖于d*，而非设计的维度d。
英文摘要
Estimation of a regression function from exponentially $β$-mixing data is considered. The $L_2$ error with integration with respect to the design is used as the error criterion. Deep neural network estimates with logistic activation function are defined, where all parameters are learned by gradient descent. The rate of convergence of the expected $L_2$ error is analyzed for $(p,C)$-smooth regression functions. In the special case that the design is concentrated on a $d^*$-dimensional manifold, it is shown that the expected $L_2$ error of the estimate achieves a rate of convergence which depends on $d^*$ and not on the dimension $d$ of the design.
