Learning Rates of Deep Nets for Geometrically Strongly Mixing Sequence

The great success of deep learning poses an urgent challenge to establish the theoretical basis for its working mechanism. Recently, research on the convergence of deep neural networks (DNNs) has made great progress. However, the existing studies are based on the assumption that the samples are independent, which is too strong to be applied to many real-world scenarios. In this brief, we establish a fast learning rate for the empirical risk minimization (ERM) on DNN regression with dependent samples, and the dependence is expressed in terms of geometrically strongly mixing sequence. To the best of our knowledge, this is the first convergence result of DNN methods based on mixing sequences. This result is a natural generalization of the independent sample case.