  Recurrent Neural Networks (RNNs) have been widely applied to sequential data
analysis. Due to their complicated modeling structures, however, the theory
behind is still largely missing. To connect theory and practice, we study the
generalization properties of vanilla RNNs as well as their variants, including
Minimal Gated Unit (MGU), Long Short Term Memory (LSTM), and Convolutional
(Conv) RNNs. Specifically, our theory is established under the PAC-Learning
framework. The generalization bound is presented in terms of the spectral norms
of the weight matrices and the total number of parameters. We also establish
refined generalization bounds with additional norm assumptions, and draw a
comparison among these bounds. We remark: (1) Our generalization bound for
vanilla RNNs is significantly tighter than the best of existing results; (2) We
are not aware of any other generalization bounds for MGU, LSTM, and Conv RNNs
in the exiting literature; (3) We demonstrate the advantages of these variants
in generalization.
