Inverse Approximation Theory for Nonlinear Recurrent Neural Networks
This work addresses fundamental limitations in RNN architectures for learning long-term sequential dependencies, which is crucial for applications in time-series analysis and natural language processing, though it is incremental as it extends prior linear results to nonlinear cases.
The paper proves an inverse approximation theorem showing that nonlinear recurrent neural networks (RNNs) can only stably approximate sequence relationships with exponentially decaying memory, extending the curse of memory to nonlinear settings and quantifying architectural limitations for learning long-term dependencies. It proposes a reparameterization method to overcome these limitations, supported by numerical experiments.
We prove an inverse approximation theorem for the approximation of nonlinear sequence-to-sequence relationships using recurrent neural networks (RNNs). This is a so-called Bernstein-type result in approximation theory, which deduces properties of a target function under the assumption that it can be effectively approximated by a hypothesis space. In particular, we show that nonlinear sequence relationships that can be stably approximated by nonlinear RNNs must have an exponential decaying memory structure - a notion that can be made precise. This extends the previously identified curse of memory in linear RNNs into the general nonlinear setting, and quantifies the essential limitations of the RNN architecture for learning sequential relationships with long-term memory. Based on the analysis, we propose a principled reparameterization method to overcome the limitations. Our theoretical results are confirmed by numerical experiments. The code has been released in https://github.com/radarFudan/Curse-of-memory