On the Role of Noise in the Sample Complexity of Learning Recurrent Neural Networks: Exponential Gaps for Long Sequences

We consider the class of noisy multi-layered sigmoid recurrent neural networks with $w$ (unbounded) weights for classification of sequences of length $T$, where independent noise distributed according to $\mathcal{N}(0,{\sigma }^2)$ is added to the output of each neuron in the network. Our main result shows that the sample complexity of PAC learning this class can be bounded by $O(w\log (T\mathrm{/}\sigma ))$. For the non-noisy version of the same class (i.e., $\sigma =0$), we prove a lower bound of $\mathrm{\Omega }(wT)$ for the sample complexity. Our results indicate an exponential gap in the dependence of sample complexity on $T$ for noisy versus non-noisy networks. Moreover, given the mild logarithmic dependence of the upper bound on $1\mathrm{/}\sigma$, this gap still holds even for numerically negligible values of $\sigma$.