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

Alireza Fathollah Pour, Hassan Ashtiani

Keywords: PAC Learning, Recurrent Neural Networks, Noise, Sample Complexity

TL;DR: We prove that the sample complexity of PAC learning noisy recurrent neural networks is logarithmic in the sequence length T while for the non-noisy networks the sample complexity grows at least linearly with T.

Abstract:

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 $O (w\log(T/\sigma))$. For the non-noisy version of the same class (i.e., $\sigma=0$), we prove a lower bound of $\Omega (wT)$ 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 $1/\sigma$, this gap still holds even for numerically negligible values of $\sigma$.

Submission Number: 8803