Learnability Window in Gated Recurrent Neural Networks

Abstract

We develop a statistical theory of temporal learnability in recurrent neural networks, quantifying the maximal temporal horizon ${\mathcal{H}}_N$ over which gradient-based learning can recover lag-dependent structure at finite sample size $N$. The theory is built on the effective learning rate envelope $f(\ell )$, a function that captures how gating mechanisms and adaptive optimizers jointly shape the coupling between state-space dynamics and parameter updates during Backpropagation Through Time. Under heavy-tailed ($\alpha$-stable) fluctuations, where empirical averages concentrate at rate $N^{-1/{\kappa }_{\alpha }}$ with ${\kappa }_{\alpha }=\alpha /(\alpha -1)$, the interplay between envelope decay and statistical concentration yields explicit scaling laws for the growth of ${\mathcal{H}}_N$: logarithmic, polynomial, and exponential temporal learning regimes emerge according to the decay law of $f(\ell )$. These results identify envelope decay as the key determinant of temporal learnability. Slower attenuation of $f(\ell )$ enlarges ${\mathcal{H}}_N$, while heavy-tailed fluctuations compress it by weakening statistical concentration. Moreover, envelope geometry outweighs dataset size: slowing the envelope's decay enlarges ${\mathcal{H}}_N$ more than adding data, so more complex architectures that realize slower-decaying envelopes can be more data-efficient than simpler ones. Experiments across multiple gated architectures and optimizers corroborate these structural predictions.

Accepted at Physical Review E

Topics & keywords