Learnability Window in Gated Recurrent Neural Networks

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 $f(\mathrm{\ell })$, a functional that captures how gating mechanisms and adaptive optimizers jointly shape the coupling between state-space transport and parameter updates during Backpropagation Through Time. Under heavy-tailed ($\alpha$-stable) fluctuations, where empirical averages concentrate at rate $N^{-1\mathrm{/}{\kappa }_{\alpha }}$ with ${\kappa }_{\alpha }=\alpha \mathrm{/}(\alpha -1)$, 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 $f(\mathrm{\ell })$. These results identify the envelope decay as the key determinant of temporal learnability: slower attenuation of $f(\mathrm{\ell })$ enlarges the learnability window ${\mathcal{H}}_N$, while heavy-tailed noise compresses temporal horizons by weakening statistical concentration. Experiments across multiple gated architectures and optimizers corroborate these structural predictions.