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 $f(\ell)$, 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 $N^{-1/\kappa_\alpha}$ 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 the envelope decay as the key determinant of temporal learnability: slower attenuation of $f(\ell)$ 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.