Learnability window in gated recurrent neural networks.

We develop a statistical theory of temporal learnability in recurrent neural networks, quantifying the maximal temporal horizon 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(ℓ), 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 (α-stable) fluctuations, where empirical averages concentrate at rate N^{-1/κ_{α}} with κ_{α}=α/(α-1), the interplay between envelope decay and statistical concentration yields explicit scaling laws for the growth of H_{N}: logarithmic, polynomial, and exponential temporal learning regimes emerge according to the decay law of f(ℓ). These results identify envelope decay as the key determinant of temporal learnability. Slower attenuation of f(ℓ) enlarges H_{N}, while heavy-tailed fluctuations compress it by weakening statistical concentration. Moreover, envelope geometry outweighs dataset size: slowing the envelope's decay enlarges 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.