Learnability Window in Gated Recurrent Neural Networks
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915895954112512 |
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| author | Livi, Lorenzo |
| author_facet | Livi, Lorenzo |
| contents | 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 ($α$-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 $\mathcal{H}_N$: 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_05790 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Learnability Window in Gated Recurrent Neural Networks Livi, Lorenzo Machine Learning Data Analysis, Statistics and Probability 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 ($α$-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 $\mathcal{H}_N$: 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. |
| title | Learnability Window in Gated Recurrent Neural Networks |
| topic | Machine Learning Data Analysis, Statistics and Probability |
| url | https://arxiv.org/abs/2512.05790 |