Time-Scale Coupling Between States and Parameters in Recurrent Neural Networks

• URL: http://arxiv.org/abs/2508.12121v3
• Date: Sun, 24 Aug 2025 17:10:20 GMT
• Title: Time-Scale Coupling Between States and Parameters in Recurrent Neural Networks
• Authors: Lorenzo Livi,
• Abstract summary: Gated neural networks (RNNs) implicitly induce adaptive learning-rate behavior.<n>Effect arises from the coupling between state-space time scales--parametrized by the gates--and parameter-space dynamics.<n> Empirical simulations corroborate these claims.
• Score: 3.924071936547547
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study how gating mechanisms in recurrent neural networks (RNNs) implicitly induce adaptive learning-rate behavior, even when training is carried out with a fixed, global learning rate. This effect arises from the coupling between state-space time scales--parametrized by the gates--and parameter-space dynamics during gradient descent. By deriving exact Jacobians for leaky-integrator and gated RNNs, we obtain a first-order expansion that makes explicit how constant, scalar, and multi-dimensional gates reshape gradient propagation, modulate effective step sizes, and introduce anisotropy in parameter updates. These findings reveal that gates not only control information flow, but also act as data-driven preconditioners that adapt optimization trajectories in parameter space. We further draw formal analogies with learning-rate schedules, momentum, and adaptive methods such as Adam. Empirical simulations corroborate these claims: in several sequence tasks, we show that gates induce lag-dependent effective learning rates and directional concentration of gradient flow, with multi-gate models matching or exceeding the anisotropic structure produced by Adam. These results highlight that optimizer-driven and gate-driven adaptivity are complementary but not equivalent mechanisms. Overall, this work provides a unified dynamical systems perspective on how gating couples state evolution with parameter updates, explaining why gated architectures achieve robust trainability and stability in practice.

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