Anti-Collapse Dynamics and the Emergence of Multi-Time-Scale Learning in Recurrent Neural Networks

Long-range learning is hard for recurrent networks trained with stochastic gradient descent, because the influence of a past input fades with the lag ℓ, and if it fades too fast the dependence cannot be learned from finite data. This fade is captured by an envelope f(ℓ). An exponential fade makes the data needed to learn a lag-ℓ dependence grow exponentially, putting long horizons out of reach; a power-law fade keeps the cost polynomial. We show that the asymptotic decay behavior of f(ℓ) is not fixed by the architecture. Instead, it emerges from the coupling between the state dynamics and parameter dynamics, settling into either a collapsed regime (fast, exponential forgetting) or an extended, anti-collapsed regime (slow, power-law forgetting). The intuition is a competition within these coupled dynamics. Training drives the network's effective time scales toward short ones, while rare, heavy-tailed fluctuations of the learning dynamics push a few of them to very long values. Along the route studied here, the extended regime survives only when these heavy-tailed pushes are strong enough to balance the pull. We make this mathematically precise with a coarse-grained stochastic process and derive an explicit threshold at which this route to the extended regime becomes available. A single exponent, the spectral exponent~β, then governs both the spread of time scales and how slowly the network forgets. Realizing the regime in practice needs one more ingredient: the joint action of the architecture and the optimizer must be able to hold such a broad spread. A network whose capacity to generate broad time-scale spectra is severely constrained still collapses, even when supplied with strong heavy-tailed forcing. Heavy-tailed fluctuations thus act not as noise to be suppressed, but as the mechanism that sustains long-range learning.