{"paper":{"title":"Anti-Collapse Dynamics and the Emergence of Multi-Time-Scale Learning in Recurrent Neural Networks","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["physics.data-an"],"primary_cat":"cs.LG","authors_text":"Lorenzo Livi","submitted_at":"2026-06-28T17:30:36Z","abstract_excerpt":"Long-range learning is hard for recurrent networks trained with stochastic gradient descent, because the influence of a past input fades with the lag $\\ell$, and if it fades too fast the dependence cannot be learned from finite data. This fade is captured by an envelope $f(\\ell)$. An exponential fade makes the data needed to learn a lag-$\\ell$ dependence grow exponentially, putting long horizons out of reach; a power-law fade keeps the cost polynomial. We show that the asymptotic decay class of $f(\\ell)$ is not fixed by the architecture. Instead, it emerges from the coupling between the state "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.29519","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.29519/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}