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[Submitted on 16 Sep 2026] Title:The Drift Contract: Spectral Updates for Depth-Robust Local Learning View a PDF of the paper titled The Drift Contract: Spectral Updates for Depth-Robust Local Learning, by Fabien Polly View PDF HTML (experimental) Abstract:Local learning trains each layer with its own auxiliary loss and no global backward pass, which makes layer updates structurally parallel. Two problems have kept it marginal: accuracy degrades as depth grows, and hyperparameters are fragile. We apply Muon-style spectral update geometry (momentum orthogonalization with spectral step scaling) to per-layer local updates, an intersection not previously studied. On CIFAR-10 MLP benchmarks with local linear heads, a single step-size setting is the best value in our tested grids from width 128 to 2048 and from depth 12 to 48, while local Adam requires re-tuning along both axes and still collapses at depth 48 (31.3 percent re-tuned per depth, 19 percent with its depth-12 setting transferred, vs 42.7 percent for the spectral update at its unchanged setting). At five seeds and width 512 the spectral update leads local Adam by a clear margin (48.9 +/- 0.5 vs 46.6 +/- 0.3). Prospectively specified controls attribute the transfer and most of the depth robustness to the spectral geometry itself rather than to any step-size rule on top of it. We additionally formulate the step size as a drift contract, lr = epsilon / RMS(input), which bounds each layer's weight-induced pre-activation change per step, conditioned on its current input. The contract yields a small gain over the best fixed learning rate where that baseline is measured, makes the step size interpretable, and provides a per-layer, input-conditioned drift bound that standard optimizers do not offer. We report one negative result: with RMSNorm and weight decay in the trunk, the stability benefit of spectral updates accrues to global rather than local training, so the local advantage concentrates precisely where normalization is absent. Comments: 7 pages, 2 figures, 2 tables. Code and raw results: this https URL Subjects: Machine Learning (cs.LG) ACM classes: I.2.6; G.1.6 Cite as: arXiv:2609.26811 [cs.LG] (or arXiv:2609.26811v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2609.26811 arXiv-issued DOI via DataCite Submission history From: Fabien Polly [view email] [v1] Wed, 16 Sep 2026 12:43:12 UTC (27 KB) Full-text links: Access Paper: View a PDF of the paper titled The Drift Contract: Spectral Updates for Depth-Robust Local Learning, by Fabien Polly View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG new | recent | 2026-09 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) IArxiv recommender toggle IArxiv Recommender (What is IArxiv?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)