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[Submitted on 7 Oct 2026] Title:Phase-HDC: Replacing Optimizer History with Gradient Thresholds in Discrete Phase Learning View a PDF of the paper titled Phase-HDC: Replacing Optimizer History with Gradient Thresholds in Discrete Phase Learning, by Ahmed Nebli View PDF HTML (experimental) Abstract:Training a compact model often needs far more memory than storing it, because the optimizer keeps its own records of past gradients. For a hyperdimensional classifier whose learned parameters are low-bit angles, which we call a \emph{phase memory}, these records take several times more memory than the model itself. We ask whether such a model can be trained while storing nothing but the model. The proposed method, Phase-HDC, turns each stored angle by at most one step per update, against the sign of its current gradient, and only when that gradient is large enough. We show that this simple rule is the exact solution of a first-order loss model in which every changed parameter pays a fixed cost. When everything except the update rule is held fixed, Phase-HDC matches the accuracy of Adam with 6-bit moments while storing three times less. Across eleven image, tabular, and text datasets, it stores 16--23$\times$ less than standard float32 Adam and 4--6$\times$ less than 8-bit Adam. The price is an average loss of about five accuracy points against float32 Adam, while Phase-HDC is more accurate than 8-bit Adam on six of the eleven datasets, including byte-level text prediction, where 8-bit Adam collapses. Instrumented training runs explain these outcomes. Once parameters must sit on a discrete grid, Adam's moments mainly decide whether a parameter moves at all, a decision that a threshold on the current gradient can make without memory, and coarse quantization of the moments breaks this decision for inputs that the data rarely contain. The storage savings are logical state rather than measured hardware memory. Subjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI) Cite as: arXiv:2610.10630 [cs.LG] (or arXiv:2610.10630v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2610.10630 arXiv-issued DOI via DataCite (pending registration) Submission history From: Ahmed Nebli [view email] [v1] Wed, 7 Oct 2026 13:05:15 UTC (256 KB) Full-text links: Access Paper: View a PDF of the paper titled Phase-HDC: Replacing Optimizer History with Gradient Thresholds in Discrete Phase Learning, by Ahmed Nebli View PDF HTML (experimental) TeX Source view license Additional Features Audio Summary Current browse context: cs.LG new | recent | 2026-10 Change to browse by: cs cs.AI 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?)