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[Submitted on 25 Jul 2026] Title:When the Preconditioning Exponent Turns Negative: Learning-Rate Coupling and Cross-Environment Generalization View a PDF of the paper titled When the Preconditioning Exponent Turns Negative: Learning-Rate Coupling and Cross-Environment Generalization, by Gongyue Zhang and Honghai Liu View PDF HTML (experimental) Abstract:Adaptive optimizers are commonly parameterized by a fixed power of the second-moment estimate. Existing partially adaptive methods study exponents between momentum-like updates and the standard Adam square root, while the interaction between this exponent and the global learning rate is less understood. We perform a controlled cross-environment study using a paired four-environment classification problem with stable sparse features, environment-dependent spurious sparse features, dense features, and high-dimensional noise. Across \NumRuns{} source-training runs covering 21 preconditioning exponents $p\in[-0.5,0.5]$ and five learning rates $\eta\in[10^{-4},10^{-2}]$, we find that the exponent maximizing cross-environment accuracy decreases almost linearly with $\log_{10}\eta$. The fitted slopes range from $-0.270$ to $-0.300$, with $R^2$ between $0.972$ and $0.996$. At $\eta=10^{-2}$, source-validation selection still prefers positive exponents in all four environments, whereas cross-environment and worst-environment criteria prefer negative exponents. Checkpoint decomposition shows that lower $p$ reduces the learned spurious-to-stable and noise-to-stable weight ratios; under reversed correlation, it also reduces the magnitude of the harmful spurious margin. Negative $p$ is therefore not a universally optimal setting. It is a high-step-size allocation regime produced by the joint action of learning rate and preconditioning. The study also exposes a model-selection conflict: source-domain validation systematically selects a different preconditioning regime from the one that maximizes robustness to environmental change. The results are a single-seed, finite-budget mechanism study rather than a broad benchmark claim. Subjects: Machine Learning (cs.LG) Cite as: arXiv:2609.30271 [cs.LG] (or arXiv:2609.30271v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2609.30271 arXiv-issued DOI via DataCite Submission history From: Gongyue Zhang [view email] [v1] Sat, 25 Jul 2026 01:24:44 UTC (328 KB) Full-text links: Access Paper: View a PDF of the paper titled When the Preconditioning Exponent Turns Negative: Learning-Rate Coupling and Cross-Environment Generalization, by Gongyue Zhang and Honghai Liu 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?)