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[Submitted on 30 May 2026] Title:How Far is Adam from Natural Gradient Descent? View a PDF of the paper titled How Far is Adam from Natural Gradient Descent?, by Vihaan Paka-Hegde View PDF HTML (experimental) Abstract:Adam is the standard optimizer in deep learning, yet its geometric relationship to natural gradient descent (NGD) contains unresolved questions. We study Adam's full update rule, including momentum, as a diagonal empirical Fisher approximation subject to diagonal truncation, empirical label substitution, and temporal lag. Using the scale-invariant $\gamma(\Delta\theta)$ metric, we measure Adam's geometric deviation from true NGD across four loss landscapes: well-conditioned linear regression, ill-conditioned linear regression, logistic regression, and a non-convex small neural network. Adam's geometric trajectory is context-dependent. Deviation remains low in well-conditioned settings but rises significantly under ill-conditioning, reaching misalignments of $\approx 10^3$ in the neural network. Higher geometric drift correlates with slower initial optimization but does not degrade final objective minimization; Adam consistently reaches low loss. Furthermore, the improved empirical Fisher (iEF) tracks more stable paths than the standard empirical Fisher (EF), which frequently oscillates or diverges. Our results suggest Adam's practical optimization power may stem from a balance of structural approximation errors and momentum smoothing rather than close tracking of the natural gradient path. Comments: 9 pages, 4 figures, 2 tables Subjects: Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE) MSC classes: 68T07 ACM classes: I.2.6 Cite as: arXiv:2610.00004 [cs.LG] (or arXiv:2610.00004v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2610.00004 arXiv-issued DOI via DataCite Related DOI: https://doi.org/10.5281/zenodo.20466393 DOI(s) linking to related resources Submission history From: Vihaan Paka-Hegde [view email] [v1] Sat, 30 May 2026 21:59:23 UTC (169 KB) Full-text links: Access Paper: View a PDF of the paper titled How Far is Adam from Natural Gradient Descent?, by Vihaan Paka-Hegde View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG new | recent | 2026-10 Change to browse by: cs cs.NE 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?)