{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/ngd-converges-to-less-degenerate-solutions","title":"NGD converges to less degenerate solutions than SGD","arxiv_id":"2409.04913","date":"2024-09-07","proceeding":null,"authors":["Moosa Saghir","N. R. Raghavendra","Zihe Liu","Evan Ryan Gunter"],"abstract":"The number of free parameters, or dimension, of a model is a straightforward way to measure its complexity: a model with more parameters can encode more information. However, this is not an accurate measure of complexity: models capable of memorizing their training data often generalize well despite their high dimension. Effective dimension aims to more directly capture the complexity of a model by counting only the number of parameters required to represent the functionality of the model. Singular learning theory (SLT) proposes the learning coefficient $ \\lambda $ as a more accurate measure of effective dimension. By describing the rate of increase of the volume of the region of parameter space around a local minimum with respect to loss, $ \\lambda $ incorporates information from higher-order terms. We compare $ \\lambda $ of models trained using natural gradient descent (NGD) and stochastic gradient descent (SGD), and find that those trained with NGD consistently have a higher effective dimension for both of our methods: the Hessian trace $ \\text{Tr}(\\mathbf{H}) $, and the estimate of the local learning coefficient (LLC) $ \\hat{\\lambda}(w^*) $.","url_abs":"https://arxiv.org/abs/2409.04913v2","url_pdf":"https://arxiv.org/pdf/2409.04913v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"ngd-converges-to-less-degenerate-solutions","repo_url":"https://github.com/cxtraa/ngd_with_slt","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok"}}],"tasks":[{"task_slug":"learning-theory","task_name":"Learning Theory"}],"methods":[{"method_slug":"natural-gradient-descent","method_name":"Natural Gradient Descent"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}