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Both preconditioners are fitted with an online updating manner minimizing a criterion that is free of line search and robust to stochastic gradient noise, and further constrained to be on certain connected Lie groups to preserve their corresponding symmetry or invariance, e.g., orientation of coordinates by the connected general linear group with positive determinants. The Lie group's equivariance property facilitates preconditioner fitting, and its invariance property saves any need of damping, which is common in second-order optimizers, but difficult to tune. The learning rate for parameter updating and step size for preconditioner fitting are naturally normalized, and their default values work well in most situations.","url_abs":"https://arxiv.org/abs/2211.04422v1","url_pdf":"https://arxiv.org/pdf/2211.04422v1.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":"black-box-lie-group-preconditioners-for-sgd","repo_url":"https://github.com/lixilinx/psgd_torch","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null},{"paper_slug":"black-box-lie-group-preconditioners-for-sgd","repo_url":"https://github.com/lixilinx/psgd_tf","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2211.04422","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2211.04422"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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