{"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/holomorphic-equilibrium-propagation-computes","title":"Holomorphic Equilibrium Propagation Computes Exact Gradients Through Finite Size Oscillations","arxiv_id":"2209.00530","date":"2022-09-01","proceeding":null,"authors":["Axel Laborieux","Friedemann Zenke"],"abstract":"Equilibrium propagation (EP) is an alternative to backpropagation (BP) that allows the training of deep neural networks with local learning rules. It thus provides a compelling framework for training neuromorphic systems and understanding learning in neurobiology. However, EP requires infinitesimal teaching signals, thereby limiting its applicability in noisy physical systems. Moreover, the algorithm requires separate temporal phases and has not been applied to large-scale problems. Here we address these issues by extending EP to holomorphic networks. We show analytically that this extension naturally leads to exact gradients even for finite-amplitude teaching signals. Importantly, the gradient can be computed as the first Fourier coefficient from finite neuronal activity oscillations in continuous time without requiring separate phases. Further, we demonstrate in numerical simulations that our approach permits robust estimation of gradients in the presence of noise and that deeper models benefit from the finite teaching signals. Finally, we establish the first benchmark for EP on the ImageNet 32x32 dataset and show that it matches the performance of an equivalent network trained with BP. Our work provides analytical insights that enable scaling EP to large-scale problems and establishes a formal framework for how oscillations could support learning in biological and neuromorphic systems.","url_abs":"https://arxiv.org/abs/2209.00530v1","url_pdf":"https://arxiv.org/pdf/2209.00530v1.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":"holomorphic-equilibrium-propagation-computes","repo_url":"https://github.com/Laborieux-Axel/holomorphic_eqprop","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"jax","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2209.00530","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2209.00530"}},"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. Samples come from repositories linked to the paper, official or community; repo_kind says which.","repos":[{"provenance":"external:paperswithcode_snapshot_2025-07-28","url":"https://github.com/Laborieux-Axel/holomorphic_eqprop","reach":{"status":"ok","spdx":"MIT"}}],"summary":{"ran_honours":1,"unverified":6},"by_repo_kind":{"official":{"samples":7,"ran":1,"repositories":1}},"repo_kind_vocabulary":{"official":"The archive marks this repository official for the paper","named_in_paper":"The archive records that the paper mentions this repository; it is not marked official","listed":"In the archive's code links for this paper, not marked official and not recorded as mentioned in the paper","found_in_text":"Syntology found this repository in the paper's own text; whether it is the authors' implementation is not asserted","community":"Not in the archive's code links for this paper; a community repository Syntology harvested"},"n_pointer_only_for_licence":0,"samples":[{"code_sha256_prefix":"9910e2fc297f8665","entry":"identity","repo":"Laborieux-Axel/holomorphic_eqprop","repo_kind":"official","path":"utils/functions.py","file_url":"https://github.com/Laborieux-Axel/holomorphic_eqprop/blob/HEAD/utils/functions.py","link_basis":"harvester_set","language":"python","status":"ran_honours","verification_level":1,"contract_check":"HONOURS","metamorphic_tier":"invariant","behaviour_fingerprint":true,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"9910e2fc297f8665"}},{"code_sha256_prefix":"840f7736c00503d1","entry":"make_result_dir","repo":"Laborieux-Axel/holomorphic_eqprop","repo_kind":"official","path":"utils/data.py","file_url":"https://github.com/Laborieux-Axel/holomorphic_eqprop/blob/HEAD/utils/data.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"840f7736c00503d1"}},{"code_sha256_prefix":"9249643e87c169c9","entry":"my_sigmoid","repo":"Laborieux-Axel/holomorphic_eqprop","repo_kind":"official","path":"utils/functions.py","file_url":"https://github.com/Laborieux-Axel/holomorphic_eqprop/blob/HEAD/utils/functions.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"9249643e87c169c9"}},{"code_sha256_prefix":"38f2f3d1795ca431","entry":"one_hot","repo":"Laborieux-Axel/holomorphic_eqprop","repo_kind":"official","path":"utils/data.py","file_url":"https://github.com/Laborieux-Axel/holomorphic_eqprop/blob/HEAD/utils/data.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"38f2f3d1795ca431"}},{"code_sha256_prefix":"c1c975a446032f3b","entry":"sfm_pool","repo":"Laborieux-Axel/holomorphic_eqprop","repo_kind":"official","path":"utils/custom.py","file_url":"https://github.com/Laborieux-Axel/holomorphic_eqprop/blob/HEAD/utils/custom.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"c1c975a446032f3b"}},{"code_sha256_prefix":"b438bb89cb4b14aa","entry":"silu","repo":"Laborieux-Axel/holomorphic_eqprop","repo_kind":"official","path":"utils/functions.py","file_url":"https://github.com/Laborieux-Axel/holomorphic_eqprop/blob/HEAD/utils/functions.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"b438bb89cb4b14aa"}},{"code_sha256_prefix":"ccf0c136a71a1e23","entry":"window_wise_max","repo":"Laborieux-Axel/holomorphic_eqprop","repo_kind":"official","path":"utils/custom.py","file_url":"https://github.com/Laborieux-Axel/holomorphic_eqprop/blob/HEAD/utils/custom.py","link_basis":"harvester_set","language":"python","status":"unverified","verification_level":0,"contract_check":null,"metamorphic_tier":null,"behaviour_fingerprint":false,"licence":"MIT","inline_ok":true,"mcp_get_code":{"code_sha256":"ccf0c136a71a1e23"}}]},"arxiv_metadata":null,"syntology_extracted_results":null}