{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,27]],"date-time":"2026-10-27T16:33:31Z","timestamp":1793118811024,"version":"4.5.0"},"reference-count":40,"publisher":"Oxford University Press (OUP)","issue":"10","license":[{"start":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T00:00:00Z","timestamp":1790121600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000002","name":"NIH","doi-asserted-by":"publisher","award":["R01NS125298"],"award-info":[{"award-number":["R01NS125298"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"NIH","doi-asserted-by":"publisher","award":["R01DC024495"],"award-info":[{"award-number":["R01DC024495"]}],"id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,9,30]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Backpropagation (BP), a foundational algorithm for training artificial neural networks, predominates in contemporary deep learning. Although highly successful, it is widely considered biologically implausible, because it relies on precise symmetry between feedforward and feedback weights to accurately propagate gradient signals that assign credit. The so-called weight transport problem concerns how biological brains learn to align feedforward and feedback paths while avoiding the nonbiological transport of feedforward weights into feedback weights. To address this, several credit assignment algorithms, such as feedback alignment and the Kolen\u2013Pollack rule, have been proposed. While they can achieve the desired weight alignment, these algorithms imply that if a neuron sends a feedforward synapse to another neuron, it should also receive an identical or at least partially correlated feedback synapse from the latter neuron, thereby forming a bidirectional symmetric connection. However, experimental studies of cortical connectivity do not support the widespread prevalence of such symmetric connections, and in fact report many unidirectional connections, thus imposing a biological constraint on credit assignment. We refer to the failure of existing algorithms to satisfy this constraint as the weight symmetry problem. To address this challenge of biological constraints on connectivity, we introduce the product feedback alignment (PFA) algorithm. We demonstrate that PFA can eliminate explicit weight symmetry entirely while closely approximating BP and achieving comparable performance in deep convolutional networks. Our results offer an approach to solve the longstanding problem of credit assignment in the brain, leading to more biologically plausible learning in deep networks compared to previous methods.<\/jats:p>","DOI":"10.1093\/pnasnexus\/pgag321","type":"journal-article","created":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T07:46:05Z","timestamp":1790149565000},"source":"Crossref","is-referenced-by-count":0,"title":["Deep learning without weight symmetry"],"prefix":"10.1093","volume":"5","author":[{"given":"Li","family":"Ji-An","sequence":"first","affiliation":[{"name":"Neurosciences Graduate Program and Department of Neurobiology, University of California San Diego , La Jolla, CA 92093 ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8100-2857","authenticated-orcid":false,"given":"Marcus K","family":"Benna","sequence":"additional","affiliation":[{"name":"Department of Neurobiology, University of California San Diego , 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Burges CJC. MNIST database of handwritten digits [dataset]. UCI Machine Learning Repository; 1998."},{"key":"2026102712065113000_pgag321-B27","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4419-0661-8","volume-title":"Spectral analysis of large dimensional random matrices","author":"Bai","year":"2010"},{"key":"2026102712065113000_pgag321-B28","volume-title":"Deep learning","author":"Goodfellow","year":"2016"},{"key":"2026102712065113000_pgag321-B29","author":"Krizhevsky"},{"key":"2026102712065113000_pgag321-B30","unstructured":"Krizhevsky A, Nair V, Hinton G. CIFAR-10 [dataset]. UCI Machine Learning Repository; 2009."},{"key":"2026102712065113000_pgag321-B31","author":"He"},{"key":"2026102712065113000_pgag321-B32","unstructured":"Deng J, et al \u00a0ImageNet [dataset]. 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