{"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/exact-solutions-to-the-nonlinear-dynamics-of","title":"Exact solutions to the nonlinear dynamics of learning in deep linear neural networks","arxiv_id":"1312.6120","date":"2013-12-20","proceeding":null,"authors":["Andrew M. Saxe","James L. McClelland","Surya Ganguli"],"abstract":"Despite the widespread practical success of deep learning methods, our\ntheoretical understanding of the dynamics of learning in deep neural networks\nremains quite sparse. We attempt to bridge the gap between the theory and\npractice of deep learning by systematically analyzing learning dynamics for the\nrestricted case of deep linear neural networks. Despite the linearity of their\ninput-output map, such networks have nonlinear gradient descent dynamics on\nweights that change with the addition of each new hidden layer. We show that\ndeep linear networks exhibit nonlinear learning phenomena similar to those seen\nin simulations of nonlinear networks, including long plateaus followed by rapid\ntransitions to lower error solutions, and faster convergence from greedy\nunsupervised pretraining initial conditions than from random initial\nconditions. We provide an analytical description of these phenomena by finding\nnew exact solutions to the nonlinear dynamics of deep learning. Our theoretical\nanalysis also reveals the surprising finding that as the depth of a network\napproaches infinity, learning speed can nevertheless remain finite: for a\nspecial class of initial conditions on the weights, very deep networks incur\nonly a finite, depth independent, delay in learning speed relative to shallow\nnetworks. We show that, under certain conditions on the training data,\nunsupervised pretraining can find this special class of initial conditions,\nwhile scaled random Gaussian initializations cannot. We further exhibit a new\nclass of random orthogonal initial conditions on weights that, like\nunsupervised pre-training, enjoys depth independent learning times. We further\nshow that these initial conditions also lead to faithful propagation of\ngradients even in deep nonlinear networks, as long as they operate in a special\nregime known as the edge of chaos.","url_abs":"http://arxiv.org/abs/1312.6120v3","url_pdf":"http://arxiv.org/pdf/1312.6120v3.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":"exact-solutions-to-the-nonlinear-dynamics-of","repo_url":"https://github.com/MatteoZambra/SM_ML__MScThesis","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}},{"paper_slug":"exact-solutions-to-the-nonlinear-dynamics-of","repo_url":"https://github.com/ducha-aiki/LSUVinit","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[{"task_slug":"unsupervised-pre-training","task_name":"Unsupervised Pre-training"}],"methods":[{"method_slug":"speed","method_name":"SPEED"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/1312.6120","atlas_url":"https://app.syntology.ai/?focus=1312.6120","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}