Low-dimensional intrinsic dimension reveals a phase transition in gradient-based learning of deep neural networks

Low-dimensional intrinsic dimension reveals a phase transition in gradient-based learning of deep neural networks | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Low-dimensional intrinsic dimension reveals a phase transition in gradient-based learning of deep neural networks Chengli Tan, Jiangshe Zhang, Junmin Liu, Zixiang Zhao This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4140354/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 11 You are reading this latest preprint version Abstract Deep neural networks complete a feature extraction task by propagating the inputs through multiple modules. However, how the representations evolve with the gradient-based optimization remains unknown. Here we leveraged the intrinsic dimension of the representations to study the learning dynamics and found that the training process underwent a phase transition from expansion to compression under disparate training regimes---a phenomenon that is ubiquitous across a wide variety of model architectures, optimizers, and data sets. We showed that the variation in the intrinsic dimension is consistent with the complexity of the learned hypothesis, which can be quantitatively assessed by the critical sample ratio that was rooted in adversarial robustness. Meanwhile, we mathematically demonstrated that this phenomenon can be analyzed in terms of the mutable correlation between neurons. Although the evoked activities obey a power-law decaying rule in biological circuits, we identified that the power-law exponent of the representations in deep neural networks predicted adversarial robustness well only at the end of the training but not during the training process. These results together suggest that deep neural networks are prone to producing robust representations by adaptively eliminating or retaining redundancies. The code is publicly available at \url{ https://github.com/cltan023/learning2022} . Deep neural networks Stochastic gradient descent Intrinsic dimension Hypothesis complexity Adversarial robustness Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 05 May, 2024 Reviews received at journal 18 Apr, 2024 Reviewers agreed at journal 15 Apr, 2024 Reviews received at journal 07 Apr, 2024 Reviews received at journal 05 Apr, 2024 Reviewers agreed at journal 01 Apr, 2024 Reviewers agreed at journal 31 Mar, 2024 Reviewers invited by journal 31 Mar, 2024 Editor assigned by journal 22 Mar, 2024 Submission checks completed at journal 21 Mar, 2024 First submitted to journal 20 Mar, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. 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