Bayesian PASA: Provably Stable AdaptiveActivation with Uncertainty Quantification

Bayesian PASA: Provably Stable AdaptiveActivation with Uncertainty Quantification | 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 Bayesian PASA: Provably Stable AdaptiveActivation with Uncertainty Quantification Mohsen Mostafa Sayed This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9032403/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract The choice of activation function is a fundamental design decision in deep learning, yet most popular options like ReLU, GELU, or Swish are static and treat all inputs uniformly. This one-size-fits-all approach breaks down in the presence of noisy or corrupted data, where the optimal non-linearity should depend on the input's statistical context. In this paper, we introduce Bayesian Probabilistic Adaptive Sigmoidal Activation (Bayesian PASA), a novel activation function that dynamically adapts its behavior based on the input's uncertainty. Bayesian PASA is not just a new function, but a new paradigm. It frames activation selection as a Bayesian model averaging problem, adaptively mixing sigmoidal, linear, and noise-aware behaviors. The mixing weights are derived from a principled variational evidence lower bound (ELBO), regularized by a stable ψ-function that guarantees bounded influence from noise estimates. We provide three formal theorems proving its Lipschitz continuity, gradient stability, and convergence under standard training assumptions. On the challenging CIFAR-100 benchmark, Bayesian PASA achieves a state-of-the-art test accuracy of 76.38% , outperforming ReLU (75.68%), GELU (75.98%), and the original PASA (75.53%). On the corrupted CIFAR-10-C dataset, the full Bayesian PASA model combined with Bayesian R-LayerNorm achieves an average accuracy of 53.91% , a + 1.87% improvement over the ReLU+LayerNorm baseline. This work provides a drop-in replacement for existing activations, offering not only improved performance but also built-in uncertainty quantification for more robust deep learning systems. Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted 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. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-9032403","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":602738485,"identity":"79469146-364d-4810-8596-2e3cc582d26e","order_by":0,"name":"Mohsen Mostafa Sayed","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABFUlEQVRIie3RsUrDQBzH8V85iEs06x0H9QmESCB0iPggLpbATQk4ZnAoFK6L1dUi6Cs4Ol4ItMth11trXiAiFB0UE3UQmlp0crgv/I//8hn+HGCz/cvcb3tWj7M1qN8TgG4mBNANcRWg/N8Q0OOfyd5oXFSPdzjyvHleqVPs7rByyis/6rIBWTyYVRLqe8EmGunkMiZUTbEvuRBU+SLgcIIgaSEmCcm2RHprCHjloCN5Etak6F/DdXg7CZ5eGzIvyLN6w6FkeiPxeachKnZoLtGX1P0kV+uI1iEbS9rcEvbycxpLV8Q9Xd/ChmtumZ0F1YuM0gsvL41aRgc3oyI3WRZ16Wy4KFvIV3RlwcdH2Ww2m+1vvQODyGPgeaUd1gAAAABJRU5ErkJggg==","orcid":"","institution":"Future Academy","correspondingAuthor":true,"prefix":"","firstName":"Mohsen","middleName":"Mostafa","lastName":"Sayed","suffix":""}],"badges":[],"createdAt":"2026-03-04 16:24:55","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9032403/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9032403/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105629632,"identity":"3d485276-73f0-4b25-9a24-e9f9bf342560","added_by":"auto","created_at":"2026-03-28 11:55:13","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":652451,"visible":true,"origin":"","legend":"","description":"","filename":"paper.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9032403/v1_covered_7d878d57-199f-4092-bc16-35706363de2d.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Bayesian PASA: Provably Stable AdaptiveActivation with Uncertainty Quantification","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"","lastPublishedDoi":"10.21203/rs.3.rs-9032403/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9032403/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe choice of activation function is a fundamental design decision in deep learning, yet most popular options like ReLU, GELU, or Swish are static and treat all inputs uniformly. This one-size-fits-all approach breaks down in the presence of noisy or corrupted data, where the optimal non-linearity should depend on the input's statistical context. In this paper, we introduce Bayesian Probabilistic Adaptive Sigmoidal Activation (Bayesian PASA), a novel activation function that dynamically adapts its behavior based on the input's uncertainty. Bayesian PASA is not just a new function, but a new paradigm. It frames activation selection as a Bayesian model averaging problem, adaptively mixing sigmoidal, linear, and noise-aware behaviors. The mixing weights are derived from a principled variational evidence lower bound (ELBO), regularized by a stable ψ-function that guarantees bounded influence from noise estimates. We provide three formal theorems proving its Lipschitz continuity, gradient stability, and convergence under standard training assumptions. 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