--> Computer Science > Machine Learning arXiv:2603.21991 (cs) [Submitted on 23 Mar 2026 ( v1 ), last revised 3 Apr 2026 (this version, v2)] Title: $λ$-GELU: Learning Gating Hardness for Controlled ReLU-ization in Deep Networks Authors: Cristian Pérez-Corral , Alberto Fernández-Hernández , Jose I. Mestre , Manuel F. Dolz , Enrique S. Quintana-Ortí View a PDF of the paper titled $\lambda$-GELU: Learning Gating Hardness for Controlled ReLU-ization in Deep Networks, by Cristian P\'erez-Corral and 4 other authors View PDF HTML (experimental) Abstract: Gaussian Error Linear Unit (GELU) is a widely used smooth alternative to Rectifier Linear Unit (ReLU), yet many deployment, compression, and analysis toolchains are most naturally expressed for piecewise-linear (ReLU-type) networks. We study a hardness-parameterized formulation of GELU, f(x;{\lambda})=x{\Phi}({\lambda} x), where {\Phi} is the Gaussian CDF and {\lambda} \in [1, infty) controls gate sharpness, with the goal of turning smooth gated training into a controlled path toward ReLU-compatible models. Learning {\lambda} is non-trivial: naive updates yield unstable dynamics and effective gradient attenuation, so we introduce a constrained reparameterization and an optimizer-aware update scheme. Empirically, across a diverse set of model--dataset pairs spanning MLPs, CNNs, and Transformers, we observe structured layerwise hardness profiles and assess their robustness under different initializations. We further study a deterministic ReLU-ization strategy in which the learned gates are progressively hardened toward a principled target, enabling a post-training substitution of {\lambda}-GELU by ReLU with reduced disruption. Overall, {\lambda}-GELU provides a minimal and interpretable knob to profile and control gating hardness, bridging smooth training with ReLU-centric downstream pipelines. Subjects: Machine Learning (cs.LG) ; Artificial Intelligence (cs.AI) Cite as: arXiv:2603.21991 [cs.LG] (or arXiv:2603.21991v2 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2603.21991 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Cristian Pérez-Corral [ view email ] [v1] Mon, 23 Mar 2026 13:58:19 UTC (734 KB) [v2] Fri, 3 Apr 2026 12:02:29 UTC (734 KB) Full-text links: Access Paper: View a PDF of the paper titled $\lambda$-GELU: Learning Gating Hardness for Controlled ReLU-ization in Deep Networks, by Cristian P\'erez-Corral and 4 other authors View PDF HTML (experimental) TeX Source view license Current browse context: cs.LG < prev | next > new | recent | 2026-03 Change to browse by: cs cs.AI References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation &times; loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) IArxiv recommender toggle IArxiv Recommender ( What is IArxiv? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? )