$\lambda$-GELU: Learning Gating Hardness for Controlled ReLU-ization in Deep Networks

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.