Softsign: Smooth Sign in Your Optimizer For Better Parameter Heterogeneity Handling

Sign-based and LMO-inspired optimizers have recently attracted substantial attention in deep learning due to their strong performance and low memory footprint. However, their fixed-magnitude updates can hurt terminal convergence: they decouple update mechanisms from gradient magnitudes and fail to account for parameter heterogeneity, often leading to oscillation rather than convergence. We propose SoftSignum, a smooth relaxation of sign-based optimization that replaces the hard sign map with a temperature-controlled soft-sign transformation, enabling a parameter-wise transition from sign-like updates to magnitude-sensitive SGD-like steps. We complement it with an adaptive quantile-based temperature schedule and extend the same principle to matrix-valued optimizers, obtaining SoftMuon. We also develop a generalized geometry-relaxation framework based on strongly convex regularizers and Fenchel conjugates, proving convergence in stochastic non-convex setting. Experiments on diverse deep learning tasks, including LLM pretraining, show that SoftSignum and SoftMuon consistently improve over their hard sign-based counterparts and standard AdamW. Recent work argues that the gradient-magnitude factor ∥g k ∥ * in (2) is not essential for deep learning and can be dropped (Bernstein & Newhouse, 2024; Pethick et al., 2025) . What remains is a pure Linear Minimization Oracle (LMO) step: Choosing ∥ • ∥ as the ℓ ∞ norm reduces (3) to a coordinatewise sign step and recovers normSGD (