AdamX: Cosine similarity meets gradient descent

We introduce SODA, a generalization of Optimistic Dual Averaging, which provides a common perspective on state-of-the-art optimizers like Muon, Lion, AdEMAMix and NAdam, showing that they can all be viewed as optimistic instances of this framework. Based on this framing, we propose a practical SODA wrapper for any base optimizer that eliminates weight decay tuning through a theoretically-grounded $1/k$ decay schedule. Empirical results across various scales and training horizons show that SODA consistently improves performance without any additional hyperparameter tuning.

Adaptive optimizers, most notably Adam, have become the default standard for training large-scale neural networks such as Transformers. These methods maintain running estimates of gradient first and second moments, incurring substantial memory overhead. We introduce PowerStep, a memory-efficient optimizer that achieves coordinate-wise adaptivity without storing second-moment statistics. Motivated by steepest descent under an $\ell_p$-norm geometry, we show that applying a nonlinear transform directly to a momentum buffer yields coordinate-wise adaptivity. We prove that PowerStep converges at the optimal $O(1/\sqrt{T})$ rate for non-convex stochastic optimization. Extensive experiments on Transformer models ranging from 124M to 235B parameters demonstrate that PowerStep matches Adam's convergence speed while halving optimizer memory. Furthermore, when combined with aggressive \texttt{int8} quantization, PowerStep remains numerically stable and reduces optimizer memory by $\sim\!8\times$ compared to full-precision Adam. PowerStep thus provides a principled, scalable and resource-efficient alternative for large-scale training. Code is available at https://github.com/yaolubrain/PowerStep.

Adaptive optimizers such as Adam and non-adaptive methods like SGD exhibit distinct generalization capabilities across different architectures. Prior tunable optimizers attempt to bridge this gap by strictly interpolating between SGD and Adam, effectively confining adaptivity within the 0-to-1 bound. However, this restricted interpolation is fundamentally insufficient: we reveal that optimal adaptivity often requires extrapolation, such as negative adaptivity for classical CNNs and adaptivity of at least one ($γ\geq 1$) for Transformers. Extrapolating adaptivity theoretically violates the strict non-decreasing pre-conditioner assumption, often leading to divergence in existing methods. To break this barrier, we propose Anon, an optimizer that achieves fully continuous adaptivity extrapolation across the entire real-number spectrum. To guarantee provable stability in these out-of-bound regimes, we introduce Incremental Delay Update (IDU), a novel mechanism that bypasses hard max-tracking strategies. We theoretically establish Anon's convergence in both convex and non-convex settings. Empirically, by exploring previously unreachable adaptivity landscapes, Anon demonstrates highly competitive and scalable performance among state-of-the-art element-wise optimizers on representative image classification, diffusion, and large language modeling tasks.