We introduce AdamS, a simple yet effective alternative to Adam for large language model (LLM) pretraining and post-training. By leveraging a novel denominator, i.e., the root of weighted sum of squares of the momentum and the current gradient, AdamS eliminates the need for second-moment estimates. Hence, AdamS is efficient, matching the memory and compute footprint of SGD with momentum while delivering superior optimization performance. Moreover, AdamS is easy to adopt: it can directly inherit hyperparameters of AdamW, and is entirely model-agnostic, integrating seamlessly into existing pipelines without modifications to optimizer APIs or architectures. The motivation behind AdamS stems from the observed smoothness properties in transformer objectives, where local smoothness is governed by gradient magnitudes that can be further approximated by momentum magnitudes. We establish rigorous theoretical convergence guarantees and provide practical guidelines for hyperparameter selection. Empirically, AdamS demonstrates strong performance in various tasks, including pre-training runs on GPT-2 and Llama2 (up to 13B parameters) and reinforcement learning in post-training regimes. With its efficiency, simplicity, and theoretical grounding, AdamS stands as a compelling alternative to existing optimizers.

Recent studies (CITATION) highlight a fundamental dichotomy in deep learning optimization: Although parameter updates along the top eigendirections of the loss Hessian (Dom-space) capture most of the update magnitude, they often contribute minimally to loss reduction. In contrast, updates in the orthogonal component (Bulk-space) have smaller magnitudes but drive most learning progress.In this work, we further advance the understanding of this phenomenon and introduce the Bulk-Space-Filtration-Accelerator (BSFA), a novel plug-and-play framework. BSFA accelerates training by differentially scaling update components projected onto these distinct subspaces, simultaneously enhancing stability by moderating updates in the dominant subspace and boosting convergence speed by amplifying those in the bulk-space.To ensure BSFA is both practical and scalable for contemporary large models, we introduce two key innovations: an efficient estimator using Principal Component Analysis (PCA) on historical updates for fast subspace estimation, and a block-wise strategy that applies this estimation on a per-parameter-block basis. These designs make BSFA computationally tractable and highly effective.We demonstrate BSFA’s acceleration across various tasks, notably achieving approximately 2× speedup when pre-training LLaMA-72M on WikiText-103 and LLaMA-134M on OpenWebText compared to vanilla AdamW.