SAGE: Sign-Adaptive Gradient for Memory-Efficient LLM Optimization

Training large language models requires optimization algorithms that are not only statistically effective, but also computationally and memory efficient at extreme scale. Although Adam remains the dominant optimizer for large-scale language-model pretraining and fine-tuning, recent work has revisited nearly every component of the optimization stack: adaptive moment estimation, decoupled weight decay, memory footprint, curvature approximation, sign-based updates, large-batch stability, low-rank gradient structure, and matrix-wise orthogonalized updates. This survey reviews optimizer design for large language models through a systems-and-optimization lens. We organize the literature into classical first-order optimizers, adaptive optimizers, memory-efficient variants, second-order and curvature-aware methods, sign-based and discovered optimizers, low-rank and projection-based methods, and matrix-based optimizers such as Muon. We also discuss benchmarking methodology, including hyperparameter fairness, scale dependence, wall-clock efficiency, token efficiency, memory overhead, and downstream evaluation. We argue that optimizer research for LLMs is entering a new phase: moving from single-algorithm speedup claims toward rigorous, scale-aware comparisons that jointly evaluate convergence, stability, memory, and implementation complexity.

It is widely believed that stochastic gradient descent (SGD) performs significantly worse than adaptive optimizers such as Adam in pre-training Large Language Models (LLMs). Yet the underlying reason for this gap remains unclear. In this work, we attribute a large part of the discrepancy to SGD's inability to sustain learning rates comparable to Adam's much larger effective learning rates. Through empirical and theoretical analysis of LLM pre-training dynamics, we identify that training is characterized by small gradient norms and large weight-to-gradient ratios, an effect that becomes more pronounced with larger batch sizes typical in pre-training, necessitating such large effective learning rates. However, we find that output-layer gradient magnitudes become highly uneven across token classes, and that large gradient spikes frequently occur during training. Together, these effects severely restrict the admissible learning rate of SGD. Guided by this understanding, we show that simple clipping mechanisms that stabilize SGD at large learning rates enable it to recover most of Adam's performance. In our large-scale experiments, the validation loss gap between large-learning-rate SGD and Adam shrinks from more than 50% to only about 3.5% when pre-training a 1B-parameter LLaMA model with a 1M-token batch size.

Training large language models (LLMs) entails a fundamental trade-off: memory-efficient optimizers such as Adam discard cross-parameter curvature, whereas full-curvature methods such as SOAP can accelerate convergence at prohibitive memory costs. We introduce Clean, a memory-efficient and full-curvature optimizer designed to resolve this bottleneck. Clean leverages the randomized Nystrom method to accurately approximate the left and right preconditioners in SOAP, and to reduce the optimizer's memory complexity from quadratic to linear in terms of model dimensions. We subsequently reintegrate the off-subspace components to capture curvature information beyond the low-rank approximation, preserving rich curvature at minimal memory cost. We further propose Q-Clean, a low-precision variant that aggressively compresses optimizer states. Q-Clean reduces optimizer memory consumption by over 50% compared to Muon when pre-training a LLaMA-1.3B architecture, all while maintaining strong and competitive predictive performance. Notably, Clean operates with a smaller optimizer-state footprint than standard AdamW while reaching AdamW's final performance 26% faster in wall-clock time. Furthermore, our methods uniquely enable the pre-training of a 13B-parameter model on a single 80GB GPU, providing a scalable, efficient, and accessible approach to large-scale model optimization.