Navigating LLM Valley: From AdamW to Memory-Efficient and Matrix-Based Optimizers

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.

Symmetric Rank-One Quasi-Newton Methods for Deep Learning Using Cubic Regularization

Stochastic gradient descent and other first-order variants, such as Adam and AdaGrad, are commonly used in the field of deep learning due to their computational efficiency and low-storage memory requirements. However, these methods do not exploit curvature information. Consequently, iterates can converge to saddle points or poor local minima. On the other hand, Quasi-Newton methods compute Hessian approximations which exploit this information with a comparable computational budget. Quasi-Newton methods re-use previously computed iterates and gradients to compute a low-rank structured update. The most widely used quasi-Newton update is the L-BFGS, which guarantees a positive semi-definite Hessian approximation, making it suitable in a line search setting. However, the loss functions in DNNs are non-convex, where the Hessian is potentially non-positive definite. In this paper, we propose using a limited-memory symmetric rank-one quasi-Newton approach which allows for indefinite Hessian approximations, enabling directions of negative curvature to be exploited. Furthermore, we use a modified adaptive regularized cubics approach, which generates a sequence of cubic subproblems that have closed-form solutions with suitable regularization choices. We investigate the performance of our proposed method on autoencoders and feed-forward neural network models and compare our approach to state-of-the-art first-order adaptive stochastic methods as well as other quasi-Newton methods.x