Token Geometry

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.

Large language models (LLMs) are built from structured high-dimensional objects such as token representations, weights, adaptation updates, caches, and activations, whose multilinear structure is underexploited by the conventional matrix-centric view. Tensor decompositions and tensor networks provide a principled algebraic language for this structure, yet the literature often treats them as isolated compression mechanisms. This survey organizes tensor methods for LLMs through two complementary views: a seven-stage lifecycle taxonomy covering tokenization, embeddings, pre-training, adaptation, compression, inference, and interpretability, and a component view covering embeddings, attention, and feed-forward networks. We provide unified notation and theoretical foundations, analyze tensorization strategies for individual Transformer components, and compare methods at each lifecycle stage while making differences in evaluation protocols and model scales explicit. We further connect tensor methods to neighboring efficiency techniques and probabilistic tensor networks. Finally, we synthesize open challenges and introduce $ρ_{\rm gap}$, a metric for the compression-realization gap between theoretical memory reduction and measured system-level speedup. By treating tensorization as a common structural principle, the survey provides a structured entry point to tensorized language models and clarifies when parameter savings can plausibly translate into memory efficiency, computational efficiency, or interpretability. The GitHub page dedicated to this paper is accessible at \href{https://github.com/ma-tt-a/awesome-tensor-methods-for-llms}{this https URL}.

We present MetaTT, a Tensor Train (TT) adapter framework for fine-tuning of pre-trained transformers. MetaTT enables flexible and parameter-efficient model adaptation by using a single shared TT to factorize transformer sub-modules. This factorization indexes key structural dimensions, including layer and matrix type, and can optionally incorporate heads and tasks. This design allows MetaTT's parameter count to scale with the sum, rather than the product, of the modes, resulting in a substantially more compact adapter. Our benchmarks compare MetaTT with LoRA along with recent state-of-the-art matrix and tensor decomposition based fine-tuning methods. We observe that when tested on single-task standard language modeling benchmarks, MetaTT achieves competitive parameter efficiency to accuracy tradeoff. We further demonstrate that MetaTT performs competitively when compared to state-of-the-art methods on multi-task learning. Finally, we leverage the TT decomposition to design a rank adaptive optimizer inspired by the DMRG method from many-body physics. Our results demonstrate that integrating this approach with AdamW enhances optimization performance for a specified target rank.