# CoLA: Compute-Efficient Pre-Training of LLMs via Low-Rank Activation

arXiv: https://arxiv.org/abs/2502.10940 · HTML: https://mlsanity.com/paper/2502.10940/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 35 of its 49 references that are in the map.
- First posted on arXiv 16 Feb 2025.

## Builds on (most cited first, 30 of 35)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [GLUE: A Multi-Task Benchmark and Analysis Platform for Natural Language Understanding](https://mlsanity.com/paper/1804.07461/) (2018)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Training Deep Nets with Sublinear Memory Cost](https://mlsanity.com/paper/1604.06174/) (2016)
- [Large Batch Optimization for Deep Learning: Training BERT in 76 minutes](https://mlsanity.com/paper/1904.00962/) (2019)
- [GaLore: Memory-Efficient LLM Training by Gradient Low-Rank Projection](https://mlsanity.com/paper/2403.03507/) (2024)
- [Speeding up Convolutional Neural Networks with Low Rank Expansions](https://mlsanity.com/paper/1405.3866/) (2014)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [ReLoRA: High-Rank Training Through Low-Rank Updates](https://mlsanity.com/paper/2307.05695/) (2023)
- [Speeding-up Convolutional Neural Networks Using Fine-tuned CP-Decomposition](https://mlsanity.com/paper/1412.6553/) (2014)
- [Scaling Laws for Fine-Grained Mixture of Experts](https://mlsanity.com/paper/2402.07871/) (2024)
- [The Low-Rank Simplicity Bias in Deep Networks](https://mlsanity.com/paper/2103.10427/) (2021)
- [Flora: Low-Rank Adapters Are Secretly Gradient Compressors](https://mlsanity.com/paper/2402.03293/) (2024)
- [Pixelated Butterfly: Simple and Efficient Sparse training for Neural Network Models](https://mlsanity.com/paper/2112.00029/) (2021)
- [Initialization and Regularization of Factorized Neural Layers](https://mlsanity.com/paper/2105.01029/) (2021)
- [APOLLO: SGD-like Memory, AdamW-level Performance](https://mlsanity.com/paper/2412.05270/) (2024)
- [SLTrain: a sparse plus low-rank approach for parameter and memory efficient pretraining](https://mlsanity.com/paper/2406.02214/) (2024)
- [Fira: Can We Achieve Full-rank Training of LLMs Under Low-rank Constraint?](https://mlsanity.com/paper/2410.01623/) (2024)
- [Compressing Recurrent Neural Network with Tensor Train](https://mlsanity.com/paper/1705.08052/) (2017)
- [SLoPe: Double-Pruned Sparse Plus Lazy Low-Rank Adapter Pretraining of LLMs](https://mlsanity.com/paper/2405.16325/) (2024)
- [Accelerating Transformer Pre-training with 2:4 Sparsity](https://mlsanity.com/paper/2404.01847/) (2024)
- [Training Neural Networks from Scratch with Parallel Low-Rank Adapters](https://mlsanity.com/paper/2402.16828/) (2024)
- [CoMERA: Computing- and Memory-Efficient Training via Rank-Adaptive Tensor Optimization](https://mlsanity.com/paper/2405.14377/) (2024)

## Built on by (most cited first, 28 of 28)

- [AuroRA: Breaking Low-Rank Bottleneck of LoRA with Nonlinear Mapping](https://mlsanity.com/paper/2505.18738/) (2025)
- [Muon2: Boosting Muon via Adaptive Second-Moment Preconditioning](https://mlsanity.com/paper/2604.09967/) (2026)
- [Greedy Low-Rank Gradient Compression for Distributed Learning with Convergence Guarantees](https://mlsanity.com/paper/2507.08784/) (2025)
- [LOST: Low-rank and Sparse Pre-training for Large Language Models](https://mlsanity.com/paper/2508.02668/) (2025)
- [RefLoRA: Refactored Low-Rank Adaptation for Efficient Fine-Tuning of Large Models](https://mlsanity.com/paper/2505.18877/) (2025)
- [Reparameterized LLM Training via Orthogonal Equivalence Transformation](https://mlsanity.com/paper/2506.08001/) (2025)
- [CR-Net: Scaling Parameter-Efficient Training with Cross-Layer Low-Rank Structure](https://mlsanity.com/paper/2509.18993/) (2025)
- [LaX: Boosting Low-Rank Training of Foundation Models via Latent Crossing](https://mlsanity.com/paper/2505.21732/) (2025)
- [TEON: Tensorized Orthonormalization Beyond Layer-Wise Muon for Large Language Model Pre-Training](https://mlsanity.com/paper/2601.23261/) (2026)
- [Stabilizing Native Low-Rank LLM Pretraining](https://mlsanity.com/paper/2602.12429/) (2026)
- [MUON+: Towards More Effective Muon via One Additional Normalization Step for LLM Pre-training](https://mlsanity.com/paper/2602.21545/) (2026)
- [Scalable Parameter and Memory Efficient Pretraining for LLM: Recent Algorithmic Advances and Benchmarking](https://mlsanity.com/paper/2505.22922/) (2025)
- [POET-X: Memory-efficient LLM Training by Scaling Orthogonal Transformation](https://mlsanity.com/paper/2603.05500/) (2026)
- [DiaBlo: Diagonal Blocks Are Sufficient For Finetuning](https://mlsanity.com/paper/2506.03230/) (2025)
- [Memory-Efficient Fine-Tuning via Low-Rank Activation Compression](https://mlsanity.com/paper/2509.23472/) (2025)
- [BOOST: BOttleneck-Optimized Scalable Training Framework for Low-Rank Large Language Models](https://mlsanity.com/paper/2512.12131/) (2025)
- [SALAAD: Sparse And Low-Rank Adaptation via ADMM for Large Language Model Inference](https://mlsanity.com/paper/2602.00942/) (2026)
- [OASIS: Online Activation Subspace Learning for Memory-Efficient Training](https://mlsanity.com/paper/2604.09406/) (2026)
- [DLR: Zero-Inference-Cost Latent Residuals for Low-Rank Pre-Training](https://mlsanity.com/paper/2606.28932/) (2026)
- [QuantWM: Temporally Consistent 2-Bit KV Cache Quantization for World Models and Video Generation](https://mlsanity.com/paper/2609.26425/) (2026)
- [When Does Context Help? Error Dynamics of Contextual Information in Large Language Models](https://mlsanity.com/paper/2602.08294/) (2026)
- [CeRA: Breaking the Linear Ceiling of Low-Rank Adaptation with Non-linearity Retained at Inference](https://mlsanity.com/paper/2602.22911/) (2026)
- [ELAS: Efficient Pre-Training of Low-Rank Large Language Models via 2:4 Activation Sparsity](https://mlsanity.com/paper/2605.03667/) (2026)
- [BROS: Bias-Corrected Randomized Subspaces for Memory-Efficient Single-Loop Bilevel Optimization](https://mlsanity.com/paper/2605.10288/) (2026)
- [Beyond Perplexity: A Geometric and Spectral Study of Low-Rank Pre-Training](https://mlsanity.com/paper/2605.13652/) (2026)
- [Efficient Pre-Training of LLMs through Truncated SVD Representations](https://mlsanity.com/paper/2605.28573/) (2026)
- [GRZO: Group-Relative Zeroth-Order Optimization for Large Language Model Fine-Tuning](https://mlsanity.com/paper/2606.02857/) (2026)
- [MoARa: Module-Aware Rank Allocation and Structure-Preserving Decomposition for Low-Rank LLM Pre-training](https://mlsanity.com/paper/2609.15037/) (2026)

## Abstract (arXiv)

The full-size MLPs and the projection layers in attention introduce tremendous model sizes of large language models (LLMs), consuming extensive computational resources in pre-training. We empirically observe that the activations of pre-trained LLMs exhibit low-rank property. Motivated by such observations, we propose CoLA and its memory-efficient implementation, CoLA-M, to replace these full-size layers with compute-efficient auto-encoders that naturally enforce low-rank activations throughout training. This fundamental architectural change eliminates the activation redundancy and significantly boosts model capacity and training efficiency. Experiments on LLaMA models with 60 million to 7 billion parameters show that CoLA reduces the computing cost by bf 2pmb× and improves training throughput by bf 1.86pmb× while maintaining full-rank level performance. CoLA-M further squeezes memory cost without sacrificing throughput, offering a pre-training approach with collectively superior parameter, computing, and memory efficiency. The LLMs produced are also bf 2pmb× smaller, enabling faster inference with lower memory cost on resource-constrained platforms.