# The Fine-Grained Complexity of Gradient Computation for Training Large Language Models

arXiv: https://arxiv.org/abs/2402.04497 · HTML: https://mlsanity.com/paper/2402.04497/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 58 of its 99 references that are in the map.
- First posted on arXiv 7 Feb 2024.

## Builds on (most cited first, 30 of 58)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [LoRA: Low-Rank Adaptation of Large Language Models](https://mlsanity.com/paper/2106.09685/) (2021)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Prefix-Tuning: Optimizing Continuous Prompts for Generation](https://mlsanity.com/paper/2101.00190/) (2021)
- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [XLNet: Generalized Autoregressive Pretraining for Language Understanding](https://mlsanity.com/paper/1906.08237/) (2019)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [LaMDA: Language Models for Dialog Applications](https://mlsanity.com/paper/2201.08239/) (2022)
- [SparseGPT: Massive Language Models Can Be Accurately Pruned in One-Shot](https://mlsanity.com/paper/2301.00774/) (2023)
- [A Simple and Effective Pruning Approach for Large Language Models](https://mlsanity.com/paper/2306.11695/) (2023)
- [Fine-Tuning Language Models with Just Forward Passes](https://mlsanity.com/paper/2305.17333/) (2023)
- [Q8BERT: Quantized 8Bit BERT](https://mlsanity.com/paper/1910.06188/) (2019)
- [DNABERT-2: Efficient Foundation Model and Benchmark For Multi-Species Genome](https://mlsanity.com/paper/2306.15006/) (2023)
- [On The Computational Complexity of Self-Attention](https://mlsanity.com/paper/2209.04881/) (2022)
- [The Hedgehog & the Porcupine: Expressive Linear Attentions with Softmax Mimicry](https://mlsanity.com/paper/2402.04347/) (2024)
- [Fast Attention Requires Bounded Entries](https://mlsanity.com/paper/2302.13214/) (2023)
- [The Expressive Power of Low-Rank Adaptation](https://mlsanity.com/paper/2310.17513/) (2023)
- [HyperAttention: Long-context Attention in Near-Linear Time](https://mlsanity.com/paper/2310.05869/) (2023)
- [Training Multi-Layer Over-Parametrized Neural Network in Subquadratic Time](https://mlsanity.com/paper/2112.07628/) (2021)
- [Training (Overparametrized) Neural Networks in Near-Linear Time](https://mlsanity.com/paper/2006.11648/) (2020)
- [KDEformer: Accelerating Transformers via Kernel Density Estimation](https://mlsanity.com/paper/2302.02451/) (2023)
- [Relative Error Tensor Low Rank Approximation](https://mlsanity.com/paper/1704.08246/) (2017)
- [How to Capture Higher-order Correlations? Generalizing Matrix Softmax Attention to Kronecker Computation](https://mlsanity.com/paper/2310.04064/) (2023)
- [On Sparse Modern Hopfield Model](https://mlsanity.com/paper/2309.12673/) (2023)

## Built on by (most cited first, 30 of 30)

- [On Statistical Rates and Provably Efficient Criteria of Latent Diffusion Transformers (DiTs)](https://mlsanity.com/paper/2407.01079/) (2024)
- [Uniform Memory Retrieval with Larger Capacity for Modern Hopfield Models](https://mlsanity.com/paper/2404.03827/) (2024)
- [Outlier-Efficient Hopfield Layers for Large Transformer-Based Models](https://mlsanity.com/paper/2404.03828/) (2024)
- [Conv-Basis: A New Paradigm for Efficient Attention Inference and Gradient Computation in Transformers](https://mlsanity.com/paper/2405.05219/) (2024)
- [Computational Limits of Low-Rank Adaptation (LoRA) Fine-Tuning for Transformer Models](https://mlsanity.com/paper/2406.03136/) (2024)
- [Training Tensor Attention Efficiently: From Cubic to Almost Linear Time](https://mlsanity.com/paper/2405.16411/) (2024)
- [Multi-Layer Transformers Gradient Can be Approximated in Almost Linear Time](https://mlsanity.com/paper/2408.13233/) (2024)
- [HSR-Enhanced Sparse Attention Acceleration](https://mlsanity.com/paper/2410.10165/) (2024)
- [T2VWorldBench: A Benchmark for Evaluating World Knowledge in Text-to-Video Generation](https://mlsanity.com/paper/2507.18107/) (2025)
- [Your Vision-Language Model Can't Even Count to 20: Exposing the Failures of VLMs in Compositional Counting](https://mlsanity.com/paper/2510.04401/) (2025)
- [Exploring the Frontiers of Softmax: Provable Optimization, Applications in Diffusion Model, and Beyond](https://mlsanity.com/paper/2405.03251/) (2024)
- [Provable Differentially Private Computation of the Cross-Attention Mechanism](https://mlsanity.com/paper/2407.14717/) (2024)
- [Quantum Adaptive Self-Attention for Quantum Transformer Models](https://mlsanity.com/paper/2504.05336/) (2025)
- [FZOO: Fast Zeroth-Order Optimizer for Fine-Tuning Large Language Models towards Adam-Scale Speed](https://mlsanity.com/paper/2506.09034/) (2025)
- [RoPE Attention Can Be Trained in Almost Linear Time](https://mlsanity.com/paper/2412.17316/) (2024)
- [Video Latent Flow Matching: Optimal Polynomial Projections for Video Interpolation and Extrapolation](https://mlsanity.com/paper/2502.00500/) (2025)
- [Towards Infinite-Long Prefix in Transformer](https://mlsanity.com/paper/2406.14036/) (2024)
- [On Computational Limits of FlowAR Models: Expressivity and Efficiency](https://mlsanity.com/paper/2502.16490/) (2025)
- [Modern Hopfield Networks Require Chain-of-Thought to Solve mathsfNC1-Hard Problems](https://mlsanity.com/paper/2412.05562/) (2024)
- [Towards High-Order Mean Flow Generative Models: Feasibility, Expressivity, and Provably Efficient Criteria](https://mlsanity.com/paper/2508.07102/) (2025)
- [Advancing the Understanding of Fixed Point Iterations in Deep Neural Networks: A Detailed Analytical Study](https://mlsanity.com/paper/2410.11279/) (2024)
- [Unlocking the Theory Behind Scaling 1-Bit Neural Networks](https://mlsanity.com/paper/2411.01663/) (2024)
- [Too Easily Fooled? Prompt Injection Breaks LLMs on Frustratingly Simple Multiple-Choice Questions](https://mlsanity.com/paper/2508.13214/) (2025)
- [Time and Memory Trade-off of KV-Cache Compression in Tensor Transformer Decoding](https://mlsanity.com/paper/2503.11108/) (2025)
- [Unifying Learning Dynamics and Generalization in Transformers Scaling Law](https://mlsanity.com/paper/2512.22088/) (2025)
- [kNN Attention Demystified: A Theoretical Exploration for Scalable Transformers](https://mlsanity.com/paper/2411.04013/) (2024)
- [Towards Sampling Data Structures for Tensor Products in Turnstile Streams](https://mlsanity.com/paper/2510.03678/) (2025)
- [Fundamental Limits of Crystalline Equivariant Graph Neural Networks: A Circuit Complexity Perspective](https://mlsanity.com/paper/2510.05494/) (2025)
- [Tracking High-order Evolutions via Cascading Low-rank Fitting](https://mlsanity.com/paper/2604.10980/) (2026)
- [Beyond Sparse Weights: When Is Attention Compressible?](https://mlsanity.com/paper/2608.21541/) (2026)

## Abstract (arXiv)

Large language models (LLMs) have made fundamental contributions over the last a few years. To train an LLM, one needs to alternatingly run `forward' computations and `backward' computations. The forward computation can be viewed as attention function evaluation, and the backward computation can be viewed as a gradient computation. In previous work by [Alman and Song, NeurIPS 2023], it was proved that the forward step can be performed in almost-linear time in certain parameter regimes, but that there is no truly sub-quadratic time algorithm in the remaining parameter regimes unless the popular hypothesis SETH is false. In this work, we show nearly identical results for the harder-seeming problem of computing the gradient of loss function of one layer attention network, and thus for the entire process of LLM training. This completely characterizes the fine-grained complexity of every step of LLM training.