# Why Transformers Need Adam: A Hessian Perspective

arXiv: https://arxiv.org/abs/2402.16788 · HTML: https://mlsanity.com/paper/2402.16788/

## Key facts

- Cited by 124 papers in the map (data through 14 Oct 2026).
- Builds on 74 of its 114 references that are in the map.
- First posted on arXiv 26 Feb 2024.

## Builds on (most cited first, 30 of 74)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [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)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [OPT: Open Pre-trained Transformer Language Models](https://mlsanity.com/paper/2205.01068/) (2022)
- [MLP-Mixer: An all-MLP Architecture for Vision](https://mlsanity.com/paper/2105.01601/) (2021)
- [On Layer Normalization in the Transformer Architecture](https://mlsanity.com/paper/2002.04745/) (2020)
- [GLM-130B: An Open Bilingual Pre-trained Model](https://mlsanity.com/paper/2210.02414/) (2022)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [On the Variance of the Adaptive Learning Rate and Beyond](https://mlsanity.com/paper/1908.03265/) (2019)
- [Scaling Vision Transformers to 22 Billion Parameters](https://mlsanity.com/paper/2302.05442/) (2023)
- [Baichuan 2: Open Large-scale Language Models](https://mlsanity.com/paper/2309.10305/) (2023)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [Attention is Not All You Need: Pure Attention Loses Rank Doubly Exponentially with Depth](https://mlsanity.com/paper/2103.03404/) (2021)
- [Early Convolutions Help Transformers See Better](https://mlsanity.com/paper/2106.14881/) (2021)
- [Entropy-SGD: Biasing Gradient Descent Into Wide Valleys](https://mlsanity.com/paper/1611.01838/) (2016)
- [Fine-Tuning Language Models with Just Forward Passes](https://mlsanity.com/paper/2305.17333/) (2023)
- [Why gradient clipping accelerates training: A theoretical justification for adaptivity](https://mlsanity.com/paper/1905.11881/) (2019)
- [Learning Deep Transformer Models for Machine Translation](https://mlsanity.com/paper/1906.01787/) (2019)
- [An Investigation into Neural Net Optimization via Hessian Eigenvalue Density](https://mlsanity.com/paper/1901.10159/) (2019)
- [PyHessian: Neural Networks Through the Lens of the Hessian](https://mlsanity.com/paper/1912.07145/) (2019)
- [Empirical Analysis of the Hessian of Over-Parametrized Neural Networks](https://mlsanity.com/paper/1706.04454/) (2017)
- [Tensor Programs V: Tuning Large Neural Networks via Zero-Shot Hyperparameter Transfer](https://mlsanity.com/paper/2203.03466/) (2022)

## Built on by (most cited first, 30 of 124)

- [Adam-mini: Use Fewer Learning Rates To Gain More](https://mlsanity.com/paper/2406.16793/) (2024)
- [Heavy-Tailed Class Imbalance and Why Adam Outperforms Gradient Descent on Language Models](https://mlsanity.com/paper/2402.19449/) (2024)
- [Preserving Diversity in Supervised Fine-Tuning of Large Language Models](https://mlsanity.com/paper/2408.16673/) (2024)
- [ASGO: Adaptive Structured Gradient Optimization](https://mlsanity.com/paper/2503.20762/) (2025)
- [APOLLO: SGD-like Memory, AdamW-level Performance](https://mlsanity.com/paper/2412.05270/) (2024)
- [Muon Outperforms Adam in Tail-End Associative Memory Learning](https://mlsanity.com/paper/2509.26030/) (2025)
- [Deconstructing What Makes a Good Optimizer for Language Models](https://mlsanity.com/paper/2407.07972/) (2024)
- [Implicit Bias of Spectral Descent and Muon on Multiclass Separable Data](https://mlsanity.com/paper/2502.04664/) (2025)
- [Small Batch Size Training for Language Models: When Vanilla SGD Works, and Why Gradient Accumulation Is Wasteful](https://mlsanity.com/paper/2507.07101/) (2025)
- [In Search of Adam's Secret Sauce](https://mlsanity.com/paper/2505.21829/) (2025)
- [Isotropic Curvature Model for Understanding Deep Learning Optimization: Is Gradient Orthogonalization Optimal?](https://mlsanity.com/paper/2511.00674/) (2025)
- [Does SGD really happen in tiny subspaces?](https://mlsanity.com/paper/2405.16002/) (2024)
- [Peri-LN: Revisiting Normalization Layer in the Transformer Architecture](https://mlsanity.com/paper/2502.02732/) (2025)
- [COSMOS: A Hybrid Adaptive Optimizer for Memory-Efficient Training of LLMs](https://mlsanity.com/paper/2502.17410/) (2025)
- [The Sharpness Disparity Principle in Transformers for Accelerating Language Model Pre-Training](https://mlsanity.com/paper/2502.19002/) (2025)
- [Preconditioning Benefits of Spectral Orthogonalization in Muon](https://mlsanity.com/paper/2601.13474/) (2026)
- [The emergence of sparse attention: impact of data distribution and benefits of repetition](https://mlsanity.com/paper/2505.17863/) (2025)
- [Adam with model exponential moving average is effective for nonconvex optimization](https://mlsanity.com/paper/2405.18199/) (2024)
- [Adam Exploits ℓ_∞-geometry of Loss Landscape via Coordinate-wise Adaptivity](https://mlsanity.com/paper/2410.08198/) (2024)
- [AdaGrad under Anisotropic Smoothness](https://mlsanity.com/paper/2406.15244/) (2024)
- [SWAN: SGD with Normalization and Whitening Enables Stateless LLM Training](https://mlsanity.com/paper/2412.13148/) (2024)
- [Encryption-Friendly LLM Architecture](https://mlsanity.com/paper/2410.02486/) (2024)
- [Sculpting Subspaces: Constrained Full Fine-Tuning in LLMs for Continual Learning](https://mlsanity.com/paper/2504.07097/) (2025)
- [Is your batch size the problem? Revisiting the Adam-SGD gap in language modeling](https://mlsanity.com/paper/2506.12543/) (2025)
- [FedAdamW: A Communication-Efficient Optimizer with Convergence and Generalization Guarantees for Federated Large Models](https://mlsanity.com/paper/2510.27486/) (2025)
- [RMNP: Row-Momentum Normalized Preconditioning for Scalable Matrix-Based Optimization](https://mlsanity.com/paper/2603.20527/) (2026)
- [AdaFisher: Adaptive Second Order Optimization via Fisher Information](https://mlsanity.com/paper/2405.16397/) (2024)
- [MoFO: Momentum-Filtered Optimizer for Mitigating Forgetting in LLM Fine-Tuning](https://mlsanity.com/paper/2407.20999/) (2024)
- [A Defense of the Quadratic Model](https://mlsanity.com/paper/2607.21716/) (2026)
- [Towards Quantifying the Hessian Structure of Neural Networks](https://mlsanity.com/paper/2505.02809/) (2025)

## Abstract (arXiv)

SGD performs worse than Adam by a significant margin on Transformers, but the reason remains unclear. In this work, we provide an explanation through the lens of Hessian: (i) Transformers are "heterogeneous": the Hessian spectrum across parameter blocks vary dramatically, a phenomenon we call "block heterogeneity"; (ii) Heterogeneity hampers SGD: SGD performs worse than Adam on problems with block heterogeneity. To validate (i) and (ii), we check various Transformers, CNNs, MLPs, and quadratic problems, and find that SGD can perform on par with Adam on problems without block heterogeneity, but performs worse than Adam when the heterogeneity exists. Our initial theoretical analysis indicates that SGD performs worse because it applies one single learning rate to all blocks, which cannot handle the heterogeneity among blocks. This limitation could be ameliorated if we use coordinate-wise learning rates, as designed in Adam.