# On the Principles Behind Neural Network Optimizers

arXiv: https://arxiv.org/abs/2608.16760 · HTML: https://mlsanity.com/paper/2608.16760/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 40 of its 60 references that are in the map.
- First posted on arXiv 17 Aug 2026.

## Builds on (most cited first, 30 of 40)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [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)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Qwen Technical Report](https://mlsanity.com/paper/2309.16609/) (2023)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [ZeRO: Memory Optimizations Toward Training Trillion Parameter Models](https://mlsanity.com/paper/1910.02054/) (2019)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [ZeRO-Infinity: Breaking the GPU Memory Wall for Extreme Scale Deep Learning](https://mlsanity.com/paper/2104.07857/) (2021)
- [Adaptive Gradient Methods with Dynamic Bound of Learning Rate](https://mlsanity.com/paper/1902.09843/) (2019)
- [Small-scale proxies for large-scale Transformer training instabilities](https://mlsanity.com/paper/2309.14322/) (2023)
- [On the Linear Speedup Analysis of Communication Efficient Momentum SGD for Distributed Non-Convex Optimization](https://mlsanity.com/paper/1905.03817/) (2019)
- [Kimi K3: Open Frontier Intelligence](https://mlsanity.com/paper/2607.24653/) (2026)
- [An Improved Analysis of Stochastic Gradient Descent with Momentum](https://mlsanity.com/paper/2007.07989/) (2020)
- [Adam Can Converge Without Any Modification On Update Rules](https://mlsanity.com/paper/2208.09632/) (2022)
- [Fantastic Pretraining Optimizers and Where to Find Them](https://mlsanity.com/paper/2509.02046/) (2025)
- [Resolving Discrepancies in Compute-Optimal Scaling of Language Models](https://mlsanity.com/paper/2406.19146/) (2024)
- [NorMuon: Making Muon more efficient and scalable](https://mlsanity.com/paper/2510.05491/) (2025)
- [Heavy-Tailed Class Imbalance and Why Adam Outperforms Gradient Descent on Language Models](https://mlsanity.com/paper/2402.19449/) (2024)
- [A Unified Analysis of Stochastic Momentum Methods for Deep Learning](https://mlsanity.com/paper/1808.10396/) (2018)
- [APOLLO: SGD-like Memory, AdamW-level Performance](https://mlsanity.com/paper/2412.05270/) (2024)
- [Small Batch Size Training for Language Models: When Vanilla SGD Works, and Why Gradient Accumulation Is Wasteful](https://mlsanity.com/paper/2507.07101/) (2025)
- [Preconditioning Benefits of Spectral Orthogonalization in Muon](https://mlsanity.com/paper/2601.13474/) (2026)
- [A general system of differential equations to model first order adaptive algorithms](https://mlsanity.com/paper/1810.13108/) (2018)
- [Towards Quantifying the Preconditioning Effect of Adam](https://mlsanity.com/paper/2402.07114/) (2024)
- [The Potential of Second-Order Optimization for LLMs: A Study with Full Gauss-Newton](https://mlsanity.com/paper/2510.09378/) (2025)
- [Is your batch size the problem? Revisiting the Adam-SGD gap in language modeling](https://mlsanity.com/paper/2506.12543/) (2025)

## Abstract (arXiv)

Reliable optimization is central to neural network (NN) training, yet Adam, the default optimizer for modern LLMs, rests on a fragile foundation. This thesis develops a principled grounding for Adam and motivates new designs. First, we revisit Adam's divergence--convergence debate and show the existence of a problem-dependent phase transition: with properly chosen, batch-size-dependent hyperparameters, Adam converges, whereas under small-β2 regimes it can diverge. Second, we investigate why Adam substantially outperforms SGD on Transformers through Hessian structure. We find that the Hessian evolves toward a near-block-diagonal form along training, accompanied by strong block heterogeneity. We prove that this structure makes Adam's diagonal preconditioner effective. We further show that this special Hessian structure originates from consecutive multiplications of large matrix variables, and we provide a rigorous analysis based on random matrix theory. Finally, these insights motivate Adam-mini, a new optimizer that reduces Adam's memory footprint by 50% while preserving its performance. Our results also have broader implications beyond Adam: they reveal new local structures in matrix-based nonconvex problems, and also help understand and improve recent NN optimizers, such as Muon.