Benchmarking Optimizers for Large Language Model Pretraining
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Abstract (arXiv)
The recent development of Large Language Models (LLMs) has been accompanied by an effervescence of novel ideas and methods to better optimize the loss of deep learning models. Claims from those methods are myriad: from faster convergence to removing reliance on certain hyperparameters. However, the diverse experimental protocols used to validate these claims make direct comparisons between methods challenging. This study presents a comprehensive evaluation of recent optimization techniques across standardized LLM pretraining scenarios, systematically varying model size, batch size, and training duration. Through careful tuning of each method, we provide guidance to practitioners on which optimizer is best suited for each scenario. For researchers, our work highlights promising directions for future optimization research. Finally, by releasing our code and making all experiments fully reproducible, we hope our efforts can help the development and rigorous benchmarking of future methods.
Key facts. Cited by 68 papers in the map (data through 14 Oct 2026). First posted on arXiv 1 Sep 2025.
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Citing papers per year: 2025: 12; 2026: 56.
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- 2025Isotropic Curvature Model for Understanding Deep Learning Optimization: Is Gradient Orthogonalization Optimal?cited by 37
- 2025Hyperparameter Transfer Enables Consistent Gains of Matrix-Preconditioned Optimizers Across Scalescited by 21
- 2025How Muon's Spectral Design Benefits Generalization: A Study on Imbalanced Datacited by 19
- 2025What Really Matters in Matrix-Whitening Optimizers?cited by 16
- 2026Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memorycited by 14
- 2026Muon is Not That Special: Random or Inverted Spectra Work Just as Wellcited by 14
- 2026Mano: Restriking Manifold Optimization for LLM Trainingcited by 12
- 2026HTMuon: Improving Muon via Heavy-Tailed Spectral Correctioncited by 12
- 2025Preconditioned Norms: A Unified Framework for Steepest Descent, Quasi-Newton and Adaptive Methodscited by 11
- 2026Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routerscited by 11
- 2025MARS-M: When Variance Reduction Meets Matricescited by 10
- 2026Enhancing LLM Training via Spectral Clippingcited by 10
38 more
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- 2025Optimal Scaling Needs Optimal Normcited by 8
- 2026Optimal Learning Rate Schedules under Functional Scaling Laws: Power Decay and Warmup-Stable-Decaycited by 8
- 2026Clarifying Shampoo: Adapting Spectral Descent to Stochasticity and the Parameter Trajectorycited by 8
- 2025Optimization Benchmark for Diffusion Models on Dynamical Systemscited by 7
- 2025Beyond Outliers: A Study of Optimizers Under Quantizationcited by 6
- 2026Towards Robust Scaling Laws for Optimizerscited by 6
- 2026Where Does Warm-Up Come From? Adaptive Scheduling for Norm-Constrained Optimizerscited by 5
- 2026Benchmarking Optimizers for MLPs in Tabular Deep Learningcited by 5
- 2026StoSignSGD: Unbiased Structural Stochasticity Fixes SignSGD for Training Large Language Modelscited by 5
- 2026Optimizer-Model Consistency: Full Finetuning with the Same Optimizer as Pretraining Forgets Lesscited by 5
- 2026NuMuon: Nuclear-Norm-Constrained Muon for Compressible LLM Trainingcited by 4
- 2026The Agentic Researcher: A Practical Guide to AI-Assisted Research in Mathematics and Machine Learningcited by 4
- 2026Nexus: Same Pretraining Loss, Better Downstream Generalization via Common Minimacited by 4
- 2026AMUSE: Anytime Muon with Stable Gradient Evaluationcited by 4
- 2026Softsign: Smooth Sign in Your Optimizer For Better Parameter Heterogeneity Handlingcited by 4
- 2026Evolution of Optimization Methods: Algorithms, Scenarios, and Evaluationscited by 3
- 2026DynMuon: A Dynamic Spectral Shaping View of Muoncited by 3
- 2026Anytime Training with Schedule-Free Spectral Optimizationcited by 3
- 2026CMuon: Accelerating and Stabilizing Diffusion Transformer Training via Chunked Momentum Orthogonalizationcited by 3
- 2025MT-DAO: Multi-Timescale Distributed Adaptive Optimizers with Local Updatescited by 2
- 2026When and Why SignSGD Outperforms SGD: A Theoretical Study Based on ℓ1-norm Lower Boundscited by 2
- 2026Navigating LLM Valley: From AdamW to Memory-Efficient and Matrix-Based Optimizerscited by 2
- 2026Revisiting the Adam-SGD Gap in LLM Pre-Training: The Role of Large Effective Learning Ratescited by 2
- 2026LionMuon: Alternating Spectral and Sign Descent for Efficient Trainingcited by 2
- 2026Denoise First, Orthogonalize Later: Understanding Momentum in Muon via Spectral Filteringcited by 2
- 2026When to use what Schatten-p norm in deep learning?cited by 2
- 2026Hierarchical Muon: Tiled Newton-Schulz Updates for Efficient Muon Optimizationcited by 2
- 2026PoLoRA: A Preconditioned Orthogonalized LoRA Optimizercited by 2
- 2026Scale Weight Decay and Train Bettercited by 2
- 2025DyKAF: Dynamical Kronecker Approximation of the Fisher Information Matrix for Gradient Preconditioningcited by 1
- 2026LoRDO: Distributed Low-Rank Optimization with Infrequent Communicationcited by 1
- 2026Open Problem: Is AdamW Effective Under Heavy-Tailed Noise?cited by 1
- 2026M+Adam: Low-Precision Training via Additive-Multiplicative Optimizationcited by 1
- 2026Optimizer Memory Schedules for Outscaling the Overtraining Axiscited by 1
- 2026Towards a Principled Muon under μmathsfP: Ensuring Spectral Conditions throughout Trainingcited by 0
- 2026Variance-Adaptive Muon: Pre-Orthogonalization Variance Modulation for Efficient Language Model Pretrainingcited by 0
- 2026Optimal Sample Complexity for Single Time-Scale Actor-Critic with Momentumcited by 0
50 most cited of 68 All 68 (JSON)
- 2026A Refresh Interval Does Not Fully Specify a Refresh Schedule: Learning-Rate-Dependent Placement in Kronecker-Preconditioned Language Model Training13 Oct 2026
- 2026Which Geometry on Which Layer? Measured Sign-Spectral Assignment in Mixed-Optimizer Training12 Oct 2026
- 2026Minimax Optimality of Shampoo for Nonstationary Stochastic Optimization12 Oct 2026
- 2026From Parallel to Sequential: Small-Batch Noise Restores Simplicity Bias of Muon12 Oct 2026
- 2026Towards Easier-to-Tune Training Algorithms: New Results from AlgoPerf9 Oct 2026
- 2026The Best Optimizer Depends on Batch Size6 Oct 2026
- 2026OptiSelect: How does the Optimizer Shape Data Curriculum?2 Oct 2026
- 2026LionMuon: Alternating Spectral and Sign Descent for Efficient Training28 Sep 2026
- 2026Relative Generalization Invariance of LLM Pretraining26 Sep 2026
- 2026Optimizers for Diffusion Models: A Controlled Benchmark19 Sep 2026
- 2026Optimizer Memory Schedules for Outscaling the Overtraining Axis4 Sep 2026
- 2026Momentum in large-batch training: Polyak enlarges the critical batch size, Nesterov improves data efficiency2 Sep 2026
38 more
- 2026Blog: Survey of Optimizers28 Aug 2026
- 2026CMuon: Accelerating and Stabilizing Diffusion Transformer Training via Chunked Momentum Orthogonalization3 Aug 2026
- 2026Scale Weight Decay and Train Better26 Jul 2026
- 2026PoLoRA: A Preconditioned Orthogonalized LoRA Optimizer20 Jul 2026
- 2026M+Adam: Low-Precision Training via Additive-Multiplicative Optimization12 Jul 2026
- 2026OmniOpt: Taxonomy, Geometry, and Benchmarking of Modern Optimizers4 Jul 2026
- 2026One-Step Gradient Delay is Not a Barrier for Large-Scale Asynchronous Pipeline Parallel LLM Pretraining29 Jun 2026
- 2026Hierarchical Muon: Tiled Newton-Schulz Updates for Efficient Muon Optimization25 Jun 2026
- 2026Open Problem: Is AdamW Effective Under Heavy-Tailed Noise?22 Jun 2026
- 2026When to use what Schatten-p norm in deep learning?13 Jun 2026
- 2026Denoise First, Orthogonalize Later: Understanding Momentum in Muon via Spectral Filtering2 Jun 2026
- 2026Softsign: Smooth Sign in Your Optimizer For Better Parameter Heterogeneity Handling29 May 2026
- 2026Reparametrizing Shampoo and SOAP for Subspace Basis Updates and BFloat16 Storage25 May 2026
- 2026Anytime Training with Schedule-Free Spectral Optimization21 May 2026
- 2026AMUSE: Anytime Muon with Stable Gradient Evaluation21 May 2026
- 2026HORST: Composing Optimizer Geometries for Sparse Transformer Training20 May 2026
- 2026LionMuon: Alternating Spectral and Sign Descent for Efficient Training19 May 2026
- 2026Learn-by-Wire Training Control Governance: Bounded Autonomous Training Under Stress for Stability and Efficiency18 May 2026
- 2026Symmetry-Compatible Principle for Optimizer Design: Embeddings, LM Heads, SwiGLU MLPs, and MoE Routers18 May 2026
- 2026Revisiting the Adam-SGD Gap in LLM Pre-Training: The Role of Large Effective Learning Rates18 May 2026
- 2026DynMuon: A Dynamic Spectral Shaping View of Muon16 May 2026
- 2026Muon is Not That Special: Random or Inverted Spectra Work Just as Well11 May 2026
- 2026Navigating LLM Valley: From AdamW to Memory-Efficient and Matrix-Based Optimizers9 May 2026
- 2026Optimizer-Model Consistency: Full Finetuning with the Same Optimizer as Pretraining Forgets Less7 May 2026
- 2026When and Why SignSGD Outperforms SGD: A Theoretical Study Based on ℓ1-norm Lower Bounds7 May 2026
- 2026StoSignSGD: Unbiased Structural Stochasticity Fixes SignSGD for Training Large Language Models16 Apr 2026
- 2026Benchmarking Optimizers for MLPs in Tabular Deep Learning16 Apr 2026
- 2026Evolution of Optimization Methods: Algorithms, Scenarios, and Evaluations14 Apr 2026
- 2026Nexus: Same Pretraining Loss, Better Downstream Generalization via Common Minima10 Apr 2026
- 2026Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memory27 Mar 2026
- 2026The Agentic Researcher: A Practical Guide to AI-Assisted Research in Mathematics and Machine Learning16 Mar 2026
- 2026Enhancing LLM Training via Spectral Clipping15 Mar 2026
- 2026HTMuon: Improving Muon via Heavy-Tailed Spectral Correction10 Mar 2026
- 2026NuMuon: Nuclear-Norm-Constrained Muon for Compressible LLM Training4 Mar 2026
- 2026Zero-Order Optimization for LLM Fine-Tuning via Learnable Direction Sampling14 Feb 2026
- 2026Clarifying Shampoo: Adapting Spectral Descent to Stochasticity and the Parameter Trajectory10 Feb 2026
- 2026Towards Robust Scaling Laws for Optimizers7 Feb 2026
- 2026Optimal Learning Rate Schedules under Functional Scaling Laws: Power Decay and Warmup-Stable-Decay6 Feb 2026
50 newest of 68 All 68 (JSON)
- 2025Isotropic Curvature Model for Understanding Deep Learning Optimization: Is Gradient Orthogonalization Optimal?cited by 37
- 2025Hyperparameter Transfer Enables Consistent Gains of Matrix-Preconditioned Optimizers Across Scalescited by 21
- 2025How Muon's Spectral Design Benefits Generalization: A Study on Imbalanced Datacited by 19
- 2025What Really Matters in Matrix-Whitening Optimizers?cited by 16
- 2026Sharp Capacity Scaling of Spectral Optimizers in Learning Associative Memorycited by 14
- 2026Muon is Not That Special: Random or Inverted Spectra Work Just as Wellcited by 14
- 2026Mano: Restriking Manifold Optimization for LLM Trainingcited by 12
- 2026HTMuon: Improving Muon via Heavy-Tailed Spectral Correctioncited by 12
- 2025Preconditioned Norms: A Unified Framework for Steepest Descent, Quasi-Newton and Adaptive Methodscited by 11
- 2025MARS-M: When Variance Reduction Meets Matricescited by 10
- 2026Enhancing LLM Training via Spectral Clippingcited by 10
- 2025FedMuon: Federated Learning with Bias-corrected LMO-based Optimizationcited by 8
12 most cited of 62 All 62 (JSON)
- 2026A Refresh Interval Does Not Fully Specify a Refresh Schedule: Learning-Rate-Dependent Placement in Kronecker-Preconditioned Language Model Training13 Oct 2026
- 2026Which Geometry on Which Layer? Measured Sign-Spectral Assignment in Mixed-Optimizer Training12 Oct 2026
- 2026Minimax Optimality of Shampoo for Nonstationary Stochastic Optimization12 Oct 2026
- 2026From Parallel to Sequential: Small-Batch Noise Restores Simplicity Bias of Muon12 Oct 2026
- 2026Towards Easier-to-Tune Training Algorithms: New Results from AlgoPerf9 Oct 2026
- 2026The Best Optimizer Depends on Batch Size6 Oct 2026
- 2026OptiSelect: How does the Optimizer Shape Data Curriculum?2 Oct 2026
- 2026LionMuon: Alternating Spectral and Sign Descent for Efficient Training28 Sep 2026
- 2026Relative Generalization Invariance of LLM Pretraining26 Sep 2026
- 2026Optimizer Memory Schedules for Outscaling the Overtraining Axis4 Sep 2026
- 2026Momentum in large-batch training: Polyak enlarges the critical batch size, Nesterov improves data efficiency2 Sep 2026
- 2026Blog: Survey of Optimizers28 Aug 2026
12 newest of 62 All 62 (JSON)