# Scaling Law with Learning Rate Annealing

arXiv: https://arxiv.org/abs/2408.11029 · HTML: https://mlsanity.com/paper/2408.11029/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 36 references that are in the map.
- First posted on arXiv 20 Aug 2024.

## Builds on (most cited first, 29 of 29)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [BLOOM: A 176B-Parameter Open-Access Multilingual Language Model](https://mlsanity.com/paper/2211.05100/) (2022)
- [The FineWeb Datasets: Decanting the Web for the Finest Text Data at Scale](https://mlsanity.com/paper/2406.17557/) (2024)
- [On the Variance of the Adaptive Learning Rate and Beyond](https://mlsanity.com/paper/1908.03265/) (2019)
- [MiniCPM: Unveiling the Potential of Small Language Models with Scalable Training Strategies](https://mlsanity.com/paper/2404.06395/) (2024)
- [DeepSeek LLM: Scaling Open-Source Language Models with Longtermism](https://mlsanity.com/paper/2401.02954/) (2024)
- [Cyclical Learning Rates for Training Neural Networks](https://mlsanity.com/paper/1506.01186/) (2015)
- [OLMo: Accelerating the Science of Language Models](https://mlsanity.com/paper/2402.00838/) (2024)
- [Deep Learning Scaling is Predictable, Empirically](https://mlsanity.com/paper/1712.00409/) (2017)
- [Scaling Laws for Autoregressive Generative Modeling](https://mlsanity.com/paper/2010.14701/) (2020)
- [Scaling Data-Constrained Language Models](https://mlsanity.com/paper/2305.16264/) (2023)
- [Explaining Neural Scaling Laws](https://mlsanity.com/paper/2102.06701/) (2021)
- [Don't Decay the Learning Rate, Increase the Batch Size](https://mlsanity.com/paper/1711.00489/) (2017)
- [Scaling Laws for Transfer](https://mlsanity.com/paper/2102.01293/) (2021)
- [Scaling Laws and Compute-Optimal Training Beyond Fixed Training Durations](https://mlsanity.com/paper/2405.18392/) (2024)
- [The Quantization Model of Neural Scaling](https://mlsanity.com/paper/2303.13506/) (2023)
- [Continual Pre-Training of Large Language Models: How to (re)warm your model?](https://mlsanity.com/paper/2308.04014/) (2023)
- [Simple and Scalable Strategies to Continually Pre-train Large Language Models](https://mlsanity.com/paper/2403.08763/) (2024)
- [Broken Neural Scaling Laws](https://mlsanity.com/paper/2210.14891/) (2022)
- [Understanding Emergent Abilities of Language Models from the Loss Perspective](https://mlsanity.com/paper/2403.15796/) (2024)
- [Reuse, Don't Retrain: A Recipe for Continued Pretraining of Language Models](https://mlsanity.com/paper/2407.07263/) (2024)

## Built on by (most cited first, 29 of 29)

- [Intern-S1: A Scientific Multimodal Foundation Model](https://mlsanity.com/paper/2508.15763/) (2025)
- [Scaling Laws for Predicting Downstream Performance in LLMs](https://mlsanity.com/paper/2410.08527/) (2024)
- [Adaptive Data Optimization: Dynamic Sample Selection with Scaling Laws](https://mlsanity.com/paper/2410.11820/) (2024)
- [Scaling Laws for Forgetting during Finetuning with Pretraining Data Injection](https://mlsanity.com/paper/2502.06042/) (2025)
- [Straight to Zero: Why Linearly Decaying the Learning Rate to Zero Works Best for LLMs](https://mlsanity.com/paper/2502.15938/) (2025)
- [Scaling Law for Language Models Training Considering Batch Size](https://mlsanity.com/paper/2412.01505/) (2024)
- [A Multi-Power Law for Loss Curve Prediction Across Learning Rate Schedules](https://mlsanity.com/paper/2503.12811/) (2025)
- [Scaling Collapse Reveals Universal Dynamics in Compute-Optimally Trained Neural Networks](https://mlsanity.com/paper/2507.02119/) (2025)
- [SMELT: Scaling Laws for Compute-Matched MoE Looped Transformers](https://mlsanity.com/paper/2609.01343/) (2026)
- [Training Dynamics Impact Post-Training Quantization Robustness](https://mlsanity.com/paper/2510.06213/) (2025)
- [How Learning Rate Decay Wastes Your Best Data in Curriculum-Based LLM Pretraining](https://mlsanity.com/paper/2511.18903/) (2025)
- [Pre-training LLM without Learning Rate Decay Enhances Supervised Fine-Tuning](https://mlsanity.com/paper/2603.16127/) (2026)
- [Anytime Pretraining: Horizon-Free Learning-Rate Schedules with Weight Averaging](https://mlsanity.com/paper/2602.03702/) (2026)
- [Scaling with Collapse: Efficient and Predictable Training of LLM Families](https://mlsanity.com/paper/2509.25087/) (2025)
- [Optimal Learning Rate Schedules under Functional Scaling Laws: Power Decay and Warmup-Stable-Decay](https://mlsanity.com/paper/2602.06797/) (2026)
- [YuLan-Mini: An Open Data-efficient Language Model](https://mlsanity.com/paper/2412.17743/) (2024)
- [Fast Catch-Up, Late Switching: Optimal Batch Size Scheduling via Functional Scaling Laws](https://mlsanity.com/paper/2602.14208/) (2026)
- [Learning Dynamics in Continual Pre-Training for Large Language Models](https://mlsanity.com/paper/2505.07796/) (2025)
- [Training Dynamics Underlying Language Model Scaling Laws: Loss Deceleration and Zero-Sum Learning](https://mlsanity.com/paper/2506.05447/) (2025)
- [How to Set the Learning Rate for Large-Scale Pre-training?](https://mlsanity.com/paper/2601.05049/) (2026)
- [Agentic Discovery of Neural Architectures: AIRA-Compose and AIRA-Design](https://mlsanity.com/paper/2605.15871/) (2026)
- [Efficient Hyperparameter Tuning via Trajectory Invariance Principle](https://mlsanity.com/paper/2509.25049/) (2025)
- [Predicting Training Re-evaluation Curves Enables Effective Data Curriculums for LLMs](https://mlsanity.com/paper/2509.25380/) (2025)
- [Reusing Overtrained Language Models Saturates Scaling](https://mlsanity.com/paper/2510.06548/) (2025)
- [Configuration-to-Performance Scaling Law with Neural Ansatz](https://mlsanity.com/paper/2602.10300/) (2026)
- [On the Nonlinearity of Learning Rate Scaling for LLM Training](https://mlsanity.com/paper/2606.29158/) (2026)
- [Scaling and Transferability of Annealing Strategies in Large Language Model Training](https://mlsanity.com/paper/2512.13705/) (2025)
- [Complete-muE: Optimal Hyperparameter Transfer and Scaling for MoE Models](https://mlsanity.com/paper/2605.23893/) (2026)
- [Balancing Early Performance Sacrifices with Long-Term Gains: Scaling Learning-Rate Warmup Duration Across Training Horizons](https://mlsanity.com/paper/2609.33041/) (2026)

## Abstract (arXiv)

We find that the cross-entropy loss curves of neural language models empirically adhere to a scaling law with learning rate (LR) annealing over training steps: L(s) = L0 + A· S1-α - C· S2, where L(s) is the validation loss at step s, S1 is the area under the LR curve, S2 is the LR annealing area, and L0, A, C, α are constant parameters. This formulation takes into account two factors: (1) power-law scaling over data size, and (2) the additional loss reduction during LR annealing. Therefore, this formulation can describe the full loss curve at each step, rather than the single loss point at the end of training. Applying the scaling law with LR annealing and fitting only one or two training curves, we can accurately predict the loss at any given step across any learning rate scheduler (LRS). This approach significantly reduces computational cost in formulating scaling laws while providing more accuracy and expressiveness for training dynamics. Extensive experiments demonstrate that our findings hold across a range of hyper-parameters and model architectures, and our equation can extend to scaling effect of model sizes. Moreover, our formulation provides accurate theoretical verification and explanation for empirical results observed in numerous previous studies, particularly those focusing on LR schedule and annealing. We believe that this work is promising to enhance the understanding of LLM training dynamics while greatly democratizing scaling laws, and it can guide researchers in refining training strategies (e.g. critical LRS) for further LLMs.