# Predicting Large Model Test Losses with a Noisy Quadratic System

arXiv: https://arxiv.org/abs/2605.09154 · HTML: https://mlsanity.com/paper/2605.09154/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 46 references that are in the map.
- First posted on arXiv 9 May 2026.

## Builds on (most cited first, 28 of 28)

- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [DeepSeek LLM: Scaling Open-Source Language Models with Longtermism](https://mlsanity.com/paper/2401.02954/) (2024)
- [Deep Learning Scaling is Predictable, Empirically](https://mlsanity.com/paper/1712.00409/) (2017)
- [One Billion Word Benchmark for Measuring Progress in Statistical Language Modeling](https://mlsanity.com/paper/1312.3005/) (2013)
- [An Empirical Model of Large-Batch Training](https://mlsanity.com/paper/1812.06162/) (2018)
- [Measuring the Effects of Data Parallelism on Neural Network Training](https://mlsanity.com/paper/1811.03600/) (2018)
- [L2 Regularization versus Batch and Weight Normalization](https://mlsanity.com/paper/1706.05350/) (2017)
- [Which Algorithmic Choices Matter at Which Batch Sizes? Insights From a Noisy Quadratic Model](https://mlsanity.com/paper/1907.04164/) (2019)
- [A Solvable Model of Neural Scaling Laws](https://mlsanity.com/paper/2210.16859/) (2022)
- [A Dynamical Model of Neural Scaling Laws](https://mlsanity.com/paper/2402.01092/) (2024)
- [Don't be lazy: CompleteP enables compute-efficient deep transformers](https://mlsanity.com/paper/2505.01618/) (2025)
- [How Does Critical Batch Size Scale in Pre-training?](https://mlsanity.com/paper/2410.21676/) (2024)
- [Chinchilla Scaling: A replication attempt](https://mlsanity.com/paper/2404.10102/) (2024)
- [4+3 Phases of Compute-Optimal Neural Scaling Laws](https://mlsanity.com/paper/2405.15074/) (2024)
- [Norm matters: efficient and accurate normalization schemes in deep networks](https://mlsanity.com/paper/1803.01814/) (2018)
- [Reconciling Modern Deep Learning with Traditional Optimization Analyses: The Intrinsic Learning Rate](https://mlsanity.com/paper/2010.02916/) (2020)
- [How Feature Learning Can Improve Neural Scaling Laws](https://mlsanity.com/paper/2409.17858/) (2024)
- [Scaling Laws for Optimal Data Mixtures](https://mlsanity.com/paper/2507.09404/) (2025)
- [Power Lines: Scaling Laws for Weight Decay and Batch Size in LLM Pre-training](https://mlsanity.com/paper/2505.13738/) (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)
- [Scaling Optimal LR Across Token Horizons](https://mlsanity.com/paper/2409.19913/) (2024)
- [Emergence and scaling laws in SGD learning of shallow neural networks](https://mlsanity.com/paper/2504.19983/) (2025)
- [Learning quadratic neural networks in high dimensions: SGD dynamics and scaling laws](https://mlsanity.com/paper/2508.03688/) (2025)
- [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)

## Abstract (arXiv)

We introduce a predictive model that estimates the pre-training loss of large models from model size (N), batch size (B) and number of weight updates (K). This is the first loss prediction model that can handle changing batch size. The model outperforms Chinchilla's loss model, a model of the test loss using the batch size and number of tokens, in terms of projecting the loss at extrapolated compute budgets (up to 1000 folds). A natural use of the model is to find optimal N, B, K configurations under explicit and compound resource constraints like time, memory and compute. In our experiments, the model-selected configurations are close to ground-truth optimal. Our work advocates for loss prediction as a better alternative to heuristic-based laws, which are growing in complexity. The implementation is available on https://github.com/chuningxdy/Noisy-Quadratic-System.