# WSqD: A Horizon-Free Learning Rate Schedule for Large Model Training

arXiv: https://arxiv.org/abs/2607.10959 · HTML: https://mlsanity.com/paper/2607.10959/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 35 of its 47 references that are in the map.
- First posted on arXiv 12 Jul 2026.

## Builds on (most cited first, 30 of 35)

- [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)
- [The Llama 3 Herd of Models](https://mlsanity.com/paper/2407.21783/) (2024)
- [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)
- [Don't Stop Pretraining: Adapt Language Models to Domains and Tasks](https://mlsanity.com/paper/2004.10964/) (2020)
- [Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time](https://mlsanity.com/paper/2203.05482/) (2022)
- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [MiniCPM: Unveiling the Potential of Small Language Models with Scalable Training Strategies](https://mlsanity.com/paper/2404.06395/) (2024)
- [Continual Pre-training of Language Models](https://mlsanity.com/paper/2302.03241/) (2023)
- [Scaling Laws and Compute-Optimal Training Beyond Fixed Training Durations](https://mlsanity.com/paper/2405.18392/) (2024)
- [The Road Less Scheduled](https://mlsanity.com/paper/2405.15682/) (2024)
- [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)
- [Tight Analyses for Non-Smooth Stochastic Gradient Descent](https://mlsanity.com/paper/1812.05217/) (2018)
- [Understanding Warmup-Stable-Decay Learning Rates: A River Valley Loss Landscape Perspective](https://mlsanity.com/paper/2410.05192/) (2024)
- [Reuse, Don't Retrain: A Recipe for Continued Pretraining of Language Models](https://mlsanity.com/paper/2407.07263/) (2024)
- [Continual evaluation for lifelong learning: Identifying the stability gap](https://mlsanity.com/paper/2205.13452/) (2022)
- [Optimal Linear Decay Learning Rate Schedules and Further Refinements](https://mlsanity.com/paper/2310.07831/) (2023)
- [Scaling Optimal LR Across Token Horizons](https://mlsanity.com/paper/2409.19913/) (2024)
- [Revisiting the Last-Iterate Convergence of Stochastic Gradient Methods](https://mlsanity.com/paper/2312.08531/) (2023)
- [Early Weight Averaging meets High Learning Rates for LLM Pre-training](https://mlsanity.com/paper/2306.03241/) (2023)
- [Straight to Zero: Why Linearly Decaying the Learning Rate to Zero Works Best for LLMs](https://mlsanity.com/paper/2502.15938/) (2025)
- [The Surprising Agreement Between Convex Optimization Theory and Learning-Rate Scheduling for Large Model Training](https://mlsanity.com/paper/2501.18965/) (2025)
- [Power Scheduler: A Batch Size and Token Number Agnostic Learning Rate Scheduler](https://mlsanity.com/paper/2408.13359/) (2024)
- [Efficient Continual Pre-training by Mitigating the Stability Gap](https://mlsanity.com/paper/2406.14833/) (2024)
- [WSM: Decay-Free Learning Rate Schedule via Checkpoint Merging for LLM Pre-training](https://mlsanity.com/paper/2507.17634/) (2025)
- [Mid-Training of Large Language Models: A Survey](https://mlsanity.com/paper/2510.06826/) (2025)

## Abstract (arXiv)

Standard learning rate schedules such as cosine annealing are tied to a fixed training horizon, limiting their ability to accommodate post hoc horizon extension. Warmup-stable-decay (WSD) partially addresses this issue by maintaining a long constant-rate phase before a short linear cooldown, allowing training to resume from a pre-decay checkpoint. However, its peak learning rate is still tuned based on the original training horizon and can become suboptimal when training is extended. Motivated by stochastic convex optimization, we propose WSqD (Warmup with Square-root base and linear Decay), a learning rate schedule that replaces WSD's constant stable phase with a shifted inverse-square-root base while retaining the final linear cooldown. In the stochastic convex setting, WSqD provably attains the minimax-optimal O(1/√T) last-iterate convergence rate. Importantly, its base learning rate schedule is horizon-independent, and the training horizon is needed only to determine when to begin the final cooldown. Empirically, on language-model pretraining using the SlimPajama corpus, WSqD matches or outperforms carefully tuned WSD and other baselines across multiple training horizons while reusing a single peak learning rate.